%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : SWW473+6 : TPTP v9.3.1. Released v5.3.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 27 09:10:20 AM UTC 2026
% Result : Theorem 112.65s 31.43s
% Output : Refutation 118.29s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 609
% Syntax : Number of formulae : 1220 ( 358 unt; 0 typ; 0 def)
% Number of atoms : 3226 ( 891 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 55990 ( 329 ~; 94 |; 237 &;53655 @)
% ( 103 <=>;1572 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 12 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 85 ( 83 usr; 22 con; 0-4 aty)
% Number of variables : 3421 ( 0 ^;3368 !; 53 ?;3421 :)
% Comments :
%------------------------------------------------------------------------------
thf(hBOOL_decl,type,
hBOOL: $i > $o ).
thf(hAPP_decl,type,
hAPP: $i > $i > $i > $i > $i ).
thf(fun_decl,type,
fun: $i > $i > $i ).
thf(x_a_decl,type,
x_a: $i ).
thf(bool_decl,type,
bool: $i ).
thf(ord_less_eq_decl,type,
ord_less_eq: $i > $i ).
thf(insert_decl,type,
insert: $i > $i ).
thf(pname_decl,type,
pname: $i ).
thf(mgt_call_decl,type,
mgt_call: $i ).
thf(pn_decl,type,
pn: $i ).
thf(g_decl,type,
g: $i ).
thf(image_decl,type,
image: $i > $i > $i ).
thf(u_decl,type,
u: $i ).
thf(one_decl,type,
one: $i > $o ).
thf(nat_decl,type,
nat: $i ).
thf(zero_decl,type,
zero: $i > $o ).
thf(zero_neq_one_decl,type,
zero_neq_one: $i > $o ).
thf(minus_decl,type,
minus: $i > $o ).
thf(ti_decl,type,
ti: $i > $i > $i ).
thf(one_one_decl,type,
one_one: $i > $i ).
thf(zero_zero_decl,type,
zero_zero: $i > $i ).
thf(monoid_add_decl,type,
monoid_add: $i > $o ).
thf(undefined_decl,type,
undefined: $i > $i ).
thf(bot_decl,type,
bot: $i > $o ).
thf(bot_bot_decl,type,
bot_bot: $i > $i ).
thf(order_decl,type,
order: $i > $o ).
thf(cancel_semigroup_add_decl,type,
cancel_semigroup_add: $i > $o ).
thf(cancel146912293up_add_decl,type,
cancel146912293up_add: $i > $o ).
thf(preorder_decl,type,
preorder: $i > $o ).
thf(ordere223160158up_add_decl,type,
ordere223160158up_add: $i > $o ).
thf(ordere236663937imp_le_decl,type,
ordere236663937imp_le: $i > $o ).
thf(na_decl,type,
na: $i ).
thf(ordere216010020id_add_decl,type,
ordere216010020id_add: $i > $o ).
thf(ordere779506340up_add_decl,type,
ordere779506340up_add: $i > $o ).
thf(ord_decl,type,
ord: $i > $o ).
thf(finite_finite_1_decl,type,
finite_finite_1: $i > $o ).
thf(sum_sum_decl,type,
sum_sum: $i > $i > $i ).
thf(linorder_decl,type,
linorder: $i > $o ).
thf(linordered_semidom_decl,type,
linordered_semidom: $i > $o ).
thf(ab_semigroup_add_decl,type,
ab_semigroup_add: $i > $o ).
thf(comm_monoid_add_decl,type,
comm_monoid_add: $i > $o ).
thf(collect_decl,type,
collect: $i > $i ).
thf(combs_decl,type,
combs: $i > $i > $i > $i ).
thf(combb_decl,type,
combb: $i > $i > $i > $i ).
thf(fconj_decl,type,
fconj: $i ).
thf(fequal_decl,type,
fequal: $i > $i ).
thf(finite_finite_decl,type,
finite_finite: $i > $i ).
thf(member_decl,type,
member: $i > $i ).
thf(combc_decl,type,
combc: $i > $i > $i > $i ).
thf(ord_less_decl,type,
ord_less: $i > $i ).
thf(plus_plus_decl,type,
plus_plus: $i > $i ).
thf(minus_minus_decl,type,
minus_minus: $i > $i ).
thf(suc_decl,type,
suc: $i ).
thf(fFalse_decl,type,
fFalse: $i ).
thf(sum_Plus_decl,type,
sum_Plus: $i > $i > $i ).
thf(fNot_decl,type,
fNot: $i ).
thf(finite_card_decl,type,
finite_card: $i > $i ).
thf(finite2073411215e_idem_decl,type,
finite2073411215e_idem: $i > $i ).
thf(nat_case_decl,type,
nat_case: $i > $i ).
thf(linord219039673up_add_decl,type,
linord219039673up_add: $i > $o ).
thf(fdisj_decl,type,
fdisj: $i ).
thf(the_decl,type,
the: $i > $i ).
thf(com_decl,type,
com: $i ).
thf(wt_decl,type,
wt: $i ).
thf(p_decl,type,
p: $i ).
thf(mgt_decl,type,
mgt: $i ).
thf(ordered_ab_group_add_decl,type,
ordered_ab_group_add: $i > $o ).
thf(finite1357897459simple_decl,type,
finite1357897459simple: $i > $i > $i ).
thf(dense_linorder_decl,type,
dense_linorder: $i > $o ).
thf(fimplies_decl,type,
fimplies: $i ).
thf(ab_group_add_decl,type,
ab_group_add: $i > $o ).
thf(group_add_decl,type,
group_add: $i > $o ).
thf(combi_decl,type,
combi: $i > $i ).
thf(combk_decl,type,
combk: $i > $i > $i ).
thf(finite_folding_one_decl,type,
finite_folding_one: $i > $i ).
thf(the_elem_decl,type,
the_elem: $i > $i ).
thf(option_decl,type,
option: $i > $i ).
thf(the_1_decl,type,
the_1: $i > $i ).
thf(body_decl,type,
body: $i ).
thf(finite_comp_fun_idem_decl,type,
finite_comp_fun_idem: $i > $i > $i ).
thf(partial_flat_lub_decl,type,
partial_flat_lub: $i > $i ).
thf(fTrue_decl,type,
fTrue: $i ).
thf(linordered_idom_decl,type,
linordered_idom: $i > $o ).
thf(17,axiom,
nat @ cancel_semigroup_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(645,plain,
nat @ cancel_semigroup_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(204,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
=> ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_finite__induct) ).
thf(1322,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
=> ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(606,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_insert__mono) ).
thf(2822,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[606]) ).
thf(510,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_335_linorder__neqE) ).
thf(2452,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[510]) ).
thf(208,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_390_less__add__Suc2) ).
thf(1337,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(412,axiom,
! [A: $i,B: $i] :
( ( A = B )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_eq__imp__le) ).
thf(2094,plain,
! [A: $i,B: $i] :
( ( A = B )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[412]) ).
thf(587,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_folding__image__simple_Oremove) ).
thf(2744,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[587]) ).
thf(596,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( ( A
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
| ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_le__Suc__eq) ).
thf(2778,plain,
! [A: $i,B: $i] :
( ( ( ( A
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
| ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
| ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[596]) ).
thf(321,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [E: $i,F: $i] :
( ( F @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( F @ ( E @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( F @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( F @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) )
=> ( D @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_finite__subset__induct) ).
thf(1767,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [E: $i,F: $i] :
( ( F @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( F @ ( E @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( F @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( F @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) )
=> ( D @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[321]) ).
thf(258,axiom,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_add__left__mono) ).
thf(1559,plain,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(129,axiom,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( C @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) )
<=> ( ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) )
& ( ( C @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_462_add__nonneg__eq__0__iff) ).
thf(1031,plain,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( C @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) )
& ( ( C @ ( A @ ti ) )
= ( A @ zero_zero ) ) )
=> ( ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) )
& ( ( ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) )
=> ( ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) )
& ( ( C @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(96,axiom,
! [A: $i] :
( ( A @ dense_linorder )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( C @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_431_dense__le) ).
thf(893,plain,
! [A: $i] :
( ( A @ dense_linorder )
=> ! [B: $i,C: $i] :
( ! [D: $i] :
( ( C @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(519,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( A @ hBOOL )
| ~ ( B @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_1_1_U) ).
thf(2480,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( A @ hBOOL )
| ~ ( B @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[519]) ).
thf(108,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_diff__add__inverse) ).
thf(941,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(65,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_373_leD) ).
thf(777,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(335,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_299_trans__less__add2) ).
thf(1818,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[335]) ).
thf(460,axiom,
! [A: $i] :
( ( A @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ( A @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fNot_2_1_U) ).
thf(2280,plain,
! [A: $i] :
( ( A @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ( A @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[460]) ).
thf(302,axiom,
! [A: $i] :
( ( A @ ( bool @ ti ) @ hBOOL )
<=> ( A @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hBOOL_arg1) ).
thf(1686,plain,
! [A: $i] :
( ( ( A @ hBOOL )
=> ( A @ ( bool @ ti ) @ hBOOL ) )
& ( ( A @ ( bool @ ti ) @ hBOOL )
=> ( A @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[302]) ).
thf(553,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( fFalse @ ( bool @ nat_case @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_less__eq__nat_Osimps_I2_J) ).
thf(2612,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( fFalse @ ( bool @ nat_case @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( fFalse @ ( bool @ nat_case @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[553]) ).
thf(236,axiom,
bool @ finite_finite_1,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Finite__Set_Ofinite) ).
thf(1469,plain,
bool @ finite_finite_1,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(162,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_441_finite__Plus__iff) ).
thf(1168,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(58,axiom,
! [A: $i] :
~ ( A @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_353_less__not__refl) ).
thf(760,plain,
! [A: $i] :
~ ( A @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(477,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_422_lift__Suc__mono__less__iff) ).
thf(2341,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[477]) ).
thf(415,axiom,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( C @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_461_add__nonneg__nonneg) ).
thf(2109,plain,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( C @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[415]) ).
thf(418,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_image__constant) ).
thf(2131,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[418]) ).
thf(547,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_375_linorder__antisym__conv2) ).
thf(2595,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[547]) ).
thf(524,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_diff__commute) ).
thf(2499,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[524]) ).
thf(225,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( F @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( G @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_439_order__less__le__subst2) ).
thf(1406,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( F @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( G @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(48,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( bool @ ( A @ fun ) @ bot_bot )
= ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) )
<=> ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_empty__is__image) ).
thf(730,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( bool @ ( A @ fun ) @ bot_bot )
= ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot )
= ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) )
=> ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(602,axiom,
( ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fconj ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fconj_res) ).
thf(2808,plain,
( ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fconj ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[602]) ).
thf(489,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_diff__Suc__Suc) ).
thf(2381,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[489]) ).
thf(492,axiom,
( ( suc @ ( nat @ ( nat @ fun ) @ ti ) )
= suc ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Nat_OSuc_res) ).
thf(2395,plain,
( ( suc @ ( nat @ ( nat @ fun ) @ ti ) )
= suc ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[492]) ).
thf(566,axiom,
bool @ bot,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Obot) ).
thf(2659,plain,
bool @ bot,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[566]) ).
thf(279,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_269_diff__add__assoc2) ).
thf(1618,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(530,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ bot_bot ) )
<=> ( ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
& ( ( C @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_445_Plus__eq__empty__conv) ).
thf(2527,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
& ( ( C @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) )
=> ( ( D @ ( C @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ bot_bot ) ) )
& ( ( ( D @ ( C @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ bot_bot ) )
=> ( ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
& ( ( C @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[530]) ).
thf(545,axiom,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_477_diff__0__right) ).
thf(2590,plain,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[545]) ).
thf(374,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
& ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_416_psubset__insert__iff) ).
thf(1939,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
& ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
& ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[374]) ).
thf(50,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_315_less__trans__Suc) ).
thf(741,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(255,axiom,
! [A: $i] :
( ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) )
= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_404_less__eq__Suc__le__raw) ).
thf(1544,plain,
! [A: $i] :
( ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) )
= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(461,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( B @ ( A @ hAPP ) ) ) @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_le__funE) ).
thf(2282,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( B @ ( A @ hAPP ) ) ) @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[461]) ).
thf(152,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_DiffE) ).
thf(1142,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(24,axiom,
nat @ ordere216010020id_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__comm__monoid__add) ).
thf(655,plain,
nat @ ordere216010020id_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(241,axiom,
~ ( g @ ( pn @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ member @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_5) ).
thf(1486,plain,
~ ( g @ ( pn @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ member @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(368,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ! [B: $i] : ( A @ one_one @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_423_less__add__one) ).
thf(1924,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ! [B: $i] : ( A @ one_one @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[368]) ).
thf(440,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_singleton__inject) ).
thf(2210,plain,
! [A: $i,B: $i,C: $i] :
( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[440]) ).
thf(446,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( G @ ( F @ ( E @ ( D @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [H: $i] :
( ( C @ ( H @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( H @ ( F @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( B @ ti ) ) ) )
=> ( ( C @ ( G @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( B @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_folding__image__simple_Oneutral) ).
thf(2231,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i] :
( ( G @ ( F @ ( E @ ( D @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [H: $i] :
( ( C @ ( H @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( H @ ( F @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( B @ ti ) ) ) )
=> ( ( C @ ( G @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( B @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[446]) ).
thf(555,axiom,
( ( g @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) )
= ( na @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_3) ).
thf(2622,plain,
( ( g @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) )
= ( na @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[555]) ).
thf(136,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
<=> ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_insert__ident) ).
thf(1068,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
=> ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(5,axiom,
nat @ zero_neq_one,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Ozero__neq__one) ).
thf(614,plain,
nat @ zero_neq_one,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(303,axiom,
! [A: $i] : ( A @ insert @ ( bool @ ( A @ fun ) @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_268_comp__fun__idem__insert) ).
thf(1692,plain,
! [A: $i] : ( A @ insert @ ( bool @ ( A @ fun ) @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[303]) ).
thf(433,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_342_order__less__asym_H) ).
thf(2195,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[433]) ).
thf(292,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_insertE) ).
thf(1655,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(150,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_add__Suc__right) ).
thf(1137,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(576,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
!= ( B @ ( A @ ti ) ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_378_xt1_I11_J) ).
thf(2686,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
!= ( B @ ( A @ ti ) ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[576]) ).
thf(201,axiom,
! [A: $i,B: $i] : ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_389_less__add__Suc1) ).
thf(1310,plain,
! [A: $i,B: $i] : ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(422,axiom,
! [A: $i] :
( ( nat @ zero_zero @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_475_minus__nat_Odiff__0) ).
thf(2141,plain,
! [A: $i] :
( ( nat @ zero_zero @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[422]) ).
thf(351,axiom,
bool @ minus,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Groups_Ominus) ).
thf(1872,plain,
bool @ minus,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[351]) ).
thf(178,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_287_termination__basic__simps_I4_J) ).
thf(1233,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(541,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_372_xt1_I12_J) ).
thf(2580,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[541]) ).
thf(397,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( C @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) )
=> ( B @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_406_inc__induct) ).
thf(2042,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( C @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) )
=> ( B @ ( A @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[397]) ).
thf(215,axiom,
! [A: $i,B: $i] :
~ ( A @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_303_not__add__less1) ).
thf(1354,plain,
! [A: $i,B: $i] :
~ ( A @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(234,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_add__Suc__shift) ).
thf(1458,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(366,axiom,
! [A: $i] : ( A @ ( nat @ zero_zero @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_490_less__eq__nat_Osimps_I1_J) ).
thf(1919,plain,
! [A: $i] : ( A @ ( nat @ zero_zero @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[366]) ).
thf(383,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( H @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_433_xt4) ).
thf(1988,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( H @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[383]) ).
thf(53,axiom,
! [A: $i] :
( ( A @ minus )
=> ( ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ minus_minus ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ominus__class_Ominus_res) ).
thf(747,plain,
! [A: $i] :
( ( A @ minus )
=> ( ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ minus_minus ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(101,axiom,
! [A: $i] :
( ( A @ ( nat @ zero_zero @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_476_diff__0__eq__0) ).
thf(913,plain,
! [A: $i] :
( ( A @ ( nat @ zero_zero @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(140,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_334_linorder__antisym__conv3) ).
thf(1095,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(340,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_336_less__imp__neq) ).
thf(1835,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[340]) ).
thf(505,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_Diff__insert2) ).
thf(2432,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[505]) ).
thf(117,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
<=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_471_diff__is__0__eq) ).
thf(973,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) )
& ( ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(373,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_subset__insert) ).
thf(1935,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[373]) ).
thf(367,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_le__SucE) ).
thf(1921,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[367]) ).
thf(311,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ? [F: $i] :
( ( ( C @ ( A @ ti ) )
= ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
& ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_image__iff) ).
thf(1724,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( ( C @ ( A @ ti ) )
= ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
& ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [F: $i] :
( ( ( C @ ( A @ ti ) )
= ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
& ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[311]) ).
thf(356,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( C @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_subsetI) ).
thf(1887,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i] :
( ( C @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[356]) ).
thf(21,axiom,
nat @ ordere223160158up_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(651,plain,
nat @ ordere223160158up_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(445,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_finite__Diff2) ).
thf(2227,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[445]) ).
thf(265,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_294_termination__basic__simps_I2_J) ).
thf(1576,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(605,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_insert__absorb) ).
thf(2819,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[605]) ).
thf(331,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_403_less__diff__iff) ).
thf(1803,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[331]) ).
thf(362,axiom,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_254_diff__cancel2) ).
thf(1904,plain,
! [A: $i,B: $i,C: $i] :
( ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[362]) ).
thf(188,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ finite1357897459simple ) @ ( bool @ ( A @ ( bool @ ( B @ fun ) @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ fun ) @ ( A @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ finite1357897459simple ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__image__simple_res) ).
thf(1265,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ finite1357897459simple ) @ ( bool @ ( A @ ( bool @ ( B @ fun ) @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ fun ) @ ( A @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ finite1357897459simple ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(176,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_407_less__imp__Suc__add) ).
thf(1228,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(281,axiom,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_311_diff__eq__diff__less) ).
thf(1623,plain,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(184,axiom,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_463_add__increasing) ).
thf(1254,plain,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(564,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_equals0D) ).
thf(2654,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[564]) ).
thf(584,axiom,
! [A: $i] :
( ( u @ ( A @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( body @ ( com @ option @ ( pname @ hAPP ) ) ) @ ( com @ the_1 @ ( com @ ( com @ option @ hAPP ) ) ) @ ( wt @ ( bool @ ( com @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_assms_I4_J) ).
thf(2735,plain,
! [A: $i] :
( ( u @ ( A @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( body @ ( com @ option @ ( pname @ hAPP ) ) ) @ ( com @ the_1 @ ( com @ ( com @ option @ hAPP ) ) ) @ ( wt @ ( bool @ ( com @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[584]) ).
thf(318,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_313_Suc__lessD) ).
thf(1760,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[318]) ).
thf(67,axiom,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_464_add__increasing2) ).
thf(785,plain,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(456,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_subsetD) ).
thf(2269,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[456]) ).
thf(363,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
=> ( ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_473_diffs0__imp__equal) ).
thf(1907,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
=> ( ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[363]) ).
thf(8,axiom,
! [A: $i] :
( ( A @ zero_neq_one )
=> ( ( A @ one_one )
!= ( A @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_457_one__neq__zero) ).
thf(619,plain,
! [A: $i] :
( ( A @ zero_neq_one )
=> ( ( A @ one_one )
!= ( A @ zero_zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(71,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_262_add__le__mono) ).
thf(802,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(384,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( B @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_425_bounded__nat__set__is__finite) ).
thf(1992,plain,
! [A: $i,B: $i] :
( ! [C: $i] :
( ( B @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[384]) ).
thf(514,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_273_le__add__diff__inverse2) ).
thf(2466,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[514]) ).
thf(482,axiom,
! [A: $i,B: $i] :
( ( B @ bot )
=> ( B @ ( A @ fun ) @ bot ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Orderings_Obot) ).
thf(2356,plain,
! [A: $i,B: $i] :
( ( B @ bot )
=> ( B @ ( A @ fun ) @ bot ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[482]) ).
thf(249,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ! [D: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( E @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ~ ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) )
=> ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_finite__ne__induct) ).
thf(1511,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ! [D: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( E @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ~ ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) )
=> ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(458,axiom,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_355_linorder__neqE__nat) ).
thf(2274,plain,
! [A: $i,B: $i] :
( ( A != B )
=> ( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[458]) ).
thf(76,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_xt1_I3_J) ).
thf(817,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(306,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_309_add__less__cancel__left) ).
thf(1704,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[306]) ).
thf(592,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_283_diff__Suc__diff__eq2) ).
thf(2767,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[592]) ).
thf(276,axiom,
! [A: $i] :
( ( A @ ord )
=> ( ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ ord_less_eq ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless__eq_res) ).
thf(1609,plain,
! [A: $i] :
( ( A @ ord )
=> ( ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ ord_less_eq ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(145,axiom,
( ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ti ) )
= p ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_P_res) ).
thf(1115,plain,
( ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ti ) )
= p ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(426,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_317_less__SucI) ).
thf(2158,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[426]) ).
thf(392,axiom,
! [A: $i,B: $i,C: $i] : ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_subset__insertI) ).
thf(2024,plain,
! [A: $i,B: $i,C: $i] : ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[392]) ).
thf(467,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_412_psubset__eq) ).
thf(2301,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[467]) ).
thf(488,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
<=> ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_Collect__empty__eq) ).
thf(2373,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[488]) ).
thf(264,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_358_less__not__refl3) ).
thf(1572,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(371,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fdisj_1_1_U) ).
thf(1931,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[371]) ).
thf(262,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ combc ) ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ collect @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ finite_finite @ ( bool @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1_finite__Collect__subsets) ).
thf(1568,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ combc ) ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ collect @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ finite_finite @ ( bool @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(353,axiom,
! [A: $i,B: $i] :
( ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_323_less__or__eq__imp__le) ).
thf(1875,plain,
! [A: $i,B: $i] :
( ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[353]) ).
thf(323,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_278_le__diff__conv) ).
thf(1778,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[323]) ).
thf(443,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fequal_2_1_T) ).
thf(2222,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[443]) ).
thf(447,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_401_less__diff__conv) ).
thf(2237,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[447]) ).
thf(471,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_set__mp) ).
thf(2324,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[471]) ).
thf(300,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_imageI) ).
thf(1682,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(298,axiom,
! [A: $i,B: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_Diff__empty) ).
thf(1677,plain,
! [A: $i,B: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(73,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_350_order__less__asym) ).
thf(806,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(475,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_card__image__le) ).
thf(2335,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[475]) ).
thf(133,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( ( A != B )
& ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_327_nat__less__le) ).
thf(1051,plain,
! [A: $i,B: $i] :
( ( ( ( A != B )
& ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A != B )
& ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(198,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( B @ ( C @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_folding__one__idem_Oidem) ).
thf(1302,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( B @ ( C @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(295,axiom,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( ( A @ zero_zero )
= ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
<=> ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_494_double__zero__sym) ).
thf(1664,plain,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) )
=> ( ( A @ zero_zero )
= ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( A @ zero_zero )
= ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(414,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_481_Suc__neq__Zero) ).
thf(2105,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[414]) ).
thf(333,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_320_Suc__less__eq) ).
thf(1811,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[333]) ).
thf(144,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_finite__insert) ).
thf(1109,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(283,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_291_Suc__mono) ).
thf(1632,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(515,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_295_add__lessD1) ).
thf(2469,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[515]) ).
thf(207,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_Nat_Odiff__diff__eq) ).
thf(1334,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(18,axiom,
nat @ cancel146912293up_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).
thf(646,plain,
nat @ cancel146912293up_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(345,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_339_order__less__imp__not__eq) ).
thf(1858,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[345]) ).
thf(347,axiom,
( ( mgt_call @ ( x_a @ ( pname @ fun ) @ ti ) )
= mgt_call ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_mgt__call_res) ).
thf(1864,plain,
( ( mgt_call @ ( x_a @ ( pname @ fun ) @ ti ) )
= mgt_call ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[347]) ).
thf(155,axiom,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_306_add__strict__mono) ).
thf(1151,plain,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(107,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_484_Suc__not__Zero) ).
thf(937,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(166,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fNot @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_set__diff__eq) ).
thf(1183,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fNot @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(404,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_417_card__psubset) ).
thf(2073,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[404]) ).
thf(486,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_nat__add__assoc) ).
thf(2367,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[486]) ).
thf(554,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_274_le__diff__conv2) ).
thf(2618,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[554]) ).
thf(12,axiom,
! [A: $i] :
( ( A @ undefined @ ( A @ ti ) )
= ( A @ undefined ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_HOL_Oundefined_res) ).
thf(634,plain,
! [A: $i] :
( ( A @ undefined @ ( A @ ti ) )
= ( A @ undefined ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(502,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_263_add__leD2) ).
thf(2424,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[502]) ).
thf(594,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_image__empty) ).
thf(2772,plain,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[594]) ).
thf(214,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_card__insert__disjoint) ).
thf(1351,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(28,axiom,
nat @ linorder,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Olinorder) ).
thf(660,plain,
nat @ linorder,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(589,axiom,
( ( u @ ( bool @ ( pname @ fun ) @ ti ) )
= u ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_U_res) ).
thf(2750,plain,
( ( u @ ( bool @ ( pname @ fun ) @ ti ) )
= u ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[589]) ).
thf(272,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_order__trans) ).
thf(1600,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(491,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_equalityD1) ).
thf(2392,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[491]) ).
thf(118,axiom,
( ( mgt @ ( x_a @ ( com @ fun ) @ ti ) )
= mgt ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_mgt_res) ).
thf(981,plain,
( ( mgt @ ( x_a @ ( com @ fun ) @ ti ) )
= mgt ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(60,axiom,
! [A: $i,B: $i,C: $i] :
( ( nat @ zero_zero @ ( C @ ( B @ ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( A @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_491_nat__case__0) ).
thf(766,plain,
! [A: $i,B: $i,C: $i] :
( ( nat @ zero_zero @ ( C @ ( B @ ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( A @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(93,axiom,
! [A: $i,B: $i] :
( ? [C: $i] :
( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
& ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_theI_H) ).
thf(877,plain,
! [A: $i,B: $i] :
( ? [C: $i] :
( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
& ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(37,axiom,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_308_add__strict__right__mono) ).
thf(692,plain,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(251,axiom,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_459_zero__le__double__add__iff__zero__le__single__add) ).
thf(1526,plain,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(420,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_finite__Diff) ).
thf(2137,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[420]) ).
thf(599,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ! [F: $i] :
( ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( F @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_image__subsetI) ).
thf(2799,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ! [F: $i] :
( ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( F @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[599]) ).
thf(560,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_322_not__less__eq) ).
thf(2639,plain,
! [A: $i,B: $i] :
( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[560]) ).
thf(142,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_insert__Diff) ).
thf(1104,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(305,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ~ ( C @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_414_less__fun__def) ).
thf(1697,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ( ( ~ ( C @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[305]) ).
thf(581,axiom,
! [A: $i,B: $i] :
( ( B @ bot )
=> ! [C: $i] :
( ( C @ ( B @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ hAPP ) ) ) )
= ( B @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_bot__fun__def) ).
thf(2718,plain,
! [A: $i,B: $i] :
( ( B @ bot )
=> ! [C: $i] :
( ( C @ ( B @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ hAPP ) ) ) )
= ( B @ bot_bot ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[581]) ).
thf(438,axiom,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i] :
( ( B @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_478_diff__self) ).
thf(2205,plain,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i] :
( ( B @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[438]) ).
thf(171,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ combi @ ( A @ ( A @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_image__ident) ).
thf(1206,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ combi @ ( A @ ( A @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(29,axiom,
nat @ linordered_semidom,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Rings_Olinordered__semidom) ).
thf(661,plain,
nat @ linordered_semidom,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(69,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_363_linorder__not__le) ).
thf(793,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(199,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( E @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_Diff__mono) ).
thf(1306,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( E @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(485,axiom,
! [A: $i,B: $i] :
~ ( B @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_302_not__add__less2) ).
thf(2364,plain,
! [A: $i,B: $i] :
~ ( B @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[485]) ).
thf(156,axiom,
! [A: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_finite_OemptyI) ).
thf(1153,plain,
! [A: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(398,axiom,
! [A: $i] :
( ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ member ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_member_res) ).
thf(2047,plain,
! [A: $i] :
( ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ member ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[398]) ).
thf(237,axiom,
! [A: $i] :
~ ( nat @ zero_zero @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_447_less__zeroE) ).
thf(1470,plain,
! [A: $i] :
~ ( nat @ zero_zero @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(585,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( B = C )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_order__eq__refl) ).
thf(2737,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( B = C )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[585]) ).
thf(131,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_singletonE) ).
thf(1047,plain,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(137,axiom,
! [A: $i] :
( ( A @ ab_group_add )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
<=> ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_479_eq__iff__diff__eq__0) ).
thf(1074,plain,
! [A: $i] :
( ( A @ ab_group_add )
=> ! [B: $i,C: $i] :
( ( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
& ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(274,axiom,
! [A: $i] :
( ( nat @ one_one @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_diff__Suc__1) ).
thf(1604,plain,
! [A: $i] :
( ( nat @ one_one @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(408,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( C @ ( A @ combc ) ) @ ( B @ ( A @ fun ) @ ( C @ fun ) @ ( B @ ( C @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( F @ ( D @ ( B @ ( C @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( C @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBC_1_1_U) ).
thf(2083,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( C @ ( A @ combc ) ) @ ( B @ ( A @ fun ) @ ( C @ fun ) @ ( B @ ( C @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( F @ ( D @ ( B @ ( C @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( C @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[408]) ).
thf(313,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( H @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_436_xt7) ).
thf(1736,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( H @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[313]) ).
thf(567,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ fequal @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ ( A @ fun ) @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_the__elem__def) ).
thf(2660,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ fequal @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ ( A @ fun ) @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[567]) ).
thf(44,axiom,
! [A: $i] :
( ( A @ monoid_add )
=> ! [B: $i] :
( ( B @ ( A @ zero_zero @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_492_add__0__left) ).
thf(714,plain,
! [A: $i] :
( ( A @ monoid_add )
=> ! [B: $i] :
( ( B @ ( A @ zero_zero @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(349,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_finite__subset) ).
thf(1868,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[349]) ).
thf(386,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( B @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_card__Diff__insert) ).
thf(2000,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( B @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[386]) ).
thf(285,axiom,
u @ ( pn @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_4) ).
thf(1636,plain,
u @ ( pn @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(498,axiom,
( ( fTrue @ ( bool @ ti ) )
= fTrue ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fTrue_res) ).
thf(2409,plain,
( ( fTrue @ ( bool @ ti ) )
= fTrue ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[498]) ).
thf(548,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_281_Suc__eq__plus1__left) ).
thf(2601,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[548]) ).
thf(125,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
| ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_nat__le__linear) ).
thf(1021,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
| ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(459,axiom,
! [A: $i] :
( ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ the ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_HOL_OThe_res) ).
thf(2277,plain,
! [A: $i] :
( ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ the ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[459]) ).
thf(263,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ( A @ one_one @ ( A @ zero_zero @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_452_zero__less__one) ).
thf(1570,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ( A @ one_one @ ( A @ zero_zero @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(424,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( B = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_nat__add__left__cancel) ).
thf(2147,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B = C )
=> ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[424]) ).
thf(550,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_305_add__less__imp__less__right) ).
thf(2606,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[550]) ).
thf(180,axiom,
! [A: $i] :
( ( bool @ ( A @ fun ) @ bot_bot )
= ( fFalse @ ( A @ ( bool @ combk ) @ ( bool @ ( A @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_empty__def) ).
thf(1238,plain,
! [A: $i] :
( ( bool @ ( A @ fun ) @ bot_bot )
= ( fFalse @ ( A @ ( bool @ combk ) @ ( bool @ ( A @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(190,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_DiffI) ).
thf(1274,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(572,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_442_finite__Plus) ).
thf(2673,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[572]) ).
thf(221,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_367_order__le__less) ).
thf(1387,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(100,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_279_diff__diff__right) ).
thf(910,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(113,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_insert__commute) ).
thf(957,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(174,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
=> ( F @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_rev__image__eqI) ).
thf(1223,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
=> ( F @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(211,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_318_less__antisym) ).
thf(1343,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B = A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(601,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( B @ ( B @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) )
= ( D @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_comp__fun__idem_Ofun__left__idem) ).
thf(2805,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( B @ ( B @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) )
= ( D @ ( C @ ( E @ ( B @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[601]) ).
thf(316,axiom,
! [A: $i,B: $i] :
( ( B @ ( bool @ ( A @ fun ) @ bot_bot @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_empty__Diff) ).
thf(1751,plain,
! [A: $i,B: $i] :
( ( B @ ( bool @ ( A @ fun ) @ bot_bot @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[316]) ).
thf(77,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_image__mono) ).
thf(820,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(338,axiom,
! [A: $i,B: $i] :
( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_Suc__inject) ).
thf(1824,plain,
! [A: $i,B: $i] :
( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[338]) ).
thf(218,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
<=> ( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_331_linorder__neq__iff) ).
thf(1370,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) ) )
& ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(513,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_289_card__Diff1__less) ).
thf(2464,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[513]) ).
thf(273,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_finite__surj) ).
thf(1602,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(164,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_le__SucI) ).
thf(1178,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(32,axiom,
! [A: $i] :
( ( A @ zero )
=> ! [B: $i] :
( ( ( A @ zero_zero )
= ( B @ ( A @ ti ) ) )
<=> ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_487_zero__reorient) ).
thf(665,plain,
! [A: $i] :
( ( A @ zero )
=> ! [B: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) )
=> ( ( A @ zero_zero )
= ( B @ ( A @ ti ) ) ) )
& ( ( ( A @ zero_zero )
= ( B @ ( A @ ti ) ) )
=> ( ( B @ ( A @ ti ) )
= ( A @ zero_zero ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(402,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_Suc__diff__le) ).
thf(2068,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[402]) ).
thf(159,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fNot @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fimplies @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_insert__Collect) ).
thf(1159,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fNot @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fimplies @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(441,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_order__antisym__conv) ).
thf(2213,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[441]) ).
thf(478,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( ( C @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_265_add__leE) ).
thf(2345,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( ( C @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[478]) ).
thf(284,axiom,
! [A: $i] :
( ( A @ finite_finite_1 )
=> ! [B: $i] : ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_finite) ).
thf(1634,plain,
! [A: $i] :
( ( A @ finite_finite_1 )
=> ! [B: $i] : ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(42,axiom,
~ ( fFalse @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fFalse_1_1_U) ).
thf(710,plain,
~ ( fFalse @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(102,axiom,
! [A: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ zero_zero @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_449_zero__less__Suc) ).
thf(916,plain,
! [A: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ zero_zero @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(508,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_insertCI) ).
thf(2445,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[508]) ).
thf(558,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_flat__lub__def) ).
thf(2628,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[558]) ).
thf(571,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Collect__def) ).
thf(2670,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( bool @ ( A @ fun ) @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[571]) ).
thf(10,axiom,
nat @ monoid_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Omonoid__add) ).
thf(625,plain,
nat @ monoid_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(518,axiom,
! [A: $i] :
( ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_card @ ( nat @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) )
= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_card__Collect__le__nat) ).
thf(2477,plain,
! [A: $i] :
( ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_card @ ( nat @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) )
= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[518]) ).
thf(568,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_259_trans__le__add1) ).
thf(2663,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[568]) ).
thf(474,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_450_card__infinite) ).
thf(2332,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[474]) ).
thf(608,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_419_lift__Suc__mono__less) ).
thf(2826,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[608]) ).
thf(436,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ( B @ combk ) @ ( B @ ( A @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( C @ ( B @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBK_1_1_U) ).
thf(2200,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ( B @ combk ) @ ( B @ ( A @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( C @ ( B @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[436]) ).
thf(80,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_assms_I1_J) ).
thf(830,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(35,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [E: $i] :
( ~ ( E @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ fequal @ ( B @ ( bool @ ( A @ fun ) @ ( A @ combb ) ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ ( A @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ combc ) ) @ ( bool @ ( B @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ ( B @ combc ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( fconj @ ( B @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ combs ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ collect @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( E @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_pigeonhole__infinite) ).
thf(682,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ~ ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [E: $i] :
( ~ ( E @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ fequal @ ( B @ ( bool @ ( A @ fun ) @ ( A @ combb ) ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ ( A @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ combc ) ) @ ( bool @ ( B @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ ( B @ combc ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( fconj @ ( B @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ combs ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ collect @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( E @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(169,axiom,
! [A: $i,B: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_subset__empty) ).
thf(1195,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(193,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_371_order__neq__le__trans) ).
thf(1281,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(586,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_card__Diff__singleton) ).
thf(2741,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[586]) ).
thf(314,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_Suc__n__not__n) ).
thf(1740,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[314]) ).
thf(364,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( C @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( C @ fun ) @ fun ) @ ( A @ ( C @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( C @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ combb ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ hAPP ) ) ) @ ( B @ ( C @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( C @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( C @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_image__image) ).
thf(1910,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( C @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( C @ fun ) @ fun ) @ ( A @ ( C @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( C @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( F @ ( E @ ( D @ ( C @ ( B @ ( A @ combb ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ hAPP ) ) ) @ ( B @ ( C @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( C @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( C @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[364]) ).
thf(224,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
| ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_finite__Collect__conjI) ).
thf(1402,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
| ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(270,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_338_order__less__imp__not__less) ).
thf(1592,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(254,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_470_diff__is__0__eq_H) ).
thf(1541,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(381,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_card_Oinsert__remove) ).
thf(1983,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[381]) ).
thf(319,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_insert__compr__raw) ).
thf(1762,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[319]) ).
thf(186,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ( B @ ( A @ fun ) @ ord ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Orderings_Oord) ).
thf(1259,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ( B @ ( A @ fun ) @ ord ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(6,axiom,
nat @ minus,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ominus) ).
thf(615,plain,
nat @ minus,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(205,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ! [E: $i,F: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( F @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( F @ ( E @ ( C @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( B @ ( D @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_folding__one_Oclosed) ).
thf(1328,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ! [E: $i,F: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( F @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( F @ ( E @ ( C @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( B @ ( D @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(46,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_xt1_I4_J) ).
thf(725,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(268,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_set__rev__mp) ).
thf(1587,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(578,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ? [C: $i,D: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
| ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_finite_Osimps) ).
thf(2694,plain,
! [A: $i,B: $i] :
( ( ( ? [C: $i,D: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
| ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ? [C: $i,D: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
| ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[578]) ).
thf(120,axiom,
! [A: $i,B: $i] :
( ( B @ minus )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( C @ ( B @ ( A @ fun ) @ minus_minus @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ minus_minus @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_minus__apply) ).
thf(989,plain,
! [A: $i,B: $i] :
( ( B @ minus )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( C @ ( B @ ( A @ fun ) @ minus_minus @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ minus_minus @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(222,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_add__le__imp__le__left) ).
thf(1397,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(540,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
!= ( B @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_340_order__less__imp__not__eq2) ).
thf(2577,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
!= ( B @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[540]) ).
thf(135,axiom,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D = B )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_346_ord__less__eq__trans) ).
thf(1064,plain,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D = B )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(526,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_card__Diff1__le) ).
thf(2509,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[526]) ).
thf(296,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ fun ) @ ti ) )
= ( B @ ( A @ combk ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBK_res) ).
thf(1672,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ fun ) @ ti ) )
= ( B @ ( A @ combk ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(583,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( ( B = C )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( ( B @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_359_nat__less__cases) ).
thf(2724,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( ( B = C )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( ( B @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( C @ ( A @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[583]) ).
thf(309,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combs ) ) @ ( C @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( C @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combs ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBS_res) ).
thf(1719,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combs ) ) @ ( C @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( C @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combs ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[309]) ).
thf(531,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
<=> ( ( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) ) )
| ( ( ( C @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_doubleton__eq__iff) ).
thf(2537,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ( ( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) ) )
| ( ( ( C @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( E @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
=> ( ( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) ) )
| ( ( ( C @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) )
& ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[531]) ).
thf(244,axiom,
! [A: $i] :
( ( A @ monoid_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_495_add__0__right) ).
thf(1493,plain,
! [A: $i] :
( ( A @ monoid_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(377,axiom,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
<=> ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_add__right__cancel) ).
thf(1964,plain,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[377]) ).
thf(534,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_275_add__diff__assoc) ).
thf(2561,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[534]) ).
thf(235,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( A = C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_nat__add__right__cancel) ).
thf(1461,plain,
! [A: $i,B: $i,C: $i] :
( ( ( A = C )
=> ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(407,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_469_diff__add__0) ).
thf(2080,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[407]) ).
thf(342,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot )
!= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_empty__not__insert) ).
thf(1842,plain,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot )
!= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[342]) ).
thf(479,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_finite__imageI) ).
thf(2349,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[479]) ).
thf(128,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_DiffD1) ).
thf(1028,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(138,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
<=> ? [C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_nonempty__iff) ).
thf(1082,plain,
! [A: $i,B: $i] :
( ( ? [C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ? [C: $i,D: $i] :
( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(98,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_insert__Diff1) ).
thf(901,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(36,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ! [E: $i] : ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_le__fun__def) ).
thf(686,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ( ! [E: $i] : ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ! [E: $i] : ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(228,axiom,
! [A: $i,B: $i] :
( ! [C: $i] :
~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_all__not__in__conv) ).
thf(1424,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ! [C: $i] :
~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ! [C: $i] :
~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(511,axiom,
! [A: $i] :
( ( A @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_card @ ( nat @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_385_card__Collect__less__nat) ).
thf(2455,plain,
! [A: $i] :
( ( A @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_card @ ( nat @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[511]) ).
thf(13,axiom,
! [A: $i] :
( ( A @ zero )
=> ( ( A @ zero_zero @ ( A @ ti ) )
= ( A @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Ozero__class_Ozero_res) ).
thf(637,plain,
! [A: $i] :
( ( A @ zero )
=> ( ( A @ zero_zero @ ( A @ ti ) )
= ( A @ zero_zero ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(267,axiom,
! [A: $i,B: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_the__elem__eq) ).
thf(1584,plain,
! [A: $i,B: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(277,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_le__trans) ).
thf(1612,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(130,axiom,
! [A: $i,B: $i] :
( ? [C: $i] : ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_ex__in__conv) ).
thf(1039,plain,
! [A: $i,B: $i] :
( ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ? [C: $i] : ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ? [C: $i] : ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(157,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_352_termination__basic__simps_I5_J) ).
thf(1155,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(315,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_finite__Collect__disjI) ).
thf(1744,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[315]) ).
thf(19,axiom,
nat @ preorder,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Opreorder) ).
thf(647,plain,
nat @ preorder,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(509,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_diff__le__self) ).
thf(2450,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[509]) ).
thf(528,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_454_pos__add__strict) ).
thf(2519,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( A @ zero_zero @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[528]) ).
thf(610,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_381_order__le__less__trans) ).
thf(2836,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[610]) ).
thf(522,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ~ ( A @ zero_zero @ ( A @ one_one @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_455_not__one__le__zero) ).
thf(2487,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ~ ( A @ zero_zero @ ( A @ one_one @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[522]) ).
thf(177,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_415_psubset__card__mono) ).
thf(1231,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(379,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_301_nat__add__left__cancel__less) ).
thf(1974,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[379]) ).
thf(448,axiom,
! [A: $i] :
( ( A @ comm_monoid_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_496_add_Ocomm__neutral) ).
thf(2243,plain,
! [A: $i] :
( ( A @ comm_monoid_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[448]) ).
thf(132,axiom,
u @ ( pname @ finite_finite @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(1050,plain,
u @ ( pname @ finite_finite @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(329,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_subset__trans) ).
thf(1799,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[329]) ).
thf(520,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ ti ) @ ( C @ ( B @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_arg2) ).
thf(2482,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ ti ) @ ( C @ ( B @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[520]) ).
thf(66,axiom,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( A @ zero_zero @ ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( A @ zero_zero @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_460_double__add__le__zero__iff__single__add__le__zero) ).
thf(779,plain,
! [A: $i] :
( ( A @ linord219039673up_add )
=> ! [B: $i] :
( ( ( A @ zero_zero @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( A @ zero_zero @ ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( A @ zero_zero @ ( B @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( A @ zero_zero @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(487,axiom,
! [A: $i] :
( ( A @ ab_semigroup_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_1_res) ).
thf(2370,plain,
! [A: $i] :
( ( A @ ab_semigroup_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[487]) ).
thf(9,axiom,
! [A: $i] :
( ( A @ one )
=> ( ( A @ one_one @ ( A @ ti ) )
= ( A @ one_one ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oone__class_Oone_res) ).
thf(622,plain,
! [A: $i] :
( ( A @ one )
=> ( ( A @ one_one @ ( A @ ti ) )
= ( A @ one_one ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(83,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ? [C: $i,D: $i] :
( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_418_finite__conv__nat__seg__image) ).
thf(837,plain,
! [A: $i,B: $i] :
( ( ? [C: $i,D: $i] :
( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) ) )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i,D: $i] :
( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(233,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( C = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_eq__diff__iff) ).
thf(1451,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C = A )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( C = A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(49,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ image ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_Oimage_res) ).
thf(738,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ image ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(437,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_293_termination__basic__simps_I1_J) ).
thf(2203,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[437]) ).
thf(181,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ~ ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fequal_1_1_T) ).
thf(1241,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ~ ( C @ ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(439,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_382_xt1_I8_J) ).
thf(2208,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[439]) ).
thf(343,axiom,
( ( com @ the_1 @ ( com @ ( com @ option @ fun ) @ ti ) )
= ( com @ the_1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Option_Othe_res) ).
thf(1846,plain,
( ( com @ the_1 @ ( com @ ( com @ option @ fun ) @ ti ) )
= ( com @ the_1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[343]) ).
thf(195,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_325_less__imp__le__nat) ).
thf(1286,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(495,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_277_le__add__diff) ).
thf(2402,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[495]) ).
thf(34,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_Collect__conv__if2) ).
thf(674,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(410,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_253_diff__cancel) ).
thf(2089,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[410]) ).
thf(359,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_add__Suc) ).
thf(1895,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[359]) ).
thf(561,axiom,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_413_not__psubset__empty) ).
thf(2645,plain,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[561]) ).
thf(30,axiom,
( ( nat @ bot_bot )
= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_488_bot__nat__def) ).
thf(662,plain,
( ( nat @ bot_bot )
= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(597,axiom,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_empty__iff) ).
thf(2788,plain,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[597]) ).
thf(203,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( D @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) ) )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_384_nat__seg__image__imp__finite) ).
thf(1318,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( D @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) ) )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(260,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( B @ ( A @ hAPP ) ) ) @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_le__funD) ).
thf(1563,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( B @ ( A @ hAPP ) ) ) @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(26,axiom,
nat @ ord,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oord) ).
thf(657,plain,
nat @ ord,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(63,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_260_trans__le__add2) ).
thf(773,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(385,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_le__diff__iff) ).
thf(1996,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( A @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[385]) ).
thf(423,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ fun ) @ ti ) @ ( B @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_arg1) ).
thf(2144,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ fun ) @ ti ) @ ( B @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[423]) ).
thf(192,axiom,
! [A: $i] :
( ( A @ monoid_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_0_res) ).
thf(1278,plain,
! [A: $i] :
( ( A @ monoid_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(54,axiom,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_add__mono) ).
thf(750,plain,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(500,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ insert @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_image__insert) ).
thf(2413,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ insert @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[500]) ).
thf(574,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( D @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_296_less__add__eq__less) ).
thf(2677,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( D @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[574]) ).
thf(299,axiom,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] : ( B @ ( A @ bot_bot @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_bot__least) ).
thf(1680,plain,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] : ( B @ ( A @ bot_bot @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(375,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( C @ ( A @ combs ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( F @ ( E @ ( C @ ( A @ hAPP ) ) ) @ ( F @ ( D @ ( B @ ( C @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( C @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBS_1_1_U) ).
thf(1959,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( C @ ( A @ combs ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( F @ ( E @ ( C @ ( A @ hAPP ) ) ) @ ( F @ ( D @ ( B @ ( C @ fun ) @ ( A @ hAPP ) ) ) @ ( B @ ( C @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[375]) ).
thf(450,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_395_le__imp__less__Suc) ).
thf(2253,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[450]) ).
thf(543,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_345_xt1_I1_J) ).
thf(2585,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[543]) ).
thf(240,axiom,
! [A: $i] :
~ ( A @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_Suc__n__not__le__n) ).
thf(1483,plain,
! [A: $i] :
~ ( A @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(537,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ one_one @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_280_Suc__eq__plus1) ).
thf(2570,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ one_one @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[537]) ).
thf(52,axiom,
( ( fNot @ ( bool @ ( bool @ fun ) @ ti ) )
= fNot ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fNot_res) ).
thf(745,plain,
( ( fNot @ ( bool @ ( bool @ fun ) @ ti ) )
= fNot ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(394,axiom,
! [A: $i] : ( A @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_le__refl) ).
thf(2028,plain,
! [A: $i] : ( A @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[394]) ).
thf(153,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_card__insert__le) ).
thf(1146,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(317,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_258_nat__add__left__cancel__le) ).
thf(1754,plain,
! [A: $i,B: $i,C: $i] :
( ( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[317]) ).
thf(99,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_not__less__eq__eq) ).
thf(904,plain,
! [A: $i,B: $i] :
( ( ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(38,axiom,
! [A: $i,B: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_399_diff__less__Suc) ).
thf(694,plain,
! [A: $i,B: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(324,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_421_psubset__trans) ).
thf(1784,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[324]) ).
thf(598,axiom,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( ( B @ ( A @ ti ) )
!= ( A @ bot_bot ) )
<=> ( B @ ( A @ bot_bot @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_361_bot__less) ).
thf(2791,plain,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( ( B @ ( A @ bot_bot @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
!= ( A @ bot_bot ) ) )
& ( ( ( B @ ( A @ ti ) )
!= ( A @ bot_bot ) )
=> ( B @ ( A @ bot_bot @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[598]) ).
thf(565,axiom,
( ( g @ ( bool @ ( x_a @ fun ) @ ti ) )
= g ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_G_res) ).
thf(2657,plain,
( ( g @ ( bool @ ( x_a @ fun ) @ ti ) )
= g ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[565]) ).
thf(523,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_326_le__eq__less__or__eq) ).
thf(2489,plain,
! [A: $i,B: $i] :
( ( ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[523]) ).
thf(51,axiom,
! [A: $i,B: $i] : ( B @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( A @ ( fconj @ ( nat @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ combs ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_386_finite__M__bounded__by__nat) ).
thf(743,plain,
! [A: $i,B: $i] : ( B @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( A @ ( fconj @ ( nat @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ combs ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(327,axiom,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ~ ( B @ ( A @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_3_1_U) ).
thf(1795,plain,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ~ ( B @ ( A @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[327]) ).
thf(419,axiom,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Groups_Oplus__class_Oplus_2_res) ).
thf(2134,plain,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ( ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ plus_plus ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[419]) ).
thf(444,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_348_order__less__trans) ).
thf(2225,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[444]) ).
thf(355,axiom,
! [A: $i] :
( ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ the_elem ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_Othe__elem_res) ).
thf(1884,plain,
! [A: $i] :
( ( A @ the_elem @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ the_elem ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[355]) ).
thf(291,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_440_card__Plus) ).
thf(1652,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(370,axiom,
! [A: $i] : ( A @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_292_finite__Collect__less__nat) ).
thf(1929,plain,
! [A: $i] : ( A @ ( nat @ ord_less @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[370]) ).
thf(92,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( E @ ( D @ ( bool @ ( B @ fun ) @ minus_minus @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_image__diff__subset) ).
thf(875,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] : ( E @ ( D @ ( bool @ ( B @ fun ) @ minus_minus @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(79,axiom,
! [A: $i,B: $i] :
( ( B @ ( wt @ ( bool @ ( com @ hAPP ) ) ) @ hBOOL )
=> ( ! [C: $i] :
( ( u @ ( C @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( C @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( A @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( B @ ( mgt @ ( x_a @ ( com @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( A @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_assms_I3_J) ).
thf(826,plain,
! [A: $i,B: $i] :
( ( B @ ( wt @ ( bool @ ( com @ hAPP ) ) ) @ hBOOL )
=> ( ! [C: $i] :
( ( u @ ( C @ ( pname @ member @ ( bool @ ( bool @ ( pname @ fun ) @ fun ) @ ( pname @ hAPP ) ) ) @ ( bool @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( C @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( A @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( B @ ( mgt @ ( x_a @ ( com @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( A @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(185,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_folding__image__simple_Oinsert__remove) ).
thf(1256,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(175,axiom,
! [A: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_290_lessI) ).
thf(1226,plain,
! [A: $i] : ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(242,axiom,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_387_add__le__less__mono) ).
thf(1488,plain,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(472,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_equalityI) ).
thf(2326,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[472]) ).
thf(202,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_finite__empty__induct) ).
thf(1312,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i,E: $i] :
( ( E @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( D @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(312,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combb ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combb ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBB_res) ).
thf(1733,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combb ) ) @ ( B @ ( C @ fun ) @ ( A @ ( C @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combb ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[312]) ).
thf(143,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_364_linorder__le__less__linear) ).
thf(1107,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(139,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_Diff__insert) ).
thf(1092,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(320,axiom,
! [A: $i,B: $i] : ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_le__add2) ).
thf(1765,plain,
! [A: $i,B: $i] : ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[320]) ).
thf(72,axiom,
( ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fdisj ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fdisj_res) ).
thf(804,plain,
( ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fdisj ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(40,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( E @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_427_less__mono__imp__le__mono) ).
thf(704,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( E @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(297,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_288_card__Diff2__less) ).
thf(1675,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(432,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_349_xt1_I10_J) ).
thf(2193,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[432]) ).
thf(325,axiom,
! [A: $i,B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_Diff__insert__absorb) ).
thf(1786,plain,
! [A: $i,B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[325]) ).
thf(1,conjecture,
u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( g @ ( pn @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_6) ).
thf(2,negated_conjecture,
~ ( u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( g @ ( pn @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(611,plain,
~ ( u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( g @ ( pn @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(68,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_394_Suc__le__eq) ).
thf(787,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(517,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( B @ ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( A @ ( nat @ hAPP ) ) ) )
= ( D @ ( C @ ( A @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_282_nat__case__Suc) ).
thf(2474,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( B @ ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ hAPP ) ) ) @ ( A @ ( nat @ hAPP ) ) ) )
= ( D @ ( C @ ( A @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[517]) ).
thf(110,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_insert__not__empty) ).
thf(947,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(538,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_equalityD2) ).
thf(2573,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[538]) ).
thf(483,axiom,
! [A: $i] :
~ ( A @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_356_less__irrefl__nat) ).
thf(2358,plain,
! [A: $i] :
~ ( A @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[483]) ).
thf(469,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_337_order__less__not__sym) ).
thf(2319,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[469]) ).
thf(484,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
| ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_subset__singletonD) ).
thf(2361,plain,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
| ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[484]) ).
thf(529,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ~ ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_equalityE) ).
thf(2521,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ~ ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[529]) ).
thf(556,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_368_leI) ).
thf(2624,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[556]) ).
thf(146,axiom,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_267_add__diff__cancel) ).
thf(1117,plain,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(310,axiom,
! [A: $i,B: $i] :
( ( B @ order )
=> ( B @ ( A @ fun ) @ order ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Orderings_Oorder) ).
thf(1722,plain,
! [A: $i,B: $i] :
( ( B @ order )
=> ( B @ ( A @ fun ) @ order ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[310]) ).
thf(330,axiom,
( ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fimplies ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fimplies_res) ).
thf(1801,plain,
( ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ ti ) )
= fimplies ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[330]) ).
thf(7,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ti ) @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_ti_idem) ).
thf(616,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ti ) @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(434,axiom,
bool @ ord,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oord) ).
thf(2197,plain,
bool @ ord,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[434]) ).
thf(163,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_card__Diff__subset) ).
thf(1175,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(172,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) )
& ( ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_446_card__Plus__conv__if) ).
thf(1209,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ) )
& ( ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( ( C @ ( D @ ( A @ ( B @ sum_Plus ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ ( B @ sum_sum ) @ finite_card @ ( nat @ ( bool @ ( A @ ( B @ sum_sum ) @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( B @ finite_card @ ( nat @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(582,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_folding__one_Oinsert) ).
thf(2721,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[582]) ).
thf(269,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_nat__add__commute) ).
thf(1589,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(462,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_singleton__conv2) ).
thf(2284,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[462]) ).
thf(372,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_408_subset__psubset__trans) ).
thf(1933,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[372]) ).
thf(449,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_319_not__less__less__Suc__eq) ).
thf(2246,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( A = B )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[449]) ).
thf(466,axiom,
! [A: $i] : ( A @ ( nat @ zero_zero @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_448_le0) ).
thf(2299,plain,
! [A: $i] : ( A @ ( nat @ zero_zero @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[466]) ).
thf(607,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_finite_OinsertI) ).
thf(2824,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[607]) ).
thf(75,axiom,
! [A: $i,B: $i,C: $i] :
( ? [D: $i] :
( ! [E: $i] :
( ( E @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) )
& ( D @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_the1__equality) ).
thf(811,plain,
! [A: $i,B: $i,C: $i] :
( ? [D: $i] :
( ! [E: $i] :
( ( E @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) )
& ( D @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(88,axiom,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_diff__eq__diff__less__eq) ).
thf(858,plain,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(416,axiom,
! [A: $i,B: $i] :
( ( ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_468_one__is__add) ).
thf(2111,plain,
! [A: $i,B: $i] :
( ( ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) )
=> ( ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[416]) ).
thf(455,axiom,
( ( nat @ one_one )
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_472_One__nat__def) ).
thf(2267,plain,
( ( nat @ one_one )
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[455]) ).
thf(453,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( F @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( G @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_438_order__le__less__subst2) ).
thf(2261,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( F @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( G @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[453]) ).
thf(341,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_357_less__not__refl2) ).
thf(1838,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[341]) ).
thf(546,axiom,
! [A: $i,B: $i] :
( ( B @ minus )
=> ( B @ ( A @ fun ) @ minus ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Groups_Ominus) ).
thf(2593,plain,
! [A: $i,B: $i] :
( ( B @ minus )
=> ( B @ ( A @ fun ) @ minus ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[546]) ).
thf(11,axiom,
! [A: $i] :
( ( A @ one )
=> ! [B: $i] :
( ( ( A @ one_one )
= ( B @ ( A @ ti ) ) )
<=> ( ( B @ ( A @ ti ) )
= ( A @ one_one ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_one__reorient) ).
thf(626,plain,
! [A: $i] :
( ( A @ one )
=> ! [B: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( A @ one_one ) )
=> ( ( A @ one_one )
= ( B @ ( A @ ti ) ) ) )
& ( ( ( A @ one_one )
= ( B @ ( A @ ti ) ) )
=> ( ( B @ ( A @ ti ) )
= ( A @ one_one ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(219,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [F: $i,G: $i] :
( ( G @ ( F @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( A @ ( A @ hAPP ) ) ) )
= ( G @ ( C @ ( A @ ( A @ hAPP ) ) ) @ ( F @ ( C @ ( A @ ( A @ hAPP ) ) ) @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( B @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( A @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_folding__one__idem_Ohom__commute) ).
thf(1381,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ! [F: $i,G: $i] :
( ( G @ ( F @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( A @ ( A @ hAPP ) ) ) )
= ( G @ ( C @ ( A @ ( A @ hAPP ) ) ) @ ( F @ ( C @ ( A @ ( A @ hAPP ) ) ) @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( B @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( C @ ( A @ ( A @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(95,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_double__diff) ).
thf(890,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(430,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_insert__image) ).
thf(2180,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( E @ ( C @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[430]) ).
thf(57,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_in__mono) ).
thf(758,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(209,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_402_diff__less__mono) ).
thf(1339,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(552,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_380_xt1_I7_J) ).
thf(2610,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[552]) ).
thf(70,axiom,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_add__left__imp__eq) ).
thf(799,plain,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(160,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Diff__idemp) ).
thf(1162,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(61,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_329_less__imp__diff__less) ).
thf(769,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(464,axiom,
! [A: $i] :
( ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_finite__nat__set__iff__bounded__le) ).
thf(2289,plain,
! [A: $i] :
( ( ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[464]) ).
thf(473,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_card__seteq) ).
thf(2329,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[473]) ).
thf(134,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_xt1_I5_J) ).
thf(1061,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(580,axiom,
! [A: $i] :
( ( nat @ zero_zero @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( A
= ( nat @ zero_zero ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_489_le__0__eq) ).
thf(2710,plain,
! [A: $i] :
( ( ( A
= ( nat @ zero_zero ) )
=> ( nat @ zero_zero @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( nat @ zero_zero @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A
= ( nat @ zero_zero ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[580]) ).
thf(336,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_linorder__le__cases) ).
thf(1820,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[336]) ).
thf(369,axiom,
! [A: $i] :
( ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ insert ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_Oinsert_res) ).
thf(1926,plain,
! [A: $i] :
( ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ insert ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[369]) ).
thf(271,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_393_less__Suc__eq__le) ).
thf(1594,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(591,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_420_psubsetD) ).
thf(2765,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[591]) ).
thf(409,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( B @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ finite_comp_fun_idem ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ocomp__fun__idem_res) ).
thf(2086,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( B @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ finite_comp_fun_idem ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[409]) ).
thf(421,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ( A @ one_one @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_456_zero__le__one) ).
thf(2139,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ( A @ one_one @ ( A @ zero_zero @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[421]) ).
thf(123,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( A @ ( B @ ( C @ combb ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( F @ ( E @ ( C @ ( A @ hAPP ) ) ) @ ( D @ ( B @ ( C @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBB_1_1_U) ).
thf(1015,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( A @ ( B @ ( C @ combb ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ fun ) @ ( B @ ( C @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( C @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( F @ ( E @ ( C @ ( A @ hAPP ) ) ) @ ( D @ ( B @ ( C @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(590,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_subset__insert__iff) ).
thf(2752,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[590]) ).
thf(245,axiom,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) )
<=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_480_right__minus__eq) ).
thf(1496,plain,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) ) )
& ( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( A @ zero_zero ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(85,axiom,
! [A: $i,B: $i] :
( ( B @ minus )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( C @ ( B @ ( A @ fun ) @ minus_minus @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ minus_minus @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_fun__diff__def) ).
thf(851,plain,
! [A: $i,B: $i] :
( ( B @ minus )
=> ! [C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( C @ ( B @ ( A @ fun ) @ minus_minus @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ minus_minus @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(23,axiom,
( ( na @ ( nat @ ti ) )
= na ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_na_res) ).
thf(653,plain,
( ( na @ ( nat @ ti ) )
= na ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(304,axiom,
! [A: $i,B: $i] :
( ( A @ hBOOL )
| ~ ( B @ ( A @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fconj_2_1_U) ).
thf(1694,plain,
! [A: $i,B: $i] :
( ( A @ hBOOL )
| ~ ( B @ ( A @ ( fconj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[304]) ).
thf(27,axiom,
! [A: $i,B: $i] :
( ( ( A @ finite_finite_1 )
& ( B @ finite_finite_1 ) )
=> ( B @ ( A @ sum_sum ) @ finite_finite_1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_sum___Finite__Set_Ofinite) ).
thf(658,plain,
! [A: $i,B: $i] :
( ( ( A @ finite_finite_1 )
& ( B @ finite_finite_1 ) )
=> ( B @ ( A @ sum_sum ) @ finite_finite_1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(593,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [D: $i] : ( B @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_426_psubset__imp__ex__mem) ).
thf(2770,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [D: $i] : ( B @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[593]) ).
thf(119,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ~ ! [C: $i] :
( ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B
!= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_429_lessE) ).
thf(983,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ~ ! [C: $i] :
( ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B
!= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(405,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_333_linorder__less__linear) ).
thf(2075,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[405]) ).
thf(352,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_insertI2) ).
thf(1873,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[352]) ).
thf(170,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_order__antisym) ).
thf(1203,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(457,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_the__sym__eq__trivial) ).
thf(2271,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[457]) ).
thf(516,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_diff__diff__left) ).
thf(2471,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[516]) ).
thf(227,axiom,
! [A: $i] :
( A
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_n__not__Suc__n) ).
thf(1420,plain,
! [A: $i] :
( A
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(301,axiom,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ( A @ hBOOL )
| ~ ( B @ ( A @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fdisj_3_1_U) ).
thf(1684,plain,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ( A @ hBOOL )
| ~ ( B @ ( A @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(56,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( F @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( G @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_437_order__less__le__subst1) ).
thf(754,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( F @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( G @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(109,axiom,
! [A: $i] :
( ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ nat_case ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Nat_Onat_Onat__case_res) ).
thf(944,plain,
! [A: $i] :
( ( A @ nat_case @ ( A @ ( nat @ fun ) @ ( A @ ( nat @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ nat_case ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(569,axiom,
! [A: $i] :
( ( A @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_474_diff__self__eq__0) ).
thf(2665,plain,
! [A: $i] :
( ( A @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[569]) ).
thf(149,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_folding__one__idem_Oin__idem) ).
thf(1134,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(20,axiom,
! [A: $i] :
( ( A @ zero_neq_one )
=> ( ( A @ zero_zero )
!= ( A @ one_one ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_458_zero__neq__one) ).
thf(648,plain,
! [A: $i] :
( ( A @ zero_neq_one )
=> ( ( A @ zero_zero )
!= ( A @ one_one ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(124,axiom,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_400_add__diff__inverse) ).
thf(1018,plain,
! [A: $i,B: $i] :
( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(535,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_insert__compr) ).
thf(2564,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fdisj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[535]) ).
thf(536,axiom,
! [A: $i] :
( ( ( A @ ( bool @ ti ) )
= fFalse )
| ( ( A @ ( bool @ ti ) )
= fTrue ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fFalse_1_1_T) ).
thf(2567,plain,
! [A: $i] :
( ( ( A @ ( bool @ ti ) )
= fFalse )
| ( ( A @ ( bool @ ti ) )
= fTrue ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[536]) ).
thf(256,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_270_add__diff__assoc2) ).
thf(1547,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(43,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_286_termination__basic__simps_I3_J) ).
thf(712,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(87,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_276_le__add__diff__inverse) ).
thf(855,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(14,axiom,
! [A: $i] :
( ( A @ bot )
=> ( ( A @ bot_bot @ ( A @ ti ) )
= ( A @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Orderings_Obot__class_Obot_res) ).
thf(640,plain,
! [A: $i] :
( ( A @ bot )
=> ( ( A @ bot_bot @ ( A @ ti ) )
= ( A @ bot_bot ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(116,axiom,
! [A: $i,B: $i] : ( B @ ( bool @ ( A @ fun ) @ bot_bot @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_empty__subsetI) ).
thf(971,plain,
! [A: $i,B: $i] : ( B @ ( bool @ ( A @ fun ) @ bot_bot @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(389,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_card__insert__if) ).
thf(2012,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[389]) ).
thf(544,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_261_add__le__mono1) ).
thf(2588,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[544]) ).
thf(289,axiom,
! [A: $i,B: $i] :
( ( bool @ ( x_a @ fun ) @ bot_bot @ ( A @ ( body @ ( com @ option @ ( pname @ hAPP ) ) ) @ ( com @ the_1 @ ( com @ ( com @ option @ hAPP ) ) ) @ ( mgt @ ( x_a @ ( com @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( A @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( B @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_assms_I2_J) ).
thf(1647,plain,
! [A: $i,B: $i] :
( ( bool @ ( x_a @ fun ) @ bot_bot @ ( A @ ( body @ ( com @ option @ ( pname @ hAPP ) ) ) @ ( com @ the_1 @ ( com @ ( com @ option @ hAPP ) ) ) @ ( mgt @ ( x_a @ ( com @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( bool @ ( x_a @ fun ) @ bot_bot @ ( A @ ( mgt_call @ ( x_a @ ( pname @ hAPP ) ) ) @ ( x_a @ insert @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ fun ) @ ( x_a @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( B @ ( p @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(183,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i] : ( B @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_order__refl) ).
thf(1252,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i] : ( B @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(427,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ~ ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_Diff__iff) ).
thf(2160,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ~ ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[427]) ).
thf(577,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ! [F: $i] :
( ( ( C @ ( A @ ti ) )
= ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
=> ~ ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_imageE) ).
thf(2689,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ! [F: $i] :
( ( ( C @ ( A @ ti ) )
= ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) ) )
=> ~ ( E @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[577]) ).
thf(493,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_343_xt1_I9_J) ).
thf(2397,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[493]) ).
thf(127,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( D @ ( A @ ti ) )
= ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) ) )
=> ( ( C @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_image__eqI) ).
thf(1025,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( ( D @ ( A @ ti ) )
= ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) ) )
=> ( ( C @ ( F @ ( B @ member @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(588,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_376_order__le__imp__less__or__eq) ).
thf(2747,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[588]) ).
thf(229,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_insert__Diff__if) ).
thf(1432,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(286,axiom,
! [A: $i] :
( ~ ( A @ hBOOL )
| ~ ( A @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fNot_1_1_U) ).
thf(1637,plain,
! [A: $i] :
( ~ ( A @ hBOOL )
| ~ ( A @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(167,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_Suc__diff__diff) ).
thf(1186,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(45,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_257_le__iff__add) ).
thf(717,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(504,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_377_order__le__neq__trans) ).
thf(2429,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[504]) ).
thf(179,axiom,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_diff__add__inverse2) ).
thf(1235,plain,
! [A: $i,B: $i] :
( ( B @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(476,axiom,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_482_Zero__neq__Suc) ).
thf(2337,plain,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[476]) ).
thf(361,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_271_diff__add__assoc) ).
thf(1901,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[361]) ).
thf(293,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_folding__one__idem_Osubset__idem) ).
thf(1658,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(15,axiom,
nat @ order,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Oorder) ).
thf(643,plain,
nat @ order,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(173,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_285_card_Oinsert) ).
thf(1220,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(481,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_409_psubset__subset__trans) ).
thf(2354,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[481]) ).
thf(33,axiom,
nat @ comm_monoid_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(673,plain,
nat @ comm_monoid_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(411,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_subset__insertI2) ).
thf(2092,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[411]) ).
thf(442,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_folding__one_Osingleton) ).
thf(2219,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( D @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[442]) ).
thf(380,axiom,
! [A: $i,B: $i] :
( ( B @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_Diff__cancel) ).
thf(1980,plain,
! [A: $i,B: $i] :
( ( B @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[380]) ).
thf(417,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_351_linorder__cases) ).
thf(2128,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[417]) ).
thf(253,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_insert__absorb2) ).
thf(1538,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(111,axiom,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_483_nat_Osimps_I3_J) ).
thf(951,plain,
! [A: $i] :
( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(74,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i] :
~ ( B @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_330_order__less__irrefl) ).
thf(808,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i] :
~ ( B @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(499,axiom,
! [A: $i,B: $i,C: $i] : ( B @ ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_Diff__subset) ).
thf(2411,plain,
! [A: $i,B: $i,C: $i] : ( B @ ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[499]) ).
thf(360,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_nat__add__left__commute) ).
thf(1898,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[360]) ).
thf(533,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_card__insert) ).
thf(2558,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[533]) ).
thf(503,axiom,
! [A: $i] :
( ( A @ linordered_idom )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_424_linorder__neqE__linordered__idom) ).
thf(2426,plain,
! [A: $i] :
( ( A @ linordered_idom )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
=> ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[503]) ).
thf(4,axiom,
nat @ zero,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Ozero) ).
thf(613,plain,
nat @ zero,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(194,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_diff__card__le__card__Diff) ).
thf(1284,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(89,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_folding__image__simple_Oinsert) ).
thf(864,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i,G: $i,H: $i] :
( ( H @ ( G @ ( F @ ( E @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( D @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( H @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( G @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( B @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( B @ ( B @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(246,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_300_trans__less__add1) ).
thf(1504,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(328,axiom,
! [A: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_comp__fun__idem__remove) ).
thf(1797,plain,
! [A: $i] : ( bool @ ( A @ fun ) @ bot_bot @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ combc ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( A @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ finite_comp_fun_idem ) @ ( bool @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[328]) ).
thf(395,axiom,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
<=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_set__eq__subset) ).
thf(2030,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) ) )
& ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
=> ( ( B @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[395]) ).
thf(573,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_diff__single__insert) ).
thf(2675,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[573]) ).
thf(400,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_332_not__less__iff__gr__or__eq) ).
thf(2056,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
| ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[400]) ).
thf(25,axiom,
nat @ ordere779506340up_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(656,plain,
nat @ ordere779506340up_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(126,axiom,
! [A: $i,B: $i] :
( ( B @ preorder )
=> ( B @ ( A @ fun ) @ preorder ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Orderings_Opreorder) ).
thf(1023,plain,
! [A: $i,B: $i] :
( ( B @ preorder )
=> ( B @ ( A @ fun ) @ preorder ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(378,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_Suc__leD) ).
thf(1972,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[378]) ).
thf(247,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_341_order__less__imp__triv) ).
thf(1506,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(151,axiom,
! [A: $i,B: $i] : ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_256_le__add1) ).
thf(1140,plain,
! [A: $i,B: $i] : ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(527,axiom,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( A @ bot_bot @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( A @ ti ) )
= ( A @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_bot__unique) ).
thf(2511,plain,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( A @ bot_bot ) )
=> ( A @ bot_bot @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( A @ bot_bot @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( A @ bot_bot ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[527]) ).
thf(354,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_Suc__le__D) ).
thf(1881,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[354]) ).
thf(3,axiom,
nat @ one,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oone) ).
thf(612,plain,
nat @ one,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(512,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_392_less__eq__Suc__le) ).
thf(2458,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[512]) ).
thf(165,axiom,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_266_diff__add__cancel) ).
thf(1180,plain,
! [A: $i] :
( ( A @ group_add )
=> ! [B: $i,C: $i] :
( ( C @ ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(609,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A )
<=> ( C
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_272_le__imp__diff__is__add) ).
thf(2828,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ) )
& ( ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A )
=> ( C
= ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[609]) ).
thf(334,axiom,
bool @ preorder,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Opreorder) ).
thf(1817,plain,
bool @ preorder,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[334]) ).
thf(468,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_365_order__less__le) ).
thf(2310,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( A @ ti ) )
!= ( C @ ( A @ ti ) ) )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[468]) ).
thf(148,axiom,
! [A: $i,B: $i] :
( ( ( bool @ ( A @ fun ) @ bot_bot )
= ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
<=> ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_empty__Collect__eq) ).
thf(1126,plain,
! [A: $i,B: $i] :
( ( ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot )
= ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot )
= ( B @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
=> ! [C: $i] :
~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(506,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_284_diff__Suc__diff__eq1) ).
thf(2435,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[506]) ).
thf(84,axiom,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( A @ zero_zero @ ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_466_le__iff__diff__le__0) ).
thf(845,plain,
! [A: $i] :
( ( A @ ordered_ab_group_add )
=> ! [B: $i,C: $i] :
( ( ( A @ zero_zero @ ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( A @ zero_zero @ ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(557,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_304_add__less__imp__less__left) ).
thf(2626,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[557]) ).
thf(290,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_singleton__conv) ).
thf(1649,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(158,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_diff__le__mono) ).
thf(1157,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(406,axiom,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_388_add__less__le__mono) ).
thf(2078,plain,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( E @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[406]) ).
thf(365,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_card__Diff__singleton__if) ).
thf(1913,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ~ ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ one_one @ ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[365]) ).
thf(231,axiom,
bool @ order,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_HOL_Obool___Orderings_Oorder) ).
thf(1448,plain,
bool @ order,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(168,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_Suc__le__mono) ).
thf(1189,plain,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(348,axiom,
! [A: $i,B: $i] : ( B @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_subset__refl) ).
thf(1866,plain,
! [A: $i,B: $i] : ( B @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[348]) ).
thf(403,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_298_add__less__mono1) ).
thf(2071,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( B @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[403]) ).
thf(595,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_314_less__SucE) ).
thf(2775,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[595]) ).
thf(332,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( H @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_435_xt5) ).
thf(1807,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( H @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( D @ ( A @ ( B @ hAPP ) ) ) @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[332]) ).
thf(62,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_374_order__less__imp__le) ).
thf(771,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(191,axiom,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ! [E: $i] : ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_le__funI) ).
thf(1276,plain,
! [A: $i,B: $i] :
( ( B @ ord )
=> ! [C: $i,D: $i] :
( ! [E: $i] : ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( B @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( B @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(579,axiom,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( A @ bot_bot @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( A @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_le__bot) ).
thf(2707,plain,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
( ( A @ bot_bot @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( A @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[579]) ).
thf(287,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_card_Oremove) ).
thf(1639,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ one_one @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(497,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combc ) ) @ ( C @ ( A @ fun ) @ ( B @ fun ) @ ( C @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combc ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBC_res) ).
thf(2406,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ combc ) ) @ ( C @ ( A @ fun ) @ ( B @ fun ) @ ( C @ ( B @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( C @ ( B @ ( A @ combc ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[497]) ).
thf(532,axiom,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( E @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_405_mono__nat__linear__lb) ).
thf(2554,plain,
! [A: $i,B: $i,C: $i] :
( ! [D: $i,E: $i] :
( ( E @ ( D @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( E @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( A @ ( C @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[532]) ).
thf(248,axiom,
! [A: $i,B: $i] :
( ( B @ bot )
=> ! [C: $i] :
( ( C @ ( B @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ hAPP ) ) ) )
= ( B @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_bot__apply) ).
thf(1508,plain,
! [A: $i,B: $i] :
( ( B @ bot )
=> ! [C: $i] :
( ( C @ ( B @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ hAPP ) ) ) )
= ( B @ bot_bot ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(239,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fdisj_2_1_U) ).
thf(1481,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( fdisj @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(141,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_diff__diff__cancel) ).
thf(1101,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(22,axiom,
nat @ ordere236663937imp_le,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(652,plain,
nat @ ordere236663937imp_le,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(282,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_insert__Diff__single) ).
thf(1629,plain,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(393,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_linorder__linear) ).
thf(2026,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[393]) ).
thf(551,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_328_diff__less__mono2) ).
thf(2608,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[551]) ).
thf(250,axiom,
! [A: $i] :
( ( A @ combi @ ( A @ ( A @ fun ) @ ti ) )
= ( A @ combi ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_COMBI_res) ).
thf(1523,plain,
! [A: $i] :
( ( A @ combi @ ( A @ ( A @ fun ) @ ti ) )
= ( A @ combi ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(501,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
!= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( D @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( D @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_image__constant__conv) ).
thf(2416,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
!= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( D @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( ( D @ ( bool @ ( B @ fun ) @ ti ) )
= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( D @ ( C @ ( B @ ( A @ combk ) @ ( A @ ( B @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[501]) ).
thf(115,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( D @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_If__def) ).
thf(963,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ~ ( D @ hBOOL )
=> ( ( C @ ( A @ ti ) )
= ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) )
& ( ( D @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fNot @ ( bool @ ( bool @ hAPP ) ) ) @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( D @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( A @ ( bool @ ( bool @ combb ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(428,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_347_xt1_I2_J) ).
thf(2167,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( A @ ti ) )
= ( B @ ( A @ ti ) ) )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[428]) ).
thf(210,axiom,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ~ ( A @ hBOOL )
| ~ ( B @ ( A @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fimplies_3_1_U) ).
thf(1341,plain,
! [A: $i,B: $i] :
( ( B @ hBOOL )
| ~ ( A @ hBOOL )
| ~ ( B @ ( A @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(413,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ~ ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_366_less__le__not__le) ).
thf(2098,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i] :
( ( ( ~ ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ~ ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[413]) ).
thf(182,axiom,
! [A: $i,B: $i] :
( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_nat_Oinject) ).
thf(1244,plain,
! [A: $i,B: $i] :
( ( ( A = B )
=> ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(337,axiom,
( ( wt @ ( bool @ ( com @ fun ) @ ti ) )
= wt ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_wt_res) ).
thf(1822,plain,
( ( wt @ ( bool @ ( com @ fun ) @ ti ) )
= wt ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[337]) ).
thf(490,axiom,
! [A: $i,B: $i,C: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_singleton__iff) ).
thf(2384,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[490]) ).
thf(494,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ sum_Plus ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Sum__Type_OPlus_res) ).
thf(2399,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( B @ ( A @ sum_Plus ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[494]) ).
thf(562,axiom,
! [A: $i] :
( ( A @ cancel146912293up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_add__imp__eq) ).
thf(2648,plain,
! [A: $i] :
( ( A @ cancel146912293up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[562]) ).
thf(294,axiom,
! [A: $i] :
( ( A @ comm_monoid_add )
=> ! [B: $i] :
( ( B @ ( A @ zero_zero @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_493_add__0) ).
thf(1661,plain,
! [A: $i] :
( ( A @ comm_monoid_add )
=> ! [B: $i] :
( ( B @ ( A @ zero_zero @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(81,axiom,
! [A: $i] :
( ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ collect ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Set_OCollect_res) ).
thf(832,plain,
! [A: $i] :
( ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ collect ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(358,axiom,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_379_order__less__le__trans) ).
thf(1893,plain,
! [A: $i] :
( ( A @ preorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[358]) ).
thf(278,axiom,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D = B )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_ord__le__eq__trans) ).
thf(1614,plain,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D = B )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(388,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ti ) )
= ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_hAPP_res) ).
thf(2009,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ti ) )
= ( D @ ( C @ ( A @ ( B @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[388]) ).
thf(346,axiom,
! [A: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_451_card_Oempty) ).
thf(1861,plain,
! [A: $i] :
( ( bool @ ( A @ fun ) @ bot_bot @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( nat @ zero_zero ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[346]) ).
thf(230,axiom,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
<=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_add__left__cancel) ).
thf(1440,plain,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) )
=> ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(105,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
=> ( ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_the__equality) ).
thf(929,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
=> ( ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(200,axiom,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_307_add__strict__left__mono) ).
thf(1308,plain,
! [A: $i] :
( ( A @ ordere223160158up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(435,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_264_add__leD1) ).
thf(2198,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[435]) ).
thf(539,axiom,
u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( g @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).
thf(2576,plain,
u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( g @ ( bool @ ( x_a @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( x_a @ fun ) @ fun ) @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ hBOOL,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[539]) ).
thf(401,axiom,
! [A: $i,B: $i] :
( ( ( A @ finite_finite_1 )
& ( B @ finite_finite_1 ) )
=> ( B @ ( A @ fun ) @ finite_finite_1 ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_fun___Finite__Set_Ofinite) ).
thf(2066,plain,
! [A: $i,B: $i] :
( ( ( A @ finite_finite_1 )
& ( B @ finite_finite_1 ) )
=> ( B @ ( A @ fun ) @ finite_finite_1 ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[401]) ).
thf(575,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_370_linorder__antisym__conv1) ).
thf(2680,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[575]) ).
thf(431,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_insert__iff) ).
thf(2183,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[431]) ).
thf(275,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_xt1_I6_J) ).
thf(1607,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(59,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_folding__one__idem_Oinsert__idem) ).
thf(763,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(387,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_folding__one_Oinsert__remove) ).
thf(2003,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[387]) ).
thf(452,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( nat @ one_one @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_diff__Suc__eq__diff__pred) ).
thf(2258,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( B @ ( nat @ one_one @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[452]) ).
thf(213,axiom,
! [A: $i] :
( ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_finite ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofinite_res) ).
thf(1348,plain,
! [A: $i] :
( ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_finite ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(463,axiom,
! [A: $i,B: $i] :
( ! [C: $i] : ( C @ ( B @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less_eq @ ( nat @ ( bool @ ( nat @ fun ) @ ( nat @ combb ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( nat @ ( nat @ fun ) @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( nat @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_finite__less__ub) ).
thf(2287,plain,
! [A: $i,B: $i] :
( ! [C: $i] : ( C @ ( B @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A @ ( B @ ( nat @ ord_less_eq @ ( nat @ ( bool @ ( nat @ fun ) @ ( nat @ combb ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( nat @ ( nat @ fun ) @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( nat @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[463]) ).
thf(322,axiom,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_ord__eq__le__trans) ).
thf(1774,plain,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[322]) ).
thf(266,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_362_linorder__not__less) ).
thf(1578,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ~ ( C @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(82,axiom,
( ( fFalse @ ( bool @ ti ) )
= fFalse ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fFalse_res) ).
thf(835,plain,
( ( fFalse @ ( bool @ ti ) )
= fFalse ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(549,axiom,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ( B @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fimplies_1_1_U) ).
thf(2604,plain,
! [A: $i,B: $i] :
( ( A @ ( B @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ( B @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[549]) ).
thf(154,axiom,
! [A: $i] :
( ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ fequal ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_fequal_res) ).
thf(1148,plain,
! [A: $i] :
( ( A @ fequal @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ fequal ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(399,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_add__le__cancel__left) ).
thf(2050,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[399]) ).
thf(90,axiom,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_344_ord__eq__less__trans) ).
thf(867,plain,
! [A: $i] :
( ( A @ ord )
=> ! [B: $i,C: $i,D: $i] :
( ( C = D )
=> ( ( B @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(288,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ! [C: $i] :
( ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B
!= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_428_Suc__lessE) ).
thf(1642,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ~ ! [C: $i] :
( ( C @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B
!= ( C @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(55,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_card__mono) ).
thf(752,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(425,axiom,
! [A: $i] :
( ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ partial_flat_lub ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Partial__Function_Oflat__lub_res) ).
thf(2155,plain,
! [A: $i] :
( ( A @ partial_flat_lub @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ partial_flat_lub ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[425]) ).
thf(451,axiom,
! [A: $i] :
( ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite2073411215e_idem ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__one__idem_res) ).
thf(2255,plain,
! [A: $i] :
( ( A @ finite2073411215e_idem @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite2073411215e_idem ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[451]) ).
thf(604,axiom,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_486_Zero__not__Suc) ).
thf(2815,plain,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[604]) ).
thf(86,axiom,
u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( na @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_2) ).
thf(854,plain,
u @ ( mgt_call @ ( x_a @ ( pname @ image ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ fun ) @ ( x_a @ ( pname @ fun ) @ hAPP ) ) ) @ ( bool @ ( x_a @ fun ) @ ( bool @ ( pname @ fun ) @ hAPP ) ) ) @ ( x_a @ finite_card @ ( nat @ ( bool @ ( x_a @ fun ) @ hAPP ) ) ) @ ( na @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(382,axiom,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( A @ zero_zero @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ zero_zero @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( A @ zero_zero @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_465_add__nonpos__nonpos) ).
thf(1986,plain,
! [A: $i] :
( ( A @ ordere216010020id_add )
=> ! [B: $i,C: $i] :
( ( A @ zero_zero @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ zero_zero @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( A @ zero_zero @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[382]) ).
thf(559,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_le__antisym) ).
thf(2636,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[559]) ).
thf(103,axiom,
! [A: $i] :
( ( A @ dense_linorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [E: $i] :
( ( E @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( E @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( E @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) )
=> ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_432_dense__le__bounded) ).
thf(918,plain,
! [A: $i] :
( ( A @ dense_linorder )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [E: $i] :
( ( E @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( E @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( E @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) )
=> ( B @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(259,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_312_Suc__less__SucD) ).
thf(1561,plain,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(496,axiom,
( ( body @ ( com @ option @ ( pname @ fun ) @ ti ) )
= body ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Com_Obody_res) ).
thf(2404,plain,
( ( body @ ( com @ option @ ( pname @ fun ) @ ti ) )
= body ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[496]) ).
thf(357,axiom,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_add__right__mono) ).
thf(1891,plain,
! [A: $i] :
( ( A @ ordere779506340up_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[357]) ).
thf(112,axiom,
! [A: $i,B: $i,C: $i] : ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_insertI1) ).
thf(955,plain,
! [A: $i,B: $i,C: $i] : ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(563,axiom,
! [A: $i] :
( ( A @ ab_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(2651,plain,
! [A: $i] :
( ( A @ ab_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( D @ ( C @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[563]) ).
thf(97,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_zero__induct__lemma) ).
thf(897,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( A @ ( C @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(226,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( ( D @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_insert__code) ).
thf(1410,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) )
=> ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
| ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(326,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_310_add__less__cancel__right) ).
thf(1789,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[326]) ).
thf(31,axiom,
nat @ ab_semigroup_add,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(664,plain,
nat @ ab_semigroup_add,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(196,axiom,
! [A: $i,B: $i] :
( ( A != B )
<=> ( ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_354_nat__neq__iff) ).
thf(1288,plain,
! [A: $i,B: $i] :
( ( ( ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( A != B ) )
& ( ( A != B )
=> ( ( A @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(307,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ combi @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_COMBI_1_1_U) ).
thf(1710,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ combi @ ( A @ ( A @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[307]) ).
thf(114,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( B @ ( A @ fun ) @ ti ) )
= ( D @ ( B @ ( A @ fun ) @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_ext) ).
thf(960,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ! [E: $i] :
( ( E @ ( C @ ( B @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) ) )
=> ( ( C @ ( B @ ( A @ fun ) @ ti ) )
= ( D @ ( B @ ( A @ fun ) @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(238,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_391_less__iff__Suc__add) ).
thf(1473,plain,
! [A: $i,B: $i] :
( ( ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ? [C: $i] :
( B
= ( C @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(396,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_396_Suc__leI) ).
thf(2040,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[396]) ).
thf(39,axiom,
! [A: $i,B: $i,C: $i] :
( ( ~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_Collect__conv__if) ).
thf(696,plain,
! [A: $i,B: $i,C: $i] :
( ( ~ ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) )
& ( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( C @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( fconj @ ( A @ ( bool @ ( bool @ fun ) @ ( bool @ combb ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( bool @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ combs ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( bool @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ collect @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(570,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_add__le__imp__le__right) ).
thf(2668,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[570]) ).
thf(161,axiom,
! [A: $i] :
( ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_card ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ocard_res) ).
thf(1165,plain,
! [A: $i] :
( ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_card ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(391,axiom,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_add__right__imp__eq) ).
thf(2021,plain,
! [A: $i] :
( ( A @ cancel_semigroup_add )
=> ! [B: $i,C: $i,D: $i] :
( ( ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( B @ ( A @ ti ) )
= ( D @ ( A @ ti ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[391]) ).
thf(212,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_398_Suc__le__lessD) ).
thf(1346,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(47,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_444_finite__PlusD_I2_J) ).
thf(728,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(525,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_insert__subset) ).
thf(2502,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[525]) ).
thf(603,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
=> ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_theI) ).
thf(2810,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [D: $i] :
( ( D @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
=> ( B @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[603]) ).
thf(189,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( B @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_mem__def) ).
thf(1268,plain,
! [A: $i,B: $i,C: $i] :
( ( ( B @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(454,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_48_lift__Suc__mono__le) ).
thf(2265,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i,D: $i] :
( ! [E: $i] : ( E @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( E @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( nat @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[454]) ).
thf(121,axiom,
! [A: $i,B: $i] :
( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
<=> ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_467_add__is__1) ).
thf(992,plain,
! [A: $i,B: $i] :
( ( ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) )
=> ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) )
& ( ( ( B @ ( A @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) )
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
=> ( ( ( B
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) )
& ( A
= ( nat @ zero_zero ) ) )
| ( ( B
= ( nat @ zero_zero ) )
& ( A
= ( nat @ zero_zero @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(78,axiom,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_485_nat_Osimps_I2_J) ).
thf(822,plain,
! [A: $i] :
( ( nat @ zero_zero )
!= ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(308,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [F: $i] :
( ( ( E @ ( bool @ ( B @ fun ) @ ti ) )
= ( F @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
& ( F @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( F @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_finite__subset__image) ).
thf(1713,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( B @ finite_finite @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( D @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [F: $i] :
( ( ( E @ ( bool @ ( B @ fun ) @ ti ) )
= ( F @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) )
& ( F @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
& ( D @ ( F @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[308]) ).
thf(197,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A != B )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_324_le__neq__implies__less) ).
thf(1299,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A != B )
=> ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(91,axiom,
! [A: $i,B: $i] :
( ( ( A @ ord )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( H @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_430_xt6) ).
thf(871,plain,
! [A: $i,B: $i] :
( ( ( A @ ord )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less_eq @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ( E @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( G @ ( H @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( G @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( H @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) )
=> ( F @ ( C @ ( D @ ( B @ ( A @ hAPP ) ) ) @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(147,axiom,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
<=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_add__le__cancel__right) ).
thf(1120,plain,
! [A: $i] :
( ( A @ ordere236663937imp_le )
=> ! [B: $i,C: $i,D: $i] :
( ( ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( D @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( C @ ( B @ ( A @ plus_plus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(339,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( B @ fun ) @ bot_bot ) )
<=> ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_image__is__empty) ).
thf(1827,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( D @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( B @ fun ) @ bot_bot ) ) )
& ( ( ( D @ ( C @ ( B @ ( A @ image ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( B @ fun ) @ bot_bot ) )
=> ( ( D @ ( bool @ ( A @ fun ) @ ti ) )
= ( bool @ ( A @ fun ) @ bot_bot ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[339]) ).
thf(187,axiom,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( F @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( G @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_434_order__le__less__subst1) ).
thf(1261,plain,
! [A: $i,B: $i] :
( ( ( A @ order )
& ( B @ order ) )
=> ! [C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( F @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( ! [G: $i,H: $i] :
( ( H @ ( G @ ( B @ ord_less @ ( bool @ ( B @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ hAPP ) ) ) @ hBOOL )
=> ( H @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( G @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( E @ ( A @ ( B @ hAPP ) ) ) @ ( D @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(600,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',help_fimplies_2_1_U) ).
thf(2803,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( fimplies @ ( bool @ ( bool @ fun ) @ ( bool @ hAPP ) ) ) @ ( bool @ ( bool @ hAPP ) ) ) @ hBOOL )
| ~ ( B @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[600]) ).
thf(252,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_folding__one_Oremove) ).
thf(1532,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
!= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( D @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) ) )
& ( ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( bool @ ( A @ fun ) @ bot_bot ) )
=> ( ( C @ ( E @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(220,axiom,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_emptyE) ).
thf(1384,plain,
! [A: $i,B: $i] :
~ ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(521,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_rev__finite__subset) ).
thf(2485,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[521]) ).
thf(16,axiom,
nat @ bot,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',arity_Nat_Onat___Orderings_Obot) ).
thf(644,plain,
nat @ bot,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(257,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ? [F: $i] :
( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
= ( F @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) )
& ( E @ ( F @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_subset__image__iff) ).
thf(1550,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i] :
( ( ? [F: $i] :
( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
= ( F @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) )
& ( E @ ( F @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( E @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [F: $i] :
( ( ( C @ ( bool @ ( A @ fun ) @ ti ) )
= ( F @ ( D @ ( A @ ( B @ image ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( A @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) ) )
& ( E @ ( F @ ( bool @ ( B @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(216,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
| ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_411_subset__iff__psubset__eq) ).
thf(1357,plain,
! [A: $i,B: $i,C: $i] :
( ( ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
| ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B @ ( bool @ ( A @ fun ) @ ti ) )
= ( C @ ( bool @ ( A @ fun ) @ ti ) ) )
| ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(542,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_297_add__less__mono) ).
thf(2583,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( D @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( C @ ( nat @ plus_plus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[542]) ).
thf(223,axiom,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
~ ( A @ bot_bot @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_360_not__less__bot) ).
thf(1399,plain,
! [A: $i] :
( ( A @ bot )
=> ! [B: $i] :
~ ( A @ bot_bot @ ( B @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(280,axiom,
! [A: $i] :
( ( A @ linordered_semidom )
=> ~ ( A @ zero_zero @ ( A @ one_one @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_453_not__one__less__zero) ).
thf(1621,plain,
! [A: $i] :
( ( A @ linordered_semidom )
=> ~ ( A @ zero_zero @ ( A @ one_one @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(206,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_card__Suc__Diff1) ).
thf(1331,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( C @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( B @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
= ( C @ ( A @ finite_card @ ( nat @ ( bool @ ( A @ fun ) @ hAPP ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(376,axiom,
! [A: $i] : ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_finite__Collect__le__nat) ).
thf(1962,plain,
! [A: $i] : ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ ( nat @ combc ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ ( nat @ fun ) @ hAPP ) ) ) @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ collect @ ( bool @ ( nat @ fun ) @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[376]) ).
thf(104,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_finite__Diff__insert) ).
thf(923,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( D @ ( C @ ( A @ insert @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( D @ ( B @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(261,axiom,
! [A: $i] :
( ( A @ ord )
=> ( ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ ord_less ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Orderings_Oord__class_Oless_res) ).
thf(1565,plain,
! [A: $i] :
( ( A @ ord )
=> ( ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ti ) )
= ( A @ ord_less ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(390,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_the__eq__trivial) ).
thf(2018,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ fequal @ ( bool @ ( A @ ( A @ combc ) ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ the @ ( A @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( B @ ( A @ ti ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[390]) ).
thf(344,axiom,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
<=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_order__eq__iff) ).
thf(1848,plain,
! [A: $i] :
( ( A @ order )
=> ! [B: $i,C: $i] :
( ( ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
& ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( ( B @ ( C @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
& ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[344]) ).
thf(232,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_DiffD2) ).
thf(1449,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( bool @ ( A @ fun ) @ minus_minus @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( A @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ~ ( D @ ( B @ ( A @ member @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(41,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_diff__le__mono2) ).
thf(708,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( C @ ( A @ ( nat @ minus_minus @ ( nat @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(94,axiom,
! [A: $i] :
( ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
<=> ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_383_finite__nat__set__iff__bounded) ).
thf(882,plain,
! [A: $i] :
( ( ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL ) )
& ( ( A @ ( nat @ finite_finite @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ? [B: $i] :
! [C: $i] :
( ( A @ ( C @ ( nat @ member @ ( bool @ ( bool @ ( nat @ fun ) @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( bool @ ( nat @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(350,axiom,
! [A: $i] :
( ( A @ finite_finite_1 )
=> ! [B: $i] : ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_finite__code) ).
thf(1870,plain,
! [A: $i] :
( ( A @ finite_finite_1 )
=> ! [B: $i] : ( B @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[350]) ).
thf(470,axiom,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_443_finite__PlusD_I1_J) ).
thf(2321,plain,
! [A: $i,B: $i,C: $i,D: $i] :
( ( D @ ( C @ ( B @ ( A @ sum_Plus ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ ( bool @ ( B @ fun ) @ hAPP ) ) ) @ ( B @ ( A @ sum_sum ) @ finite_finite @ ( bool @ ( bool @ ( B @ ( A @ sum_sum ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( A @ finite_finite @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[470]) ).
thf(217,axiom,
! [A: $i] :
( ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_folding_one ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_c_Finite__Set_Ofolding__one_res) ).
thf(1367,plain,
! [A: $i] :
( ( A @ finite_folding_one @ ( bool @ ( A @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( A @ ( A @ fun ) @ ( A @ fun ) @ fun ) @ ti ) )
= ( A @ finite_folding_one ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(106,axiom,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_369_not__leE) ).
thf(935,plain,
! [A: $i] :
( ( A @ linorder )
=> ! [B: $i,C: $i] :
( ~ ( C @ ( B @ ( A @ ord_less_eq @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL )
=> ( B @ ( C @ ( A @ ord_less @ ( bool @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( bool @ ( A @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(122,axiom,
! [A: $i] :
( ( A @ ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
<=> ( ( D @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_diff__eq__diff__eq) ).
thf(1009,plain,
! [A: $i] :
( ( A @ ab_group_add )
=> ! [B: $i,C: $i,D: $i,E: $i] :
( ( ( C @ ( B @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) )
= ( E @ ( D @ ( A @ minus_minus @ ( A @ ( A @ fun ) @ ( A @ hAPP ) ) ) @ ( A @ ( A @ hAPP ) ) ) ) )
=> ( ( ( ( D @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) )
=> ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) ) )
& ( ( ( B @ ( A @ ti ) )
= ( C @ ( A @ ti ) ) )
=> ( ( D @ ( A @ ti ) )
= ( E @ ( A @ ti ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(507,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( B = A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_397_le__less__Suc__eq) ).
thf(2438,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less_eq @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( B = A )
=> ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( B @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[507]) ).
thf(243,axiom,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= B )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_316_Suc__lessI) ).
thf(1490,plain,
! [A: $i,B: $i] :
( ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) )
!= B )
=> ( B @ ( A @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(465,axiom,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_410_psubset__imp__subset) ).
thf(2297,plain,
! [A: $i,B: $i,C: $i] :
( ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( C @ ( B @ ( bool @ ( A @ fun ) @ ord_less_eq @ ( bool @ ( bool @ ( A @ fun ) @ fun ) @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( bool @ ( A @ fun ) @ hAPP ) ) ) @ hBOOL ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[465]) ).
thf(64,axiom,
( ( pn @ ( pname @ ti ) )
= pn ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tsy_v_pn_res) ).
thf(775,plain,
( ( pn @ ( pname @ ti ) )
= pn ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(429,axiom,
! [A: $i,B: $i] :
( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
<=> ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_321_less__Suc__eq) ).
thf(2170,plain,
! [A: $i,B: $i] :
( ( ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
=> ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) )
& ( ( B @ ( suc @ ( nat @ ( nat @ hAPP ) ) ) @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL )
=> ( ( A = B )
| ( B @ ( A @ ( nat @ ord_less @ ( bool @ ( nat @ fun ) @ ( nat @ hAPP ) ) ) @ ( bool @ ( nat @ hAPP ) ) ) @ hBOOL ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[429]) ).
thf(480,axiom,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( C @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( F @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ti ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_folding__image__simple_Oempty) ).
thf(2351,plain,
! [A: $i,B: $i,C: $i,D: $i,E: $i,F: $i] :
( ( F @ ( E @ ( D @ ( C @ ( A @ ( B @ finite1357897459simple ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ fun ) @ ( B @ ( B @ fun ) @ ( B @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ fun ) @ ( B @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ fun ) @ ( B @ ( A @ fun ) @ hAPP ) ) ) @ ( bool @ ( B @ ( bool @ ( A @ fun ) @ fun ) @ hAPP ) ) ) @ hBOOL )
=> ( ( bool @ ( A @ fun ) @ bot_bot @ ( F @ ( B @ ( bool @ ( A @ fun ) @ hAPP ) ) ) )
= ( D @ ( B @ ti ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[480]) ).
thf(6554,plain,
$false,
inference(e,[status(thm)],[645,1322,2822,2452,1337,2094,2744,2778,1767,1559,1031,893,2480,941,777,1818,2280,1686,2612,1469,1168,760,2341,2109,2131,2595,2499,1406,730,2808,2381,2395,2659,1618,2527,2590,1939,741,1544,2282,1142,655,1486,1924,2210,2231,2622,1068,614,1692,2195,1655,1137,2686,1310,2141,1872,1233,2580,2042,1354,1458,1919,1988,747,913,1095,1835,2432,973,1935,1921,1724,1887,651,2227,1576,2819,1803,1904,1265,1228,1623,1254,2654,2735,1760,785,2269,1907,619,802,1992,2466,2356,1511,2274,817,1704,2767,1609,1115,2158,2024,2301,2373,1572,1931,1568,1875,1778,2222,2237,2324,1682,1677,806,2335,1051,1302,1664,2105,1811,1109,1632,2469,1334,646,1858,1864,1151,937,1183,2073,2367,2618,634,2424,2772,1351,660,2750,1600,2392,981,766,877,692,1526,2137,2799,2639,1104,1697,2718,2205,1206,661,793,1306,2364,1153,2047,1470,2737,1047,1074,1604,2083,1736,2660,714,1868,2000,1636,2409,2601,1021,2277,1570,2147,2606,1238,1274,2673,1387,910,957,1223,1343,2805,1751,820,1824,1370,2464,1602,1178,665,2068,1159,2213,2345,1634,710,916,2445,2628,2670,625,2477,2663,2332,2826,2200,830,682,1195,1281,2741,1740,1910,1402,1592,1541,1983,1762,1259,615,1328,725,1587,2694,989,1397,2577,1064,2509,1672,2724,1719,2537,1493,1964,2561,1461,2080,1842,2349,1028,1082,901,686,1424,2455,637,1584,1612,1039,1155,1744,647,2450,2519,2836,2487,1231,1974,2243,1050,1799,2482,779,2370,622,837,1451,738,2203,1241,2208,1846,1286,2402,674,2089,1895,2645,662,2788,1318,1563,657,773,1996,2144,1278,750,2413,2677,1680,1959,2253,2585,1483,2570,745,2028,1146,1754,904,694,1784,2791,2657,2489,743,1795,2134,2225,1884,1652,1929,875,826,1256,1226,1488,2326,1312,1733,1107,1092,1765,804,704,1675,2193,1786,611,787,2474,947,2573,2358,2319,2361,2521,2624,1117,1722,1801,616,2197,1175,1209,2721,1589,2284,1933,2246,2299,2824,811,858,2111,2267,2261,1838,2593,626,1381,890,2180,758,1339,2610,799,1162,769,2289,2329,1061,2710,1820,1926,1594,2765,2086,2139,1015,2752,1496,851,653,1694,658,2770,983,2075,1873,1203,2271,2471,1420,1684,754,944,2665,1134,648,1018,2564,2567,1547,712,855,640,971,2012,2588,1647,1252,2160,2689,2397,1025,2747,1432,1637,1186,717,2429,1235,2337,1901,1658,643,1220,2354,673,2092,2219,1980,2128,1538,951,808,2411,1898,2558,2426,613,1284,864,1504,1797,2030,2675,2056,656,1023,1972,1506,1140,2511,1881,612,2458,1180,2828,1817,2310,1126,2435,845,2626,1649,1157,2078,1913,1448,1189,1866,2071,2775,1807,771,1276,2707,1639,2406,2554,1508,1481,1101,652,1629,2026,2608,1523,2416,963,2167,1341,2098,1244,1822,2384,2399,2648,1661,832,1893,1614,2009,1861,1440,929,1308,2198,2576,2066,2680,2183,1607,763,2003,2258,1348,2287,1774,1578,835,2604,1148,2050,867,1642,752,2155,2255,2815,854,1986,2636,918,1561,2404,1891,955,2651,897,1410,1789,664,1288,1710,960,1473,2040,696,2668,1165,2021,1346,728,2502,2810,1268,2265,992,822,1713,1299,871,1120,1827,1261,2803,1532,1384,2485,644,1550,1357,2583,1399,1621,1331,1962,923,1565,2018,1848,1449,708,882,1870,2321,1367,935,1009,2438,1490,2297,775,2170,2351]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SWW473+6 : TPTP v9.3.1. Released v5.3.0.
% 0.00/0.07 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.11/0.38 % Computer : n014.cluster.edu
% 0.11/0.38 % Model : x86_64 x86_64
% 0.11/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.38 % Memory : 8046.5625MB
% 0.11/0.38 % OS : Linux 6.8.0-71-generic
% 0.11/0.38 % CPULimit : 300
% 0.11/0.38 % WCLimit : 300
% 0.11/0.38 % DateTime : Sat Sep 26 16:02:01 UTC 2026
% 0.11/0.38 % CPUTime :
% 0.11/0.38 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.87/1.03 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 3.30/1.79 % [INFO] Parsing done (749ms).
% 3.74/1.81 % [INFO] Running in sequential loop mode.
% 5.73/2.41 % [INFO] eprover registered as external prover.
% 5.73/2.42 % [INFO] Scanning for conjecture ...
% 7.17/2.84 % [INFO] Found a conjecture (or negated_conjecture) and 608 axioms. Running axiom selection ...
% 8.33/3.15 % [INFO] Axiom selection finished. Selected 608 axioms (removed 0 axioms).
% 12.22/3.98 % [INFO] Problem is first-order (TPTP FOF).
% 12.22/4.05 % [INFO] Type checking passed.
% 12.90/4.05 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 112.65/31.41 % External prover 'e' found a proof!
% 112.65/31.41 % [INFO] Killing All external provers ...
% 112.65/31.41 % Time passed: 30893ms (effective reasoning time: 29595ms)
% 112.65/31.41 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 112.65/31.43 % Axioms used in derivation (608): fact_327_nat__less__le, fact_44_diff__le__mono2, fact_273_le__add__diff__inverse2, fact_212_fun__diff__def, fact_488_bot__nat__def, fact_117_bot__least, fact_48_lift__Suc__mono__le, arity_HOL_Obool___Finite__Set_Ofinite, fact_250_diff__add__inverse2, fact_407_less__imp__Suc__add, fact_351_linorder__cases, fact_332_not__less__iff__gr__or__eq, fact_92_subset__image__iff, fact_450_card__infinite, fact_55_zero__induct__lemma, help_fimplies_2_1_U, arity_Nat_Onat___Orderings_Obot, tsy_c_fconj_res, fact_161_the__elem__def, arity_HOL_Obool___Orderings_Oorder, fact_224_card_Oinsert__remove, fact_93_image__mono, fact_320_Suc__less__eq, help_COMBS_1_1_U, fact_155_le__funI, fact_104_finite_OemptyI, tsy_c_Orderings_Obot__class_Obot_res, fact_47_finite__subset__image, fact_73_set__mp, fact_86_insert__subset, fact_460_double__add__le__zero__iff__single__add__le__zero, tsy_v_na_res, arity_HOL_Obool___Orderings_Oord, fact_243_nat__add__commute, tsy_c_Nat_OSuc_res, fact_393_less__Suc__eq__le, fact_426_psubset__imp__ex__mem, fact_118_singleton__inject, fact_437_order__less__le__subst1, fact_99_assms_I3_J, arity_fun___Orderings_Obot, fact_333_linorder__less__linear, fact_444_finite__PlusD_I2_J, fact_343_xt1_I9_J, fact_366_less__le__not__le, fact_301_nat__add__left__cancel__less, fact_61_insert__commute, fact_284_diff__Suc__diff__eq1, arity_HOL_Obool___Orderings_Opreorder, fact_317_less__SucI, tsy_c_COMBK_res, help_fconj_3_1_U, fact_13_card__Collect__le__nat, fact_269_diff__add__assoc2, fact_82_insert__compr__raw, fact_149_assms_I2_J, fact_367_order__le__less, fact_101_le__funD, fact_38_diff__Suc__Suc, fact_227_less__eq__nat_Osimps_I2_J, fact_336_less__imp__neq, fact_260_trans__le__add2, fact_331_linorder__neq__iff, fact_365_order__less__le, fact_315_less__trans__Suc, fact_68_set__eq__subset, tsy_c_member_res, fact_291_Suc__mono, fact_210_the__equality, fact_439_order__less__le__subst2, fact_466_le__iff__diff__le__0, fact_115_le__bot, fact_15_finite, tsy_c_fimplies_res, fact_330_order__less__irrefl, fact_352_termination__basic__simps_I5_J, conj_3, tsy_c_Groups_Oplus__class_Oplus_0_res, fact_163_finite__Diff, fact_442_finite__Plus, fact_39_le__diff__iff, arity_fun___Orderings_Oord, fact_421_psubset__trans, fact_314_less__SucE, fact_494_double__zero__sym, fact_5_card__mono, fact_37_Suc__diff__diff, help_COMBI_1_1_U, fact_271_diff__add__assoc, fact_443_finite__PlusD_I1_J, fact_282_nat__case__Suc, fact_484_Suc__not__Zero, fact_53_insertCI, fact_490_less__eq__nat_Osimps_I1_J, arity_Nat_Onat___Orderings_Opreorder, fact_334_linorder__antisym__conv3, fact_464_add__increasing2, fact_96_image__subsetI, fact_189_diff__card__le__card__Diff, fact_18_Suc__n__not__n, fact_275_add__diff__assoc, fact_102_le__funE, fact_424_linorder__neqE__linordered__idom, tsy_c_Nat_Onat_Onat__case_res, fact_106_equals0D, fact_308_add__strict__right__mono, fact_2_finite__imageI, fact_396_Suc__leI, fact_353_less__not__refl, fact_390_less__add__Suc2, fact_246_add__left__imp__eq, arity_HOL_Obool___Groups_Ominus, fact_298_add__less__mono1, arity_sum___Finite__Set_Ofinite, fact_248_add__left__cancel, fact_19_n__not__Suc__n, fact_403_less__diff__iff, fact_90_image__insert, fact_69_equalityD1, fact_467_add__is__1, fact_385_card__Collect__less__nat, fact_75_mem__def, fact_126_image__empty, fact_29_rev__finite__subset, fact_495_add__0__right, fact_256_le__add1, fact_307_add__strict__left__mono, fact_436_xt7, fact_172_Diff__mono, fact_297_add__less__mono, fact_83_image__image, fact_485_nat_Osimps_I2_J, fact_340_order__less__imp__not__eq2, fact_119_singletonE, fact_230_add__right__mono, tsy_c_Set_OCollect_res, fact_237_add__Suc__shift, fact_382_xt1_I8_J, fact_412_psubset__eq, fact_196_folding__image__simple_Oremove, fact_153_the__elem__eq, fact_268_comp__fun__idem__insert, fact_71_in__mono, fact_422_lift__Suc__mono__less__iff, tsy_c_hAPP_res, arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_483_nat_Osimps_I3_J, fact_337_order__less__not__sym, fact_258_nat__add__left__cancel__le, fact_286_termination__basic__simps_I3_J, fact_164_folding__one__idem_Oidem, tsy_c_Groups_Oplus__class_Oplus_1_res, fact_30_Suc__leD, fact_486_Zero__not__Suc, fact_97_order__refl, fact_67_subset__refl, fact_46_finite__surj, tsy_c_hAPP_arg1, fact_449_zero__less__Suc, fact_262_add__le__mono, fact_124_subset__empty, tsy_c_Orderings_Oord__class_Oless__eq_res, fact_58_insert__compr, fact_302_not__add__less2, fact_213_minus__apply, fact_379_order__less__le__trans, fact_151_finite_Osimps, fact_478_diff__self, fact_122_insert__not__empty, fact_295_add__lessD1, fact_88_subset__insertI2, fact_434_order__le__less__subst1, fact_91_insert__image, fact_369_not__leE, help_fNot_2_1_U, fact_31_le__SucE, fact_391_less__iff__Suc__add, fact_158_folding__one__idem_Osubset__idem, fact_399_diff__less__Suc, fact_398_Suc__le__lessD, fact_280_Suc__eq__plus1, fact_22_eq__imp__le, tsy_c_COMBC_res, fact_477_diff__0__right, fact_169_insert__Diff1, fact_324_le__neq__implies__less, fact_386_finite__M__bounded__by__nat, fact_312_Suc__less__SucD, fact_278_le__diff__conv, fact_217_flat__lub__def, fact_160_finite__ne__induct, fact_59_insert__Collect, fact_401_less__diff__conv, help_fNot_1_1_U, tsy_v_mgt_res, fact_123_empty__not__insert, arity_Nat_Onat___Groups_Ocomm__monoid__add, fact_140_xt1_I4_J, tsy_c_COMBS_res, fact_263_add__leD2, fact_150_finite__induct, fact_338_order__less__imp__not__less, fact_325_less__imp__le__nat, fact_321_less__Suc__eq, fact_481_Suc__neq__Zero, fact_438_order__le__less__subst2, fact_487_zero__reorient, fact_66_insert__absorb, tsy_v_U_res, fact_166_Diff__empty, fact_231_add__left__mono, fact_202_DiffD1, fact_468_one__is__add, fact_270_add__diff__assoc2, fact_204_Diff__iff, fact_480_right__minus__eq, fact_54_insertE, fact_209_the__eq__trivial, help_fFalse_1_1_T, fact_41_eq__diff__iff, tsy_v_wt_res, fact_311_diff__eq__diff__less, fact_279_diff__diff__right, tsy_c_fFalse_res, fact_74_ext, fact_433_xt4, fact_453_not__one__less__zero, fact_458_zero__neq__one, tsy_c_hBOOL_arg1, fact_165_Diff__cancel, fact_313_Suc__lessD, fact_492_add__0__left, fact_265_add__leE, fact_135_linorder__le__cases, fact_233_add__le__imp__le__right, fact_84_image__ident, fact_36_Suc__n__not__le__n, fact_10_finite__Collect__conjI, fact_383_finite__nat__set__iff__bounded, fact_475_minus__nat_Odiff__0, fact_410_psubset__imp__subset, fact_455_not__one__le__zero, fact_251_diff__add__inverse, fact_221_diff__Suc__eq__diff__pred, fact_131_singleton__conv2, fact_306_add__strict__mono, tsy_c_Finite__Set_Ofinite_res, help_COMBB_1_1_U, fact_199_DiffE, fact_397_le__less__Suc__eq, fact_144_order__antisym__conv, fact_370_linorder__antisym__conv1, fact_62_insert__iff, fact_7_card__insert__le, tsy_c_Finite__Set_Ocomp__fun__idem_res, fact_64_insert__ident, fact_261_add__le__mono1, fact_462_add__nonneg__eq__0__iff, conj_2, fact_188_card__Diff__subset, arity_fun___Orderings_Oorder, fact_244_add__right__imp__eq, fact_134_image__constant__conv, fact_23_nat__le__linear, fact_247_add__right__cancel, help_fdisj_1_1_U, fact_423_less__add__one, fact_225_card_Oremove, fact_304_add__less__imp__less__left, fact_198_the__sym__eq__trivial, fact_203_Diff__idemp, fact_157_diff__eq__diff__less__eq, arity_Nat_Onat___Orderings_Olinorder, tsy_c_Set_Othe__elem_res, fact_70_equalityD2, fact_430_xt6, fact_136_xt1_I6_J, tsy_c_hAPP_arg2, fact_76_Collect__def, fact_326_le__eq__less__or__eq, fact_394_Suc__le__eq, fact_293_termination__basic__simps_I1_J, fact_377_order__le__neq__trans, fact_105_empty__subsetI, fact_253_diff__cancel, fact_257_le__iff__add, fact_78_equalityE, tsy_c_Finite__Set_Ofolding__one__idem_res, fact_139_order__antisym, fact_417_card__psubset, fact_357_less__not__refl2, arity_fun___Orderings_Opreorder, fact_103_emptyE, fact_322_not__less__eq, tsy_c_COMBB_res, tsy_c_fNot_res, tsy_c_HOL_OThe_res, tsy_c_Set_Oimage_res, fact_459_zero__le__double__add__iff__zero__le__single__add, fact_347_xt1_I2_J, fact_363_linorder__not__le, help_fconj_2_1_U, fact_376_order__le__imp__less__or__eq, fact_432_dense__le__bounded, fact_12_finite__Collect__le__nat, fact_373_leD, fact_319_not__less__less__Suc__eq, fact_65_insertI2, fact_427_less__mono__imp__le__mono, fact_42_diff__diff__cancel, fact_81_rev__image__eqI, fact_182_image__diff__subset, fact_378_xt1_I11_J, fact_384_nat__seg__image__imp__finite, fact_200_DiffI, fact_87_subset__insert, fact_27_finite__insert, fact_392_less__eq__Suc__le, fact_425_bounded__nat__set__is__finite, fact_358_less__not__refl3, fact_98_finite__nat__set__iff__bounded__le, fact_252_diff__diff__left, fact_402_diff__less__mono, fact_360_not__less__bot, fact_232_add__mono, fact_406_inc__induct, fact_121_singleton__iff, fact_388_add__less__le__mono, tsy_c_HOL_Oundefined_res, tsy_c_Finite__Set_Ofolding__one_res, fact_21_le__trans, fact_489_le__0__eq, fact_219_card__Diff__singleton__if, fact_288_card__Diff2__less, fact_420_psubsetD, fact_162_folding__one_Oinsert, fact_235_add__Suc__right, fact_185_subset__insert__iff, fact_174_folding__one_Oremove, help_ti_idem, fact_43_diff__le__mono, fact_159_folding__one__idem_Oinsert__idem, fact_323_less__or__eq__imp__le, arity_Nat_Onat___Rings_Olinordered__semidom, fact_381_order__le__less__trans, fact_389_less__add__Suc1, fact_20_le__antisym, fact_414_less__fun__def, fact_110_ex__in__conv, fact_77_subset__trans, fact_335_linorder__neqE, help_fequal_1_1_T, fact_241_nat__add__assoc, fact_342_order__less__asym_H, fact_267_add__diff__cancel, fact_245_add__imp__eq, fact_255_le__add2, fact_184_diff__single__insert, fact_100_le__fun__def, arity_fun___Finite__Set_Ofinite, fact_411_subset__iff__psubset__eq, fact_52_subsetD, tsy_c_fequal_res, fact_94_imageE, fact_25_diff__commute, fact_413_not__psubset__empty, fact_368_leI, fact_362_linorder__not__less, fact_419_lift__Suc__mono__less, fact_344_ord__eq__less__trans, fact_130_singleton__conv, arity_Nat_Onat___Groups_Oordered__comm__monoid__add, tsy_c_Groups_Oplus__class_Oplus_2_res, fact_236_add__Suc, fact_228_add__le__cancel__right, fact_276_le__add__diff__inverse, fact_479_eq__iff__diff__eq__0, fact_229_add__le__cancel__left, fact_175_folding__one_Oinsert__remove, tsy_v_P_res, arity_Nat_Onat___Groups_Ozero, arity_Nat_Onat___Orderings_Oord, fact_197_folding__image__simple_Oinsert__remove, fact_14_finite__code, fact_441_finite__Plus__iff, help_fdisj_2_1_U, help_COMBK_1_1_U, tsy_c_Set_Oinsert_res, fact_9_card__insert__disjoint, tsy_v_pn_res, fact_428_Suc__lessE, fact_435_xt5, fact_292_finite__Collect__less__nat, fact_45_diff__le__self, fact_16_Suc__inject, arity_Nat_Onat___Groups_Oordered__ab__semigroup__add, fact_33_Suc__le__mono, fact_254_diff__cancel2, fact_305_add__less__imp__less__right, fact_80_imageI, tsy_c_Com_Obody_res, fact_239_nat__add__right__cancel, fact_190_card__Suc__Diff1, fact_448_le0, fact_277_le__add__diff, fact_469_diff__add__0, tsy_c_Finite__Set_Ocard_res, fact_222_diff__Suc__1, fact_350_order__less__asym, fact_491_nat__case__0, fact_63_insert__code, fact_274_le__diff__conv2, fact_496_add_Ocomm__neutral, fact_445_Plus__eq__empty__conv, fact_183_folding__one__idem_Oin__idem, fact_328_diff__less__mono2, fact_8_card__insert__if, fact_56_Suc__le__D, fact_452_zero__less__one, fact_132_subset__singletonD, fact_472_One__nat__def, fact_125_image__is__empty, fact_72_set__rev__mp, fact_375_linorder__antisym__conv2, fact_408_subset__psubset__trans, fact_34_le__Suc__eq, fact_51_equalityI, tsy_v_mgt__call_res, fact_179_Diff__insert__absorb, fact_454_pos__add__strict, arity_HOL_Obool___Orderings_Obot, help_COMBC_1_1_U, fact_310_add__less__cancel__right, fact_17_nat_Oinject, fact_108_empty__iff, fact_128_Collect__conv__if, fact_178_insert__Diff__single, fact_300_trans__less__add1, fact_264_add__leD1, conj_1, fact_215_theI_H, fact_133_image__constant, fact_137_xt1_I5_J, help_fequal_2_1_T, fact_1_finite__Collect__subsets, fact_95_subsetI, arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_409_psubset__subset__trans, fact_146_order__eq__iff, fact_240_nat__add__left__cancel, fact_141_ord__le__eq__trans, fact_50_image__eqI, fact_49_pigeonhole__infinite, fact_329_less__imp__diff__less, fact_107_Collect__empty__eq, fact_120_doubleton__eq__iff, fact_226_comp__fun__idem__remove, fact_205_set__diff__eq, fact_474_diff__self__eq__0, fact_493_add__0, help_fconj_1_1_U, fact_451_card_Oempty, help_fdisj_3_1_U, fact_206_folding__image__simple_Oempty, fact_476_diff__0__eq__0, fact_143_ord__eq__le__trans, fact_296_less__add__eq__less, tsy_c_Groups_Ozero__class_Ozero_res, fact_176_Diff__insert, fact_294_termination__basic__simps_I2_J, tsy_c_Option_Othe_res, fact_109_empty__Collect__eq, fact_191_card__insert, fact_461_add__nonneg__nonneg, fact_89_insert__mono, fact_395_le__imp__less__Suc, fact_431_dense__le, fact_355_linorder__neqE__nat, fact_194_folding__one_Oclosed, fact_112_empty__def, fact_168_finite__Diff2, help_fFalse_1_1_U, fact_242_nat__add__left__commute, fact_6_card__seteq, fact_416_psubset__insert__iff, fact_147_linorder__linear, arity_Nat_Onat___Groups_Oone, fact_259_trans__le__add1, tsy_v_G_res, fact_129_Collect__conv__if2, fact_201_DiffD2, fact_193_folding__one__idem_Ohom__commute, fact_28_finite__subset, fact_272_le__imp__diff__is__add, fact_138_order__trans, fact_440_card__Plus, tsy_c_Groups_Oone__class_Oone_res, fact_116_bot__unique, fact_473_diffs0__imp__equal, fact_111_all__not__in__conv, fact_79_image__iff, fact_40_Nat_Odiff__diff__eq, fact_404_less__eq__Suc__le__raw, arity_Nat_Onat___Orderings_Oorder, fact_186_diff__eq__diff__eq, fact_283_diff__Suc__diff__eq2, fact_4_card__image__le, fact_359_nat__less__cases, fact_24_le__refl, fact_181_finite__Diff__insert, fact_145_order__eq__refl, fact_465_add__nonpos__nonpos, fact_316_Suc__lessI, fact_214_theI, fact_309_add__less__cancel__left, fact_180_insert__Diff, fact_318_less__antisym, fact_35_not__less__eq__eq, fact_400_add__diff__inverse, fact_356_less__irrefl__nat, conj_4, fact_167_empty__Diff, fact_152_finite__less__ub, tsy_c_Finite__Set_Ofolding__image__simple_res, fact_195_finite__empty__induct, fact_177_Diff__insert2, tsy_c_COMBI_res, fact_447_less__zeroE, fact_26_finite__Collect__disjI, fact_339_order__less__imp__not__eq, fact_216_the1__equality, fact_374_order__less__imp__le, fact_354_nat__neq__iff, fact_463_add__increasing, fact_470_diff__is__0__eq_H, fact_148_finite__subset__induct, help_fimplies_1_1_U, fact_32_le__SucI, fact_341_order__less__imp__triv, fact_187_folding__one_Osingleton, fact_154_nonempty__iff, fact_234_add__le__imp__le__left, arity_Nat_Onat___Groups_Omonoid__add, tsy_c_Partial__Function_Oflat__lub_res, fact_303_not__add__less1, fact_456_zero__le__one, tsy_c_Groups_Ominus__class_Ominus_res, fact_446_card__Plus__conv__if, fact_207_folding__image__simple_Oinsert, fact_211_folding__image__simple_Oneutral, tsy_c_Sum__Type_OPlus_res, fact_238_comp__fun__idem_Ofun__left__idem, fact_11_Suc__diff__le, arity_Nat_Onat___Groups_Oab__semigroup__add, fact_0_assms_I1_J, fact_220_one__reorient, fact_289_card__Diff1__less, fact_429_lessE, fact_361_bot__less, fact_380_xt1_I7_J, conj_5, help_fimplies_3_1_U, fact_85_subset__insertI, conj_0, fact_266_diff__add__cancel, fact_371_order__neq__le__trans, fact_249_ab__semigroup__add__class_Oadd__ac_I1_J, arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add, fact_192_card__Diff1__le, fact_364_linorder__le__less__linear, fact_482_Zero__neq__Suc, fact_60_insert__absorb2, fact_405_mono__nat__linear__lb, arity_Nat_Onat___Groups_Ominus, fact_156_assms_I4_J, fact_471_diff__is__0__eq, arity_Nat_Onat___Rings_Ozero__neq__one, fact_285_card_Oinsert, fact_281_Suc__eq__plus1__left, fact_218_card__Diff__singleton, fact_372_xt1_I12_J, fact_208_If__def, fact_290_lessI, fact_57_insertI1, fact_173_Diff__subset, fact_348_order__less__trans, tsy_c_fdisj_res, fact_3_finite_OinsertI, fact_287_termination__basic__simps_I4_J, fact_415_psubset__card__mono, fact_114_bot__apply, fact_127_empty__is__image, tsy_c_Orderings_Oord__class_Oless_res, fact_170_insert__Diff__if, fact_171_double__diff, fact_346_ord__less__eq__trans, fact_457_one__neq__zero, fact_299_trans__less__add2, fact_418_finite__conv__nat__seg__image, arity_fun___Groups_Ominus, arity_Nat_Onat___Groups_Ocancel__semigroup__add, fact_387_add__le__less__mono, fact_142_xt1_I3_J, fact_113_bot__fun__def, tsy_c_fTrue_res, fact_223_card__Diff__insert, fact_349_xt1_I10_J, fact_345_xt1_I1_J
% 112.65/31.43 % No. of inferences in proof: 1220
% 112.65/31.43 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 30893 ms resp. 29595 ms w/o parsing
% 118.29/32.72 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 118.29/32.74 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------