%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP113^2 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:06:48 AM UTC 2026
% Result : Theorem 85.98s 63.41s
% Output : Refutation 86.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 181
% Syntax : Number of formulae : 364 ( 140 unt; 0 typ; 0 def)
% Number of atoms : 1234 ( 202 equ; 2 cnn)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 4500 ( 118 ~; 14 |; 104 &;3806 @)
% ( 0 <=>; 458 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 116 ( 116 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 36 usr; 9 con; 0-5 aty)
% Number of variables : 1230 ( 50 ^;1108 !; 72 ?;1230 :)
% Comments :
%------------------------------------------------------------------------------
thf(extended_ereal_type,type,
extended_ereal: $tType ).
thf(real_type,type,
real: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(a_type,type,
a: $tType ).
thf(euclid925273238_space_decl,type,
euclid925273238_space:
!>[TA: $tType] : $o ).
thf(zero_decl,type,
zero:
!>[TA: $tType] : $o ).
thf(top_decl,type,
top:
!>[TA: $tType] : $o ).
thf(no_bot_decl,type,
no_bot:
!>[TA: $tType] : $o ).
thf(no_top_decl,type,
no_top:
!>[TA: $tType] : $o ).
thf(order_top_decl,type,
order_top:
!>[TA: $tType] : $o ).
thf(dense_order_decl,type,
dense_order:
!>[TA: $tType] : $o ).
thf(linordered_field_decl,type,
linordered_field:
!>[TA: $tType] : $o ).
thf(dense_linorder_decl,type,
dense_linorder:
!>[TA: $tType] : $o ).
thf(real_V2090557954_space_decl,type,
real_V2090557954_space:
!>[TA: $tType] : $o ).
thf(topolo890362671_space_decl,type,
topolo890362671_space:
!>[TA: $tType] : $o ).
thf(real_V1759440057_space_decl,type,
real_V1759440057_space:
!>[TA: $tType] : $o ).
thf(linord1659791738miring_decl,type,
linord1659791738miring:
!>[TA: $tType] : $o ).
thf(real_V55928688vector_decl,type,
real_V55928688vector:
!>[TA: $tType] : $o ).
thf(condit1656338222tinuum_decl,type,
condit1656338222tinuum:
!>[TA: $tType] : $o ).
thf(condit1037483654norder_decl,type,
condit1037483654norder:
!>[TA: $tType] : $o ).
thf(elementary_closure_decl,type,
elementary_closure:
!>[TA: $tType] : ( ( set @ TA ) > ( set @ TA ) ) ).
thf(zero_zero_decl,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(lower_261390618igraph_decl,type,
lower_261390618igraph:
!>[TA: $tType] : ( ( set @ TA ) > ( TA > extended_ereal ) > ( set @ ( product_prod @ TA @ real ) ) ) ).
thf(ord_less_decl,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_decl,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(top_top_decl,type,
top_top:
!>[TA: $tType] : TA ).
thf(product_Pair_decl,type,
product_Pair:
!>[TA: $tType,TB: $tType] : ( TB > TA > ( product_prod @ TB @ TA ) ) ).
thf(product_rec_prod_decl,type,
product_rec_prod:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( TC > TB > TA ) > ( product_prod @ TC @ TB ) > TA ) ).
thf(real_V2000881966t_dist_decl,type,
real_V2000881966t_dist:
!>[TA: $tType] : ( TA > TA > real ) ).
thf(collect_decl,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(unifor760693226chy_on_decl,type,
unifor760693226chy_on:
!>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( nat > TB > TA ) > $o ) ).
thf(member_decl,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(e_decl,type,
e: real ).
thf(f_decl,type,
f: a > extended_ereal ).
thf(thesis_decl,type,
thesis: $o ).
thf(x_decl,type,
x: a ).
thf(y_decl,type,
y: real ).
thf(z_decl,type,
z: real ).
thf(141,axiom,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( A
!= ( top_top @ TA ) )
= ( ord_less @ TA @ A @ ( top_top @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_top_Onot__eq__extremum) ).
thf(760,plain,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( A
!= ( top_top @ TA ) )
= ( ord_less @ TA @ A @ ( top_top @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(72,axiom,
! [TA: $tType,TB: $tType] :
( ( ( zero @ TA )
& ( zero @ TB ) )
=> ( ( zero_zero @ ( product_prod @ TB @ TA ) )
= ( product_Pair @ TB @ TA @ ( zero_zero @ TB ) @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_zero__prod__def) ).
thf(479,plain,
! [TA: $tType,TB: $tType] :
( ( ( zero @ TA )
& ( zero @ TB ) )
=> ( ( zero_zero @ ( product_prod @ TB @ TA ) )
= ( product_Pair @ TB @ TA @ ( zero_zero @ TB ) @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(40,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_nat__le__linear) ).
thf(347,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(69,axiom,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ ( top_top @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_top__greatest) ).
thf(468,plain,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ ( top_top @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(148,axiom,
dense_linorder @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Orderings_Odense__linorder) ).
thf(777,plain,
dense_linorder @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(42,axiom,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_less__or__eq__imp__le) ).
thf(352,plain,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(39,axiom,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ ( zero_zero @ nat ) )
=> ? [C: nat] :
( ( ord_less_eq @ nat @ C @ B )
& ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_ex__least__nat__le) ).
thf(333,plain,
! [A: nat > $o,B: nat] :
( ( A @ B )
=> ( ~ ( A @ ( zero_zero @ nat ) )
=> ? [C: nat] :
( ( ord_less_eq @ nat @ C @ B )
& ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ~ ( A @ D ) )
& ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(21,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
( ( A @ B )
=> ? [D: TA] :
( ( A @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_ex__has__least__nat) ).
thf(249,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
( ( A @ B )
=> ? [D: TA] :
( ( A @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(114,axiom,
! [TA: $tType,TB: $tType] :
( ( ( real_V1759440057_space @ TB )
& ( real_V1759440057_space @ TA ) )
=> ( real_V1759440057_space @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30) ).
thf(683,plain,
! [TA: $tType,TB: $tType] :
( ( ( real_V1759440057_space @ TB )
& ( real_V1759440057_space @ TA ) )
=> ( real_V1759440057_space @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(59,axiom,
top @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Otop_17) ).
thf(440,plain,
top @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(104,axiom,
no_bot @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Orderings_Ono__bot) ).
thf(655,plain,
no_bot @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(81,axiom,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( top_top @ TA ) @ A )
= ( A
= ( top_top @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_top_Oextremum__unique) ).
thf(533,plain,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( top_top @ TA ) @ A )
= ( A
= ( top_top @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(156,axiom,
condit1656338222tinuum @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum) ).
thf(797,plain,
condit1656338222tinuum @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(164,axiom,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_UNIV__I) ).
thf(814,plain,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(136,axiom,
! [TA: $tType] :
( ( ord_less @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_psubset__eq) ).
thf(747,plain,
! [TA: $tType] :
( ( ord_less @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(175,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
~ ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ ( zero_zero @ real ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_dist__not__less__zero) ).
thf(841,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
~ ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ ( zero_zero @ real ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(92,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_Collect__mono) ).
thf(565,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(103,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_prod__induct4) ).
thf(651,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(171,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_Pair__inject) ).
thf(829,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(49,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_le__0__eq) ).
thf(404,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(127,axiom,
condit1037483654norder @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19) ).
thf(724,plain,
condit1037483654norder @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(3,axiom,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_infinite__descent0__measure) ).
thf(184,plain,
! [TA: $tType,A: TA > nat,B: TA > $o,C: TA] :
( ! [D: TA] :
( ( ( A @ D )
= ( zero_zero @ nat ) )
=> ( B @ D ) )
=> ( ! [D: TA] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ D ) )
=> ( ~ ( B @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( A @ E ) @ ( A @ D ) )
& ~ ( B @ E ) ) ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(125,axiom,
( ( ord_less_eq @ real )
= ( ^ [A: real,B: real] :
( ( ord_less @ real @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_less__eq__real__def) ).
thf(719,plain,
( ( ord_less_eq @ real )
= ( ^ [A: real,B: real] :
( ( ord_less @ real @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(173,axiom,
! [TA: $tType] :
( ( condit1656338222tinuum @ TA )
=> ! [A: TA] :
? [B: TA] :
( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_ex__gt__or__lt) ).
thf(838,plain,
! [TA: $tType] :
( ( condit1656338222tinuum @ TA )
=> ! [A: TA] :
? [B: TA] :
( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(65,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_zero__reorient) ).
thf(460,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(89,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Groups_Ozero_11) ).
thf(559,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ( zero @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(179,axiom,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_gt__ex) ).
thf(853,plain,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(26,axiom,
linord1659791738miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).
thf(284,plain,
linord1659791738miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(35,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_less__imp__le__nat) ).
thf(325,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(28,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_le__eq__less__or__eq) ).
thf(289,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(62,axiom,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ! [A: set @ TA,B: nat > TA > TB] :
( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less @ nat @ F @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) )
=> ( unifor760693226chy_on @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_uniformly__Cauchy__onI_H) ).
thf(448,plain,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ! [A: set @ TA,B: nat > TA > TB] :
( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less @ nat @ F @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) )
=> ( unifor760693226chy_on @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(71,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_dense__ge) ).
thf(475,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ A @ C )
=> ( ord_less_eq @ TA @ B @ C ) )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(45,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_eq__imp__le) ).
thf(376,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(17,axiom,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_nat__neq__iff) ).
thf(238,plain,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(95,axiom,
! [TA: $tType,TB: $tType] :
( ( real_V1759440057_space @ TA )
=> ! [A: TB > $o,B: nat > TB > TA,C: TB > TA] :
( ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: nat,H: TB] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( ord_less_eq @ nat @ E @ G )
& ( A @ H ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ F @ H ) @ ( B @ G @ H ) ) @ D ) ) )
=> ( ! [D: TB] :
( ( A @ D )
=> ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [F: nat] :
! [G: nat] :
( ( ord_less_eq @ nat @ F @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ G @ D ) @ ( C @ D ) ) @ E ) ) ) )
=> ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: TB] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ F @ G ) @ ( C @ G ) ) @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_uniformly__cauchy__imp__uniformly__convergent) ).
thf(574,plain,
! [TA: $tType,TB: $tType] :
( ( real_V1759440057_space @ TA )
=> ! [A: TB > $o,B: nat > TB > TA,C: TB > TA] :
( ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: nat,H: TB] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( ord_less_eq @ nat @ E @ G )
& ( A @ H ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ F @ H ) @ ( B @ G @ H ) ) @ D ) ) )
=> ( ! [D: TB] :
( ( A @ D )
=> ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [F: nat] :
! [G: nat] :
( ( ord_less_eq @ nat @ F @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ G @ D ) @ ( C @ D ) ) @ E ) ) ) )
=> ! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: TB] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ ( B @ F @ G ) @ ( C @ G ) ) @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(161,axiom,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_lt__ex) ).
thf(806,plain,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(167,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_subset__psubset__trans) ).
thf(821,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(117,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_in__mono) ).
thf(692,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(131,axiom,
topolo890362671_space @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Topological__Spaces_Operfect__space) ).
thf(734,plain,
topolo890362671_space @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(47,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_gr0I) ).
thf(396,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(178,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_prod__cases) ).
thf(849,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(15,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_less__nat__zero__code) ).
thf(233,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(20,axiom,
no_top @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top) ).
thf(248,plain,
no_top @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(46,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_Nat_Oex__has__greatest__nat) ).
thf(380,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(107,axiom,
! [TA: $tType,TB: $tType] :
( ( ( euclid925273238_space @ TB )
& ( euclid925273238_space @ TA ) )
=> ( euclid925273238_space @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space) ).
thf(661,plain,
! [TA: $tType,TB: $tType] :
( ( ( euclid925273238_space @ TB )
& ( euclid925273238_space @ TA ) )
=> ( euclid925273238_space @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(86,axiom,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ! [A: set @ TA,B: nat > TA > TB] :
( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less_eq @ nat @ D @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) )
=> ( unifor760693226chy_on @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_uniformly__Cauchy__onI) ).
thf(546,plain,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ! [A: set @ TA,B: nat > TA > TB] :
( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less_eq @ nat @ D @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) )
=> ( unifor760693226chy_on @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(6,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_linorder__neqE__nat) ).
thf(201,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(11,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_bot__nat__0_Onot__eq__extremum) ).
thf(220,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(123,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_prod_Oinject) ).
thf(714,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(157,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( ord_less @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_psubset__trans) ).
thf(798,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( ord_less @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(169,axiom,
linord1659791738miring @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21) ).
thf(825,plain,
linord1659791738miring @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(149,axiom,
no_top @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Orderings_Ono__top_24) ).
thf(778,plain,
no_top @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(41,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_le__neq__implies__less) ).
thf(349,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(90,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_le__numeral__extra_I3_J) ).
thf(561,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(50,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ C @ D ) ) )
=> ( ord_less_eq @ TA @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_dense__ge__bounded) ).
thf(407,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ C @ D ) ) )
=> ( ord_less_eq @ TA @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(25,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_nat__less__induct) ).
thf(275,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(153,axiom,
! [TA: $tType,TB: $tType] :
( ( ( top @ TA )
& ( top @ TB ) )
=> ( ( top_top @ ( product_prod @ TB @ TA ) )
= ( product_Pair @ TB @ TA @ ( top_top @ TB ) @ ( top_top @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_top__prod__def) ).
thf(788,plain,
! [TA: $tType,TB: $tType] :
( ( ( top @ TA )
& ( top @ TB ) )
=> ( ( top_top @ ( product_prod @ TB @ TA ) )
= ( product_Pair @ TB @ TA @ ( top_top @ TB ) @ ( top_top @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(14,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_gr__implies__not0) ).
thf(229,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B
!= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(44,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
( ( A @ B )
=> ( ! [E: TA] :
( ( A @ E )
=> ( ord_less @ nat @ ( C @ E ) @ D ) )
=> ? [E: TA] :
( ( A @ E )
& ! [F: TA] :
( ( A @ F )
=> ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_Lattices__Big_Oex__has__greatest__nat) ).
thf(361,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
( ( A @ B )
=> ( ! [E: TA] :
( ( A @ E )
=> ( ord_less @ nat @ ( C @ E ) @ D ) )
=> ? [E: TA] :
( ( A @ E )
& ! [F: TA] :
( ( A @ F )
=> ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(140,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_old_Oprod_Oexhaust) ).
thf(757,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(122,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_prod__induct6) ).
thf(710,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(27,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_less__not__refl3) ).
thf(285,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(120,axiom,
! [A: set @ real] :
( ? [B: real] : ( member @ real @ B @ A )
=> ( ? [B: real] :
! [C: real] :
( ( member @ real @ C @ A )
=> ( ord_less_eq @ real @ C @ B ) )
=> ? [B: real] :
( ! [C: real] :
( ( member @ real @ C @ A )
=> ( ord_less_eq @ real @ C @ B ) )
& ! [C: real] :
( ! [D: real] :
( ( member @ real @ D @ A )
=> ( ord_less_eq @ real @ D @ C ) )
=> ( ord_less_eq @ real @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_complete__real) ).
thf(697,plain,
! [A: set @ real] :
( ? [B: real] : ( member @ real @ B @ A )
=> ( ? [B: real] :
! [C: real] :
( ( member @ real @ C @ A )
=> ( ord_less_eq @ real @ C @ B ) )
=> ? [B: real] :
( ! [C: real] :
( ( member @ real @ C @ A )
=> ( ord_less_eq @ real @ C @ B ) )
& ! [C: real] :
( ! [D: real] :
( ( member @ real @ D @ A )
=> ( ord_less_eq @ real @ D @ C ) )
=> ( ord_less_eq @ real @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(32,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_le__refl) ).
thf(317,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(128,axiom,
! [TA: $tType,TB: $tType] :
( ( ( top @ TB )
& ( top @ TA ) )
=> ( top @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Otop_36) ).
thf(725,plain,
! [TA: $tType,TB: $tType] :
( ( ( top @ TB )
& ( top @ TA ) )
=> ( top @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(151,axiom,
real_V2090557954_space @ a,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tfree_0) ).
thf(783,plain,
real_V2090557954_space @ a,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(22,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_bot__nat__0_Oextremum__uniqueI) ).
thf(256,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(75,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less_eq @ real @ ( real_V2000881966t_dist @ TA @ D @ A ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_closure__approachable__le) ).
thf(488,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less_eq @ real @ ( real_V2000881966t_dist @ TA @ D @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(110,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_prod__induct5) ).
thf(669,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(98,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_less__numeral__extra_I3_J) ).
thf(637,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(102,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA,C: real] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ D ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_closure__approachableD) ).
thf(647,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA,C: real] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ D ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(133,axiom,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ ( top_top @ TA ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_top_Oextremum__strict) ).
thf(738,plain,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ ( top_top @ TA ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(64,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA] :
( ( real_V2000881966t_dist @ TA @ A @ A )
= ( zero_zero @ real ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_dist__self) ).
thf(457,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA] :
( ( real_V2000881966t_dist @ TA @ A @ A )
= ( zero_zero @ real ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(126,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_dist__pos__lt) ).
thf(721,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(93,axiom,
! [TA: $tType,TB: $tType] :
( ( top @ TA )
=> ( top @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Otop) ).
thf(569,plain,
! [TA: $tType,TB: $tType] :
( ( top @ TA )
=> ( top @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(150,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_prod__induct7) ).
thf(779,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(19,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_less__not__refl2) ).
thf(244,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(31,axiom,
! [A: nat,B: nat > real,C: nat > real] :
( ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less @ real @ ( B @ D ) @ ( C @ D ) ) )
=> ( ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less_eq @ real @ ( C @ D ) @ ( C @ A ) ) )
=> ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less @ real @ ( B @ D ) @ ( C @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_seq__mono__lemma) ).
thf(311,plain,
! [A: nat,B: nat > real,C: nat > real] :
( ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less @ real @ ( B @ D ) @ ( C @ D ) ) )
=> ( ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less_eq @ real @ ( C @ D ) @ ( C @ A ) ) )
=> ! [D: nat] :
( ( ord_less_eq @ nat @ A @ D )
=> ( ord_less @ real @ ( B @ D ) @ ( C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(84,axiom,
! [TA: $tType] :
( ( top_top @ ( set @ TA ) )
= ( collect @ TA @ ( top_top @ ( TA > $o ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_top__set__def) ).
thf(541,plain,
! [TA: $tType] :
( ( top_top @ ( set @ TA ) )
= ( collect @ TA @ ( top_top @ ( TA > $o ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(144,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA,C: TA] :
( ( member @ TA @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( A = C )
= ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( ( real_V2000881966t_dist @ TA @ A @ D )
= ( real_V2000881966t_dist @ TA @ C @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_metric__eq__thm) ).
thf(765,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA,C: TA] :
( ( member @ TA @ A @ B )
=> ( ( member @ TA @ C @ B )
=> ( ( A = C )
= ( ! [D: TA] :
( ( member @ TA @ D @ B )
=> ( ( real_V2000881966t_dist @ TA @ A @ D )
= ( real_V2000881966t_dist @ TA @ C @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(155,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_prod__cases6) ).
thf(794,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(51,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_subset__antisym) ).
thf(412,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(137,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_psubset__subset__trans) ).
thf(750,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(116,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_subset__eq) ).
thf(689,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(61,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_equalityD2) ).
thf(444,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(166,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_subsetD) ).
thf(819,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(53,axiom,
! [TA: $tType,A: TA > $o,B: TA > real,C: TA > $o] :
( ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ( ord_less @ real @ ( B @ E ) @ D )
=> ( C @ E ) ) ) ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [E: TA] :
( ( ord_less_eq @ real @ ( B @ E ) @ D )
=> ( ( A @ E )
=> ( C @ E ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_approachable__lt__le2) ).
thf(423,plain,
! [TA: $tType,A: TA > $o,B: TA > real,C: TA > $o] :
( ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ( ord_less @ real @ ( B @ E ) @ D )
=> ( C @ E ) ) ) ) )
= ( ? [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
& ! [E: TA] :
( ( ord_less_eq @ real @ ( B @ E ) @ D )
=> ( ( A @ E )
=> ( C @ E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(118,axiom,
! [TA: $tType,TB: $tType] :
( ( ( real_V55928688vector @ TB )
& ( real_V55928688vector @ TA ) )
=> ( real_V55928688vector @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28) ).
thf(694,plain,
! [TA: $tType,TB: $tType] :
( ( ( real_V55928688vector @ TB )
& ( real_V55928688vector @ TA ) )
=> ( real_V55928688vector @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(176,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA,C: real] :
( ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ C )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ B @ A ) @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_dist__commute__lessI) ).
thf(843,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA,C: real] :
( ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ C )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ B @ A ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(146,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_prod__cases7) ).
thf(772,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(135,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ! [C: TA,D: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_subrelI) ).
thf(743,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ! [C: TA,D: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(170,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( A != B )
=> ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_psubsetI) ).
thf(826,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( A != B )
=> ( ord_less @ ( set @ TA ) @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(29,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_less__irrefl__nat) ).
thf(291,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(7,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_infinite__descent) ).
thf(204,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(23,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_not__less0) ).
thf(259,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(68,axiom,
! [TA: $tType,TB: $tType] :
( ( order_top @ TA )
=> ( order_top @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder__top) ).
thf(466,plain,
! [TA: $tType,TB: $tType] :
( ( order_top @ TA )
=> ( order_top @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(16,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_bot__nat__0_Oextremum) ).
thf(236,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(143,axiom,
real_V55928688vector @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector) ).
thf(764,plain,
real_V55928688vector @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(152,axiom,
? [A: product_prod @ a @ real] :
( ( member @ ( product_prod @ a @ real ) @ A @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ A @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062) ).
thf(784,plain,
? [A: product_prod @ a @ real] :
( ( member @ ( product_prod @ a @ real ) @ A @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ A @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(57,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).
thf(434,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(163,axiom,
! [TA: $tType,TB: $tType] :
( ( top @ TB )
=> ( ( top_top @ ( TA > TB ) )
= ( ^ [A: TA] : ( top_top @ TB ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_top__apply) ).
thf(811,plain,
! [TA: $tType,TB: $tType] :
( ( top @ TB )
=> ( ( top_top @ ( TA > TB ) )
= ( ^ [A: TA] : ( top_top @ TB ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(77,axiom,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( top_top @ TA ) @ A )
=> ( A
= ( top_top @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_top_Oextremum__uniqueI) ).
thf(494,plain,
! [TA: $tType] :
( ( order_top @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ ( top_top @ TA ) @ A )
=> ( A
= ( top_top @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(181,axiom,
euclid925273238_space @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1) ).
thf(858,plain,
euclid925273238_space @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(165,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ D @ A ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_closure__approachable) ).
thf(816,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: set @ TA] :
( ( member @ TA @ A @ ( elementary_closure @ TA @ B ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: TA] :
( ( member @ TA @ D @ B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ D @ A ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(8,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_le0) ).
thf(213,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(37,axiom,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_nat__less__le) ).
thf(330,plain,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(115,axiom,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_dense) ).
thf(685,plain,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(48,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_less__mono__imp__le__mono) ).
thf(399,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(145,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_old_Oprod_Oinducts) ).
thf(768,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(63,axiom,
! [TA: $tType] : ( order_top @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder__top_6) ).
thf(455,plain,
! [TA: $tType] : ( order_top @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(12,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_bot__nat__0_Oextremum__unique) ).
thf(223,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(91,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_subset__trans) ).
thf(563,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ C )
=> ( ord_less_eq @ ( set @ TA ) @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(13,axiom,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_not__gr0) ).
thf(226,plain,
! [A: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(55,axiom,
! [TA: $tType,TB: $tType] :
( ( ( real_V2090557954_space @ TB )
& ( real_V2090557954_space @ TA ) )
=> ( real_V2090557954_space @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32) ).
thf(431,plain,
! [TA: $tType,TB: $tType] :
( ( ( real_V2090557954_space @ TB )
& ( real_V2090557954_space @ TA ) )
=> ( real_V2090557954_space @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(18,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_less__zeroE) ).
thf(241,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(54,axiom,
! [TA: $tType] :
( ( ( real_V2090557954_space @ TA )
& ( topolo890362671_space @ TA ) )
=> ! [A: real,B: TA] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A )
=> ? [C: TA] :
( ( C != B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ C @ B ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_perfect__choose__dist) ).
thf(426,plain,
! [TA: $tType] :
( ( ( real_V2090557954_space @ TA )
& ( topolo890362671_space @ TA ) )
=> ! [A: real,B: TA] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A )
=> ? [C: TA] :
( ( C != B )
& ( ord_less @ real @ ( real_V2000881966t_dist @ TA @ C @ B ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(80,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order_top @ TB )
& ( order_top @ TA ) )
=> ( order_top @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Orderings_Oorder__top_33) ).
thf(531,plain,
! [TA: $tType,TB: $tType] :
( ( ( order_top @ TB )
& ( order_top @ TA ) )
=> ( order_top @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(113,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_subset__iff) ).
thf(680,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(108,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_subsetI) ).
thf(663,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(87,axiom,
! [TA: $tType] :
( ( ( real_V55928688vector @ TA )
& ( topolo890362671_space @ TA ) )
=> ! [A: real,B: TA] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A )
=> ~ ! [C: TA] :
( ( real_V2000881966t_dist @ TA @ B @ C )
!= A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_vector__choose__dist) ).
thf(553,plain,
! [TA: $tType] :
( ( ( real_V55928688vector @ TA )
& ( topolo890362671_space @ TA ) )
=> ! [A: real,B: TA] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A )
=> ~ ! [C: TA] :
( ( real_V2000881966t_dist @ TA @ B @ C )
!= A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(168,axiom,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_iso__tuple__UNIV__I) ).
thf(823,plain,
! [TA: $tType,A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(4,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).
thf(194,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(180,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_surj__pair) ).
thf(855,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(100,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_dense__le) ).
thf(640,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA] :
( ! [C: TA] :
( ( ord_less @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(121,axiom,
! [TA: $tType] :
( ( top_top @ ( TA > $o ) )
= ( ^ [A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_top__empty__eq) ).
thf(707,plain,
! [TA: $tType] :
( ( top_top @ ( TA > $o ) )
= ( ^ [A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(36,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_bot__nat__0_Oextremum__strict) ).
thf(327,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(111,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_prod__cases5) ).
thf(673,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(154,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_prod__cases3) ).
thf(791,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(60,axiom,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_set__eq__subset) ).
thf(441,plain,
! [TA: $tType] :
( ( (=) @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(124,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( top_top @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_subset__UNIV) ).
thf(717,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ ( top_top @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(79,axiom,
! [TA: $tType] :
( ( condit1037483654norder @ TA )
=> ! [A: TA,B: TA,C: TA > $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( C @ A )
=> ( ~ ( C @ B )
=> ? [D: TA] :
( ( ord_less_eq @ TA @ A @ D )
& ( ord_less_eq @ TA @ D @ B )
& ! [E: TA] :
( ( ( ord_less_eq @ TA @ A @ E )
& ( ord_less @ TA @ E @ D ) )
=> ( C @ E ) )
& ! [E: TA] :
( ! [F: TA] :
( ( ( ord_less_eq @ TA @ A @ F )
& ( ord_less @ TA @ F @ E ) )
=> ( C @ F ) )
=> ( ord_less_eq @ TA @ E @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_complete__interval) ).
thf(502,plain,
! [TA: $tType] :
( ( condit1037483654norder @ TA )
=> ! [A: TA,B: TA,C: TA > $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( C @ A )
=> ( ~ ( C @ B )
=> ? [D: TA] :
( ( ord_less_eq @ TA @ A @ D )
& ( ord_less_eq @ TA @ D @ B )
& ! [E: TA] :
( ( ( ord_less_eq @ TA @ A @ E )
& ( ord_less @ TA @ E @ D ) )
=> ( C @ E ) )
& ! [E: TA] :
( ! [F: TA] :
( ( ( ord_less_eq @ TA @ A @ F )
& ( ord_less @ TA @ F @ E ) )
=> ( C @ F ) )
=> ( ord_less_eq @ TA @ E @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(174,axiom,
ord_less @ real @ ( zero_zero @ real ) @ e,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__092_060open_0620_A_060_Ae_092_060close_062) ).
thf(840,plain,
ord_less @ real @ ( zero_zero @ real ) @ e,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(66,axiom,
! [TA: $tType,A: TA] : ( top_top @ ( TA > $o ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_top1I) ).
thf(463,plain,
! [TA: $tType,A: TA] : ( top_top @ ( TA > $o ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(70,axiom,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ D @ C ) ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_dense__le__bounded) ).
thf(470,plain,
! [TA: $tType] :
( ( dense_linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ! [D: TA] :
( ( ord_less @ TA @ A @ D )
=> ( ( ord_less @ TA @ D @ B )
=> ( ord_less_eq @ TA @ D @ C ) ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(162,axiom,
! [TA: $tType] :
( ( ord_less @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_subset__not__subset__eq) ).
thf(808,plain,
! [TA: $tType] :
( ( ord_less @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
& ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(85,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_zero__le__dist) ).
thf(544,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(105,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).
thf(656,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(129,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_psubsetE) ).
thf(727,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ( ( unifor760693226chy_on @ TA @ TB )
= ( ^ [A: set @ TA,B: nat > TA > TB] :
! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less_eq @ nat @ D @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_uniformly__Cauchy__on__def) ).
thf(571,plain,
! [TA: $tType,TB: $tType] :
( ( real_V2090557954_space @ TB )
=> ( ( unifor760693226chy_on @ TA @ TB )
= ( ^ [A: set @ TA,B: nat > TA > TB] :
! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: TA] :
( ( member @ TA @ E @ A )
=> ! [F: nat] :
( ( ord_less_eq @ nat @ D @ F )
=> ! [G: nat] :
( ( ord_less_eq @ nat @ D @ G )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ E ) @ ( B @ G @ E ) ) @ C ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(78,axiom,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
& ( ord_less @ TA @ C @ A )
& ( ord_less @ TA @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_field__lbound__gt__zero) ).
thf(497,plain,
! [TA: $tType] :
( ( linordered_field @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less @ TA @ ( zero_zero @ TA ) @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ C )
& ( ord_less @ TA @ C @ A )
& ( ord_less @ TA @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(182,axiom,
! [A: a,B: real] :
( ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A @ B ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A @ B ) @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) )
=> thesis ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(859,plain,
! [A: a,B: real] :
( ( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A @ B ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) )
& ( ord_less @ real @ ( real_V2000881966t_dist @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ A @ B ) @ ( product_Pair @ a @ real @ x @ y ) ) @ e ) )
=> thesis ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(177,axiom,
( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ x @ y ) @ ( elementary_closure @ ( product_prod @ a @ real ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) ) )
& ( ord_less_eq @ real @ y @ z ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_xy) ).
thf(845,plain,
( ( member @ ( product_prod @ a @ real ) @ ( product_Pair @ a @ real @ x @ y ) @ ( elementary_closure @ ( product_prod @ a @ real ) @ ( lower_261390618igraph @ a @ ( top_top @ ( set @ a ) ) @ f ) ) )
& ( ord_less_eq @ real @ y @ z ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(67,axiom,
order_top @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder__top_13) ).
thf(465,plain,
order_top @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(88,axiom,
! [TA: $tType,TB: $tType] :
( ( real_V1759440057_space @ TB )
=> ! [A: TA > $o,B: nat > TA > TB] :
( ( ? [C: TA > TB] :
! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: TA] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ G ) @ ( C @ G ) ) @ D ) ) ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: nat,F: nat,G: TA] :
( ( ( ord_less_eq @ nat @ D @ E )
& ( ord_less_eq @ nat @ D @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ E @ G ) @ ( B @ F @ G ) ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_uniformly__convergent__eq__cauchy) ).
thf(556,plain,
! [TA: $tType,TB: $tType] :
( ( real_V1759440057_space @ TB )
=> ! [A: TA > $o,B: nat > TA > TB] :
( ( ? [C: TA > TB] :
! [D: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D )
=> ? [E: nat] :
! [F: nat,G: TA] :
( ( ( ord_less_eq @ nat @ E @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ F @ G ) @ ( C @ G ) ) @ D ) ) ) )
= ( ! [C: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C )
=> ? [D: nat] :
! [E: nat,F: nat,G: TA] :
( ( ( ord_less_eq @ nat @ D @ E )
& ( ord_less_eq @ nat @ D @ F )
& ( A @ G ) )
=> ( ord_less @ real @ ( real_V2000881966t_dist @ TB @ ( B @ E @ G ) @ ( B @ F @ G ) ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(83,axiom,
! [TA: $tType] : ( top @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Otop_9) ).
thf(539,plain,
! [TA: $tType] : ( top @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(73,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_equalityD1) ).
thf(482,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(10,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_less__eq__nat_Osimps_I1_J) ).
thf(218,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(33,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_neq0__conv) ).
thf(319,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(96,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ ( zero_zero @ real ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_dist__le__zero__iff) ).
thf(632,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ real @ ( real_V2000881966t_dist @ TA @ A @ B ) @ ( zero_zero @ real ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(130,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) )
= ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_zero__less__dist__iff) ).
thf(731,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V2000881966t_dist @ TA @ A @ B ) )
= ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(56,axiom,
zero @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Groups_Ozero_27) ).
thf(433,plain,
zero @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(109,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: real,C: TA > extended_ereal,D: real] :
( ( ( member @ ( product_prod @ TA @ real ) @ ( product_Pair @ TA @ real @ A @ B ) @ ( lower_261390618igraph @ TA @ ( top_top @ ( set @ TA ) ) @ C ) )
& ( ord_less_eq @ real @ B @ D ) )
=> ( member @ ( product_prod @ TA @ real ) @ ( product_Pair @ TA @ real @ A @ D ) @ ( lower_261390618igraph @ TA @ ( top_top @ ( set @ TA ) ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_epigraph__mono) ).
thf(667,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: real,C: TA > extended_ereal,D: real] :
( ( ( member @ ( product_prod @ TA @ real ) @ ( product_Pair @ TA @ real @ A @ B ) @ ( lower_261390618igraph @ TA @ ( top_top @ ( set @ TA ) ) @ C ) )
& ( ord_less_eq @ real @ B @ D ) )
=> ( member @ ( product_prod @ TA @ real ) @ ( product_Pair @ TA @ real @ A @ D ) @ ( lower_261390618igraph @ TA @ ( top_top @ ( set @ TA ) ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(142,axiom,
real_V1759440057_space @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space) ).
thf(763,plain,
real_V1759440057_space @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(82,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ( real_V2000881966t_dist @ TA @ A @ B )
= ( zero_zero @ real ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_dist__eq__0__iff) ).
thf(536,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ! [A: TA,B: TA] :
( ( ( real_V2000881966t_dist @ TA @ A @ B )
= ( zero_zero @ real ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(158,axiom,
linordered_field @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Fields_Olinordered__field) ).
thf(800,plain,
linordered_field @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(138,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_prod__cases4) ).
thf(752,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(1,conjecture,
thesis,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_1) ).
thf(2,negated_conjecture,
~ thesis,
inference(neg_conjecture,[status(cth)],[1]) ).
thf(183,plain,
~ thesis,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(132,axiom,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_subset__iff__psubset__eq) ).
thf(735,plain,
! [TA: $tType] :
( ( ord_less_eq @ ( set @ TA ) )
= ( ^ [A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(172,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_old_Oprod_Orec) ).
thf(835,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(24,axiom,
! [A: nat,B: nat > $o,C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ A @ D )
=> ( B @ D ) )
=> ( ! [D: nat] :
( ( ord_less_eq @ nat @ D @ A )
=> ( ! [E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( B @ E ) )
=> ( B @ D ) ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_nat__descend__induct) ).
thf(262,plain,
! [A: nat,B: nat > $o,C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ A @ D )
=> ( B @ D ) )
=> ( ! [D: nat] :
( ( ord_less_eq @ nat @ D @ A )
=> ( ! [E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( B @ E ) )
=> ( B @ D ) ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(43,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_less__not__refl) ).
thf(358,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(5,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_infinite__descent__measure) ).
thf(195,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(58,axiom,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ( ( real_V2000881966t_dist @ TA )
= ( ^ [A: TA,B: TA] : ( real_V2000881966t_dist @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_dist__commute) ).
thf(437,plain,
! [TA: $tType] :
( ( real_V2090557954_space @ TA )
=> ( ( real_V2000881966t_dist @ TA )
= ( ^ [A: TA,B: TA] : ( real_V2000881966t_dist @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(97,axiom,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_subset__refl) ).
thf(635,plain,
! [TA: $tType,A: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(119,axiom,
dense_order @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Orderings_Odense__order) ).
thf(696,plain,
dense_order @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(52,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_equalityE) ).
thf(415,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( A = B )
=> ~ ( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ~ ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(30,axiom,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_infinite__descent0) ).
thf(294,plain,
! [A: nat > $o,B: nat] :
( ( A @ ( zero_zero @ nat ) )
=> ( ! [C: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C )
=> ( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(159,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_psubsetD) ).
thf(801,plain,
! [TA: $tType,A: set @ TA,B: set @ TA,C: TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(134,axiom,
! [TA: $tType,A: set @ TA] :
( ! [B: TA] : ( member @ TA @ B @ A )
=> ( ( top_top @ ( set @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_UNIV__eq__I) ).
thf(740,plain,
! [TA: $tType,A: set @ TA] :
( ! [B: TA] : ( member @ TA @ B @ A )
=> ( ( top_top @ ( set @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(38,axiom,
condit1037483654norder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder) ).
thf(332,plain,
condit1037483654norder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(101,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).
thf(644,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(34,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_le__trans) ).
thf(322,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(74,axiom,
! [TA: $tType,TB: $tType] :
( ( ( euclid925273238_space @ TB )
& ( euclid925273238_space @ TA ) )
=> ( topolo890362671_space @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31) ).
thf(486,plain,
! [TA: $tType,TB: $tType] :
( ( ( euclid925273238_space @ TB )
& ( euclid925273238_space @ TA ) )
=> ( topolo890362671_space @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(160,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_old_Oprod_Oinject) ).
thf(803,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(106,axiom,
! [TA: $tType,TB: $tType] :
( ( ( zero @ TB )
& ( zero @ TA ) )
=> ( zero @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Groups_Ozero_38) ).
thf(659,plain,
! [TA: $tType,TB: $tType] :
( ( ( zero @ TB )
& ( zero @ TA ) )
=> ( zero @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(9,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_le__antisym) ).
thf(215,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(147,axiom,
! [TA: $tType] :
? [A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_UNIV__witness) ).
thf(775,plain,
! [TA: $tType] :
? [A: TA] : ( member @ TA @ A @ ( top_top @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(99,axiom,
real_V2090557954_space @ real,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space) ).
thf(639,plain,
real_V2090557954_space @ real,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(76,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_Collect__mono__iff) ).
thf(491,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ord_less_eq @ ( set @ TA ) @ ( collect @ TA @ A ) @ ( collect @ TA @ B ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(139,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_psubset__imp__subset) ).
thf(755,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less @ ( set @ TA ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(112,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_prod__induct3) ).
thf(676,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(865,plain,
$false,
inference(e,[status(thm)],[760,479,347,468,777,352,333,249,683,440,655,533,797,814,747,841,565,651,829,404,724,184,719,838,460,559,853,284,325,289,448,475,376,238,574,806,821,692,734,396,849,233,248,380,661,546,201,220,714,798,825,778,349,561,407,275,788,229,361,757,710,285,697,317,725,783,256,488,669,637,647,738,457,721,569,779,244,311,541,765,794,412,750,689,444,819,423,694,843,772,743,826,291,204,259,466,236,764,784,434,811,494,858,816,213,330,685,399,768,455,223,563,226,431,241,426,531,680,663,553,823,194,855,640,707,327,673,791,441,717,502,840,463,470,808,544,656,727,571,497,859,845,465,556,539,482,218,319,632,731,433,667,763,536,800,752,183,735,835,262,358,195,437,635,696,415,294,801,740,332,644,322,486,803,659,215,775,639,491,755,676]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ITP113^2 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.10/0.23 % Computer : n016.cluster.edu
% 0.10/0.23 % Model : x86_64 x86_64
% 0.10/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.23 % Memory : 8046.5625MB
% 0.10/0.23 % OS : Linux 6.8.0-71-generic
% 0.10/0.23 % CPULimit : 300
% 0.10/0.23 % WCLimit : 300
% 0.10/0.23 % DateTime : Tue Sep 29 20:00:19 UTC 2026
% 0.10/0.23 % CPUTime :
% 0.10/0.23 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 1.03/0.83 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 2.31/1.24 % [INFO] Parsing done (406ms).
% 2.31/1.25 % [INFO] Running in sequential loop mode.
% 3.80/1.64 % [INFO] eprover registered as external prover.
% 3.80/1.65 % [INFO] Scanning for conjecture ...
% 4.04/1.79 % [INFO] Found a conjecture (or negated_conjecture) and 319 axioms. Running axiom selection ...
% 4.56/1.94 % [INFO] Axiom selection finished. Selected 180 axioms (removed 139 axioms).
% 5.60/2.23 % [INFO] Problem is higher-order (TPTP THF).
% 5.60/2.26 % [INFO] Type checking passed.
% 5.60/2.27 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 85.98/63.41 % External prover 'e' found a proof!
% 85.98/63.41 % [INFO] Killing All external provers ...
% 85.98/63.41 % Time passed: 63031ms (effective reasoning time: 62151ms)
% 85.98/63.41 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 85.98/63.41 % Axioms used in derivation (180): fact_181_equalityD2, tcon_Real_Oreal___Orderings_Ono__top_24, fact_150_closure__approachableD, fact_180_equalityD1, tcon_Product__Type_Oprod___Groups_Ozero_38, fact_233_linorder__neqE__nat, tcon_Set_Oset___Orderings_Oorder__top_6, tfree_0, tcon_Nat_Onat___Orderings_Ono__top, fact_184_subset__refl, fact_250_complete__interval, fact_189_psubset__imp__subset, fact_213_bot__nat__0_Oextremum__uniqueI, fact_193_subset__iff__psubset__eq, fact_186_subset__trans, fact_90_top__greatest, fact_151_closure__approachable, fact_215_less__mono__imp__le__mono, fact_208_infinite__descent0__measure, tcon_Product__Type_Oprod___Real__Vector__Spaces_Oreal__normed__vector_28, fact_10_top_Oextremum__strict, fact_128_old_Oprod_Oinducts, tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder, fact_200_bot__nat__0_Onot__eq__extremum, fact_209_psubset__trans, fact_154_vector__choose__dist, fact_2_xy, fact_18_dist__le__zero__iff, fact_216_le__neq__implies__less, tcon_Product__Type_Oprod___Orderings_Otop_36, tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19, fact_179_subset__eq, fact_217_Nat_Oex__has__greatest__nat, fact_136_prod__cases6, fact_16_dist__eq__0__iff, tcon_Real_Oreal___Topological__Spaces_Operfect__space, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_178_equalityE, fact_244_ex__has__least__nat, fact_132_prod__induct5, fact_131_prod__induct6, fact_246_zero__prod__def, fact_224_le__antisym, fact_143_UNIV__witness, fact_6_old_Oprod_Oinject, fact_227_le__refl, fact_174_subset__antisym, fact_175_in__mono, fact_5_prod_Oinject, fact_141_prod__cases, fact_0__092_060open_062_092_060exists_062ya_092_060in_062Epigraph_AUNIV_Af_O_Adist_Aya_A_Ix_M_Ay_J_A_060_Ae_092_060close_062, tcon_Set_Oset___Orderings_Otop_9, fact_196_le0, fact_133_prod__induct4, fact_207_bot__nat__0_Oextremum__strict, fact_46_Collect__mem__eq, fact_237_less__not__refl3, fact_70_dense__le, fact_92_dist__pos__lt, fact_88_top_Oextremum__uniqueI, tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space, fact_235_nat__less__induct, fact_142_surj__pair, tcon_Product__Type_Oprod___Euclidean__Space_Oeuclidean__space, fact_17_dist__self, fact_230_uniformly__Cauchy__onI, tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector, fact_247_complete__real, tcon_Product__Type_Oprod___Real__Vector__Spaces_Ometric__space_32, fact_183_subset__iff, fact_172_subsetI, fact_187_set__eq__subset, fact_153_perfect__choose__dist, tcon_Nat_Onat___Groups_Ozero, fact_162_subset__UNIV, fact_221_ex__least__nat__le, fact_228_le__0__eq, fact_118_dense, fact_1__092_060open_0620_A_060_Ae_092_060close_062, fact_152_old_Oprod_Orec, fact_45_mem__Collect__eq, fact_138_prod__cases4, fact_129_old_Oprod_Oexhaust, fact_241_uniformly__Cauchy__onI_H, fact_7_top__apply, fact_219_le__eq__less__or__eq, tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum, tcon_Product__Type_Oprod___Real__Vector__Spaces_Ocomplete__space_30, fact_222_nat__le__linear, fact_137_prod__cases5, fact_51_zero__le__dist, tcon_HOL_Obool___Orderings_Oorder__top_13, fact_182_psubset__eq, fact_9_top__prod__def, tcon_Product__Type_Oprod___Orderings_Oorder__top_33, tcon_Real_Oreal___Orderings_Odense__order, fact_232_infinite__descent__measure, tcon_Real_Oreal___Orderings_Odense__linorder, fact_190_psubset__subset__trans, fact_155_approachable__lt__le2, fact_252_le__numeral__extra_I3_J, tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_21, fact_52_top__set__def, tcon_Real_Oreal___Orderings_Ono__bot, fact_59_dense__ge__bounded, fact_218_less__or__eq__imp__le, fact_47_Collect__cong, fact_123_lt__ex, fact_139_prod__cases3, fact_140_Pair__inject, fact_144_UNIV__eq__I, fact_147_closure__approachable__le, tcon_fun___Orderings_Otop, fact_245_less__eq__real__def, fact_214_bot__nat__0_Oextremum__unique, fact_243_Lattices__Big_Oex__has__greatest__nat, fact_199_less__nat__zero__code, fact_202_not__gr0, fact_4_iso__tuple__UNIV__I, fact_19_zero__less__dist__iff, fact_58_dense__le__bounded, fact_135_prod__cases7, fact_170_uniformly__cauchy__imp__uniformly__convergent, tcon_Real_Oreal___Groups_Ozero_27, fact_188_Collect__mono__iff, fact_197_bot__nat__0_Oextremum, fact_220_less__imp__le__nat, fact_173_psubsetI, tcon_Product__Type_Oprod___Topological__Spaces_Operfect__space_31, fact_177_psubsetE, fact_205_gr__implies__not0, fact_242_nat__descend__induct, tcon_fun___Orderings_Oorder__top, fact_157_top1I, fact_226_le__trans, tcon_Real_Oreal___Fields_Olinordered__field, fact_240_nat__neq__iff, fact_248_field__lbound__gt__zero, fact_231_uniformly__Cauchy__on__def, fact_169_uniformly__convergent__eq__cauchy, fact_185_Collect__mono, fact_225_eq__imp__le, fact_71_dense__ge, fact_3_UNIV__I, fact_203_not__less0, fact_176_subsetD, fact_234_infinite__descent, fact_130_prod__induct7, fact_204_less__zeroE, fact_236_less__irrefl__nat, fact_239_less__not__refl, fact_192_subset__psubset__trans, fact_91_dist__not__less__zero, tcon_HOL_Obool___Orderings_Otop_17, fact_195_seq__mono__lemma, fact_89_top_Oextremum__unique, fact_171_top__empty__eq, fact_201_gr0I, fact_122_gt__ex, fact_11_top_Onot__eq__extremum, fact_249_less__numeral__extra_I3_J, fact_206_infinite__descent0, tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space, fact_194_subrelI, fact_191_subset__not__subset__eq, fact_134_prod__induct3, fact_251_ex__gt__or__lt, conj_0, fact_238_less__not__refl2, fact_223_nat__less__le, fact_12_epigraph__mono, fact_13_metric__eq__thm, fact_8_dist__commute__lessI, fact_198_neq0__conv, tcon_Real_Oreal___Euclidean__Space_Oeuclidean__space_1, fact_163_zero__reorient, fact_229_less__eq__nat_Osimps_I1_J, fact_210_psubsetD, fact_145_dist__commute, tcon_Set_Oset___Groups_Ozero_11
% 85.98/63.41 % No. of inferences in proof: 364
% 85.98/63.41 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 63031 ms resp. 62151 ms w/o parsing
% 86.49/63.50 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 86.49/63.50 % [INFO] Killing All external provers ...
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