%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : DAT162^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:44:59 AM UTC 2026
% Result : Theorem 2.88s 1.63s
% Output : Refutation 2.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : DAT162^1 : TPTP v9.2.1. Released v7.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n029.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 06:09:59 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : DAT162^1 : TPTP v9.2.1. Released v7.0.0.
% 0.32/0.47 % Domain : Data Structures
% 0.32/0.47 % Problem : Hamming stream 305
% 0.32/0.47 % Version : [Bla16] axioms : Especial.
% 0.32/0.47 % English :
% 0.32/0.47
% 0.32/0.47 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.32/0.47 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.32/0.47 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.32/0.47 % Source : [Bla16]
% 0.32/0.47 % Names : hamming_stream__305.p [Bla16]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.33 v9.1.0, 0.00 v7.5.0, 1.00 v7.3.0, 0.67 v7.2.0, 0.50 v7.1.0
% 0.32/0.47 % Syntax : Number of formulae : 330 ( 77 unt; 50 typ; 0 def)
% 0.32/0.47 % Number of atoms : 845 ( 181 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 13 ( 3 avg)
% 0.32/0.47 % Number of connectives : 3236 ( 117 ~; 36 |; 52 &;2589 @)
% 0.32/0.47 % ( 0 <=>; 442 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 17 ( 8 avg)
% 0.32/0.47 % Number of types : 2 ( 1 usr)
% 0.32/0.47 % Number of type conns : 137 ( 137 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 50 ( 49 usr; 5 con; 0-4 aty)
% 0.32/0.47 % Number of variables : 855 ( 25 ^; 751 !; 40 ?; 855 :)
% 0.32/0.47 % ( 39 !>; 0 ?*; 0 @-; 0 @+)
% 0.32/0.47 % SPC : TH1_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.32/0.47 % 2016-07-13 14:43:25.429
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 %----Could-be-implicit typings (4)
% 0.32/0.47 thf(ty_t_Coinductive__List_Ollist,type,
% 0.32/0.47 coinductive_llist: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Set_Oset,type,
% 0.32/0.47 set: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_Nat_Onat,type,
% 0.32/0.47 nat: $tType ).
% 0.32/0.47
% 0.32/0.47 thf(ty_t_itself,type,
% 0.32/0.47 itself: $tType > $tType ).
% 0.32/0.47
% 0.32/0.47 %----Explicit typings (46)
% 0.32/0.47 thf(sy_cl_HOL_Otype,type,
% 0.32/0.47 type:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Odvd,type,
% 0.32/0.47 dvd:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oone,type,
% 0.32/0.47 one:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oord,type,
% 0.32/0.47 ord:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.32/0.47 order:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_GCD_Osemiring__gcd,type,
% 0.32/0.47 semiring_gcd:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Ono__bot,type,
% 0.32/0.47 no_bot:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Ono__top,type,
% 0.32/0.47 no_top:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.32/0.47 monoid_mult:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/0.47 linorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/0.47 preorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.32/0.47 wellorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.32/0.47 semigroup_mult:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Odense__order,type,
% 0.32/0.47 dense_order:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.32/0.47 comm_semiring_1:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.32/0.47 linordered_idom:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.32/0.47 comm_monoid_mult:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 0.32/0.47 algebraic_semidom:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.32/0.47 ab_semigroup_mult:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.32/0.47 dense_linorder:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.32/0.47 linordered_semidom:
% 0.32/0.47 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Oldistinct,type,
% 0.32/0.47 coindu351974385stinct:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.32/0.47 coinductive_lfinite:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.32/0.47 coinductive_LNil:
% 0.32/0.47 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.32/0.47 coinductive_lnull:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.32/0.47 coinductive_lset:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Olmember,type,
% 0.32/0.47 coinductive_lmember:
% 0.32/0.47 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Olsetp,type,
% 0.32/0.47 coinductive_lsetp:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 0.32/0.47 coindu63249387sorted:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.32/0.47 one_one:
% 0.32/0.47 !>[A: $tType] : A ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.32/0.47 times_times:
% 0.32/0.47 !>[A: $tType] : ( A > A > A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Ohamming,type,
% 0.32/0.47 hammin2015774020amming: coinductive_llist @ nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
% 0.32/0.47 hammin1328233080lmerge:
% 0.32/0.47 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord__class_Olmerge,type,
% 0.32/0.47 hammin1138290136lmerge:
% 0.32/0.47 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth,type,
% 0.32/0.47 hammin1195055315smooth: nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/0.47 ord_less:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/0.47 ord_less_eq:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Primes_Oprime,type,
% 0.32/0.47 prime: nat > $o ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Pure_Otype,type,
% 0.32/0.47 type2:
% 0.32/0.47 !>[A: $tType] : ( itself @ A ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.32/0.47 dvd_dvd:
% 0.32/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_Set_OCollect,type,
% 0.32/0.47 collect:
% 0.32/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_c_member,type,
% 0.32/0.47 member:
% 0.32/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_n,type,
% 0.32/0.47 n: nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_n_H____,type,
% 0.32/0.47 n2: nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_na____,type,
% 0.32/0.47 na: nat ).
% 0.32/0.47
% 0.32/0.47 thf(sy_v_p____,type,
% 0.32/0.47 p: nat ).
% 0.32/0.47
% 0.32/0.47 %----Relevant facts (253)
% 0.32/0.47 thf(fact_0__092_060open_062smooth_An_H_092_060close_062,axiom,
% 0.32/0.47 hammin1195055315smooth @ n2 ).
% 0.32/0.47
% 0.32/0.47 % \<open>smooth n'\<close>
% 0.32/0.47 thf(fact_1__092_060open_062n_H_A_060_An_092_060close_062,axiom,
% 0.32/0.47 ord_less @ nat @ n2 @ na ).
% 0.32/0.47
% 0.32/0.47 % \<open>n' < n\<close>
% 0.32/0.47 thf(fact_2_less_OIH,axiom,
% 0.32/0.47 ! [Y: nat] :
% 0.32/0.47 ( ( ord_less @ nat @ Y @ na )
% 0.32/0.47 => ( ( hammin1195055315smooth @ Y )
% 0.32/0.47 => ( member @ nat @ Y @ ( coinductive_lset @ nat @ hammin2015774020amming ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less.IH
% 0.32/0.47 thf(fact_3_n,axiom,
% 0.32/0.47 ( na
% 0.32/0.47 = ( times_times @ nat @ p @ n2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % n
% 0.32/0.47 thf(fact_4_hamming__eq__LNil__iff,axiom,
% 0.32/0.47 ( hammin2015774020amming
% 0.32/0.47 != ( coinductive_LNil @ nat ) ) ).
% 0.32/0.47
% 0.32/0.47 % hamming_eq_LNil_iff
% 0.32/0.47 thf(fact_5_ldistinct__hamming,axiom,
% 0.32/0.47 coindu351974385stinct @ nat @ hammin2015774020amming ).
% 0.32/0.47
% 0.32/0.47 % ldistinct_hamming
% 0.32/0.47 thf(fact_6_lsorted__hamming,axiom,
% 0.32/0.47 coindu63249387sorted @ nat @ hammin2015774020amming ).
% 0.32/0.47
% 0.32/0.47 % lsorted_hamming
% 0.32/0.47 thf(fact_7_True,axiom,
% 0.32/0.47 ord_less @ nat @ ( one_one @ nat ) @ na ).
% 0.32/0.47
% 0.32/0.47 % True
% 0.32/0.47 thf(fact_8_lnull__hamming,axiom,
% 0.32/0.47 ~ ( coinductive_lnull @ nat @ hammin2015774020amming ) ).
% 0.32/0.47
% 0.32/0.47 % lnull_hamming
% 0.32/0.47 thf(fact_9_lset__lmember,axiom,
% 0.32/0.47 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.32/0.47 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.47 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 0.32/0.47
% 0.32/0.47 % lset_lmember
% 0.32/0.47 thf(fact_10_lfinite__hamming,axiom,
% 0.32/0.47 ~ ( coinductive_lfinite @ nat @ hammin2015774020amming ) ).
% 0.32/0.47
% 0.32/0.47 % lfinite_hamming
% 0.32/0.47 thf(fact_11_in__lset__lmergeD,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) )
% 0.32/0.47 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.47 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % in_lset_lmergeD
% 0.32/0.47 thf(fact_12_ord_Oin__lset__lmergeD,axiom,
% 0.32/0.47 ! [A: $tType,X: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 0.32/0.47 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.47 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.in_lset_lmergeD
% 0.32/0.47 thf(fact_13_Coinductive__List_Olset__into__lsetp,axiom,
% 0.32/0.47 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.32/0.47 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.47 => ( coinductive_lsetp @ A @ Xs @ X ) ) ).
% 0.32/0.47
% 0.32/0.47 % Coinductive_List.lset_into_lsetp
% 0.32/0.47 thf(fact_14_assms,axiom,
% 0.32/0.47 hammin1195055315smooth @ n ).
% 0.32/0.47
% 0.32/0.47 % assms
% 0.32/0.47 thf(fact_15_less_Oprems,axiom,
% 0.32/0.47 hammin1195055315smooth @ na ).
% 0.32/0.47
% 0.32/0.47 % less.prems
% 0.32/0.47 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_An_A_061_Ap_A_K_An_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/0.47 ~ ! [N: nat] :
% 0.32/0.47 ( na
% 0.32/0.47 != ( times_times @ nat @ p @ N ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<open>\<And>thesis. (\<And>n'. n = p * n' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.32/0.47 thf(fact_17__092_060open_062prime_Ap_092_060close_062,axiom,
% 0.32/0.47 prime @ p ).
% 0.32/0.47
% 0.32/0.47 % \<open>prime p\<close>
% 0.32/0.47 thf(fact_18_lfinite__code_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % lfinite_code(1)
% 0.32/0.47 thf(fact_19_lmerge_Odisc__iff_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ~ ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) )
% 0.32/0.47 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge.disc_iff(2)
% 0.32/0.47 thf(fact_20_lnull__lmerge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) )
% 0.32/0.47 = ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lnull_lmerge
% 0.32/0.47 thf(fact_21_lmerge__LNil_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A] :
% 0.32/0.47 ( ( hammin1138290136lmerge @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge_LNil(2)
% 0.32/0.47 thf(fact_22_lmerge__LNil_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( hammin1138290136lmerge @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge_LNil(1)
% 0.32/0.47 thf(fact_23_lsorted__code_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsorted_code(1)
% 0.32/0.47 thf(fact_24_ldistinct__LNil__code,axiom,
% 0.32/0.47 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ldistinct_LNil_code
% 0.32/0.47 thf(fact_25_smooth__times,axiom,
% 0.32/0.47 ! [X: nat,Y: nat] :
% 0.32/0.47 ( ( hammin1195055315smooth @ ( times_times @ nat @ X @ Y ) )
% 0.32/0.47 = ( ( hammin1195055315smooth @ X )
% 0.32/0.47 & ( hammin1195055315smooth @ Y ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % smooth_times
% 0.32/0.47 thf(fact_26__092_060open_062p_Advd_An_092_060close_062,axiom,
% 0.32/0.47 dvd_dvd @ nat @ p @ na ).
% 0.32/0.47
% 0.32/0.47 % \<open>p dvd n\<close>
% 0.32/0.47 thf(fact_27_llist_Odisc_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % llist.disc(1)
% 0.32/0.47 thf(fact_28_llist_OdiscI_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.32/0.47 ( ( Llist
% 0.32/0.47 = ( coinductive_LNil @ A ) )
% 0.32/0.47 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.32/0.47
% 0.32/0.47 % llist.discI(1)
% 0.32/0.47 thf(fact_29_lmember__code_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,X: A] :
% 0.32/0.47 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmember_code(1)
% 0.32/0.47 thf(fact_30_llist_Ocollapse_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lnull @ A @ Llist )
% 0.32/0.47 => ( Llist
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % llist.collapse(1)
% 0.32/0.47 thf(fact_31_ord_Olmerge_Octr_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge.ctr(1)
% 0.32/0.47 thf(fact_32_ord_Olmerge_Odisc_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/0.47 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( ~ ( coinductive_lnull @ A @ Ys )
% 0.32/0.47 => ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge.disc(2)
% 0.32/0.47 thf(fact_33_ord_Olmerge_Odisc_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge.disc(1)
% 0.32/0.47 thf(fact_34_ord_Olmerge__LNil_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A] :
% 0.32/0.47 ( ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LNil @ A ) )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge_LNil(2)
% 0.32/0.47 thf(fact_35_ord_Olmerge__LNil_I1_J,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LNil @ A ) @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge_LNil(1)
% 0.32/0.47 thf(fact_36_ord_Olmerge_Odisc__iff_I2_J,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 0.32/0.47 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge.disc_iff(2)
% 0.32/0.47 thf(fact_37_lmerge_Octr_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ( ( hammin1138290136lmerge @ A @ Xs @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge.ctr(1)
% 0.32/0.47 thf(fact_38_lmerge_Odisc_I2_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( ~ ( coinductive_lnull @ A @ Ys )
% 0.32/0.47 => ~ ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge.disc(2)
% 0.32/0.47 thf(fact_39_lmerge_Odisc_I1_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge.disc(1)
% 0.32/0.47 thf(fact_40_lzip_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.32/0.47 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ B @ Ys ) )
% 0.32/0.47 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lzip.exhaust
% 0.32/0.47 thf(fact_41_ldistinct_OLNil,axiom,
% 0.32/0.47 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % ldistinct.LNil
% 0.32/0.47 thf(fact_42_lappend_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lappend.exhaust
% 0.32/0.47 thf(fact_43_lnull__def,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( coinductive_lnull @ A )
% 0.32/0.47 = ( ^ [Llist2: coinductive_llist @ A] :
% 0.32/0.47 ( Llist2
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lnull_def
% 0.32/0.47 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.32/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.32/0.47 = ( P @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: $tType,A3: set @ A] :
% 0.32/0.47 ( ( collect @ A
% 0.32/0.47 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.32/0.47 = A3 ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_46_Collect__cong,axiom,
% 0.32/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/0.47 ( ! [X3: A] :
% 0.32/0.47 ( ( P @ X3 )
% 0.32/0.47 = ( Q @ X3 ) )
% 0.32/0.47 => ( ( collect @ A @ P )
% 0.32/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_cong
% 0.32/0.47 thf(fact_47_lnull__imp__lfinite,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.32/0.47
% 0.32/0.47 % lnull_imp_lfinite
% 0.32/0.47 thf(fact_48_lfinite__LNil,axiom,
% 0.32/0.47 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % lfinite_LNil
% 0.32/0.47 thf(fact_49_LNil,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % LNil
% 0.32/0.47 thf(fact_50_ord_Olnull__lmerge,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 0.32/0.47 = ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lnull_lmerge
% 0.32/0.47 thf(fact_51_ord_Olmerge_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge.exhaust
% 0.32/0.47 thf(fact_52_ord_Olfinite__lmergeD,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 0.32/0.47 => ( ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 | ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lfinite_lmergeD
% 0.32/0.47 thf(fact_53_ord_Olfinite__lmergeI,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/0.47 ( ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 => ( ( coinductive_lfinite @ A @ Ys )
% 0.32/0.47 => ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lfinite_lmergeI
% 0.32/0.47 thf(fact_54_ord_Olinfinite__lmerge,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/0.47 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/0.47 => ~ ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.linfinite_lmerge
% 0.32/0.47 thf(fact_55_ord_Olmerge__eq__LNil__iff,axiom,
% 0.32/0.47 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) )
% 0.32/0.47 = ( ( Xs
% 0.32/0.47 = ( coinductive_LNil @ A ) )
% 0.32/0.47 | ( Ys
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ord.lmerge_eq_LNil_iff
% 0.32/0.47 thf(fact_56_lmerge_Oexhaust,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/0.47 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/0.47 => ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge.exhaust
% 0.32/0.47 thf(fact_57_lsorted__ldistinct__lset__unique,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/0.47 => ( ( coindu351974385stinct @ A @ Xs )
% 0.32/0.47 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/0.47 => ( ( coindu351974385stinct @ A @ Ys )
% 0.32/0.47 => ( ( ( coinductive_lset @ A @ Xs )
% 0.32/0.47 = ( coinductive_lset @ A @ Ys ) )
% 0.32/0.47 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsorted_ldistinct_lset_unique
% 0.32/0.47 thf(fact_58_lfinite__lmergeD,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) )
% 0.32/0.47 => ( ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 | ( coinductive_lfinite @ A @ Ys ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lfinite_lmergeD
% 0.32/0.47 thf(fact_59_lfinite__lmergeI,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 => ( ( coinductive_lfinite @ A @ Ys )
% 0.32/0.47 => ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lfinite_lmergeI
% 0.32/0.47 thf(fact_60_linfinite__lmerge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/0.47 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/0.47 => ~ ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % linfinite_lmerge
% 0.32/0.47 thf(fact_61_lmerge__eq__LNil__iff,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( ( hammin1138290136lmerge @ A @ Xs @ Ys )
% 0.32/0.47 = ( coinductive_LNil @ A ) )
% 0.32/0.47 = ( ( Xs
% 0.32/0.47 = ( coinductive_LNil @ A ) )
% 0.32/0.47 | ( Ys
% 0.32/0.47 = ( coinductive_LNil @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lmerge_eq_LNil_iff
% 0.32/0.47 thf(fact_62_lsorted__lmerge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/0.47 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/0.47 => ( coindu63249387sorted @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsorted_lmerge
% 0.32/0.47 thf(fact_63_ldistinct__lmerge,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.47 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/0.47 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/0.47 => ( ( coindu351974385stinct @ A @ Xs )
% 0.32/0.47 => ( ( coindu351974385stinct @ A @ Ys )
% 0.32/0.47 => ( coindu351974385stinct @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % ldistinct_lmerge
% 0.32/0.47 thf(fact_64_lsetp__into__lset,axiom,
% 0.32/0.47 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.32/0.47 ( ( coinductive_lsetp @ A @ Xs @ X )
% 0.32/0.47 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % lsetp_into_lset
% 0.32/0.47 thf(fact_65_nat__1__eq__mult__iff,axiom,
% 0.32/0.47 ! [M: nat,N2: nat] :
% 0.32/0.47 ( ( ( one_one @ nat )
% 0.32/0.47 = ( times_times @ nat @ M @ N2 ) )
% 0.32/0.47 = ( ( M
% 0.32/0.47 = ( one_one @ nat ) )
% 0.32/0.47 & ( N2
% 0.32/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_1_eq_mult_iff
% 0.32/0.47 thf(fact_66_nat__mult__eq__1__iff,axiom,
% 0.32/0.47 ! [M: nat,N2: nat] :
% 0.32/0.47 ( ( ( times_times @ nat @ M @ N2 )
% 0.32/0.47 = ( one_one @ nat ) )
% 0.32/0.47 = ( ( M
% 0.32/0.47 = ( one_one @ nat ) )
% 0.32/0.47 & ( N2
% 0.32/0.47 = ( one_one @ nat ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_mult_eq_1_iff
% 0.32/0.47 thf(fact_67_mult_Oleft__neutral,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult.left_neutral
% 0.32/0.47 thf(fact_68_mult_Oright__neutral,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mult.right_neutral
% 0.32/0.47 thf(fact_69_less__1__mult,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [M: A,N2: A] :
% 0.32/0.47 ( ( ord_less @ A @ ( one_one @ A ) @ M )
% 0.32/0.47 => ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
% 0.32/0.47 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % less_1_mult
% 0.32/0.47 thf(fact_70_nat__mult__1,axiom,
% 0.32/0.47 ! [N2: nat] :
% 0.32/0.47 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
% 0.32/0.47 = N2 ) ).
% 0.32/0.47
% 0.32/0.47 % nat_mult_1
% 0.32/0.47 thf(fact_71_nat__mult__1__right,axiom,
% 0.32/0.47 ! [N2: nat] :
% 0.32/0.47 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
% 0.32/0.47 = N2 ) ).
% 0.32/0.47
% 0.32/0.47 % nat_mult_1_right
% 0.32/0.47 thf(fact_72_semiring__normalization__rules_I12_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % semiring_normalization_rules(12)
% 0.32/0.47 thf(fact_73_semiring__normalization__rules_I11_J,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % semiring_normalization_rules(11)
% 0.32/0.47 thf(fact_74_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] :
% 0.32/0.47 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/0.47 = A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % comm_monoid_mult_class.mult_1
% 0.32/0.47 thf(fact_75__092_060open_062_092_060exists_062p_O_Aprime_Ap_A_092_060and_062_Ap_Advd_An_092_060close_062,axiom,
% 0.32/0.47 ? [P2: nat] :
% 0.32/0.47 ( ( prime @ P2 )
% 0.32/0.47 & ( dvd_dvd @ nat @ P2 @ na ) ) ).
% 0.32/0.47
% 0.32/0.47 % \<open>\<exists>p. prime p \<and> p dvd n\<close>
% 0.32/0.47 thf(fact_76_nat__dvd__1__iff__1,axiom,
% 0.32/0.47 ! [M: nat] :
% 0.32/0.47 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 0.32/0.47 = ( M
% 0.32/0.47 = ( one_one @ nat ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % nat_dvd_1_iff_1
% 0.32/0.47 thf(fact_77_unit__prod,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.47 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/0.47 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % unit_prod
% 0.32/0.47 thf(fact_78_dvd__trans,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A,B2: A,C: A] :
% 0.32/0.47 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/0.47 => ( ( dvd_dvd @ A @ B2 @ C )
% 0.32/0.47 => ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % dvd_trans
% 0.32/0.47 thf(fact_79_dvd__refl,axiom,
% 0.32/0.47 ! [A: $tType] :
% 0.32/0.47 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.47 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_refl
% 0.32/0.48 thf(fact_80_dvd__antisym,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( dvd_dvd @ nat @ M @ N2 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ N2 @ M )
% 0.32/0.48 => ( M = N2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_antisym
% 0.32/0.48 thf(fact_81_prime__nat__dvdD,axiom,
% 0.32/0.48 ! [P3: nat,N2: nat] :
% 0.32/0.48 ( ( prime @ P3 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ N2 @ P3 )
% 0.32/0.48 => ( ( N2
% 0.32/0.48 = ( one_one @ nat ) )
% 0.32/0.48 | ( N2 = P3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_nat_dvdD
% 0.32/0.48 thf(fact_82_dvd__triv__right,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_triv_right
% 0.32/0.48 thf(fact_83_dvd__mult__right,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/0.48 => ( dvd_dvd @ A @ B2 @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_mult_right
% 0.32/0.48 thf(fact_84_mult__dvd__mono,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A,D: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( dvd_dvd @ A @ C @ D )
% 0.32/0.48 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_dvd_mono
% 0.32/0.48 thf(fact_85_dvd__triv__left,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_triv_left
% 0.32/0.48 thf(fact_86_dvd__mult__left,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/0.48 => ( dvd_dvd @ A @ A2 @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_mult_left
% 0.32/0.48 thf(fact_87_dvd__mult2,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/0.48 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_mult2
% 0.32/0.48 thf(fact_88_dvd__mult,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,C: A,B2: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ C )
% 0.32/0.48 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_mult
% 0.32/0.48 thf(fact_89_dvd__def,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( dvd_dvd @ A )
% 0.32/0.48 = ( ^ [B3: A,A4: A] :
% 0.32/0.48 ? [K: A] :
% 0.32/0.48 ( A4
% 0.32/0.48 = ( times_times @ A @ B3 @ K ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_def
% 0.32/0.48 thf(fact_90_dvdI,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,K2: A] :
% 0.32/0.48 ( ( A2
% 0.32/0.48 = ( times_times @ A @ B2 @ K2 ) )
% 0.32/0.48 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvdI
% 0.32/0.48 thf(fact_91_dvdE,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ B2 @ A2 )
% 0.32/0.48 => ~ ! [K3: A] :
% 0.32/0.48 ( A2
% 0.32/0.48 != ( times_times @ A @ B2 @ K3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvdE
% 0.32/0.48 thf(fact_92_dvd__unit__imp__unit,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/0.48 => ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_unit_imp_unit
% 0.32/0.48 thf(fact_93_unit__imp__dvd,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/0.48 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unit_imp_dvd
% 0.32/0.48 thf(fact_94_one__dvd,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % one_dvd
% 0.32/0.48 thf(fact_95_unit__mult__right__cancel,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.48 => ( ( ( times_times @ A @ B2 @ A2 )
% 0.32/0.48 = ( times_times @ A @ C @ A2 ) )
% 0.32/0.48 = ( B2 = C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unit_mult_right_cancel
% 0.32/0.48 thf(fact_96_unit__mult__left__cancel,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.48 => ( ( ( times_times @ A @ A2 @ B2 )
% 0.32/0.48 = ( times_times @ A @ A2 @ C ) )
% 0.32/0.48 = ( B2 = C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % unit_mult_left_cancel
% 0.32/0.48 thf(fact_97_mult__unit__dvd__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/0.48 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/0.48 = ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_unit_dvd_iff
% 0.32/0.48 thf(fact_98_dvd__mult__unit__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/0.48 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C @ B2 ) )
% 0.32/0.48 = ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dvd_mult_unit_iff
% 0.32/0.48 thf(fact_99_is__unit__mult__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
% 0.32/0.48 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.48 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % is_unit_mult_iff
% 0.32/0.48 thf(fact_100_linorder__neqE__linordered__idom,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( X != Y )
% 0.32/0.48 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_neqE_linordered_idom
% 0.32/0.48 thf(fact_101_mult_Oleft__commute,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A,C: A] :
% 0.32/0.48 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C ) )
% 0.32/0.48 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult.left_commute
% 0.32/0.48 thf(fact_102_mult_Ocommute,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( times_times @ A )
% 0.32/0.48 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult.commute
% 0.32/0.48 thf(fact_103_mult_Oassoc,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/0.48 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult.assoc
% 0.32/0.48 thf(fact_104_semiring__normalization__rules_I7_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( times_times @ A )
% 0.32/0.48 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(7)
% 0.32/0.48 thf(fact_105_semiring__normalization__rules_I13_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/0.48 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(13)
% 0.32/0.48 thf(fact_106_semiring__normalization__rules_I14_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/0.48 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(14)
% 0.32/0.48 thf(fact_107_semiring__normalization__rules_I15_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/0.48 = ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(15)
% 0.32/0.48 thf(fact_108_semiring__normalization__rules_I16_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Ly: A,Rx: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.32/0.48 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(16)
% 0.32/0.48 thf(fact_109_semiring__normalization__rules_I17_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Ly: A,Rx: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.32/0.48 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(17)
% 0.32/0.48 thf(fact_110_semiring__normalization__rules_I18_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Rx: A,Ry: A] :
% 0.32/0.48 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/0.48 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(18)
% 0.32/0.48 thf(fact_111_semiring__normalization__rules_I19_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Lx: A,Rx: A,Ry: A] :
% 0.32/0.48 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/0.48 = ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_normalization_rules(19)
% 0.32/0.48 thf(fact_112_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/0.48 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.48 thf(fact_113_one__reorient,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( one @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A] :
% 0.32/0.48 ( ( ( one_one @ A )
% 0.32/0.48 = X )
% 0.32/0.48 = ( X
% 0.32/0.48 = ( one_one @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % one_reorient
% 0.32/0.48 thf(fact_114_infinite__descent__measure,axiom,
% 0.32/0.48 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.32/0.48 ( ! [X3: A] :
% 0.32/0.48 ( ~ ( P @ X3 )
% 0.32/0.48 => ? [Y2: A] :
% 0.32/0.48 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 0.32/0.48 & ~ ( P @ Y2 ) ) )
% 0.32/0.48 => ( P @ X ) ) ).
% 0.32/0.48
% 0.32/0.48 % infinite_descent_measure
% 0.32/0.48 thf(fact_115_measure__induct__rule,axiom,
% 0.32/0.48 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.32/0.48 ( ! [X3: A] :
% 0.32/0.48 ( ! [Y2: A] :
% 0.32/0.48 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.32/0.48 => ( P @ Y2 ) )
% 0.32/0.48 => ( P @ X3 ) )
% 0.32/0.48 => ( P @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % measure_induct_rule
% 0.32/0.48 thf(fact_116_linorder__neqE__nat,axiom,
% 0.32/0.48 ! [X: nat,Y: nat] :
% 0.32/0.48 ( ( X != Y )
% 0.32/0.48 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.32/0.48 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_neqE_nat
% 0.32/0.48 thf(fact_117_infinite__descent,axiom,
% 0.32/0.48 ! [P: nat > $o,N2: nat] :
% 0.32/0.48 ( ! [N3: nat] :
% 0.32/0.48 ( ~ ( P @ N3 )
% 0.32/0.48 => ? [M2: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ M2 @ N3 )
% 0.32/0.48 & ~ ( P @ M2 ) ) )
% 0.32/0.48 => ( P @ N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % infinite_descent
% 0.32/0.48 thf(fact_118_nat__less__induct,axiom,
% 0.32/0.48 ! [P: nat > $o,N2: nat] :
% 0.32/0.48 ( ! [N3: nat] :
% 0.32/0.48 ( ! [M2: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ M2 @ N3 )
% 0.32/0.48 => ( P @ M2 ) )
% 0.32/0.48 => ( P @ N3 ) )
% 0.32/0.48 => ( P @ N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_less_induct
% 0.32/0.48 thf(fact_119_less__irrefl__nat,axiom,
% 0.32/0.48 ! [N2: nat] :
% 0.32/0.48 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.32/0.48
% 0.32/0.48 % less_irrefl_nat
% 0.32/0.48 thf(fact_120_measure__induct,axiom,
% 0.32/0.48 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.32/0.48 ( ! [X3: A] :
% 0.32/0.48 ( ! [Y2: A] :
% 0.32/0.48 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.32/0.48 => ( P @ Y2 ) )
% 0.32/0.48 => ( P @ X3 ) )
% 0.32/0.48 => ( P @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % measure_induct
% 0.32/0.48 thf(fact_121_less__not__refl3,axiom,
% 0.32/0.48 ! [S: nat,T: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ S @ T )
% 0.32/0.48 => ( S != T ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_not_refl3
% 0.32/0.48 thf(fact_122_less__not__refl2,axiom,
% 0.32/0.48 ! [N2: nat,M: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ N2 @ M )
% 0.32/0.48 => ( M != N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_not_refl2
% 0.32/0.48 thf(fact_123_less__not__refl,axiom,
% 0.32/0.48 ! [N2: nat] :
% 0.32/0.48 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.32/0.48
% 0.32/0.48 % less_not_refl
% 0.32/0.48 thf(fact_124_nat__neq__iff,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( M != N2 )
% 0.32/0.48 = ( ( ord_less @ nat @ M @ N2 )
% 0.32/0.48 | ( ord_less @ nat @ N2 @ M ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_neq_iff
% 0.32/0.48 thf(fact_125_mult_Ocomm__neutral,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A] :
% 0.32/0.48 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/0.48 = A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult.comm_neutral
% 0.32/0.48 thf(fact_126_prime__dvd__mult__eq__nat,axiom,
% 0.32/0.48 ! [P3: nat,M: nat,N2: nat] :
% 0.32/0.48 ( ( prime @ P3 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ P3 @ ( times_times @ nat @ M @ N2 ) )
% 0.32/0.48 = ( ( dvd_dvd @ nat @ P3 @ M )
% 0.32/0.48 | ( dvd_dvd @ nat @ P3 @ N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_dvd_mult_eq_nat
% 0.32/0.48 thf(fact_127_prime__def,axiom,
% 0.32/0.48 ( prime
% 0.32/0.48 = ( ^ [P4: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ ( one_one @ nat ) @ P4 )
% 0.32/0.48 & ! [M3: nat] :
% 0.32/0.48 ( ( dvd_dvd @ nat @ M3 @ P4 )
% 0.32/0.48 => ( ( M3
% 0.32/0.48 = ( one_one @ nat ) )
% 0.32/0.48 | ( M3 = P4 ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_def
% 0.32/0.48 thf(fact_128_not__prime__eq__prod__nat,axiom,
% 0.32/0.48 ! [N2: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
% 0.32/0.48 => ( ~ ( prime @ N2 )
% 0.32/0.48 => ? [M4: nat,K3: nat] :
% 0.32/0.48 ( ( N2
% 0.32/0.48 = ( times_times @ nat @ M4 @ K3 ) )
% 0.32/0.48 & ( ord_less @ nat @ ( one_one @ nat ) @ M4 )
% 0.32/0.48 & ( ord_less @ nat @ M4 @ N2 )
% 0.32/0.48 & ( ord_less @ nat @ ( one_one @ nat ) @ K3 )
% 0.32/0.48 & ( ord_less @ nat @ K3 @ N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % not_prime_eq_prod_nat
% 0.32/0.48 thf(fact_129_prime__factor__nat,axiom,
% 0.32/0.48 ! [N2: nat] :
% 0.32/0.48 ( ( N2
% 0.32/0.48 != ( one_one @ nat ) )
% 0.32/0.48 => ? [P2: nat] :
% 0.32/0.48 ( ( prime @ P2 )
% 0.32/0.48 & ( dvd_dvd @ nat @ P2 @ N2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_factor_nat
% 0.32/0.48 thf(fact_130_prime__dvd__mult__nat,axiom,
% 0.32/0.48 ! [P3: nat,M: nat,N2: nat] :
% 0.32/0.48 ( ( prime @ P3 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ P3 @ ( times_times @ nat @ M @ N2 ) )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ P3 @ M )
% 0.32/0.48 | ( dvd_dvd @ nat @ P3 @ N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_dvd_mult_nat
% 0.32/0.48 thf(fact_131_prime__product,axiom,
% 0.32/0.48 ! [P3: nat,Q2: nat] :
% 0.32/0.48 ( ( prime @ ( times_times @ nat @ P3 @ Q2 ) )
% 0.32/0.48 => ( ( P3
% 0.32/0.48 = ( one_one @ nat ) )
% 0.32/0.48 | ( Q2
% 0.32/0.48 = ( one_one @ nat ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_product
% 0.32/0.48 thf(fact_132_bigger__prime,axiom,
% 0.32/0.48 ! [N2: nat] :
% 0.32/0.48 ? [P2: nat] :
% 0.32/0.48 ( ( prime @ P2 )
% 0.32/0.48 & ( ord_less @ nat @ N2 @ P2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % bigger_prime
% 0.32/0.48 thf(fact_133_one__not__prime__nat,axiom,
% 0.32/0.48 ~ ( prime @ ( one_one @ nat ) ) ).
% 0.32/0.48
% 0.32/0.48 % one_not_prime_nat
% 0.32/0.48 thf(fact_134_prime__gt__1__nat,axiom,
% 0.32/0.48 ! [P3: nat] :
% 0.32/0.48 ( ( prime @ P3 )
% 0.32/0.48 => ( ord_less @ nat @ ( one_one @ nat ) @ P3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_gt_1_nat
% 0.32/0.48 thf(fact_135_division__decomp,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( semiring_gcd @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) )
% 0.32/0.48 => ? [B4: A,C2: A] :
% 0.32/0.48 ( ( A2
% 0.32/0.48 = ( times_times @ A @ B4 @ C2 ) )
% 0.32/0.48 & ( dvd_dvd @ A @ B4 @ B2 )
% 0.32/0.48 & ( dvd_dvd @ A @ C2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % division_decomp
% 0.32/0.48 thf(fact_136_less__numeral__extra_I4_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_numeral_extra(4)
% 0.32/0.48 thf(fact_137_in__lset__lmergeI1,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/0.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.48 => ( ( coindu63249387sorted @ A @ Xs )
% 0.32/0.48 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/0.48 => ( ? [X4: A] :
% 0.32/0.48 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Ys ) )
% 0.32/0.48 & ( ord_less_eq @ A @ X @ X4 ) )
% 0.32/0.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % in_lset_lmergeI1
% 0.32/0.48 thf(fact_138_in__lset__lmergeI2,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.32/0.48 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.32/0.48 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/0.48 => ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/0.48 => ( ? [X4: A] :
% 0.32/0.48 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.32/0.48 & ( ord_less_eq @ A @ X @ X4 ) )
% 0.32/0.48 => ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % in_lset_lmergeI2
% 0.32/0.48 thf(fact_139_nat__less__le,axiom,
% 0.32/0.48 ( ( ord_less @ nat )
% 0.32/0.48 = ( ^ [M3: nat,N4: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ M3 @ N4 )
% 0.32/0.48 & ( M3 != N4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_less_le
% 0.32/0.48 thf(fact_140_less__imp__le__nat,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ M @ N2 )
% 0.32/0.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_le_nat
% 0.32/0.48 thf(fact_141_le__eq__less__or__eq,axiom,
% 0.32/0.48 ( ( ord_less_eq @ nat )
% 0.32/0.48 = ( ^ [M3: nat,N4: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ M3 @ N4 )
% 0.32/0.48 | ( M3 = N4 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_eq_less_or_eq
% 0.32/0.48 thf(fact_142_less__or__eq__imp__le,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( ( ord_less @ nat @ M @ N2 )
% 0.32/0.48 | ( M = N2 ) )
% 0.32/0.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_or_eq_imp_le
% 0.32/0.48 thf(fact_143_le__neq__implies__less,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/0.48 => ( ( M != N2 )
% 0.32/0.48 => ( ord_less @ nat @ M @ N2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_neq_implies_less
% 0.32/0.48 thf(fact_144_less__mono__imp__le__mono,axiom,
% 0.32/0.48 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/0.48 ( ! [I2: nat,J2: nat] :
% 0.32/0.48 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/0.48 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_mono_imp_le_mono
% 0.32/0.48 thf(fact_145_le__cube,axiom,
% 0.32/0.48 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_cube
% 0.32/0.48 thf(fact_146_le__square,axiom,
% 0.32/0.48 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_square
% 0.32/0.48 thf(fact_147_mult__le__mono,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K2: nat,L: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ K2 @ L )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_le_mono
% 0.32/0.48 thf(fact_148_mult__le__mono1,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_le_mono1
% 0.32/0.48 thf(fact_149_mult__le__mono2,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I ) @ ( times_times @ nat @ K2 @ J ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % mult_le_mono2
% 0.32/0.48 thf(fact_150_wlog__linorder__le,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.32/0.48 ( ! [A5: A,B5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/0.48 => ( P @ A5 @ B5 ) )
% 0.32/0.48 => ( ( ( P @ B2 @ A2 )
% 0.32/0.48 => ( P @ A2 @ B2 ) )
% 0.32/0.48 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % wlog_linorder_le
% 0.32/0.48 thf(fact_151_le__numeral__extra_I4_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_numeral_extra(4)
% 0.32/0.48 thf(fact_152_prime__ge__1__nat,axiom,
% 0.32/0.48 ! [P3: nat] :
% 0.32/0.48 ( ( prime @ P3 )
% 0.32/0.48 => ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 ) ) ).
% 0.32/0.48
% 0.32/0.48 % prime_ge_1_nat
% 0.32/0.48 thf(fact_153_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( A2 != B2 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.not_eq_order_implies_strict
% 0.32/0.48 thf(fact_154_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 => ( A2 != B2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_implies_not_eq
% 0.32/0.48 thf(fact_155_gcd__nat_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_implies_order
% 0.32/0.48 thf(fact_156_gcd__nat_Ostrict__iff__order,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 = ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_iff_order
% 0.32/0.48 thf(fact_157_gcd__nat_Oorder__iff__strict,axiom,
% 0.32/0.48 ( ( dvd_dvd @ nat )
% 0.32/0.48 = ( ^ [A4: nat,B3: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A4 @ B3 )
% 0.32/0.48 & ( A4 != B3 ) )
% 0.32/0.48 | ( A4 = B3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.order_iff_strict
% 0.32/0.48 thf(fact_158_gcd__nat_Ostrict__trans2,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat,C: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/0.48 & ( A2 != C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_trans2
% 0.32/0.48 thf(fact_159_gcd__nat_Ostrict__trans1,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat,C: nat] :
% 0.32/0.48 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 => ( ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/0.48 & ( B2 != C ) )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/0.48 & ( A2 != C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_trans1
% 0.32/0.48 thf(fact_160_gcd__nat_Ostrict__trans,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat,C: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 => ( ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/0.48 & ( B2 != C ) )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/0.48 & ( A2 != C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.strict_trans
% 0.32/0.48 thf(fact_161_gcd__nat_Oantisym,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ B2 @ A2 )
% 0.32/0.48 => ( A2 = B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.antisym
% 0.32/0.48 thf(fact_162_gcd__nat_Oirrefl,axiom,
% 0.32/0.48 ! [A2: nat] :
% 0.32/0.48 ~ ( ( dvd_dvd @ nat @ A2 @ A2 )
% 0.32/0.48 & ( A2 != A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.irrefl
% 0.32/0.48 thf(fact_163_gcd__nat_Otrans,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat,C: nat] :
% 0.32/0.48 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 => ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/0.48 => ( dvd_dvd @ nat @ A2 @ C ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.trans
% 0.32/0.48 thf(fact_164_gcd__nat_Orefl,axiom,
% 0.32/0.48 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.refl
% 0.32/0.48 thf(fact_165_gcd__nat_Oasym,axiom,
% 0.32/0.48 ! [A2: nat,B2: nat] :
% 0.32/0.48 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/0.48 & ( A2 != B2 ) )
% 0.32/0.48 => ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
% 0.32/0.48 & ( B2 != A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % gcd_nat.asym
% 0.32/0.48 thf(fact_166_order__refl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_refl
% 0.32/0.48 thf(fact_167_minf_I8_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(8)
% 0.32/0.48 thf(fact_168_minf_I6_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(6)
% 0.32/0.48 thf(fact_169_pinf_I8_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(8)
% 0.32/0.48 thf(fact_170_le__refl,axiom,
% 0.32/0.48 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.32/0.48
% 0.32/0.48 % le_refl
% 0.32/0.48 thf(fact_171_le__trans,axiom,
% 0.32/0.48 ! [I: nat,J: nat,K2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ J @ K2 )
% 0.32/0.48 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_trans
% 0.32/0.48 thf(fact_172_eq__imp__le,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( M = N2 )
% 0.32/0.48 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_imp_le
% 0.32/0.48 thf(fact_173_le__antisym,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/0.48 => ( ( ord_less_eq @ nat @ N2 @ M )
% 0.32/0.48 => ( M = N2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_antisym
% 0.32/0.48 thf(fact_174_nat__le__linear,axiom,
% 0.32/0.48 ! [M: nat,N2: nat] :
% 0.32/0.48 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/0.48 | ( ord_less_eq @ nat @ N2 @ M ) ) ).
% 0.32/0.48
% 0.32/0.48 % nat_le_linear
% 0.32/0.48 thf(fact_175_le__funD,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.32/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_funD
% 0.32/0.48 thf(fact_176_le__funE,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 => ! [F: A > B,G: A > B,X: A] :
% 0.32/0.48 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/0.48 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_funE
% 0.32/0.48 thf(fact_177_le__funI,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 => ! [F: A > B,G: A > B] :
% 0.32/0.48 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.32/0.48 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_funI
% 0.32/0.48 thf(fact_178_le__fun__def,axiom,
% 0.32/0.48 ! [B: $tType,A: $tType] :
% 0.32/0.48 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/0.48 = ( ^ [F2: A > B,G2: A > B] :
% 0.32/0.48 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_fun_def
% 0.32/0.48 thf(fact_179_order__subst1,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.32/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.32/0.48 => ( ! [X3: B,Y3: B] :
% 0.32/0.48 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_subst1
% 0.32/0.48 thf(fact_180_order__subst2,axiom,
% 0.32/0.48 ! [A: $tType,C3: $tType] :
% 0.32/0.48 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.32/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,B2: A,F: A > C3,C: C3] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 0.32/0.48 => ( ! [X3: A,Y3: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less_eq @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_subst2
% 0.32/0.48 thf(fact_181_ord__eq__le__subst,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/0.48 ( ( A2
% 0.32/0.48 = ( F @ B2 ) )
% 0.32/0.48 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.32/0.48 => ( ! [X3: B,Y3: B] :
% 0.32/0.48 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_eq_le_subst
% 0.32/0.48 thf(fact_182_ord__le__eq__subst,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ( F @ B2 )
% 0.32/0.48 = C )
% 0.32/0.48 => ( ! [X3: A,Y3: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_le_eq_subst
% 0.32/0.48 thf(fact_183_eq__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.32/0.48 = ( ^ [X2: A,Y5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.32/0.48 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_iff
% 0.32/0.48 thf(fact_184_antisym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.48 => ( X = Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % antisym
% 0.32/0.48 thf(fact_185_linear,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linear
% 0.32/0.48 thf(fact_186_eq__refl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( X = Y )
% 0.32/0.48 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % eq_refl
% 0.32/0.48 thf(fact_187_le__cases,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_cases
% 0.32/0.48 thf(fact_188_order_Otrans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.trans
% 0.32/0.48 thf(fact_189_le__cases3,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.32/0.48 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.32/0.48 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.32/0.48 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.32/0.48 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % le_cases3
% 0.32/0.48 thf(fact_190_antisym__conv,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Y: A,X: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/0.48 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 = ( X = Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % antisym_conv
% 0.32/0.48 thf(fact_191_ord__eq__le__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( A2 = B2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_eq_le_trans
% 0.32/0.48 thf(fact_192_ord__le__eq__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( B2 = C )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_le_eq_trans
% 0.32/0.48 thf(fact_193_order__class_Oorder_Oantisym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_class.order.antisym
% 0.32/0.48 thf(fact_194_order__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/0.48 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.32/0.48 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_trans
% 0.32/0.48 thf(fact_195_dual__order_Orefl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.refl
% 0.32/0.48 thf(fact_196_linorder__wlog,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.32/0.48 ( ! [A5: A,B5: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/0.48 => ( P @ A5 @ B5 ) )
% 0.32/0.48 => ( ! [A5: A,B5: A] :
% 0.32/0.48 ( ( P @ B5 @ A5 )
% 0.32/0.48 => ( P @ A5 @ B5 ) )
% 0.32/0.48 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_wlog
% 0.32/0.48 thf(fact_197_dual__order_Otrans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A,C: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.32/0.48 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.trans
% 0.32/0.48 thf(fact_198_dual__order_Oantisym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( A2 = B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.antisym
% 0.32/0.48 thf(fact_199_minf_I11_J,axiom,
% 0.32/0.48 ! [C3: $tType,D2: $tType] :
% 0.32/0.48 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.32/0.48 => ! [F3: D2] :
% 0.32/0.48 ? [Z: C3] :
% 0.32/0.48 ! [X4: C3] :
% 0.32/0.48 ( ( ord_less @ C3 @ X4 @ Z )
% 0.32/0.48 => ( F3 = F3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(11)
% 0.32/0.48 thf(fact_200_minf_I7_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(7)
% 0.32/0.48 thf(fact_201_minf_I5_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(5)
% 0.32/0.48 thf(fact_202_minf_I4_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( X4 != T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(4)
% 0.32/0.48 thf(fact_203_minf_I3_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( X4 != T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(3)
% 0.32/0.48 thf(fact_204_minf_I2_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/0.48 ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/0.48 => ( ( P @ X3 )
% 0.32/0.48 = ( P5 @ X3 ) ) )
% 0.32/0.48 => ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/0.48 => ( ( Q @ X3 )
% 0.32/0.48 = ( Q3 @ X3 ) ) )
% 0.32/0.48 => ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( ( ( P @ X4 )
% 0.32/0.48 | ( Q @ X4 ) )
% 0.32/0.48 = ( ( P5 @ X4 )
% 0.32/0.48 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(2)
% 0.32/0.48 thf(fact_205_minf_I1_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/0.48 ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/0.48 => ( ( P @ X3 )
% 0.32/0.48 = ( P5 @ X3 ) ) )
% 0.32/0.48 => ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/0.48 => ( ( Q @ X3 )
% 0.32/0.48 = ( Q3 @ X3 ) ) )
% 0.32/0.48 => ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ X4 @ Z )
% 0.32/0.48 => ( ( ( P @ X4 )
% 0.32/0.48 & ( Q @ X4 ) )
% 0.32/0.48 = ( ( P5 @ X4 )
% 0.32/0.48 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % minf(1)
% 0.32/0.48 thf(fact_206_pinf_I11_J,axiom,
% 0.32/0.48 ! [C3: $tType,D2: $tType] :
% 0.32/0.48 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.32/0.48 => ! [F3: D2] :
% 0.32/0.48 ? [Z: C3] :
% 0.32/0.48 ! [X4: C3] :
% 0.32/0.48 ( ( ord_less @ C3 @ Z @ X4 )
% 0.32/0.48 => ( F3 = F3 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(11)
% 0.32/0.48 thf(fact_207_pinf_I7_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(7)
% 0.32/0.48 thf(fact_208_pinf_I5_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(5)
% 0.32/0.48 thf(fact_209_pinf_I4_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( X4 != T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(4)
% 0.32/0.48 thf(fact_210_pinf_I3_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [T: A] :
% 0.32/0.48 ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( X4 != T ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(3)
% 0.32/0.48 thf(fact_211_pinf_I2_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/0.48 ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/0.48 => ( ( P @ X3 )
% 0.32/0.48 = ( P5 @ X3 ) ) )
% 0.32/0.48 => ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/0.48 => ( ( Q @ X3 )
% 0.32/0.48 = ( Q3 @ X3 ) ) )
% 0.32/0.48 => ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( ( ( P @ X4 )
% 0.32/0.48 | ( Q @ X4 ) )
% 0.32/0.48 = ( ( P5 @ X4 )
% 0.32/0.48 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(2)
% 0.32/0.48 thf(fact_212_pinf_I1_J,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/0.48 ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/0.48 => ( ( P @ X3 )
% 0.32/0.48 = ( P5 @ X3 ) ) )
% 0.32/0.48 => ( ? [Z4: A] :
% 0.32/0.48 ! [X3: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/0.48 => ( ( Q @ X3 )
% 0.32/0.48 = ( Q3 @ X3 ) ) )
% 0.32/0.48 => ? [Z: A] :
% 0.32/0.48 ! [X4: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z @ X4 )
% 0.32/0.48 => ( ( ( P @ X4 )
% 0.32/0.48 & ( Q @ X4 ) )
% 0.32/0.48 = ( ( P5 @ X4 )
% 0.32/0.48 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % pinf(1)
% 0.32/0.48 thf(fact_213_ord__eq__less__subst,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/0.48 ( ( A2
% 0.32/0.48 = ( F @ B2 ) )
% 0.32/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.32/0.48 => ( ! [X3: B,Y3: B] :
% 0.32/0.48 ( ( ord_less @ B @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_eq_less_subst
% 0.32/0.48 thf(fact_214_ord__less__eq__subst,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ( F @ B2 )
% 0.32/0.48 = C )
% 0.32/0.48 => ( ! [X3: A,Y3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_less_eq_subst
% 0.32/0.48 thf(fact_215_order__less__subst1,axiom,
% 0.32/0.48 ! [A: $tType,B: $tType] :
% 0.32/0.48 ( ( ( order @ B @ ( type2 @ B ) )
% 0.32/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.32/0.48 => ( ( ord_less @ B @ B2 @ C )
% 0.32/0.48 => ( ! [X3: B,Y3: B] :
% 0.32/0.48 ( ( ord_less @ B @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_less_subst1
% 0.32/0.48 thf(fact_216_order__less__subst2,axiom,
% 0.32/0.48 ! [A: $tType,C3: $tType] :
% 0.32/0.48 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.32/0.48 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/0.48 => ! [A2: A,B2: A,F: A > C3,C: C3] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
% 0.32/0.48 => ( ! [X3: A,Y3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X3 @ Y3 )
% 0.32/0.48 => ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/0.48 => ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order_less_subst2
% 0.32/0.48 thf(fact_217_lt__ex,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A] :
% 0.32/0.48 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.32/0.48
% 0.32/0.48 % lt_ex
% 0.32/0.48 thf(fact_218_gt__ex,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( no_top @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A] :
% 0.32/0.48 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.32/0.48
% 0.32/0.48 % gt_ex
% 0.32/0.48 thf(fact_219_neqE,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( X != Y )
% 0.32/0.48 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % neqE
% 0.32/0.48 thf(fact_220_neq__iff,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( X != Y )
% 0.32/0.48 = ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % neq_iff
% 0.32/0.48 thf(fact_221_order_Oasym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.asym
% 0.32/0.48 thf(fact_222_dense,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ? [Z: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Z )
% 0.32/0.48 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dense
% 0.32/0.48 thf(fact_223_less__imp__neq,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( X != Y ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_neq
% 0.32/0.48 thf(fact_224_less__asym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_asym
% 0.32/0.48 thf(fact_225_less__asym_H,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_asym'
% 0.32/0.48 thf(fact_226_less__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ( ord_less @ A @ Y @ Z3 )
% 0.32/0.48 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_trans
% 0.32/0.48 thf(fact_227_less__linear,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 | ( X = Y )
% 0.32/0.48 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_linear
% 0.32/0.48 thf(fact_228_less__irrefl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A] :
% 0.32/0.48 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_irrefl
% 0.32/0.48 thf(fact_229_ord__eq__less__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( A2 = B2 )
% 0.32/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.32/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_eq_less_trans
% 0.32/0.48 thf(fact_230_ord__less__eq__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( B2 = C )
% 0.32/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % ord_less_eq_trans
% 0.32/0.48 thf(fact_231_dual__order_Oasym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/0.48 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.asym
% 0.32/0.48 thf(fact_232_less__imp__not__eq,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( X != Y ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_not_eq
% 0.32/0.48 thf(fact_233_less__not__sym,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_not_sym
% 0.32/0.48 thf(fact_234_less__induct,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [P: A > $o,A2: A] :
% 0.32/0.48 ( ! [X3: A] :
% 0.32/0.48 ( ! [Y2: A] :
% 0.32/0.48 ( ( ord_less @ A @ Y2 @ X3 )
% 0.32/0.48 => ( P @ Y2 ) )
% 0.32/0.48 => ( P @ X3 ) )
% 0.32/0.48 => ( P @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_induct
% 0.32/0.48 thf(fact_235_antisym__conv3,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Y: A,X: A] :
% 0.32/0.48 ( ~ ( ord_less @ A @ Y @ X )
% 0.32/0.48 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.48 = ( X = Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % antisym_conv3
% 0.32/0.48 thf(fact_236_less__imp__not__eq2,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( Y != X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_not_eq2
% 0.32/0.48 thf(fact_237_less__imp__triv,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A,P: $o] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ( ord_less @ A @ Y @ X )
% 0.32/0.48 => P ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_triv
% 0.32/0.48 thf(fact_238_linorder__cases,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ~ ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ( X != Y )
% 0.32/0.48 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % linorder_cases
% 0.32/0.48 thf(fact_239_dual__order_Oirrefl,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A] :
% 0.32/0.48 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.irrefl
% 0.32/0.48 thf(fact_240_order_Ostrict__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( ( ord_less @ A @ B2 @ C )
% 0.32/0.48 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_trans
% 0.32/0.48 thf(fact_241_less__imp__not__less,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % less_imp_not_less
% 0.32/0.48 thf(fact_242_dual__order_Ostrict__trans,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A,C: A] :
% 0.32/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/0.48 => ( ( ord_less @ A @ C @ B2 )
% 0.32/0.48 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_trans
% 0.32/0.48 thf(fact_243_not__less__iff__gr__or__eq,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A] :
% 0.32/0.48 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/0.48 = ( ( ord_less @ A @ Y @ X )
% 0.32/0.48 | ( X = Y ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % not_less_iff_gr_or_eq
% 0.32/0.48 thf(fact_244_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( A2 != B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_implies_not_eq
% 0.32/0.48 thf(fact_245_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/0.48 => ( A2 != B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_implies_not_eq
% 0.32/0.48 thf(fact_246_order_Onot__eq__order__implies__strict,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( A2 != B2 )
% 0.32/0.48 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/0.48 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.not_eq_order_implies_strict
% 0.32/0.48 thf(fact_247_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [B2: A,A2: A] :
% 0.32/0.48 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/0.48 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_implies_order
% 0.32/0.48 thf(fact_248_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( ord_less @ A )
% 0.32/0.48 = ( ^ [B3: A,A4: A] :
% 0.32/0.48 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.32/0.48 & ( A4 != B3 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.strict_iff_order
% 0.32/0.48 thf(fact_249_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ( ( ord_less_eq @ A )
% 0.32/0.48 = ( ^ [B3: A,A4: A] :
% 0.32/0.48 ( ( ord_less @ A @ B3 @ A4 )
% 0.32/0.48 | ( A4 = B3 ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dual_order.order_iff_strict
% 0.32/0.48 thf(fact_250_order_Ostrict__implies__order,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( order @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [A2: A,B2: A] :
% 0.32/0.48 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/0.48 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % order.strict_implies_order
% 0.32/0.48 thf(fact_251_dense__le__bounded,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [X: A,Y: A,Z3: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ Y )
% 0.32/0.48 => ( ! [W: A] :
% 0.32/0.48 ( ( ord_less @ A @ X @ W )
% 0.32/0.48 => ( ( ord_less @ A @ W @ Y )
% 0.32/0.48 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dense_le_bounded
% 0.32/0.48 thf(fact_252_dense__ge__bounded,axiom,
% 0.32/0.48 ! [A: $tType] :
% 0.32/0.48 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.32/0.48 => ! [Z3: A,X: A,Y: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z3 @ X )
% 0.32/0.48 => ( ! [W: A] :
% 0.32/0.48 ( ( ord_less @ A @ Z3 @ W )
% 0.32/0.48 => ( ( ord_less @ A @ W @ X )
% 0.32/0.48 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.32/0.48 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % dense_ge_bounded
% 0.32/0.48
% 0.32/0.48 %----Type constructors (26)
% 0.32/0.48 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/0.48 ! [A6: $tType,A7: $tType] :
% 0.32/0.48 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.32/0.48 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/0.48 ! [A6: $tType,A7: $tType] :
% 0.32/0.48 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.32/0.48 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/0.48 ! [A6: $tType,A7: $tType] :
% 0.32/0.48 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.32/0.48 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.32/0.48 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.32/0.48 ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom,axiom,
% 0.32/0.48 algebraic_semidom @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.32/0.48 comm_monoid_mult @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.32/0.48 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.32/0.48 semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/0.48 wellorder @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.32/0.48 preorder @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.32/0.48 linorder @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 0.32/0.48 monoid_mult @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.32/0.48 no_top @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___GCD_Osemiring__gcd,axiom,
% 0.32/0.48 semiring_gcd @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.32/0.48 order @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.32/0.48 ord @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.32/0.48 one @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Nat_Onat___Rings_Odvd,axiom,
% 0.32/0.48 dvd @ nat @ ( type2 @ nat ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.32/0.48 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.32/0.48 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.32/0.48 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.32/0.48 preorder @ $o @ ( type2 @ $o ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.32/0.48 linorder @ $o @ ( type2 @ $o ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.32/0.48 order @ $o @ ( type2 @ $o ) ).
% 0.32/0.48
% 0.32/0.48 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.32/0.48 ord @ $o @ ( type2 @ $o ) ).
% 0.32/0.48
% 0.32/0.48 %----Conjectures (1)
% 0.32/0.48 thf(conj_0,conjecture,
% 0.32/0.48 member @ nat @ n2 @ ( coinductive_lset @ nat @ hammin2015774020amming ) ).
% 0.32/0.48
% 0.32/0.48 %------------------------------------------------------------------------------
% 0.32/0.48 ------------------------
% 0.32/1.50 ---- Embedded in THF ---
% 0.32/1.50 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.32/1.50 %%% Generated on Tue Feb 24 06:10:00 EST 2026
% 0.32/1.50 %%% using '$$dhol' embedding, version 1.3.0.
% 0.32/1.50 %%% Logic specification used:
% 0.32/1.50 %%% thf(spec, logic, $$dhol).
% 0.32/1.50
% 0.32/1.50 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2DTF18998.p
% 0.32/1.50 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 0.32/1.50 thf(ty_t_Nat_Onat, type, nat: $tType).
% 0.32/1.50 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 0.32/1.50 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Rings_Odvd, type, dvd: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_GCD_Osemiring__gcd, type, semiring_gcd: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Odense__order, type, dense_order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Rings_Oalgebraic__semidom, type, algebraic_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: (((itself @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Oldistinct, type, coindu351974385stinct: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Ollist_OLNil, type, coinductive_LNil: (!> [A:$tType]: ((coinductive_llist @ A)))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Olmember, type, coinductive_lmember: (!> [A:$tType]: ((A > ((coinductive_llist @ A) > $o))))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Olsetp, type, coinductive_lsetp: (!> [A:$tType]: (((coinductive_llist @ A) > (A > $o))))).
% 0.32/1.50 thf(sy_c_Coinductive__List_Oord__class_Olsorted, type, coindu63249387sorted: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 0.32/1.50 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 0.32/1.50 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 0.32/1.50 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Ohamming, type, hammin2015774020amming: (coinductive_llist @ nat)).
% 0.32/1.50 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge, type, hammin1328233080lmerge: (!> [A:$tType]: (((A > (A > $o)) > ((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A))))))).
% 0.32/1.50 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord__class_Olmerge, type, hammin1138290136lmerge: (!> [A:$tType]: (((coinductive_llist @ A) > ((coinductive_llist @ A) > (coinductive_llist @ A)))))).
% 0.32/1.50 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth, type, hammin1195055315smooth: (nat > $o)).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Primes_Oprime, type, prime: (nat > $o)).
% 0.32/1.50 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 0.32/1.50 thf(sy_c_Rings_Odvd__class_Odvd, type, dvd_dvd: (!> [A:$tType]: ((A > (A > $o))))).
% 0.32/1.50 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 0.32/1.50 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 0.32/1.50 thf(sy_v_n, type, n: nat).
% 0.32/1.50 thf(sy_v_n_H____, type, n2: nat).
% 0.32/1.50 thf(sy_v_na____, type, na: nat).
% 0.32/1.50 thf(sy_v_p____, type, p: nat).
% 0.32/1.50 thf(fact_0__092_060open_062smooth_An_H_092_060close_062, axiom, (hammin1195055315smooth @ n2)).
% 0.32/1.50 thf(fact_1__092_060open_062n_H_A_060_An_092_060close_062, axiom, (((ord_less @ nat) @ n2) @ na)).
% 0.32/1.50 thf(fact_2_less_OIH, axiom, (! [Y:nat]: (((((ord_less @ nat) @ Y) @ na) => ((hammin1195055315smooth @ Y) => (((member @ nat) @ Y) @ ((coinductive_lset @ nat) @ hammin2015774020amming))))))).
% 0.32/1.50 thf(fact_3_n, axiom, (na = (((times_times @ nat) @ p) @ n2))).
% 0.32/1.50 thf(fact_4_hamming__eq__LNil__iff, axiom, (hammin2015774020amming != (coinductive_LNil @ nat))).
% 0.32/1.50 thf(fact_5_ldistinct__hamming, axiom, ((coindu351974385stinct @ nat) @ hammin2015774020amming)).
% 0.32/1.50 thf(fact_6_lsorted__hamming, axiom, ((coindu63249387sorted @ nat) @ hammin2015774020amming)).
% 0.32/1.50 thf(fact_7_True, axiom, (((ord_less @ nat) @ (one_one @ nat)) @ na)).
% 0.32/1.50 thf(fact_8_lnull__hamming, axiom, (~ (((coinductive_lnull @ nat) @ hammin2015774020amming)))).
% 0.32/1.50 thf(fact_9_lset__lmember, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) = (((coinductive_lmember @ A) @ X) @ Xs))))).
% 0.32/1.50 thf(fact_10_lfinite__hamming, axiom, (~ (((coinductive_lfinite @ nat) @ hammin2015774020amming)))).
% 0.32/1.50 thf(fact_11_in__lset__lmergeD, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [X:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) | (((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)))))))))).
% 0.32/1.50 thf(fact_12_ord_Oin__lset__lmergeD, axiom, (! [A:$tType,X:A,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))) => ((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) | (((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys))))))).
% 0.32/1.50 thf(fact_13_Coinductive__List_Olset__into__lsetp, axiom, (! [A:$tType,X:A,Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((coinductive_lsetp @ A) @ Xs) @ X))))).
% 0.32/1.50 thf(fact_14_assms, axiom, (hammin1195055315smooth @ n)).
% 0.32/1.50 thf(fact_15_less_Oprems, axiom, (hammin1195055315smooth @ na)).
% 0.32/1.50 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_An_A_061_Ap_A_K_An_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [N:nat]: ((na != (((times_times @ nat) @ p) @ N))))))).
% 0.32/1.50 thf(fact_17__092_060open_062prime_Ap_092_060close_062, axiom, (prime @ p)).
% 0.32/1.50 thf(fact_18_lfinite__code_I1_J, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_19_lmerge_Odisc__iff_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (~ (((coinductive_lnull @ A) @ Ys))))))))))).
% 0.32/1.50 thf(fact_20_lnull__lmerge, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys))))))))).
% 0.32/1.50 thf(fact_21_lmerge__LNil_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A)]: (((((hammin1138290136lmerge @ A) @ Xs) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A)))))))).
% 0.32/1.50 thf(fact_22_lmerge__LNil_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Ys:(coinductive_llist @ A)]: (((((hammin1138290136lmerge @ A) @ (coinductive_LNil @ A)) @ Ys) = (coinductive_LNil @ A)))))))).
% 0.32/1.50 thf(fact_23_lsorted__code_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 0.32/1.50 thf(fact_24_ldistinct__LNil__code, axiom, (! [A:$tType]: (((coindu351974385stinct @ A) @ (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_25_smooth__times, axiom, (! [X:nat,Y:nat]: (((hammin1195055315smooth @ (((times_times @ nat) @ X) @ Y)) = ((hammin1195055315smooth @ X) & (hammin1195055315smooth @ Y)))))).
% 0.32/1.50 thf(fact_26__092_060open_062p_Advd_An_092_060close_062, axiom, (((dvd_dvd @ nat) @ p) @ na)).
% 0.32/1.50 thf(fact_27_llist_Odisc_I1_J, axiom, (! [A:$tType]: (((coinductive_lnull @ A) @ (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_28_llist_OdiscI_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: (((Llist = (coinductive_LNil @ A)) => ((coinductive_lnull @ A) @ Llist))))).
% 0.32/1.50 thf(fact_29_lmember__code_I1_J, axiom, (! [A:$tType,X:A]: ((~ ((((coinductive_lmember @ A) @ X) @ (coinductive_LNil @ A))))))).
% 0.32/1.50 thf(fact_30_llist_Ocollapse_I1_J, axiom, (! [A:$tType,Llist:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Llist) => (Llist = (coinductive_LNil @ A)))))).
% 0.32/1.50 thf(fact_31_ord_Olmerge_Octr_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)) => (((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys) = (coinductive_LNil @ A)))))).
% 0.32/1.50 thf(fact_32_ord_Olmerge_Odisc_I2_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ A) @ Ys))) => (~ (((coinductive_lnull @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))))))))).
% 0.32/1.50 thf(fact_33_ord_Olmerge_Odisc_I1_J, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)) => ((coinductive_lnull @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)))))).
% 0.32/1.50 thf(fact_34_ord_Olmerge__LNil_I2_J, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A)]: ((((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ (coinductive_LNil @ A)) = (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_35_ord_Olmerge__LNil_I1_J, axiom, (! [A:$tType,Less:(A > (A > $o)),Ys:(coinductive_llist @ A)]: ((((((hammin1328233080lmerge @ A) @ Less) @ (coinductive_LNil @ A)) @ Ys) = (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_36_ord_Olmerge_Odisc__iff_I2_J, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)))) = ((~ (((coinductive_lnull @ A) @ Xs))) & (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 0.32/1.50 thf(fact_37_lmerge_Octr_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)) => ((((hammin1138290136lmerge @ A) @ Xs) @ Ys) = (coinductive_LNil @ A))))))))).
% 0.32/1.50 thf(fact_38_lmerge_Odisc_I2_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lnull @ A) @ Xs))) => ((~ (((coinductive_lnull @ A) @ Ys))) => (~ (((coinductive_lnull @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))))))))))).
% 0.32/1.50 thf(fact_39_lmerge_Odisc_I1_J, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)) => ((coinductive_lnull @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys))))))))).
% 0.32/1.50 thf(fact_40_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 0.32/1.50 thf(fact_41_ldistinct_OLNil, axiom, (! [A:$tType]: (((coindu351974385stinct @ A) @ (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_42_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (~ (((coinductive_lnull @ A) @ Ys)))) => ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 0.32/1.50 thf(fact_43_lnull__def, axiom, (! [A:$tType]: (((coinductive_lnull @ A) = (^ [Llist2:(coinductive_llist @ A)]: ((Llist2 = (coinductive_LNil @ A)))))))).
% 0.32/1.50 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 0.32/1.50 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A3)))) = A3)))).
% 0.32/1.50 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P @ X3) = (Q @ X3)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 0.32/1.50 thf(fact_47_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 0.32/1.50 thf(fact_48_lfinite__LNil, axiom, (! [A:$tType]: (((coinductive_lfinite @ A) @ (coinductive_LNil @ A))))).
% 0.32/1.50 thf(fact_49_LNil, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => ((coindu63249387sorted @ A) @ (coinductive_LNil @ A)))))).
% 0.32/1.50 thf(fact_50_ord_Olnull__lmerge, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)) = (((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)))))).
% 0.32/1.50 thf(fact_51_ord_Olmerge_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ A) @ Ys)))))))).
% 0.32/1.50 thf(fact_52_ord_Olfinite__lmergeD, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys)) => (((coinductive_lfinite @ A) @ Xs) | ((coinductive_lfinite @ A) @ Ys)))))).
% 0.32/1.50 thf(fact_53_ord_Olfinite__lmergeI, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))))))).
% 0.32/1.50 thf(fact_54_ord_Olinfinite__lmerge, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A),Less:(A > (A > $o))]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((~ (((coinductive_lfinite @ A) @ Ys))) => (~ (((coinductive_lfinite @ A) @ ((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys))))))))).
% 0.32/1.50 thf(fact_55_ord_Olmerge__eq__LNil__iff, axiom, (! [A:$tType,Less:(A > (A > $o)),Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((((hammin1328233080lmerge @ A) @ Less) @ Xs) @ Ys) = (coinductive_LNil @ A)) = ((Xs = (coinductive_LNil @ A)) | (Ys = (coinductive_LNil @ A))))))).
% 0.32/1.50 thf(fact_56_lmerge_Oexhaust, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ A) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ A) @ Ys))))))))))).
% 0.32/1.50 thf(fact_57_lsorted__ldistinct__lset__unique, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ Xs) => (((coindu351974385stinct @ A) @ Xs) => (((coindu63249387sorted @ A) @ Ys) => (((coindu351974385stinct @ A) @ Ys) => ((((coinductive_lset @ A) @ Xs) = ((coinductive_lset @ A) @ Ys)) => (Xs = Ys)))))))))))).
% 0.32/1.50 thf(fact_58_lfinite__lmergeD, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)) => (((coinductive_lfinite @ A) @ Xs) | ((coinductive_lfinite @ A) @ Ys))))))))).
% 0.32/1.50 thf(fact_59_lfinite__lmergeI, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => (((coinductive_lfinite @ A) @ Ys) => ((coinductive_lfinite @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))))))))).
% 0.32/1.50 thf(fact_60_linfinite__lmerge, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((~ (((coinductive_lfinite @ A) @ Xs))) => ((~ (((coinductive_lfinite @ A) @ Ys))) => (~ (((coinductive_lfinite @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))))))))))).
% 0.32/1.50 thf(fact_61_lmerge__eq__LNil__iff, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((((hammin1138290136lmerge @ A) @ Xs) @ Ys) = (coinductive_LNil @ A)) = ((Xs = (coinductive_LNil @ A)) | (Ys = (coinductive_LNil @ A)))))))))).
% 0.32/1.50 thf(fact_62_lsorted__lmerge, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ Xs) => (((coindu63249387sorted @ A) @ Ys) => ((coindu63249387sorted @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))))))))).
% 0.32/1.50 thf(fact_63_ldistinct__lmerge, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: ((((coindu63249387sorted @ A) @ Xs) => (((coindu63249387sorted @ A) @ Ys) => (((coindu351974385stinct @ A) @ Xs) => (((coindu351974385stinct @ A) @ Ys) => ((coindu351974385stinct @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys)))))))))))).
% 0.32/1.50 thf(fact_64_lsetp__into__lset, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),X:A]: (((((coinductive_lsetp @ A) @ Xs) @ X) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)))))).
% 0.32/1.50 thf(fact_65_nat__1__eq__mult__iff, axiom, (! [M:nat,N2:nat]: ((((one_one @ nat) = (((times_times @ nat) @ M) @ N2)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))).
% 0.32/1.50 thf(fact_66_nat__mult__eq__1__iff, axiom, (! [M:nat,N2:nat]: ((((((times_times @ nat) @ M) @ N2) = (one_one @ nat)) = ((M = (one_one @ nat)) & (N2 = (one_one @ nat))))))).
% 0.32/1.50 thf(fact_67_mult_Oleft__neutral, axiom, (! [A:$tType]: ((((monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))).
% 0.32/1.50 thf(fact_68_mult_Oright__neutral, axiom, (! [A:$tType]: ((((monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 0.32/1.50 thf(fact_69_less__1__mult, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (! [M:A,N2:A]: (((((ord_less @ A) @ (one_one @ A)) @ M) => ((((ord_less @ A) @ (one_one @ A)) @ N2) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ M) @ N2)))))))))).
% 0.32/1.50 thf(fact_70_nat__mult__1, axiom, (! [N2:nat]: (((((times_times @ nat) @ (one_one @ nat)) @ N2) = N2)))).
% 0.32/1.50 thf(fact_71_nat__mult__1__right, axiom, (! [N2:nat]: (((((times_times @ nat) @ N2) @ (one_one @ nat)) = N2)))).
% 0.32/1.50 thf(fact_72_semiring__normalization__rules_I12_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 0.32/1.50 thf(fact_73_semiring__normalization__rules_I11_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))).
% 0.32/1.50 thf(fact_74_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ (one_one @ A)) @ A2) = A2))))))).
% 0.32/1.50 thf(fact_75__092_060open_062_092_060exists_062p_O_Aprime_Ap_A_092_060and_062_Ap_Advd_An_092_060close_062, axiom, (? [P2:nat]: (((prime @ P2) & (((dvd_dvd @ nat) @ P2) @ na))))).
% 0.32/1.50 thf(fact_76_nat__dvd__1__iff__1, axiom, (! [M:nat]: (((((dvd_dvd @ nat) @ M) @ (one_one @ nat)) = (M = (one_one @ nat)))))).
% 0.32/1.50 thf(fact_77_unit__prod, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ (one_one @ A)))))))))).
% 0.32/1.50 thf(fact_78_dvd__trans, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ B2) @ C) => (((dvd_dvd @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_79_dvd__refl, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: ((((dvd_dvd @ A) @ A2) @ A2))))))).
% 0.32/1.50 thf(fact_80_dvd__antisym, axiom, (! [M:nat,N2:nat]: (((((dvd_dvd @ nat) @ M) @ N2) => ((((dvd_dvd @ nat) @ N2) @ M) => (M = N2)))))).
% 0.32/1.50 thf(fact_81_prime__nat__dvdD, axiom, (! [P3:nat,N2:nat]: (((prime @ P3) => ((((dvd_dvd @ nat) @ N2) @ P3) => ((N2 = (one_one @ nat)) | (N2 = P3))))))).
% 0.32/1.50 thf(fact_82_dvd__triv__right, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ A2)))))))).
% 0.32/1.50 thf(fact_83_dvd__mult__right, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C) => (((dvd_dvd @ A) @ B2) @ C)))))))).
% 0.32/1.50 thf(fact_84_mult__dvd__mono, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A,D:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ C) @ D) => (((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ C)) @ (((times_times @ A) @ B2) @ D)))))))))).
% 0.32/1.50 thf(fact_85_dvd__triv__left, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ A2) @ B2)))))))).
% 0.32/1.50 thf(fact_86_dvd__mult__left, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C) => (((dvd_dvd @ A) @ A2) @ C)))))))).
% 0.32/1.50 thf(fact_87_dvd__mult2, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ A2) @ B2) => (((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C))))))))).
% 0.32/1.50 thf(fact_88_dvd__mult, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A,C:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ C) => (((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C))))))))).
% 0.32/1.50 thf(fact_89_dvd__def, axiom, (! [A:$tType]: ((((dvd @ A) @ (type2 @ A)) => ((dvd_dvd @ A) = (^ [B3:A,A4:A]: ((? [K:A]: ((A4 = (((times_times @ A) @ B3) @ K))))))))))).
% 0.32/1.50 thf(fact_90_dvdI, axiom, (! [A:$tType]: ((((dvd @ A) @ (type2 @ A)) => (! [A2:A,B2:A,K2:A]: (((A2 = (((times_times @ A) @ B2) @ K2)) => (((dvd_dvd @ A) @ B2) @ A2)))))))).
% 0.32/1.50 thf(fact_91_dvdE, axiom, (! [A:$tType]: ((((dvd @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ A2) => (~ ((! [K3:A]: ((A2 != (((times_times @ A) @ B2) @ K3))))))))))))).
% 0.32/1.50 thf(fact_92_dvd__unit__imp__unit, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ A2) @ B2) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ A2) @ (one_one @ A)))))))))).
% 0.32/1.50 thf(fact_93_unit__imp__dvd, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ B2) @ A2)))))))).
% 0.32/1.50 thf(fact_94_one__dvd, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: ((((dvd_dvd @ A) @ (one_one @ A)) @ A2))))))).
% 0.32/1.50 thf(fact_95_unit__mult__right__cancel, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((((times_times @ A) @ B2) @ A2) = (((times_times @ A) @ C) @ A2)) = (B2 = C))))))))).
% 0.32/1.50 thf(fact_96_unit__mult__left__cancel, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ A2) @ (one_one @ A)) => (((((times_times @ A) @ A2) @ B2) = (((times_times @ A) @ A2) @ C)) = (B2 = C))))))))).
% 0.32/1.50 thf(fact_97_mult__unit__dvd__iff, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ C) = (((dvd_dvd @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_98_dvd__mult__unit__iff, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ C) @ B2)) = (((dvd_dvd @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_99_is__unit__mult__iff, axiom, (! [A:$tType]: ((((algebraic_semidom @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A2) @ B2)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A2) @ (one_one @ A)) & (((dvd_dvd @ A) @ B2) @ (one_one @ A)))))))))).
% 0.32/1.50 thf(fact_100_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: ((((linordered_idom @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_101_mult_Oleft__commute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A2) @ C)) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C))))))))).
% 0.32/1.50 thf(fact_102_mult_Ocommute, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => ((times_times @ A) = (^ [A4:A,B3:A]: ((((times_times @ A) @ B3) @ A4)))))))).
% 0.32/1.50 thf(fact_103_mult_Oassoc, axiom, (! [A:$tType]: ((((semigroup_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C))))))))).
% 0.32/1.50 thf(fact_104_semiring__normalization__rules_I7_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => ((times_times @ A) = (^ [A4:A,B3:A]: ((((times_times @ A) @ B3) @ A4)))))))).
% 0.32/1.50 thf(fact_105_semiring__normalization__rules_I13_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ (((times_times @ A) @ Ly) @ Ry))))))))).
% 0.32/1.50 thf(fact_106_semiring__normalization__rules_I14_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Lx) @ (((times_times @ A) @ Ly) @ (((times_times @ A) @ Rx) @ Ry)))))))))).
% 0.32/1.50 thf(fact_107_semiring__normalization__rules_I15_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Ly:A,Rx:A,Ry:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Rx) @ (((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Ry))))))))).
% 0.32/1.50 thf(fact_108_semiring__normalization__rules_I16_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Ly:A,Rx:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Rx) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ Ly)))))))).
% 0.32/1.50 thf(fact_109_semiring__normalization__rules_I17_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Ly:A,Rx:A]: (((((times_times @ A) @ (((times_times @ A) @ Lx) @ Ly)) @ Rx) = (((times_times @ A) @ Lx) @ (((times_times @ A) @ Ly) @ Rx))))))))).
% 0.32/1.50 thf(fact_110_semiring__normalization__rules_I18_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Rx:A,Ry:A]: (((((times_times @ A) @ Lx) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ (((times_times @ A) @ Lx) @ Rx)) @ Ry)))))))).
% 0.32/1.50 thf(fact_111_semiring__normalization__rules_I19_J, axiom, (! [A:$tType]: ((((comm_semiring_1 @ A) @ (type2 @ A)) => (! [Lx:A,Rx:A,Ry:A]: (((((times_times @ A) @ Lx) @ (((times_times @ A) @ Rx) @ Ry)) = (((times_times @ A) @ Rx) @ (((times_times @ A) @ Lx) @ Ry))))))))).
% 0.32/1.50 thf(fact_112_ab__semigroup__mult__class_Omult__ac_I1_J, axiom, (! [A:$tType]: ((((ab_semigroup_mult @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((times_times @ A) @ (((times_times @ A) @ A2) @ B2)) @ C) = (((times_times @ A) @ A2) @ (((times_times @ A) @ B2) @ C))))))))).
% 0.32/1.50 thf(fact_113_one__reorient, axiom, (! [A:$tType]: ((((one @ A) @ (type2 @ A)) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 0.32/1.50 thf(fact_114_infinite__descent__measure, axiom, (! [A:$tType,P:(A > $o),V:(A > nat),X:A]: (((! [X3:A]: (((~ ((P @ X3))) => (? [Y2:A]: (((((ord_less @ nat) @ (V @ Y2)) @ (V @ X3)) & (~ ((P @ Y2))))))))) => (P @ X))))).
% 0.32/1.50 thf(fact_115_measure__induct__rule, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2))))).
% 0.32/1.50 thf(fact_116_linorder__neqE__nat, axiom, (! [X:nat,Y:nat]: (((X != Y) => ((~ ((((ord_less @ nat) @ X) @ Y))) => (((ord_less @ nat) @ Y) @ X)))))).
% 0.32/1.50 thf(fact_117_infinite__descent, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((~ ((P @ N3))) => (? [M2:nat]: (((((ord_less @ nat) @ M2) @ N3) & (~ ((P @ M2))))))))) => (P @ N2))))).
% 0.32/1.50 thf(fact_118_nat__less__induct, axiom, (! [P:(nat > $o),N2:nat]: (((! [N3:nat]: (((! [M2:nat]: (((((ord_less @ nat) @ M2) @ N3) => (P @ M2)))) => (P @ N3)))) => (P @ N2))))).
% 0.32/1.50 thf(fact_119_less__irrefl__nat, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 0.32/1.50 thf(fact_120_measure__induct, axiom, (! [A:$tType,F:(A > nat),P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ nat) @ (F @ Y2)) @ (F @ X3)) => (P @ Y2)))) => (P @ X3)))) => (P @ A2))))).
% 0.32/1.50 thf(fact_121_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 0.32/1.50 thf(fact_122_less__not__refl2, axiom, (! [N2:nat,M:nat]: (((((ord_less @ nat) @ N2) @ M) => (M != N2))))).
% 0.32/1.50 thf(fact_123_less__not__refl, axiom, (! [N2:nat]: ((~ ((((ord_less @ nat) @ N2) @ N2)))))).
% 0.32/1.50 thf(fact_124_nat__neq__iff, axiom, (! [M:nat,N2:nat]: (((M != N2) = ((((ord_less @ nat) @ M) @ N2) | (((ord_less @ nat) @ N2) @ M)))))).
% 0.32/1.50 thf(fact_125_mult_Ocomm__neutral, axiom, (! [A:$tType]: ((((comm_monoid_mult @ A) @ (type2 @ A)) => (! [A2:A]: (((((times_times @ A) @ A2) @ (one_one @ A)) = A2))))))).
% 0.32/1.50 thf(fact_126_prime__dvd__mult__eq__nat, axiom, (! [P3:nat,M:nat,N2:nat]: (((prime @ P3) => ((((dvd_dvd @ nat) @ P3) @ (((times_times @ nat) @ M) @ N2)) = ((((dvd_dvd @ nat) @ P3) @ M) | (((dvd_dvd @ nat) @ P3) @ N2))))))).
% 0.32/1.50 thf(fact_127_prime__def, axiom, (prime = (^ [P4:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ P4) & (! [M3:nat]: (((((dvd_dvd @ nat) @ M3) @ P4) => ((M3 = (one_one @ nat)) | (M3 = P4)))))))))).
% 0.32/1.50 thf(fact_128_not__prime__eq__prod__nat, axiom, (! [N2:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N2) => ((~ ((prime @ N2))) => (? [M4:nat,K3:nat]: (((N2 = (((times_times @ nat) @ M4) @ K3)) & ((((ord_less @ nat) @ (one_one @ nat)) @ M4) & ((((ord_less @ nat) @ M4) @ N2) & ((((ord_less @ nat) @ (one_one @ nat)) @ K3) & (((ord_less @ nat) @ K3) @ N2)))))))))))).
% 0.32/1.50 thf(fact_129_prime__factor__nat, axiom, (! [N2:nat]: (((N2 != (one_one @ nat)) => (? [P2:nat]: (((prime @ P2) & (((dvd_dvd @ nat) @ P2) @ N2)))))))).
% 0.32/1.50 thf(fact_130_prime__dvd__mult__nat, axiom, (! [P3:nat,M:nat,N2:nat]: (((prime @ P3) => ((((dvd_dvd @ nat) @ P3) @ (((times_times @ nat) @ M) @ N2)) => ((((dvd_dvd @ nat) @ P3) @ M) | (((dvd_dvd @ nat) @ P3) @ N2))))))).
% 0.32/1.50 thf(fact_131_prime__product, axiom, (! [P3:nat,Q2:nat]: (((prime @ (((times_times @ nat) @ P3) @ Q2)) => ((P3 = (one_one @ nat)) | (Q2 = (one_one @ nat))))))).
% 0.32/1.50 thf(fact_132_bigger__prime, axiom, (! [N2:nat]: ((? [P2:nat]: (((prime @ P2) & (((ord_less @ nat) @ N2) @ P2))))))).
% 0.32/1.50 thf(fact_133_one__not__prime__nat, axiom, (~ ((prime @ (one_one @ nat))))).
% 0.32/1.50 thf(fact_134_prime__gt__1__nat, axiom, (! [P3:nat]: (((prime @ P3) => (((ord_less @ nat) @ (one_one @ nat)) @ P3))))).
% 0.32/1.50 thf(fact_135_division__decomp, axiom, (! [A:$tType]: ((((semiring_gcd @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((dvd_dvd @ A) @ A2) @ (((times_times @ A) @ B2) @ C)) => (? [B4:A,C2:A]: (((A2 = (((times_times @ A) @ B4) @ C2)) & ((((dvd_dvd @ A) @ B4) @ B2) & (((dvd_dvd @ A) @ C2) @ C)))))))))))).
% 0.32/1.50 thf(fact_136_less__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 0.32/1.50 thf(fact_137_in__lset__lmergeI1, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Xs)) => (((coindu63249387sorted @ A) @ Xs) => ((~ (((coinductive_lfinite @ A) @ Ys))) => ((? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Ys)) & (((ord_less_eq @ A) @ X) @ X4)))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys))))))))))))).
% 0.32/1.50 thf(fact_138_in__lset__lmergeI2, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Ys:(coinductive_llist @ A),Xs:(coinductive_llist @ A)]: (((((member @ A) @ X) @ ((coinductive_lset @ A) @ Ys)) => (((coindu63249387sorted @ A) @ Ys) => ((~ (((coinductive_lfinite @ A) @ Xs))) => ((? [X4:A]: (((((member @ A) @ X4) @ ((coinductive_lset @ A) @ Xs)) & (((ord_less_eq @ A) @ X) @ X4)))) => (((member @ A) @ X) @ ((coinductive_lset @ A) @ (((hammin1138290136lmerge @ A) @ Xs) @ Ys))))))))))))).
% 0.32/1.50 thf(fact_139_nat__less__le, axiom, ((ord_less @ nat) = (^ [M3:nat,N4:nat]: (((((ord_less_eq @ nat) @ M3) @ N4) & (M3 != N4)))))).
% 0.32/1.50 thf(fact_140_less__imp__le__nat, axiom, (! [M:nat,N2:nat]: (((((ord_less @ nat) @ M) @ N2) => (((ord_less_eq @ nat) @ M) @ N2))))).
% 0.32/1.50 thf(fact_141_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M3:nat,N4:nat]: (((((ord_less @ nat) @ M3) @ N4) | (M3 = N4)))))).
% 0.32/1.50 thf(fact_142_less__or__eq__imp__le, axiom, (! [M:nat,N2:nat]: ((((((ord_less @ nat) @ M) @ N2) | (M = N2)) => (((ord_less_eq @ nat) @ M) @ N2))))).
% 0.32/1.50 thf(fact_143_le__neq__implies__less, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((M != N2) => (((ord_less @ nat) @ M) @ N2)))))).
% 0.32/1.50 thf(fact_144_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 0.32/1.50 thf(fact_145_le__cube, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ (((times_times @ nat) @ M) @ M)))))).
% 0.32/1.50 thf(fact_146_le__square, axiom, (! [M:nat]: ((((ord_less_eq @ nat) @ M) @ (((times_times @ nat) @ M) @ M))))).
% 0.32/1.50 thf(fact_147_mult__le__mono, axiom, (! [I:nat,J:nat,K2:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K2) @ L) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I) @ K2)) @ (((times_times @ nat) @ J) @ L))))))).
% 0.32/1.50 thf(fact_148_mult__le__mono1, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ I) @ K2)) @ (((times_times @ nat) @ J) @ K2)))))).
% 0.32/1.50 thf(fact_149_mult__le__mono2, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((times_times @ nat) @ K2) @ I)) @ (((times_times @ nat) @ K2) @ J)))))).
% 0.32/1.50 thf(fact_150_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B2:A,A2:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((((P @ B2) @ A2) => ((P @ A2) @ B2)) => ((P @ A2) @ B2))))))))).
% 0.32/1.50 thf(fact_151_le__numeral__extra_I4_J, axiom, (! [A:$tType]: ((((linordered_semidom @ A) @ (type2 @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 0.32/1.50 thf(fact_152_prime__ge__1__nat, axiom, (! [P3:nat]: (((prime @ P3) => (((ord_less_eq @ nat) @ (one_one @ nat)) @ P3))))).
% 0.32/1.50 thf(fact_153_gcd__nat_Onot__eq__order__implies__strict, axiom, (! [A2:nat,B2:nat]: (((A2 != B2) => ((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2))))))).
% 0.32/1.50 thf(fact_154_gcd__nat_Ostrict__implies__not__eq, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (A2 != B2))))).
% 0.32/1.50 thf(fact_155_gcd__nat_Ostrict__implies__order, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (((dvd_dvd @ nat) @ A2) @ B2))))).
% 0.32/1.50 thf(fact_156_gcd__nat_Ostrict__iff__order, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) = ((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)))))).
% 0.32/1.50 thf(fact_157_gcd__nat_Oorder__iff__strict, axiom, ((dvd_dvd @ nat) = (^ [A4:nat,B3:nat]: ((((((dvd_dvd @ nat) @ A4) @ B3) & (A4 != B3)) | (A4 = B3)))))).
% 0.32/1.50 thf(fact_158_gcd__nat_Ostrict__trans2, axiom, (! [A2:nat,B2:nat,C:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => ((((dvd_dvd @ nat) @ B2) @ C) => ((((dvd_dvd @ nat) @ A2) @ C) & (A2 != C))))))).
% 0.32/1.50 thf(fact_159_gcd__nat_Ostrict__trans1, axiom, (! [A2:nat,B2:nat,C:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => (((((dvd_dvd @ nat) @ B2) @ C) & (B2 != C)) => ((((dvd_dvd @ nat) @ A2) @ C) & (A2 != C))))))).
% 0.32/1.50 thf(fact_160_gcd__nat_Ostrict__trans, axiom, (! [A2:nat,B2:nat,C:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (((((dvd_dvd @ nat) @ B2) @ C) & (B2 != C)) => ((((dvd_dvd @ nat) @ A2) @ C) & (A2 != C))))))).
% 0.32/1.50 thf(fact_161_gcd__nat_Oantisym, axiom, (! [A2:nat,B2:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ B2) @ A2) => (A2 = B2)))))).
% 0.32/1.50 thf(fact_162_gcd__nat_Oirrefl, axiom, (! [A2:nat]: ((~ (((((dvd_dvd @ nat) @ A2) @ A2) & (A2 != A2))))))).
% 0.32/1.50 thf(fact_163_gcd__nat_Otrans, axiom, (! [A2:nat,B2:nat,C:nat]: (((((dvd_dvd @ nat) @ A2) @ B2) => ((((dvd_dvd @ nat) @ B2) @ C) => (((dvd_dvd @ nat) @ A2) @ C)))))).
% 0.32/1.50 thf(fact_164_gcd__nat_Orefl, axiom, (! [A2:nat]: ((((dvd_dvd @ nat) @ A2) @ A2)))).
% 0.32/1.50 thf(fact_165_gcd__nat_Oasym, axiom, (! [A2:nat,B2:nat]: ((((((dvd_dvd @ nat) @ A2) @ B2) & (A2 != B2)) => (~ (((((dvd_dvd @ nat) @ B2) @ A2) & (B2 != A2)))))))).
% 0.32/1.50 thf(fact_166_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 0.32/1.50 thf(fact_167_minf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (~ ((((ord_less_eq @ A) @ T) @ X4)))))))))))))).
% 0.32/1.50 thf(fact_168_minf_I6_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((ord_less_eq @ A) @ X4) @ T)))))))))))).
% 0.32/1.50 thf(fact_169_pinf_I8_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((ord_less_eq @ A) @ T) @ X4)))))))))))).
% 0.32/1.50 thf(fact_170_le__refl, axiom, (! [N2:nat]: ((((ord_less_eq @ nat) @ N2) @ N2)))).
% 0.32/1.50 thf(fact_171_le__trans, axiom, (! [I:nat,J:nat,K2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K2) => (((ord_less_eq @ nat) @ I) @ K2)))))).
% 0.32/1.50 thf(fact_172_eq__imp__le, axiom, (! [M:nat,N2:nat]: (((M = N2) => (((ord_less_eq @ nat) @ M) @ N2))))).
% 0.32/1.50 thf(fact_173_le__antisym, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) => ((((ord_less_eq @ nat) @ N2) @ M) => (M = N2)))))).
% 0.32/1.50 thf(fact_174_nat__le__linear, axiom, (! [M:nat,N2:nat]: (((((ord_less_eq @ nat) @ M) @ N2) | (((ord_less_eq @ nat) @ N2) @ M))))).
% 0.32/1.50 thf(fact_175_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.32/1.50 thf(fact_176_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 0.32/1.50 thf(fact_177_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 0.32/1.50 thf(fact_178_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F2:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F2 @ X2)) @ (G2 @ X2))))))))))).
% 0.32/1.50 thf(fact_179_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 0.32/1.50 thf(fact_180_order__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C3),C:C3]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C3) @ (F @ B2)) @ C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ C3) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ C3) @ (F @ A2)) @ C)))))))))).
% 0.32/1.50 thf(fact_181_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y3:B]: (((((ord_less_eq @ B) @ X3) @ Y3) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 0.32/1.50 thf(fact_182_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 0.32/1.50 thf(fact_183_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X2:A,Y5:A]: (((((ord_less_eq @ A) @ X2) @ Y5) & (((ord_less_eq @ A) @ Y5) @ X2))))))))).
% 0.32/1.50 thf(fact_184_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ X) => (X = Y))))))))).
% 0.32/1.50 thf(fact_185_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) | (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.32/1.50 thf(fact_186_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X = Y) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 0.32/1.50 thf(fact_187_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((ord_less_eq @ A) @ Y) @ X)))))))).
% 0.32/1.50 thf(fact_188_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_189_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z3)))) => (((((ord_less_eq @ A) @ Y) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z3)))) => (((((ord_less_eq @ A) @ X) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ Y)))) => (((((ord_less_eq @ A) @ Z3) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X)))) => (((((ord_less_eq @ A) @ Y) @ Z3) => (~ ((((ord_less_eq @ A) @ Z3) @ X)))) => (~ (((((ord_less_eq @ A) @ Z3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y))))))))))))))))).
% 0.32/1.50 thf(fact_190_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((((ord_less_eq @ A) @ Y) @ X) => ((((ord_less_eq @ A) @ X) @ Y) = (X = Y))))))))).
% 0.32/1.50 thf(fact_191_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_192_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_193_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 0.32/1.50 thf(fact_194_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less_eq @ A) @ X) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z3) => (((ord_less_eq @ A) @ X) @ Z3))))))))).
% 0.32/1.50 thf(fact_195_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 0.32/1.50 thf(fact_196_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P @ B5) @ A5) => ((P @ A5) @ B5)))) => ((P @ A2) @ B2))))))))).
% 0.32/1.50 thf(fact_197_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 0.32/1.50 thf(fact_198_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 0.32/1.50 thf(fact_199_minf_I11_J, axiom, (! [C3:$tType,D2:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F3:D2]: ((? [Z:C3]: ((! [X4:C3]: (((((ord_less @ C3) @ X4) @ Z) => (F3 = F3)))))))))))).
% 0.32/1.50 thf(fact_200_minf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (~ ((((ord_less @ A) @ T) @ X4)))))))))))))).
% 0.32/1.50 thf(fact_201_minf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((ord_less @ A) @ X4) @ T)))))))))))).
% 0.32/1.50 thf(fact_202_minf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (X4 != T)))))))))))).
% 0.32/1.50 thf(fact_203_minf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (X4 != T)))))))))))).
% 0.32/1.50 thf(fact_204_minf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P5 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q3 @ X3))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((P @ X4) | (Q @ X4)) = ((P5 @ X4) | (Q3 @ X4)))))))))))))))).
% 0.32/1.50 thf(fact_205_minf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((P @ X3) = (P5 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ X3) @ Z4) => ((Q @ X3) = (Q3 @ X3))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ X4) @ Z) => (((P @ X4) & (Q @ X4)) = ((P5 @ X4) & (Q3 @ X4)))))))))))))))).
% 0.32/1.50 thf(fact_206_pinf_I11_J, axiom, (! [C3:$tType,D2:$tType]: ((((ord @ C3) @ (type2 @ C3)) => (! [F3:D2]: ((? [Z:C3]: ((! [X4:C3]: (((((ord_less @ C3) @ Z) @ X4) => (F3 = F3)))))))))))).
% 0.32/1.50 thf(fact_207_pinf_I7_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((ord_less @ A) @ T) @ X4)))))))))))).
% 0.32/1.50 thf(fact_208_pinf_I5_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (~ ((((ord_less @ A) @ X4) @ T)))))))))))))).
% 0.32/1.50 thf(fact_209_pinf_I4_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (X4 != T)))))))))))).
% 0.32/1.50 thf(fact_210_pinf_I3_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [T:A]: ((? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (X4 != T)))))))))))).
% 0.32/1.50 thf(fact_211_pinf_I2_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P5 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q3 @ X3))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((P @ X4) | (Q @ X4)) = ((P5 @ X4) | (Q3 @ X4)))))))))))))))).
% 0.32/1.50 thf(fact_212_pinf_I1_J, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > $o),P5:(A > $o),Q:(A > $o),Q3:(A > $o)]: (((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((P @ X3) = (P5 @ X3))))))) => ((? [Z4:A]: ((! [X3:A]: (((((ord_less @ A) @ Z4) @ X3) => ((Q @ X3) = (Q3 @ X3))))))) => (? [Z:A]: ((! [X4:A]: (((((ord_less @ A) @ Z) @ X4) => (((P @ X4) & (Q @ X4)) = ((P5 @ X4) & (Q3 @ X4)))))))))))))))).
% 0.32/1.50 thf(fact_213_ord__eq__less__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X3:B,Y3:B]: (((((ord_less @ B) @ X3) @ Y3) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.32/1.50 thf(fact_214_ord__less__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ B) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ B) @ (F @ A2)) @ C)))))))))).
% 0.32/1.50 thf(fact_215_order__less__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less @ A) @ A2) @ (F @ B2)) => ((((ord_less @ B) @ B2) @ C) => ((! [X3:B,Y3:B]: (((((ord_less @ B) @ X3) @ Y3) => (((ord_less @ A) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ A) @ A2) @ (F @ C))))))))))).
% 0.32/1.50 thf(fact_216_order__less__subst2, axiom, (! [A:$tType,C3:$tType]: (((((order @ C3) @ (type2 @ C3)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B2:A,F:(A > C3),C:C3]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ C3) @ (F @ B2)) @ C) => ((! [X3:A,Y3:A]: (((((ord_less @ A) @ X3) @ Y3) => (((ord_less @ C3) @ (F @ X3)) @ (F @ Y3))))) => (((ord_less @ C3) @ (F @ A2)) @ C)))))))))).
% 0.32/1.50 thf(fact_217_lt__ex, axiom, (! [A:$tType]: ((((no_bot @ A) @ (type2 @ A)) => (! [X:A]: ((? [Y3:A]: ((((ord_less @ A) @ Y3) @ X))))))))).
% 0.32/1.50 thf(fact_218_gt__ex, axiom, (! [A:$tType]: ((((no_top @ A) @ (type2 @ A)) => (! [X:A]: ((? [X1:A]: ((((ord_less @ A) @ X) @ X1))))))))).
% 0.32/1.50 thf(fact_219_neqE, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) => ((~ ((((ord_less @ A) @ X) @ Y))) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_220_neq__iff, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((X != Y) = ((((ord_less @ A) @ X) @ Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_221_order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 0.32/1.50 thf(fact_222_dense, axiom, (! [A:$tType]: ((((dense_order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (? [Z:A]: (((((ord_less @ A) @ X) @ Z) & (((ord_less @ A) @ Z) @ Y))))))))))).
% 0.32/1.50 thf(fact_223_less__imp__neq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 0.32/1.50 thf(fact_224_less__asym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.32/1.50 thf(fact_225_less__asym_H, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ B2) @ A2)))))))))).
% 0.32/1.50 thf(fact_226_less__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ Z3) => (((ord_less @ A) @ X) @ Z3))))))))).
% 0.32/1.50 thf(fact_227_less__linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) | ((X = Y) | (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_228_less__irrefl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A]: ((~ ((((ord_less @ A) @ X) @ X))))))))).
% 0.32/1.50 thf(fact_229_ord__eq__less__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_230_ord__less__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_231_dual__order_Oasym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (~ ((((ord_less @ A) @ A2) @ B2)))))))))).
% 0.32/1.50 thf(fact_232_less__imp__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (X != Y)))))))).
% 0.32/1.50 thf(fact_233_less__not__sym, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.32/1.50 thf(fact_234_less__induct, axiom, (! [A:$tType]: ((((wellorder @ A) @ (type2 @ A)) => (! [P:(A > $o),A2:A]: (((! [X3:A]: (((! [Y2:A]: (((((ord_less @ A) @ Y2) @ X3) => (P @ Y2)))) => (P @ X3)))) => (P @ A2)))))))).
% 0.32/1.50 thf(fact_235_antisym__conv3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [Y:A,X:A]: (((~ ((((ord_less @ A) @ Y) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y))) = (X = Y))))))))).
% 0.32/1.50 thf(fact_236_less__imp__not__eq2, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (Y != X)))))))).
% 0.32/1.50 thf(fact_237_less__imp__triv, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,P:$o]: (((((ord_less @ A) @ X) @ Y) => ((((ord_less @ A) @ Y) @ X) => P)))))))).
% 0.32/1.50 thf(fact_238_linorder__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) => ((X != Y) => (((ord_less @ A) @ Y) @ X))))))))).
% 0.32/1.50 thf(fact_239_dual__order_Oirrefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 0.32/1.50 thf(fact_240_order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 0.32/1.50 thf(fact_241_less__imp__not__less, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (~ ((((ord_less @ A) @ Y) @ X)))))))))).
% 0.32/1.50 thf(fact_242_dual__order_Ostrict__trans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 0.32/1.50 thf(fact_243_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A]: (((~ ((((ord_less @ A) @ X) @ Y))) = ((((ord_less @ A) @ Y) @ X) | (X = Y))))))))).
% 0.32/1.50 thf(fact_244_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 0.32/1.50 thf(fact_245_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 0.32/1.50 thf(fact_246_order_Onot__eq__order__implies__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((A2 != B2) => ((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less @ A) @ A2) @ B2))))))))).
% 0.32/1.50 thf(fact_247_dual__order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ B2) @ A2)))))))).
% 0.32/1.50 thf(fact_248_dual__order_Ostrict__iff__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less @ A) = (^ [B3:A,A4:A]: (((((ord_less_eq @ A) @ B3) @ A4) & (A4 != B3))))))))).
% 0.32/1.50 thf(fact_249_dual__order_Oorder__iff__strict, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((ord_less_eq @ A) = (^ [B3:A,A4:A]: (((((ord_less @ A) @ B3) @ A4) | (A4 = B3))))))))).
% 0.32/1.50 thf(fact_250_order_Ostrict__implies__order, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 0.32/1.50 thf(fact_251_dense__le__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [X:A,Y:A,Z3:A]: (((((ord_less @ A) @ X) @ Y) => ((! [W:A]: (((((ord_less @ A) @ X) @ W) => ((((ord_less @ A) @ W) @ Y) => (((ord_less_eq @ A) @ W) @ Z3))))) => (((ord_less_eq @ A) @ Y) @ Z3))))))))).
% 0.32/1.50 thf(fact_252_dense__ge__bounded, axiom, (! [A:$tType]: ((((dense_linorder @ A) @ (type2 @ A)) => (! [Z3:A,X:A,Y:A]: (((((ord_less @ A) @ Z3) @ X) => ((! [W:A]: (((((ord_less @ A) @ Z3) @ W) => ((((ord_less @ A) @ W) @ X) => (((ord_less_eq @ A) @ Y) @ W))))) => (((ord_less_eq @ A) @ Y) @ Z3))))))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A6:$tType,A7:$tType]: ((((preorder @ A7) @ (type2 @ A7)) => ((preorder @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: ((((order @ A7) @ (type2 @ A7)) => ((order @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 0.32/1.50 thf(tcon_fun___Orderings_Oord, axiom, (! [A6:$tType,A7:$tType]: ((((ord @ A7) @ (type2 @ A7)) => ((ord @ (A6 > A7)) @ (type2 @ (A6 > A7))))))).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, ((linordered_semidom @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult, axiom, ((ab_semigroup_mult @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom, axiom, ((algebraic_semidom @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult, axiom, ((comm_monoid_mult @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1, axiom, ((comm_semiring_1 @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Osemigroup__mult, axiom, ((semigroup_mult @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, ((wellorder @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, ((preorder @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, ((linorder @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Omonoid__mult, axiom, ((monoid_mult @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Ono__top, axiom, ((no_top @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___GCD_Osemiring__gcd, axiom, ((semiring_gcd @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, ((order @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, ((ord @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Groups_Oone, axiom, ((one @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Nat_Onat___Rings_Odvd, axiom, ((dvd @ nat) @ (type2 @ nat))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A6:$tType]: (((preorder @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A6:$tType]: (((order @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 0.32/1.50 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A6:$tType]: (((ord @ (set @ A6)) @ (type2 @ (set @ A6)))))).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, ((order @ $o) @ (type2 @ $o))).
% 0.32/1.50 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, ((ord @ $o) @ (type2 @ $o))).
% 0.32/1.50 thf(conj_0, conjecture, (((member @ nat) @ n2) @ ((coinductive_lset @ nat) @ hammin2015774020amming))).
% 0.32/1.50 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2DTF18998.p
% 0.32/1.50 ------------------------
% 0.32/1.53 ---- Cleaned THF ---
% 0.32/1.53 thf(ty_t_Coinductive__List_Ollist,type,
% 0.32/1.53 coinductive_llist: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Set_Oset,type,
% 0.32/1.53 set: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_Nat_Onat,type,
% 0.32/1.53 nat: $tType ).
% 0.32/1.53
% 0.32/1.53 thf(ty_t_itself,type,
% 0.32/1.53 itself: $tType > $tType ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_HOL_Otype,type,
% 0.32/1.53 type:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Odvd,type,
% 0.32/1.53 dvd:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Oone,type,
% 0.32/1.53 one:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Oord,type,
% 0.32/1.53 ord:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Oorder,type,
% 0.32/1.53 order:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_GCD_Osemiring__gcd,type,
% 0.32/1.53 semiring_gcd:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Ono__bot,type,
% 0.32/1.53 no_bot:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Ono__top,type,
% 0.32/1.53 no_top:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.32/1.53 monoid_mult:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Olinorder,type,
% 0.32/1.53 linorder:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Opreorder,type,
% 0.32/1.53 preorder:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Owellorder,type,
% 0.32/1.53 wellorder:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.32/1.53 semigroup_mult:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Odense__order,type,
% 0.32/1.53 dense_order:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.32/1.53 comm_semiring_1:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.32/1.53 linordered_idom:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.32/1.53 comm_monoid_mult:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 0.32/1.53 algebraic_semidom:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 0.32/1.53 ab_semigroup_mult:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Orderings_Odense__linorder,type,
% 0.32/1.53 dense_linorder:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.32/1.53 linordered_semidom:
% 0.32/1.53 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Oldistinct,type,
% 0.32/1.53 coindu351974385stinct:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.32/1.53 coinductive_lfinite:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Ollist_OLNil,type,
% 0.32/1.53 coinductive_LNil:
% 0.32/1.53 !>[A: $tType] : ( coinductive_llist @ A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.32/1.53 coinductive_lnull:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.32/1.53 coinductive_lset:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Olmember,type,
% 0.32/1.53 coinductive_lmember:
% 0.32/1.53 !>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Olsetp,type,
% 0.32/1.53 coinductive_lsetp:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
% 0.32/1.53 coindu63249387sorted:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.32/1.53 one_one:
% 0.32/1.53 !>[A: $tType] : A ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.32/1.53 times_times:
% 0.32/1.53 !>[A: $tType] : ( A > A > A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Ohamming,type,
% 0.32/1.53 hammin2015774020amming: coinductive_llist @ nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
% 0.32/1.53 hammin1328233080lmerge:
% 0.32/1.53 !>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord__class_Olmerge,type,
% 0.32/1.53 hammin1138290136lmerge:
% 0.32/1.53 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth,type,
% 0.32/1.53 hammin1195055315smooth: nat > $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.32/1.53 ord_less:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.32/1.53 ord_less_eq:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Primes_Oprime,type,
% 0.32/1.53 prime: nat > $o ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Pure_Otype,type,
% 0.32/1.53 type2:
% 0.32/1.53 !>[A: $tType] : ( itself @ A ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.32/1.53 dvd_dvd:
% 0.32/1.53 !>[A: $tType] : ( A > A > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_Set_OCollect,type,
% 0.32/1.53 collect:
% 0.32/1.53 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_c_member,type,
% 0.32/1.53 member:
% 0.32/1.53 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_n,type,
% 0.32/1.53 n: nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_n_H____,type,
% 0.32/1.53 n2: nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_na____,type,
% 0.32/1.53 na: nat ).
% 0.32/1.53
% 0.32/1.53 thf(sy_v_p____,type,
% 0.32/1.53 p: nat ).
% 0.32/1.53
% 0.32/1.53 thf(fact_0__092_060open_062smooth_An_H_092_060close_062,axiom,
% 0.32/1.53 hammin1195055315smooth @ n2 ).
% 0.32/1.53
% 0.32/1.53 thf(fact_1__092_060open_062n_H_A_060_An_092_060close_062,axiom,
% 0.32/1.53 ord_less @ nat @ n2 @ na ).
% 0.32/1.53
% 0.32/1.53 thf(fact_2_less_OIH,axiom,
% 0.32/1.53 ! [Y: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ Y @ na )
% 0.32/1.53 => ( ( hammin1195055315smooth @ Y )
% 0.32/1.53 => ( member @ nat @ Y @ ( coinductive_lset @ nat @ hammin2015774020amming ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_3_n,axiom,
% 0.32/1.53 ( na
% 0.32/1.53 = ( times_times @ nat @ p @ n2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_4_hamming__eq__LNil__iff,axiom,
% 0.32/1.53 ( hammin2015774020amming
% 0.32/1.53 != ( coinductive_LNil @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_5_ldistinct__hamming,axiom,
% 0.32/1.53 coindu351974385stinct @ nat @ hammin2015774020amming ).
% 0.32/1.53
% 0.32/1.53 thf(fact_6_lsorted__hamming,axiom,
% 0.32/1.53 coindu63249387sorted @ nat @ hammin2015774020amming ).
% 0.32/1.53
% 0.32/1.53 thf(fact_7_True,axiom,
% 0.32/1.53 ord_less @ nat @ ( one_one @ nat ) @ na ).
% 0.32/1.53
% 0.32/1.53 thf(fact_8_lnull__hamming,axiom,
% 0.32/1.53 ~ ( coinductive_lnull @ nat @ hammin2015774020amming ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_9_lset__lmember,axiom,
% 0.32/1.53 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 = ( coinductive_lmember @ A @ X @ Xs ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_10_lfinite__hamming,axiom,
% 0.32/1.53 ~ ( coinductive_lfinite @ nat @ hammin2015774020amming ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_11_in__lset__lmergeD,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) )
% 0.32/1.53 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_12_ord_Oin__lset__lmergeD,axiom,
% 0.32/1.53 ! [A: $tType,X: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 0.32/1.53 => ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 | ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_13_Coinductive__List_Olset__into__lsetp,axiom,
% 0.32/1.53 ! [A: $tType,X: A,Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 => ( coinductive_lsetp @ A @ Xs @ X ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_14_assms,axiom,
% 0.32/1.53 hammin1195055315smooth @ n ).
% 0.32/1.53
% 0.32/1.53 thf(fact_15_less_Oprems,axiom,
% 0.32/1.53 hammin1195055315smooth @ na ).
% 0.32/1.53
% 0.32/1.53 thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_O_An_A_061_Ap_A_K_An_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/1.53 ~ ! [N: nat] :
% 0.32/1.53 ( na
% 0.32/1.53 != ( times_times @ nat @ p @ N ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_17__092_060open_062prime_Ap_092_060close_062,axiom,
% 0.32/1.53 prime @ p ).
% 0.32/1.53
% 0.32/1.53 thf(fact_18_lfinite__code_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_19_lmerge_Odisc__iff_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ~ ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) )
% 0.32/1.53 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_20_lnull__lmerge,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) )
% 0.32/1.53 = ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_21_lmerge__LNil_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( hammin1138290136lmerge @ A @ Xs @ ( coinductive_LNil @ A ) )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_22_lmerge__LNil_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( hammin1138290136lmerge @ A @ ( coinductive_LNil @ A ) @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_23_lsorted__code_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_24_ldistinct__LNil__code,axiom,
% 0.32/1.53 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_25_smooth__times,axiom,
% 0.32/1.53 ! [X: nat,Y: nat] :
% 0.32/1.53 ( ( hammin1195055315smooth @ ( times_times @ nat @ X @ Y ) )
% 0.32/1.53 = ( ( hammin1195055315smooth @ X )
% 0.32/1.53 & ( hammin1195055315smooth @ Y ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_26__092_060open_062p_Advd_An_092_060close_062,axiom,
% 0.32/1.53 dvd_dvd @ nat @ p @ na ).
% 0.32/1.53
% 0.32/1.53 thf(fact_27_llist_Odisc_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_28_llist_OdiscI_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.32/1.53 ( ( Llist
% 0.32/1.53 = ( coinductive_LNil @ A ) )
% 0.32/1.53 => ( coinductive_lnull @ A @ Llist ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_29_lmember__code_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,X: A] :
% 0.32/1.53 ~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_30_llist_Ocollapse_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,Llist: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lnull @ A @ Llist )
% 0.32/1.53 => ( Llist
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_31_ord_Olmerge_Octr_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/1.53 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_32_ord_Olmerge_Odisc_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/1.53 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( ~ ( coinductive_lnull @ A @ Ys )
% 0.32/1.53 => ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_33_ord_Olmerge_Odisc_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/1.53 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_34_ord_Olmerge__LNil_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LNil @ A ) )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_35_ord_Olmerge__LNil_I1_J,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LNil @ A ) @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_36_ord_Olmerge_Odisc__iff_I2_J,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
% 0.32/1.53 = ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 & ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_37_lmerge_Octr_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ( ( hammin1138290136lmerge @ A @ Xs @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_38_lmerge_Odisc_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( ~ ( coinductive_lnull @ A @ Ys )
% 0.32/1.53 => ~ ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_39_lmerge_Odisc_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ( coinductive_lnull @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_40_lzip_Oexhaust,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.32/1.53 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ B @ Ys ) )
% 0.32/1.53 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_41_ldistinct_OLNil,axiom,
% 0.32/1.53 ! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_42_lappend_Oexhaust,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_43_lnull__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( coinductive_lnull @ A )
% 0.32/1.53 = ( ^ [Llist2: coinductive_llist @ A] :
% 0.32/1.53 ( Llist2
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_44_mem__Collect__eq,axiom,
% 0.32/1.53 ! [A: $tType,A2: A,P: A > $o] :
% 0.32/1.53 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.32/1.53 = ( P @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_45_Collect__mem__eq,axiom,
% 0.32/1.53 ! [A: $tType,A3: set @ A] :
% 0.32/1.53 ( ( collect @ A
% 0.32/1.53 @ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
% 0.32/1.53 = A3 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_46_Collect__cong,axiom,
% 0.32/1.53 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ( P @ X3 )
% 0.32/1.53 = ( Q @ X3 ) )
% 0.32/1.53 => ( ( collect @ A @ P )
% 0.32/1.53 = ( collect @ A @ Q ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_47_lnull__imp__lfinite,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_48_lfinite__LNil,axiom,
% 0.32/1.53 ! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_49_LNil,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_50_ord_Olnull__lmerge,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 0.32/1.53 = ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_51_ord_Olmerge_Oexhaust,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_52_ord_Olfinite__lmergeD,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
% 0.32/1.53 => ( ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 | ( coinductive_lfinite @ A @ Ys ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_53_ord_Olfinite__lmergeI,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/1.53 ( ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 => ( ( coinductive_lfinite @ A @ Ys )
% 0.32/1.53 => ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_54_ord_Olinfinite__lmerge,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
% 0.32/1.53 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/1.53 => ~ ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_55_ord_Olmerge__eq__LNil__iff,axiom,
% 0.32/1.53 ! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) )
% 0.32/1.53 = ( ( Xs
% 0.32/1.53 = ( coinductive_LNil @ A ) )
% 0.32/1.53 | ( Ys
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_56_lmerge_Oexhaust,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 | ( coinductive_lnull @ A @ Ys ) )
% 0.32/1.53 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.32/1.53 => ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_57_lsorted__ldistinct__lset__unique,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/1.53 => ( ( coindu351974385stinct @ A @ Xs )
% 0.32/1.53 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/1.53 => ( ( coindu351974385stinct @ A @ Ys )
% 0.32/1.53 => ( ( ( coinductive_lset @ A @ Xs )
% 0.32/1.53 = ( coinductive_lset @ A @ Ys ) )
% 0.32/1.53 => ( Xs = Ys ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_58_lfinite__lmergeD,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) )
% 0.32/1.53 => ( ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 | ( coinductive_lfinite @ A @ Ys ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_59_lfinite__lmergeI,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 => ( ( coinductive_lfinite @ A @ Ys )
% 0.32/1.53 => ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_60_linfinite__lmerge,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/1.53 => ~ ( coinductive_lfinite @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_61_lmerge__eq__LNil__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( ( hammin1138290136lmerge @ A @ Xs @ Ys )
% 0.32/1.53 = ( coinductive_LNil @ A ) )
% 0.32/1.53 = ( ( Xs
% 0.32/1.53 = ( coinductive_LNil @ A ) )
% 0.32/1.53 | ( Ys
% 0.32/1.53 = ( coinductive_LNil @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_62_lsorted__lmerge,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/1.53 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/1.53 => ( coindu63249387sorted @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_63_ldistinct__lmerge,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( coindu63249387sorted @ A @ Xs )
% 0.32/1.53 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/1.53 => ( ( coindu351974385stinct @ A @ Xs )
% 0.32/1.53 => ( ( coindu351974385stinct @ A @ Ys )
% 0.32/1.53 => ( coindu351974385stinct @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_64_lsetp__into__lset,axiom,
% 0.32/1.53 ! [A: $tType,Xs: coinductive_llist @ A,X: A] :
% 0.32/1.53 ( ( coinductive_lsetp @ A @ Xs @ X )
% 0.32/1.53 => ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_65_nat__1__eq__mult__iff,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ( one_one @ nat )
% 0.32/1.53 = ( times_times @ nat @ M @ N2 ) )
% 0.32/1.53 = ( ( M
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 & ( N2
% 0.32/1.53 = ( one_one @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_66_nat__mult__eq__1__iff,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ( times_times @ nat @ M @ N2 )
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 = ( ( M
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 & ( N2
% 0.32/1.53 = ( one_one @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_67_mult_Oleft__neutral,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_68_mult_Oright__neutral,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_69_less__1__mult,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [M: A,N2: A] :
% 0.32/1.53 ( ( ord_less @ A @ ( one_one @ A ) @ M )
% 0.32/1.53 => ( ( ord_less @ A @ ( one_one @ A ) @ N2 )
% 0.32/1.53 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_70_nat__mult__1,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ( ( times_times @ nat @ ( one_one @ nat ) @ N2 )
% 0.32/1.53 = N2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_71_nat__mult__1__right,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ( ( times_times @ nat @ N2 @ ( one_one @ nat ) )
% 0.32/1.53 = N2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_72_semiring__normalization__rules_I12_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_73_semiring__normalization__rules_I11_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_74_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ ( one_one @ A ) @ A2 )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_75__092_060open_062_092_060exists_062p_O_Aprime_Ap_A_092_060and_062_Ap_Advd_An_092_060close_062,axiom,
% 0.32/1.53 ? [P2: nat] :
% 0.32/1.53 ( ( prime @ P2 )
% 0.32/1.53 & ( dvd_dvd @ nat @ P2 @ na ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_76_nat__dvd__1__iff__1,axiom,
% 0.32/1.53 ! [M: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M @ ( one_one @ nat ) )
% 0.32/1.53 = ( M
% 0.32/1.53 = ( one_one @ nat ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_77_unit__prod,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/1.53 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_78_dvd__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ A @ B2 @ C )
% 0.32/1.53 => ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_79_dvd__refl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_80_dvd__antisym,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M @ N2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ N2 @ M )
% 0.32/1.53 => ( M = N2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_81_prime__nat__dvdD,axiom,
% 0.32/1.53 ! [P3: nat,N2: nat] :
% 0.32/1.53 ( ( prime @ P3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ N2 @ P3 )
% 0.32/1.53 => ( ( N2
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 | ( N2 = P3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_82_dvd__triv__right,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_83_dvd__mult__right,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/1.53 => ( dvd_dvd @ A @ B2 @ C ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_84_mult__dvd__mono,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A,D: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ A @ C @ D )
% 0.32/1.53 => ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_85_dvd__triv__left,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_86_dvd__mult__left,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/1.53 => ( dvd_dvd @ A @ A2 @ C ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_87_dvd__mult2,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/1.53 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_88_dvd__mult,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,C: A,B2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ C )
% 0.32/1.53 => ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_89_dvd__def,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( dvd_dvd @ A )
% 0.32/1.53 = ( ^ [B3: A,A4: A] :
% 0.32/1.53 ? [K: A] :
% 0.32/1.53 ( A4
% 0.32/1.53 = ( times_times @ A @ B3 @ K ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_90_dvdI,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,K2: A] :
% 0.32/1.53 ( ( A2
% 0.32/1.53 = ( times_times @ A @ B2 @ K2 ) )
% 0.32/1.53 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_91_dvdE,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dvd @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ B2 @ A2 )
% 0.32/1.53 => ~ ! [K3: A] :
% 0.32/1.53 ( A2
% 0.32/1.53 != ( times_times @ A @ B2 @ K3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_92_dvd__unit__imp__unit,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/1.53 => ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_93_unit__imp__dvd,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/1.53 => ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_94_one__dvd,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_95_unit__mult__right__cancel,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 => ( ( ( times_times @ A @ B2 @ A2 )
% 0.32/1.53 = ( times_times @ A @ C @ A2 ) )
% 0.32/1.53 = ( B2 = C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_96_unit__mult__left__cancel,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 => ( ( ( times_times @ A @ A2 @ B2 )
% 0.32/1.53 = ( times_times @ A @ A2 @ C ) )
% 0.32/1.53 = ( B2 = C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_97_mult__unit__dvd__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/1.53 => ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/1.53 = ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_98_dvd__mult__unit__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.32/1.53 => ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C @ B2 ) )
% 0.32/1.53 = ( dvd_dvd @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_99_is__unit__mult__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( algebraic_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
% 0.32/1.53 = ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_100_linorder__neqE__linordered__idom,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_idom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( X != Y )
% 0.32/1.53 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_101_mult_Oleft__commute,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A,C: A] :
% 0.32/1.53 ( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C ) )
% 0.32/1.53 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_102_mult_Ocommute,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( times_times @ A )
% 0.32/1.53 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_103_mult_Oassoc,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/1.53 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_104_semiring__normalization__rules_I7_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( times_times @ A )
% 0.32/1.53 = ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_105_semiring__normalization__rules_I13_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/1.53 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ ( times_times @ A @ Ly @ Ry ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_106_semiring__normalization__rules_I14_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/1.53 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ ( times_times @ A @ Rx @ Ry ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_107_semiring__normalization__rules_I15_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Ly: A,Rx: A,Ry: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/1.53 = ( times_times @ A @ Rx @ ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Ry ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_108_semiring__normalization__rules_I16_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Ly: A,Rx: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.32/1.53 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ly ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_109_semiring__normalization__rules_I17_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Ly: A,Rx: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ Lx @ Ly ) @ Rx )
% 0.32/1.53 = ( times_times @ A @ Lx @ ( times_times @ A @ Ly @ Rx ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_110_semiring__normalization__rules_I18_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Rx: A,Ry: A] :
% 0.32/1.53 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/1.53 = ( times_times @ A @ ( times_times @ A @ Lx @ Rx ) @ Ry ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_111_semiring__normalization__rules_I19_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_semiring_1 @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Lx: A,Rx: A,Ry: A] :
% 0.32/1.53 ( ( times_times @ A @ Lx @ ( times_times @ A @ Rx @ Ry ) )
% 0.32/1.53 = ( times_times @ A @ Rx @ ( times_times @ A @ Lx @ Ry ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_112_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C )
% 0.32/1.53 = ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_113_one__reorient,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( one @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A] :
% 0.32/1.53 ( ( ( one_one @ A )
% 0.32/1.53 = X )
% 0.32/1.53 = ( X
% 0.32/1.53 = ( one_one @ A ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_114_infinite__descent__measure,axiom,
% 0.32/1.53 ! [A: $tType,P: A > $o,V: A > nat,X: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ~ ( P @ X3 )
% 0.32/1.53 => ? [Y2: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
% 0.32/1.53 & ~ ( P @ Y2 ) ) )
% 0.32/1.53 => ( P @ X ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_115_measure__induct__rule,axiom,
% 0.32/1.53 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ! [Y2: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.32/1.53 => ( P @ Y2 ) )
% 0.32/1.53 => ( P @ X3 ) )
% 0.32/1.53 => ( P @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_116_linorder__neqE__nat,axiom,
% 0.32/1.53 ! [X: nat,Y: nat] :
% 0.32/1.53 ( ( X != Y )
% 0.32/1.53 => ( ~ ( ord_less @ nat @ X @ Y )
% 0.32/1.53 => ( ord_less @ nat @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_117_infinite__descent,axiom,
% 0.32/1.53 ! [P: nat > $o,N2: nat] :
% 0.32/1.53 ( ! [N3: nat] :
% 0.32/1.53 ( ~ ( P @ N3 )
% 0.32/1.53 => ? [M2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M2 @ N3 )
% 0.32/1.53 & ~ ( P @ M2 ) ) )
% 0.32/1.53 => ( P @ N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_118_nat__less__induct,axiom,
% 0.32/1.53 ! [P: nat > $o,N2: nat] :
% 0.32/1.53 ( ! [N3: nat] :
% 0.32/1.53 ( ! [M2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M2 @ N3 )
% 0.32/1.53 => ( P @ M2 ) )
% 0.32/1.53 => ( P @ N3 ) )
% 0.32/1.53 => ( P @ N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_119_less__irrefl__nat,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_120_measure__induct,axiom,
% 0.32/1.53 ! [A: $tType,F: A > nat,P: A > $o,A2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ! [Y2: A] :
% 0.32/1.53 ( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X3 ) )
% 0.32/1.53 => ( P @ Y2 ) )
% 0.32/1.53 => ( P @ X3 ) )
% 0.32/1.53 => ( P @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_121_less__not__refl3,axiom,
% 0.32/1.53 ! [S: nat,T: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ S @ T )
% 0.32/1.53 => ( S != T ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_122_less__not__refl2,axiom,
% 0.32/1.53 ! [N2: nat,M: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ N2 @ M )
% 0.32/1.53 => ( M != N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_123_less__not__refl,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ~ ( ord_less @ nat @ N2 @ N2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_124_nat__neq__iff,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( M != N2 )
% 0.32/1.53 = ( ( ord_less @ nat @ M @ N2 )
% 0.32/1.53 | ( ord_less @ nat @ N2 @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_125_mult_Ocomm__neutral,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( comm_monoid_mult @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ( ( times_times @ A @ A2 @ ( one_one @ A ) )
% 0.32/1.53 = A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_126_prime__dvd__mult__eq__nat,axiom,
% 0.32/1.53 ! [P3: nat,M: nat,N2: nat] :
% 0.32/1.53 ( ( prime @ P3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ P3 @ ( times_times @ nat @ M @ N2 ) )
% 0.32/1.53 = ( ( dvd_dvd @ nat @ P3 @ M )
% 0.32/1.53 | ( dvd_dvd @ nat @ P3 @ N2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_127_prime__def,axiom,
% 0.32/1.53 ( prime
% 0.32/1.53 = ( ^ [P4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( one_one @ nat ) @ P4 )
% 0.32/1.53 & ! [M3: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ M3 @ P4 )
% 0.32/1.53 => ( ( M3
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 | ( M3 = P4 ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_128_not__prime__eq__prod__nat,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
% 0.32/1.53 => ( ~ ( prime @ N2 )
% 0.32/1.53 => ? [M4: nat,K3: nat] :
% 0.32/1.53 ( ( N2
% 0.32/1.53 = ( times_times @ nat @ M4 @ K3 ) )
% 0.32/1.53 & ( ord_less @ nat @ ( one_one @ nat ) @ M4 )
% 0.32/1.53 & ( ord_less @ nat @ M4 @ N2 )
% 0.32/1.53 & ( ord_less @ nat @ ( one_one @ nat ) @ K3 )
% 0.32/1.53 & ( ord_less @ nat @ K3 @ N2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_129_prime__factor__nat,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ( ( N2
% 0.32/1.53 != ( one_one @ nat ) )
% 0.32/1.53 => ? [P2: nat] :
% 0.32/1.53 ( ( prime @ P2 )
% 0.32/1.53 & ( dvd_dvd @ nat @ P2 @ N2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_130_prime__dvd__mult__nat,axiom,
% 0.32/1.53 ! [P3: nat,M: nat,N2: nat] :
% 0.32/1.53 ( ( prime @ P3 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ P3 @ ( times_times @ nat @ M @ N2 ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ P3 @ M )
% 0.32/1.53 | ( dvd_dvd @ nat @ P3 @ N2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_131_prime__product,axiom,
% 0.32/1.53 ! [P3: nat,Q2: nat] :
% 0.32/1.53 ( ( prime @ ( times_times @ nat @ P3 @ Q2 ) )
% 0.32/1.53 => ( ( P3
% 0.32/1.53 = ( one_one @ nat ) )
% 0.32/1.53 | ( Q2
% 0.32/1.53 = ( one_one @ nat ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_132_bigger__prime,axiom,
% 0.32/1.53 ! [N2: nat] :
% 0.32/1.53 ? [P2: nat] :
% 0.32/1.53 ( ( prime @ P2 )
% 0.32/1.53 & ( ord_less @ nat @ N2 @ P2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_133_one__not__prime__nat,axiom,
% 0.32/1.53 ~ ( prime @ ( one_one @ nat ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_134_prime__gt__1__nat,axiom,
% 0.32/1.53 ! [P3: nat] :
% 0.32/1.53 ( ( prime @ P3 )
% 0.32/1.53 => ( ord_less @ nat @ ( one_one @ nat ) @ P3 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_135_division__decomp,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( semiring_gcd @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C ) )
% 0.32/1.53 => ? [B4: A,C2: A] :
% 0.32/1.53 ( ( A2
% 0.32/1.53 = ( times_times @ A @ B4 @ C2 ) )
% 0.32/1.53 & ( dvd_dvd @ A @ B4 @ B2 )
% 0.32/1.53 & ( dvd_dvd @ A @ C2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_136_less__numeral__extra_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_137_in__lset__lmergeI1,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 => ( ( coindu63249387sorted @ A @ Xs )
% 0.32/1.53 => ( ~ ( coinductive_lfinite @ A @ Ys )
% 0.32/1.53 => ( ? [X4: A] :
% 0.32/1.53 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Ys ) )
% 0.32/1.53 & ( ord_less_eq @ A @ X @ X4 ) )
% 0.32/1.53 => ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_138_in__lset__lmergeI2,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
% 0.32/1.53 ( ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) )
% 0.32/1.53 => ( ( coindu63249387sorted @ A @ Ys )
% 0.32/1.53 => ( ~ ( coinductive_lfinite @ A @ Xs )
% 0.32/1.53 => ( ? [X4: A] :
% 0.32/1.53 ( ( member @ A @ X4 @ ( coinductive_lset @ A @ Xs ) )
% 0.32/1.53 & ( ord_less_eq @ A @ X @ X4 ) )
% 0.32/1.53 => ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1138290136lmerge @ A @ Xs @ Ys ) ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_139_nat__less__le,axiom,
% 0.32/1.53 ( ( ord_less @ nat )
% 0.32/1.53 = ( ^ [M3: nat,N4: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ M3 @ N4 )
% 0.32/1.53 & ( M3 != N4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_140_less__imp__le__nat,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M @ N2 )
% 0.32/1.53 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_141_le__eq__less__or__eq,axiom,
% 0.32/1.53 ( ( ord_less_eq @ nat )
% 0.32/1.53 = ( ^ [M3: nat,N4: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ M3 @ N4 )
% 0.32/1.53 | ( M3 = N4 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_142_less__or__eq__imp__le,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ( ord_less @ nat @ M @ N2 )
% 0.32/1.53 | ( M = N2 ) )
% 0.32/1.53 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_143_le__neq__implies__less,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/1.53 => ( ( M != N2 )
% 0.32/1.53 => ( ord_less @ nat @ M @ N2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_144_less__mono__imp__le__mono,axiom,
% 0.32/1.53 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/1.53 ( ! [I2: nat,J2: nat] :
% 0.32/1.53 ( ( ord_less @ nat @ I2 @ J2 )
% 0.32/1.53 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/1.53 => ( ( ord_less_eq @ nat @ I @ J )
% 0.32/1.53 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_145_le__cube,axiom,
% 0.32/1.53 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_146_le__square,axiom,
% 0.32/1.53 ! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_147_mult__le__mono,axiom,
% 0.32/1.53 ! [I: nat,J: nat,K2: nat,L: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/1.53 => ( ( ord_less_eq @ nat @ K2 @ L )
% 0.32/1.53 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_148_mult__le__mono1,axiom,
% 0.32/1.53 ! [I: nat,J: nat,K2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/1.53 => ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_149_mult__le__mono2,axiom,
% 0.32/1.53 ! [I: nat,J: nat,K2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/1.53 => ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I ) @ ( times_times @ nat @ K2 @ J ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_150_wlog__linorder__le,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > A > $o,B2: A,A2: A] :
% 0.32/1.53 ( ! [A5: A,B5: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/1.53 => ( P @ A5 @ B5 ) )
% 0.32/1.53 => ( ( ( P @ B2 @ A2 )
% 0.32/1.53 => ( P @ A2 @ B2 ) )
% 0.32/1.53 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_151_le__numeral__extra_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linordered_semidom @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_152_prime__ge__1__nat,axiom,
% 0.32/1.53 ! [P3: nat] :
% 0.32/1.53 ( ( prime @ P3 )
% 0.32/1.53 => ( ord_less_eq @ nat @ ( one_one @ nat ) @ P3 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_153_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( A2 != B2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_154_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 => ( A2 != B2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_155_gcd__nat_Ostrict__implies__order,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 => ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_156_gcd__nat_Ostrict__iff__order,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 = ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_157_gcd__nat_Oorder__iff__strict,axiom,
% 0.32/1.53 ( ( dvd_dvd @ nat )
% 0.32/1.53 = ( ^ [A4: nat,B3: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A4 @ B3 )
% 0.32/1.53 & ( A4 != B3 ) )
% 0.32/1.53 | ( A4 = B3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_158_gcd__nat_Ostrict__trans2,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat,C: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/1.53 & ( A2 != C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_159_gcd__nat_Ostrict__trans1,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat,C: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 => ( ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/1.53 & ( B2 != C ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/1.53 & ( A2 != C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_160_gcd__nat_Ostrict__trans,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat,C: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 => ( ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/1.53 & ( B2 != C ) )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ A2 @ C )
% 0.32/1.53 & ( A2 != C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_161_gcd__nat_Oantisym,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B2 @ A2 )
% 0.32/1.53 => ( A2 = B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_162_gcd__nat_Oirrefl,axiom,
% 0.32/1.53 ! [A2: nat] :
% 0.32/1.53 ~ ( ( dvd_dvd @ nat @ A2 @ A2 )
% 0.32/1.53 & ( A2 != A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_163_gcd__nat_Otrans,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat,C: nat] :
% 0.32/1.53 ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 => ( ( dvd_dvd @ nat @ B2 @ C )
% 0.32/1.53 => ( dvd_dvd @ nat @ A2 @ C ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_164_gcd__nat_Orefl,axiom,
% 0.32/1.53 ! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_165_gcd__nat_Oasym,axiom,
% 0.32/1.53 ! [A2: nat,B2: nat] :
% 0.32/1.53 ( ( ( dvd_dvd @ nat @ A2 @ B2 )
% 0.32/1.53 & ( A2 != B2 ) )
% 0.32/1.53 => ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
% 0.32/1.53 & ( B2 != A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_166_order__refl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_167_minf_I8_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_168_minf_I6_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_169_pinf_I8_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_170_le__refl,axiom,
% 0.32/1.53 ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_171_le__trans,axiom,
% 0.32/1.53 ! [I: nat,J: nat,K2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.32/1.53 => ( ( ord_less_eq @ nat @ J @ K2 )
% 0.32/1.53 => ( ord_less_eq @ nat @ I @ K2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_172_eq__imp__le,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( M = N2 )
% 0.32/1.53 => ( ord_less_eq @ nat @ M @ N2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_173_le__antisym,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/1.53 => ( ( ord_less_eq @ nat @ N2 @ M )
% 0.32/1.53 => ( M = N2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_174_nat__le__linear,axiom,
% 0.32/1.53 ! [M: nat,N2: nat] :
% 0.32/1.53 ( ( ord_less_eq @ nat @ M @ N2 )
% 0.32/1.53 | ( ord_less_eq @ nat @ N2 @ M ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_175_le__funD,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 => ! [F: A > B,G: A > B,X: A] :
% 0.32/1.53 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/1.53 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_176_le__funE,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 => ! [F: A > B,G: A > B,X: A] :
% 0.32/1.53 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.32/1.53 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_177_le__funI,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 => ! [F: A > B,G: A > B] :
% 0.32/1.53 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.32/1.53 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_178_le__fun__def,axiom,
% 0.32/1.53 ! [B: $tType,A: $tType] :
% 0.32/1.53 ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 => ( ( ord_less_eq @ ( A > B ) )
% 0.32/1.53 = ( ^ [F2: A > B,G2: A > B] :
% 0.32/1.53 ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_179_order__subst1,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( order @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.32/1.53 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.32/1.53 => ( ! [X3: B,Y3: B] :
% 0.32/1.53 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_180_order__subst2,axiom,
% 0.32/1.53 ! [A: $tType,C3: $tType] :
% 0.32/1.53 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.32/1.53 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,B2: A,F: A > C3,C: C3] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ord_less_eq @ C3 @ ( F @ B2 ) @ C )
% 0.32/1.53 => ( ! [X3: A,Y3: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less_eq @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less_eq @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_181_ord__eq__le__subst,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/1.53 ( ( A2
% 0.32/1.53 = ( F @ B2 ) )
% 0.32/1.53 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.32/1.53 => ( ! [X3: B,Y3: B] :
% 0.32/1.53 ( ( ord_less_eq @ B @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_182_ord__le__eq__subst,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ( F @ B2 )
% 0.32/1.53 = C )
% 0.32/1.53 => ( ! [X3: A,Y3: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_183_eq__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.32/1.53 = ( ^ [X2: A,Y5: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X2 @ Y5 )
% 0.32/1.53 & ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_184_antisym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 => ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.53 => ( X = Y ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_185_linear,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 | ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_186_eq__refl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( X = Y )
% 0.32/1.53 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_187_le__cases,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 => ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_188_order_Otrans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_189_le__cases3,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.53 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ Y @ Z3 ) )
% 0.32/1.53 => ( ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ X @ Z3 ) )
% 0.32/1.53 => ( ( ( ord_less_eq @ A @ X @ Z3 )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ Z3 @ Y ) )
% 0.32/1.53 => ( ( ( ord_less_eq @ A @ Z3 @ Y )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ Y @ X ) )
% 0.32/1.53 => ( ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ Z3 @ X ) )
% 0.32/1.53 => ~ ( ( ord_less_eq @ A @ Z3 @ X )
% 0.32/1.53 => ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_190_antisym__conv,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Y: A,X: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ Y @ X )
% 0.32/1.53 => ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 = ( X = Y ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_191_ord__eq__le__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( A2 = B2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_192_ord__le__eq__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( B2 = C )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_193_order__class_Oorder_Oantisym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/1.53 => ( A2 = B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_194_order__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ X @ Y )
% 0.32/1.53 => ( ( ord_less_eq @ A @ Y @ Z3 )
% 0.32/1.53 => ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_195_dual__order_Orefl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_196_linorder__wlog,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > A > $o,A2: A,B2: A] :
% 0.32/1.53 ( ! [A5: A,B5: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.32/1.53 => ( P @ A5 @ B5 ) )
% 0.32/1.53 => ( ! [A5: A,B5: A] :
% 0.32/1.53 ( ( P @ B5 @ A5 )
% 0.32/1.53 => ( P @ A5 @ B5 ) )
% 0.32/1.53 => ( P @ A2 @ B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_197_dual__order_Otrans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A,C: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.32/1.53 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_198_dual__order_Oantisym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( A2 = B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_199_minf_I11_J,axiom,
% 0.32/1.53 ! [C3: $tType,D2: $tType] :
% 0.32/1.53 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.32/1.53 => ! [F3: D2] :
% 0.32/1.53 ? [Z: C3] :
% 0.32/1.53 ! [X4: C3] :
% 0.32/1.53 ( ( ord_less @ C3 @ X4 @ Z )
% 0.32/1.53 => ( F3 = F3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_200_minf_I7_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_201_minf_I5_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_202_minf_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( X4 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_203_minf_I3_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( X4 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_204_minf_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/1.53 ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P5 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q3 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( ( ( P @ X4 )
% 0.32/1.53 | ( Q @ X4 ) )
% 0.32/1.53 = ( ( P5 @ X4 )
% 0.32/1.53 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_205_minf_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/1.53 ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P5 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Z4 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q3 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ X4 @ Z )
% 0.32/1.53 => ( ( ( P @ X4 )
% 0.32/1.53 & ( Q @ X4 ) )
% 0.32/1.53 = ( ( P5 @ X4 )
% 0.32/1.53 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_206_pinf_I11_J,axiom,
% 0.32/1.53 ! [C3: $tType,D2: $tType] :
% 0.32/1.53 ( ( ord @ C3 @ ( type2 @ C3 ) )
% 0.32/1.53 => ! [F3: D2] :
% 0.32/1.53 ? [Z: C3] :
% 0.32/1.53 ! [X4: C3] :
% 0.32/1.53 ( ( ord_less @ C3 @ Z @ X4 )
% 0.32/1.53 => ( F3 = F3 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_207_pinf_I7_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( ord_less @ A @ T @ X4 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_208_pinf_I5_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_209_pinf_I4_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( X4 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_210_pinf_I3_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [T: A] :
% 0.32/1.53 ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( X4 != T ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_211_pinf_I2_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/1.53 ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P5 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q3 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( ( ( P @ X4 )
% 0.32/1.53 | ( Q @ X4 ) )
% 0.32/1.53 = ( ( P5 @ X4 )
% 0.32/1.53 | ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_212_pinf_I1_J,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > $o,P5: A > $o,Q: A > $o,Q3: A > $o] :
% 0.32/1.53 ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/1.53 => ( ( P @ X3 )
% 0.32/1.53 = ( P5 @ X3 ) ) )
% 0.32/1.53 => ( ? [Z4: A] :
% 0.32/1.53 ! [X3: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z4 @ X3 )
% 0.32/1.53 => ( ( Q @ X3 )
% 0.32/1.53 = ( Q3 @ X3 ) ) )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ! [X4: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z @ X4 )
% 0.32/1.53 => ( ( ( P @ X4 )
% 0.32/1.53 & ( Q @ X4 ) )
% 0.32/1.53 = ( ( P5 @ X4 )
% 0.32/1.53 & ( Q3 @ X4 ) ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_213_ord__eq__less__subst,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/1.53 ( ( A2
% 0.32/1.53 = ( F @ B2 ) )
% 0.32/1.53 => ( ( ord_less @ B @ B2 @ C )
% 0.32/1.53 => ( ! [X3: B,Y3: B] :
% 0.32/1.53 ( ( ord_less @ B @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_214_ord__less__eq__subst,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( ord @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ( F @ B2 )
% 0.32/1.53 = C )
% 0.32/1.53 => ( ! [X3: A,Y3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_215_order__less__subst1,axiom,
% 0.32/1.53 ! [A: $tType,B: $tType] :
% 0.32/1.53 ( ( ( order @ B @ ( type2 @ B ) )
% 0.32/1.53 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ ( F @ B2 ) )
% 0.32/1.53 => ( ( ord_less @ B @ B2 @ C )
% 0.32/1.53 => ( ! [X3: B,Y3: B] :
% 0.32/1.53 ( ( ord_less @ B @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_216_order__less__subst2,axiom,
% 0.32/1.53 ! [A: $tType,C3: $tType] :
% 0.32/1.53 ( ( ( order @ C3 @ ( type2 @ C3 ) )
% 0.32/1.53 & ( order @ A @ ( type2 @ A ) ) )
% 0.32/1.53 => ! [A2: A,B2: A,F: A > C3,C: C3] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ord_less @ C3 @ ( F @ B2 ) @ C )
% 0.32/1.53 => ( ! [X3: A,Y3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X3 @ Y3 )
% 0.32/1.53 => ( ord_less @ C3 @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.32/1.53 => ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_217_lt__ex,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( no_bot @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A] :
% 0.32/1.53 ? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_218_gt__ex,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( no_top @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A] :
% 0.32/1.53 ? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_219_neqE,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( X != Y )
% 0.32/1.53 => ( ~ ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_220_neq__iff,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( X != Y )
% 0.32/1.53 = ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 | ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_221_order_Oasym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_222_dense,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dense_order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ? [Z: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Z )
% 0.32/1.53 & ( ord_less @ A @ Z @ Y ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_223_less__imp__neq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( X != Y ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_224_less__asym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_225_less__asym_H,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_226_less__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ( ord_less @ A @ Y @ Z3 )
% 0.32/1.53 => ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_227_less__linear,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 | ( X = Y )
% 0.32/1.53 | ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_228_less__irrefl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A] :
% 0.32/1.53 ~ ( ord_less @ A @ X @ X ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_229_ord__eq__less__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( A2 = B2 )
% 0.32/1.53 => ( ( ord_less @ A @ B2 @ C )
% 0.32/1.53 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_230_ord__less__eq__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( ord @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( B2 = C )
% 0.32/1.53 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_231_dual__order_Oasym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/1.53 => ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_232_less__imp__not__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( X != Y ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_233_less__not__sym,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_234_less__induct,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( wellorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [P: A > $o,A2: A] :
% 0.32/1.53 ( ! [X3: A] :
% 0.32/1.53 ( ! [Y2: A] :
% 0.32/1.53 ( ( ord_less @ A @ Y2 @ X3 )
% 0.32/1.53 => ( P @ Y2 ) )
% 0.32/1.53 => ( P @ X3 ) )
% 0.32/1.53 => ( P @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_235_antisym__conv3,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Y: A,X: A] :
% 0.32/1.53 ( ~ ( ord_less @ A @ Y @ X )
% 0.32/1.53 => ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/1.53 = ( X = Y ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_236_less__imp__not__eq2,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( Y != X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_237_less__imp__triv,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A,P: $o] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ( ord_less @ A @ Y @ X )
% 0.32/1.53 => P ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_238_linorder__cases,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ~ ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ( X != Y )
% 0.32/1.53 => ( ord_less @ A @ Y @ X ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_239_dual__order_Oirrefl,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A] :
% 0.32/1.53 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_240_order_Ostrict__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A,C: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( ( ord_less @ A @ B2 @ C )
% 0.32/1.53 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_241_less__imp__not__less,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( preorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ~ ( ord_less @ A @ Y @ X ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_242_dual__order_Ostrict__trans,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A,C: A] :
% 0.32/1.53 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/1.53 => ( ( ord_less @ A @ C @ B2 )
% 0.32/1.53 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_243_not__less__iff__gr__or__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A] :
% 0.32/1.53 ( ( ~ ( ord_less @ A @ X @ Y ) )
% 0.32/1.53 = ( ( ord_less @ A @ Y @ X )
% 0.32/1.53 | ( X = Y ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_244_order_Ostrict__implies__not__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( A2 != B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_245_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/1.53 => ( A2 != B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_246_order_Onot__eq__order__implies__strict,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( A2 != B2 )
% 0.32/1.53 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.32/1.53 => ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_247_dual__order_Ostrict__implies__order,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [B2: A,A2: A] :
% 0.32/1.53 ( ( ord_less @ A @ B2 @ A2 )
% 0.32/1.53 => ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_248_dual__order_Ostrict__iff__order,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( ord_less @ A )
% 0.32/1.53 = ( ^ [B3: A,A4: A] :
% 0.32/1.53 ( ( ord_less_eq @ A @ B3 @ A4 )
% 0.32/1.53 & ( A4 != B3 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_249_dual__order_Oorder__iff__strict,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ( ( ord_less_eq @ A )
% 0.32/1.53 = ( ^ [B3: A,A4: A] :
% 0.32/1.53 ( ( ord_less @ A @ B3 @ A4 )
% 0.32/1.53 | ( A4 = B3 ) ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_250_order_Ostrict__implies__order,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( order @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [A2: A,B2: A] :
% 0.32/1.53 ( ( ord_less @ A @ A2 @ B2 )
% 0.32/1.53 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_251_dense__le__bounded,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [X: A,Y: A,Z3: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ Y )
% 0.32/1.53 => ( ! [W: A] :
% 0.32/1.53 ( ( ord_less @ A @ X @ W )
% 0.32/1.53 => ( ( ord_less @ A @ W @ Y )
% 0.32/1.53 => ( ord_less_eq @ A @ W @ Z3 ) ) )
% 0.32/1.53 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(fact_252_dense__ge__bounded,axiom,
% 0.32/1.53 ! [A: $tType] :
% 0.32/1.53 ( ( dense_linorder @ A @ ( type2 @ A ) )
% 0.32/1.53 => ! [Z3: A,X: A,Y: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z3 @ X )
% 0.32/1.53 => ( ! [W: A] :
% 0.32/1.53 ( ( ord_less @ A @ Z3 @ W )
% 0.32/1.53 => ( ( ord_less @ A @ W @ X )
% 0.32/1.53 => ( ord_less_eq @ A @ Y @ W ) ) )
% 0.32/1.53 => ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.32/1.53 ! [A6: $tType,A7: $tType] :
% 0.32/1.53 ( ( preorder @ A7 @ ( type2 @ A7 ) )
% 0.32/1.53 => ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.32/1.53 ! [A6: $tType,A7: $tType] :
% 0.32/1.53 ( ( order @ A7 @ ( type2 @ A7 ) )
% 0.32/1.53 => ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_fun___Orderings_Oord,axiom,
% 0.32/1.53 ! [A6: $tType,A7: $tType] :
% 0.32/1.53 ( ( ord @ A7 @ ( type2 @ A7 ) )
% 0.32/1.53 => ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.32/1.53 linordered_semidom @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
% 0.32/1.53 ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom,axiom,
% 0.32/1.53 algebraic_semidom @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
% 0.32/1.53 comm_monoid_mult @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
% 0.32/1.53 comm_semiring_1 @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
% 0.32/1.53 semigroup_mult @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.32/1.53 wellorder @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.32/1.53 preorder @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.32/1.53 linorder @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
% 0.32/1.53 monoid_mult @ nat @ ( type2 @ nat ) ).
% 0.32/1.53
% 0.32/1.53 thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
% 0.32/1.54 no_top @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Nat_Onat___GCD_Osemiring__gcd,axiom,
% 0.32/1.54 semiring_gcd @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.32/1.54 order @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.32/1.54 ord @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.32/1.54 one @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Nat_Onat___Rings_Odvd,axiom,
% 0.32/1.54 dvd @ nat @ ( type2 @ nat ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.32/1.54 ! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.32/1.54 ! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.32/1.54 ! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.32/1.54 preorder @ $o @ ( type2 @ $o ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.32/1.54 linorder @ $o @ ( type2 @ $o ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.32/1.54 order @ $o @ ( type2 @ $o ) ).
% 0.32/1.54
% 0.32/1.54 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.32/1.54 ord @ $o @ ( type2 @ $o ) ).
% 0.32/1.54
% 0.32/1.54 thf(conj_0,conjecture,
% 0.32/1.54 member @ nat @ n2 @ ( coinductive_lset @ nat @ hammin2015774020amming ) ).
% 0.32/1.54 ------------------------
% 0.32/1.56 % (19328)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/4Mi)
% 0.32/1.56 % (19327)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/183Mi)
% 0.32/1.56 % (19330)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/2Mi)
% 0.32/1.56 % (19328)Instruction limit reached!
% 0.32/1.56 % (19328)------------------------------
% 0.32/1.56 % (19328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (19328)Termination reason: Unknown
% 0.32/1.56 % (19328)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (19328)Memory used [KB]: 1407
% 0.32/1.56 % (19328)Time elapsed: 0.003 s
% 0.32/1.56 % (19328)Instructions burned: 5 (million)
% 0.32/1.56 % (19328)------------------------------
% 0.32/1.56 % (19328)------------------------------
% 0.32/1.56 % (19330)Instruction limit reached!
% 0.32/1.56 % (19330)------------------------------
% 0.32/1.56 % (19330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.56 % (19330)Termination reason: Unknown
% 0.32/1.56 % (19330)Termination phase: shuffling
% 0.32/1.56
% 0.32/1.56 % (19330)Memory used [KB]: 1279
% 0.32/1.56 % (19330)Time elapsed: 0.002 s
% 0.32/1.56 % (19330)Instructions burned: 3 (million)
% 0.32/1.56 % (19330)------------------------------
% 0.32/1.56 % (19330)------------------------------
% 0.32/1.56 % (19333)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/18Mi)
% 0.32/1.57 % (19329)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/27Mi)
% 0.32/1.57 % (19331)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/2Mi)
% 0.32/1.57 % (19332)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/275Mi)
% 0.32/1.57 % (19331)Instruction limit reached!
% 0.32/1.57 % (19331)------------------------------
% 0.32/1.57 % (19331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (19331)Termination reason: Unknown
% 0.32/1.57 % (19331)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (19331)Memory used [KB]: 1279
% 0.32/1.57 % (19331)Time elapsed: 0.004 s
% 0.32/1.57 % (19331)Instructions burned: 3 (million)
% 0.32/1.57 % (19331)------------------------------
% 0.32/1.57 % (19331)------------------------------
% 0.32/1.57 % (19333)Instruction limit reached!
% 0.32/1.57 % (19333)------------------------------
% 0.32/1.57 % (19333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (19333)Termination reason: Unknown
% 0.32/1.57 % (19333)Termination phase: Property scanning
% 0.32/1.57
% 0.32/1.57 % (19333)Memory used [KB]: 1663
% 0.32/1.57 % (19333)Time elapsed: 0.010 s
% 0.32/1.57 % (19333)Instructions burned: 20 (million)
% 0.32/1.57 % (19333)------------------------------
% 0.32/1.57 % (19333)------------------------------
% 0.32/1.57 % (19334)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/3Mi)
% 0.32/1.57 % (19335)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/37Mi)
% 0.32/1.57 % (19334)Instruction limit reached!
% 0.32/1.57 % (19334)------------------------------
% 0.32/1.57 % (19334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.57 % (19334)Termination reason: Unknown
% 0.32/1.57 % (19334)Termination phase: shuffling
% 0.32/1.57
% 0.32/1.57 % (19334)Memory used [KB]: 1407
% 0.32/1.57 % (19334)Time elapsed: 0.003 s
% 0.32/1.57 % (19334)Instructions burned: 5 (million)
% 0.32/1.57 % (19334)------------------------------
% 0.32/1.57 % (19334)------------------------------
% 0.32/1.58 % (19337)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/3Mi)
% 0.32/1.58 % (19338)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/1041Mi)
% 0.32/1.58 % (19335)Instruction limit reached!
% 0.32/1.58 % (19335)------------------------------
% 0.32/1.58 % (19335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (19335)Termination reason: Unknown
% 0.32/1.58 % (19335)Termination phase: Property scanning
% 0.32/1.58
% 0.32/1.58 % (19335)Memory used [KB]: 2046
% 0.32/1.58 % (19335)Time elapsed: 0.013 s
% 0.32/1.58 % (19335)Instructions burned: 38 (million)
% 0.32/1.58 % (19335)------------------------------
% 0.32/1.58 % (19335)------------------------------
% 0.32/1.58 % (19337)Instruction limit reached!
% 0.32/1.58 % (19337)------------------------------
% 0.32/1.58 % (19337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (19337)Termination reason: Unknown
% 0.32/1.58 % (19337)Termination phase: shuffling
% 0.32/1.58
% 0.32/1.58 % (19337)Memory used [KB]: 1279
% 0.32/1.58 % (19337)Time elapsed: 0.003 s
% 0.32/1.58 % (19337)Instructions burned: 3 (million)
% 0.32/1.58 % (19337)------------------------------
% 0.32/1.58 % (19337)------------------------------
% 0.32/1.58 % (19329)Instruction limit reached!
% 0.32/1.58 % (19329)------------------------------
% 0.32/1.58 % (19329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.58 % (19329)Termination reason: Unknown
% 0.32/1.58 % (19329)Termination phase: Property scanning
% 0.32/1.58
% 0.32/1.58 % (19329)Memory used [KB]: 1663
% 0.32/1.58 % (19329)Time elapsed: 0.018 s
% 0.32/1.58 % (19329)Instructions burned: 27 (million)
% 0.32/1.58 % (19329)------------------------------
% 0.32/1.58 % (19329)------------------------------
% 0.32/1.59 % (19336)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/15Mi)
% 0.32/1.59 % (19339)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/7Mi)
% 0.32/1.60 % (19336)Instruction limit reached!
% 0.32/1.60 % (19336)------------------------------
% 0.32/1.60 % (19336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.32/1.60 % (19336)Termination reason: Unknown
% 2.88/1.60 % (19336)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (19336)Memory used [KB]: 1663
% 2.88/1.60 % (19336)Time elapsed: 0.010 s
% 2.88/1.60 % (19336)Instructions burned: 15 (million)
% 2.88/1.60 % (19336)------------------------------
% 2.88/1.60 % (19336)------------------------------
% 2.88/1.60 % (19339)Instruction limit reached!
% 2.88/1.60 % (19339)------------------------------
% 2.88/1.60 % (19339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (19339)Termination reason: Unknown
% 2.88/1.60 % (19339)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (19339)Memory used [KB]: 1535
% 2.88/1.60 % (19339)Time elapsed: 0.005 s
% 2.88/1.60 % (19339)Instructions burned: 9 (million)
% 2.88/1.60 % (19339)------------------------------
% 2.88/1.60 % (19339)------------------------------
% 2.88/1.60 % (19341)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/3Mi)
% 2.88/1.60 % (19340)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/16Mi)
% 2.88/1.60 % (19341)Instruction limit reached!
% 2.88/1.60 % (19341)------------------------------
% 2.88/1.60 % (19341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (19341)Termination reason: Unknown
% 2.88/1.60 % (19341)Termination phase: shuffling
% 2.88/1.60
% 2.88/1.60 % (19341)Memory used [KB]: 1407
% 2.88/1.60 % (19341)Time elapsed: 0.003 s
% 2.88/1.60 % (19341)Instructions burned: 4 (million)
% 2.88/1.60 % (19341)------------------------------
% 2.88/1.60 % (19341)------------------------------
% 2.88/1.60 % (19332)Refutation not found, incomplete strategy
% 2.88/1.60 % (19332)------------------------------
% 2.88/1.60 % (19332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (19332)Termination reason: Refutation not found, incomplete strategy
% 2.88/1.60
% 2.88/1.60
% 2.88/1.60 % (19332)Memory used [KB]: 6524
% 2.88/1.60 % (19332)Time elapsed: 0.036 s
% 2.88/1.60 % (19332)Instructions burned: 58 (million)
% 2.88/1.60 % (19332)------------------------------
% 2.88/1.60 % (19332)------------------------------
% 2.88/1.61 % (19343)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/7Mi)
% 2.88/1.61 % (19342)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/3Mi)
% 2.88/1.61 % (19342)Instruction limit reached!
% 2.88/1.61 % (19342)------------------------------
% 2.88/1.61 % (19342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (19342)Termination reason: Unknown
% 2.88/1.61 % (19342)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (19342)Memory used [KB]: 1279
% 2.88/1.61 % (19342)Time elapsed: 0.003 s
% 2.88/1.61 % (19342)Instructions burned: 4 (million)
% 2.88/1.61 % (19342)------------------------------
% 2.88/1.61 % (19342)------------------------------
% 2.88/1.61 % (19343)Instruction limit reached!
% 2.88/1.61 % (19343)------------------------------
% 2.88/1.61 % (19343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (19343)Termination reason: Unknown
% 2.88/1.61 % (19343)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (19343)Memory used [KB]: 1535
% 2.88/1.61 % (19343)Time elapsed: 0.005 s
% 2.88/1.61 % (19343)Instructions burned: 9 (million)
% 2.88/1.61 % (19343)------------------------------
% 2.88/1.61 % (19343)------------------------------
% 2.88/1.61 % (19340)Instruction limit reached!
% 2.88/1.61 % (19340)------------------------------
% 2.88/1.61 % (19340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (19340)Termination reason: Unknown
% 2.88/1.61 % (19340)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (19340)Memory used [KB]: 1663
% 2.88/1.61 % (19340)Time elapsed: 0.012 s
% 2.88/1.61 % (19340)Instructions burned: 16 (million)
% 2.88/1.61 % (19340)------------------------------
% 2.88/1.61 % (19340)------------------------------
% 2.88/1.61 % (19344)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/3Mi)
% 2.88/1.61 % (19344)Instruction limit reached!
% 2.88/1.61 % (19344)------------------------------
% 2.88/1.61 % (19344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.61 % (19344)Termination reason: Unknown
% 2.88/1.61 % (19344)Termination phase: shuffling
% 2.88/1.61
% 2.88/1.61 % (19344)Memory used [KB]: 1407
% 2.88/1.61 % (19344)Time elapsed: 0.003 s
% 2.88/1.61 % (19344)Instructions burned: 4 (million)
% 2.88/1.61 % (19344)------------------------------
% 2.88/1.61 % (19344)------------------------------
% 2.88/1.62 % (19327)Instruction limit reached!
% 2.88/1.62 % (19327)------------------------------
% 2.88/1.62 % (19327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (19327)Termination reason: Unknown
% 2.88/1.62 % (19327)Termination phase: Saturation
% 2.88/1.62
% 2.88/1.62 % (19327)Memory used [KB]: 6908
% 2.88/1.62 % (19327)Time elapsed: 0.056 s
% 2.88/1.62 % (19327)Instructions burned: 183 (million)
% 2.88/1.62 % (19327)------------------------------
% 2.88/1.62 % (19327)------------------------------
% 2.88/1.62 % (19346)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/18Mi)
% 2.88/1.62 % (19347)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/710Mi)
% 2.88/1.62 % (19345)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/4Mi)
% 2.88/1.62 % (19349)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/902Mi)
% 2.88/1.62 % (19346)Instruction limit reached!
% 2.88/1.62 % (19346)------------------------------
% 2.88/1.62 % (19346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (19346)Termination reason: Unknown
% 2.88/1.62 % (19346)Termination phase: Property scanning
% 2.88/1.62
% 2.88/1.62 % (19346)Memory used [KB]: 1663
% 2.88/1.62 % (19346)Time elapsed: 0.007 s
% 2.88/1.62 % (19346)Instructions burned: 18 (million)
% 2.88/1.62 % (19346)------------------------------
% 2.88/1.62 % (19346)------------------------------
% 2.88/1.62 % (19345)Instruction limit reached!
% 2.88/1.62 % (19345)------------------------------
% 2.88/1.62 % (19345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.62 % (19345)Termination reason: Unknown
% 2.88/1.62 % (19345)Termination phase: shuffling
% 2.88/1.62
% 2.88/1.62 % (19345)Memory used [KB]: 1407
% 2.88/1.62 % (19345)Time elapsed: 0.005 s
% 2.88/1.62 % (19345)Instructions burned: 5 (million)
% 2.88/1.62 % (19345)------------------------------
% 2.88/1.62 % (19345)------------------------------
% 2.88/1.63 % (19350)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/21Mi)
% 2.88/1.63 % (19348)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/6Mi)
% 2.88/1.63 % (19347)First to succeed.
% 2.88/1.63 % (19350)Instruction limit reached!
% 2.88/1.63 % (19350)------------------------------
% 2.88/1.63 % (19350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (19350)Termination reason: Unknown
% 2.88/1.63 % (19350)Termination phase: Property scanning
% 2.88/1.63
% 2.88/1.63 % (19350)Memory used [KB]: 1663
% 2.88/1.63 % (19350)Time elapsed: 0.008 s
% 2.88/1.63 % (19350)Instructions burned: 22 (million)
% 2.88/1.63 % (19350)------------------------------
% 2.88/1.63 % (19350)------------------------------
% 2.88/1.63 % (19348)Instruction limit reached!
% 2.88/1.63 % (19348)------------------------------
% 2.88/1.63 % (19348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (19348)Termination reason: Unknown
% 2.88/1.63 % (19348)Termination phase: shuffling
% 2.88/1.63
% 2.88/1.63 % (19348)Memory used [KB]: 1407
% 2.88/1.63 % (19348)Time elapsed: 0.006 s
% 2.88/1.63 % (19348)Instructions burned: 6 (million)
% 2.88/1.63 % (19348)------------------------------
% 2.88/1.63 % (19348)------------------------------
% 2.88/1.63 % (19351)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18998 for (2999ds/5Mi)
% 2.88/1.63 % (19347)Refutation found. Thanks to Tanya!
% 2.88/1.63 % SZS status Theorem for DTF2THF_18998
% 2.88/1.63 % SZS output start Proof for DTF2THF_18998
% 2.88/1.63 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.88/1.63 thf(type_def_7, type, coinductive_llist: ($tType) > $tType).
% 2.88/1.63 thf(type_def_8, type, set: ($tType) > $tType).
% 2.88/1.63 thf(type_def_9, type, nat: $tType).
% 2.88/1.63 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.88/1.63 thf(func_def_1, type, set: ($tType) > $tType).
% 2.88/1.63 thf(func_def_2, type, nat: $tType).
% 2.88/1.63 thf(func_def_3, type, itself: ($tType) > $tType).
% 2.88/1.63 thf(func_def_4, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_5, type, dvd: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_6, type, one: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_7, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_8, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_9, type, semiring_gcd: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_10, type, no_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_11, type, no_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_12, type, monoid_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_13, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_14, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_15, type, wellorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_16, type, semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_17, type, dense_order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_18, type, comm_semiring_1: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_19, type, linordered_idom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_20, type, comm_monoid_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_21, type, algebraic_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_22, type, ab_semigroup_mult: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_23, type, dense_linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_24, type, linordered_semidom: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.88/1.63 thf(func_def_25, type, coindu351974385stinct: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.63 thf(func_def_26, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.63 thf(func_def_27, type, coinductive_LNil: !>[X0: $tType]:(coinductive_llist @ X0)).
% 2.88/1.63 thf(func_def_28, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.63 thf(func_def_29, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.88/1.63 thf(func_def_30, type, coinductive_lmember: !>[X0: $tType]:(X0 > coinductive_llist @ X0 > $o)).
% 2.88/1.63 thf(func_def_31, type, coinductive_lsetp: !>[X0: $tType]:(coinductive_llist @ X0 > X0 > $o)).
% 2.88/1.63 thf(func_def_32, type, coindu63249387sorted: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.88/1.63 thf(func_def_33, type, one_one: !>[X0: $tType]:(X0)).
% 2.88/1.63 thf(func_def_34, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.88/1.63 thf(func_def_35, type, hammin2015774020amming: coinductive_llist @ nat).
% 2.88/1.63 thf(func_def_36, type, hammin1328233080lmerge: !>[X0: $tType]:((X0 > X0 > $o) > coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.63 thf(func_def_37, type, hammin1138290136lmerge: !>[X0: $tType]:(coinductive_llist @ X0 > coinductive_llist @ X0 > coinductive_llist @ X0)).
% 2.88/1.63 thf(func_def_38, type, hammin1195055315smooth: nat > $o).
% 2.88/1.63 thf(func_def_39, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.88/1.63 thf(func_def_40, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.88/1.63 thf(func_def_41, type, prime: nat > $o).
% 2.88/1.63 thf(func_def_42, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.88/1.63 thf(func_def_43, type, dvd_dvd: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.88/1.63 thf(func_def_44, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.88/1.63 thf(func_def_45, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.88/1.63 thf(func_def_46, type, n: nat).
% 2.88/1.63 thf(func_def_47, type, n2: nat).
% 2.88/1.63 thf(func_def_48, type, na: nat).
% 2.88/1.63 thf(func_def_49, type, p: nat).
% 2.88/1.63 thf(f851,plain,(
% 2.88/1.63 $false),
% 2.88/1.63 inference(subsumption_resolution,[status(thm)],[f850,f833])).
% 2.88/1.63 thf(f833,plain,(
% 2.88/1.63 ((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) != $true)),
% 2.88/1.63 inference(cnf_transformation,[status(thm)],[f823])).
% 2.88/1.63 thf(f823,plain,(
% 2.88/1.63 ((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) != $true)),
% 2.88/1.63 inference(flattening,[status(thm)],[f422])).
% 2.88/1.63 thf(f422,plain,(
% 2.88/1.63 ~((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true)),
% 2.88/1.63 inference(fool_elimination,[status(thm)],[f421])).
% 2.88/1.63 thf(f421,plain,(
% 2.88/1.63 ~(member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming))),
% 2.88/1.63 inference(rectify,[status(thm)],[f281])).
% 2.88/1.63 thf(f281,negated_conjecture,(
% 2.88/1.63 ~(member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming))),
% 2.88/1.63 inference(negated_conjecture,[status(cth)],[f280])).
% 2.88/1.63 thf(f280,conjecture,(
% 2.88/1.63 (member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming))),
% 2.88/1.63 file('/export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2THF_18998.p',conj_0)).
% 2.88/1.63 thf(f850,plain,(
% 2.88/1.63 ((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true)),
% 2.88/1.63 inference(subsumption_resolution,[status(thm)],[f849,f837])).
% 2.88/1.63 thf(f837,plain,(
% 2.88/1.63 ((hammin1195055315smooth @ n2) = $true)),
% 2.88/1.63 inference(cnf_transformation,[status(thm)],[f356])).
% 2.88/1.63 thf(f356,plain,(
% 2.88/1.63 ((hammin1195055315smooth @ n2) = $true)),
% 2.88/1.63 inference(fool_elimination,[status(thm)],[f355])).
% 2.88/1.63 thf(f355,plain,(
% 2.88/1.63 (hammin1195055315smooth @ n2)),
% 2.88/1.63 inference(rectify,[status(thm)],[f1])).
% 2.88/1.63 thf(f1,axiom,(
% 2.88/1.63 (hammin1195055315smooth @ n2)),
% 2.88/1.63 file('/export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2THF_18998.p',fact_0__092_060open_062smooth_An_H_092_060close_062)).
% 2.88/1.63 thf(f849,plain,(
% 2.88/1.63 ((hammin1195055315smooth @ n2) != $true) | ((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true)),
% 2.88/1.63 inference(trivial_inequality_removal,[status(thm)],[f848])).
% 2.88/1.63 thf(f848,plain,(
% 2.88/1.63 ($true != $true) | ((member @ nat @ n2 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true) | ((hammin1195055315smooth @ n2) != $true)),
% 2.88/1.63 inference(superposition,[status(thm)],[f846,f847])).
% 2.88/1.63 thf(f847,plain,(
% 2.88/1.63 ((ord_less @ nat @ n2 @ (times_times @ nat @ p @ n2)) = $true)),
% 2.88/1.63 inference(definition_unfolding,[status(thm)],[f839,f832])).
% 2.88/1.63 thf(f832,plain,(
% 2.88/1.63 (na = (times_times @ nat @ p @ n2))),
% 2.88/1.63 inference(cnf_transformation,[status(thm)],[f4])).
% 2.88/1.63 thf(f4,axiom,(
% 2.88/1.63 (na = (times_times @ nat @ p @ n2))),
% 2.88/1.63 file('/export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2THF_18998.p',fact_3_n)).
% 2.88/1.63 thf(f839,plain,(
% 2.88/1.63 ((ord_less @ nat @ n2 @ na) = $true)),
% 2.88/1.63 inference(cnf_transformation,[status(thm)],[f682])).
% 2.88/1.63 thf(f682,plain,(
% 2.88/1.63 ((ord_less @ nat @ n2 @ na) = $true)),
% 2.88/1.63 inference(fool_elimination,[status(thm)],[f681])).
% 2.88/1.63 thf(f681,plain,(
% 2.88/1.63 (ord_less @ nat @ n2 @ na)),
% 2.88/1.63 inference(rectify,[status(thm)],[f2])).
% 2.88/1.63 thf(f2,axiom,(
% 2.88/1.63 (ord_less @ nat @ n2 @ na)),
% 2.88/1.63 file('/export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2THF_18998.p',fact_1__092_060open_062n_H_A_060_An_092_060close_062)).
% 2.88/1.63 thf(f846,plain,(
% 2.88/1.63 ( ! [X0 : nat] : (((ord_less @ nat @ X0 @ (times_times @ nat @ p @ n2)) != $true) | ((member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true) | ((hammin1195055315smooth @ X0) != $true)) )),
% 2.88/1.63 inference(definition_unfolding,[status(thm)],[f835,f832])).
% 2.88/1.63 thf(f835,plain,(
% 2.88/1.63 ( ! [X0 : nat] : (((ord_less @ nat @ X0 @ na) != $true) | ((hammin1195055315smooth @ X0) != $true) | ((member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true)) )),
% 2.88/1.63 inference(cnf_transformation,[status(thm)],[f827])).
% 2.88/1.63 thf(f827,plain,(
% 2.88/1.63 ! [X0 : nat] : (((ord_less @ nat @ X0 @ na) != $true) | ((hammin1195055315smooth @ X0) != $true) | ((member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true))),
% 2.88/1.63 inference(flattening,[status(thm)],[f826])).
% 2.88/1.63 thf(f826,plain,(
% 2.88/1.63 ! [X0 : nat] : ((((member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true) | ((hammin1195055315smooth @ X0) != $true)) | ((ord_less @ nat @ X0 @ na) != $true))),
% 2.88/1.63 inference(ennf_transformation,[status(thm)],[f294])).
% 2.88/1.63 thf(f294,plain,(
% 2.88/1.63 ! [X0 : nat] : (((ord_less @ nat @ X0 @ na) = $true) => (((hammin1195055315smooth @ X0) = $true) => ((member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming)) = $true)))),
% 2.88/1.63 inference(fool_elimination,[status(thm)],[f293])).
% 2.88/1.63 thf(f293,plain,(
% 2.88/1.63 ! [X0 : nat] : ((ord_less @ nat @ X0 @ na) => ((hammin1195055315smooth @ X0) => (member @ nat @ X0 @ (coinductive_lset @ nat @ hammin2015774020amming))))),
% 2.88/1.63 inference(rectify,[status(thm)],[f3])).
% 2.88/1.63 thf(f3,axiom,(
% 2.88/1.63 ! [X1 : nat] : ((ord_less @ nat @ X1 @ na) => ((hammin1195055315smooth @ X1) => (member @ nat @ X1 @ (coinductive_lset @ nat @ hammin2015774020amming))))),
% 2.88/1.63 file('/export/starexec/sandbox/tmp/tmp.y93aJnwIWK/DTF2THF_18998.p',fact_2_less_OIH)).
% 2.88/1.63 % SZS output end Proof for DTF2THF_18998
% 2.88/1.63 % (19347)------------------------------
% 2.88/1.63 % (19347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.63 % (19347)Termination reason: Refutation
% 2.88/1.63
% 2.88/1.63 % (19347)Memory used [KB]: 6268
% 2.88/1.63 % (19347)Time elapsed: 0.014 s
% 2.88/1.63 % (19347)Instructions burned: 33 (million)
% 2.88/1.63 % (19347)------------------------------
% 2.88/1.63 % (19347)------------------------------
% 2.88/1.63 % (19326)Success in time 0.086 s
%------------------------------------------------------------------------------