TSTP Solution File: DAT048_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT048_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n021.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 14:36:31 EDT 2022
% Result : Theorem 0.20s 0.44s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 123
% Syntax : Number of formulae : 287 ( 82 unt; 5 typ; 0 def)
% Number of atoms : 1024 ( 364 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 1063 ( 349 ~; 260 |; 17 &)
% ( 434 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of FOOLs : 28 ( 28 fml; 0 var)
% Number arithmetic : 2622 ( 62 atm; 534 fun;1716 num; 310 var)
% Number of types : 2 ( 0 usr; 1 ari)
% Number of type conns : 7 ( 4 >; 3 *; 0 +; 0 <<)
% Number of predicates : 9 ( 4 usr; 3 prp; 0-2 aty)
% Number of functors : 13 ( 4 usr; 7 con; 0-2 aty)
% Number of variables : 545 ( 484 !; 0 ?; 545 :)
% Comments :
%------------------------------------------------------------------------------
tff(count_type,type,
count: collection > $int ).
tff(add_type,type,
add: ( $int * collection ) > collection ).
tff(remove_type,type,
remove: ( $int * collection ) > collection ).
tff(empty_type,type,
empty: collection ).
tff(in_type,type,
in: ( $int * collection ) > $o ).
tff(1,plain,
^ [X6: collection] :
refl(
( $greatereq(count(X6),0)
<=> $greatereq(count(X6),0) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X6: collection] : $greatereq(count(X6),0)
<=> ! [X6: collection] : $greatereq(count(X6),0) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X6: collection] : $greatereq(count(X6),0)
<=> ! [X6: collection] : $greatereq(count(X6),0) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X6: collection] : $greatereq(count(X6),0),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax1) ).
tff(5,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X6: collection] : $greatereq(count(X6),0)
| $greatereq(count(empty),0) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
$greatereq(count(empty),0),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X8: $int,X9: collection] :
refl(
( ( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
<=> ( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
<=> ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
^ [X8: $int,X9: collection] :
rewrite(
( ( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) )
<=> ( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ) )),
inference(bind,[status(th)],]) ).
tff(13,plain,
( ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) )
<=> ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ) ),
inference(quant_intro,[status(thm)],[12]) ).
tff(14,plain,
^ [X8: $int,X9: collection] :
rewrite(
( ( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) )
<=> ( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(15,plain,
( ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) )
<=> ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[14]) ).
tff(16,plain,
( ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) )
<=> ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(17,plain,
^ [X8: $int,X9: collection] :
rewrite(
( ( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(count(X9),1) ) )
<=> ( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ) )),
inference(bind,[status(th)],]) ).
tff(18,plain,
( ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(count(X9),1) ) )
<=> ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ) ),
inference(quant_intro,[status(thm)],[17]) ).
tff(19,axiom,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(count(X9),1) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax3) ).
tff(20,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ),
inference(modus_ponens,[status(thm)],[20,16]) ).
tff(22,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) ),
inference(modus_ponens,[status(thm)],[21,15]) ).
tff(23,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[22,13]) ).
tff(24,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[24,11]) ).
tff(26,plain,
( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(2,empty)
<=> ( $sum(count(empty),$product(-1,count(add(2,empty)))) = -1 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
( ~ in(2,empty)
<=> ( $sum(count(empty),$product(-1,count(add(2,empty)))) = -1 ) ),
inference(unit_resolution,[status(thm)],[26,25]) ).
tff(28,plain,
^ [U: $int] :
refl(
( ~ in(U,empty)
<=> ~ in(U,empty) )),
inference(bind,[status(th)],]) ).
tff(29,plain,
( ! [U: $int] : ~ in(U,empty)
<=> ! [U: $int] : ~ in(U,empty) ),
inference(quant_intro,[status(thm)],[28]) ).
tff(30,plain,
( ! [U: $int] : ~ in(U,empty)
<=> ! [U: $int] : ~ in(U,empty) ),
inference(rewrite,[status(thm)],]) ).
tff(31,axiom,
! [U: $int] : ~ in(U,empty),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=0.ax',ax1) ).
tff(32,plain,
! [U: $int] : ~ in(U,empty),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [U: $int] : ~ in(U,empty),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [U: $int] : ~ in(U,empty),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
( ~ ! [U: $int] : ~ in(U,empty)
| ~ in(2,empty) ),
inference(quant_inst,[status(thm)],]) ).
tff(36,plain,
~ in(2,empty),
inference(unit_resolution,[status(thm)],[35,34]) ).
tff(37,plain,
( ~ ( ~ in(2,empty)
<=> ( $sum(count(empty),$product(-1,count(add(2,empty)))) = -1 ) )
| in(2,empty)
| ( $sum(count(empty),$product(-1,count(add(2,empty)))) = -1 ) ),
inference(tautology,[status(thm)],]) ).
tff(38,plain,
$sum(count(empty),$product(-1,count(add(2,empty)))) = -1,
inference(unit_resolution,[status(thm)],[37,36,27]) ).
tff(39,plain,
( ( $sum(count(empty),$product(-1,count(add(2,empty)))) != -1 )
| $lesseq($sum(count(empty),$product(-1,count(add(2,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(40,plain,
$lesseq($sum(count(empty),$product(-1,count(add(2,empty)))),-1),
inference(unit_resolution,[status(thm)],[39,38]) ).
tff(41,plain,
^ [X14: $int,X15: collection] :
refl(
( ( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) )
<=> ( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(42,plain,
( ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) )
<=> ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[41]) ).
tff(43,plain,
^ [X14: $int,X15: collection] :
rewrite(
( ( ~ in(X14,X15)
<=> ( $sum(count(remove(X14,X15)),$product(-1,count(X15))) = 0 ) )
<=> ( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(44,plain,
( ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(remove(X14,X15)),$product(-1,count(X15))) = 0 ) )
<=> ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[43]) ).
tff(45,plain,
^ [X14: $int,X15: collection] :
rewrite(
( ( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) )
<=> ( ~ in(X14,X15)
<=> ( $sum(count(remove(X14,X15)),$product(-1,count(X15))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(46,plain,
( ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) )
<=> ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(remove(X14,X15)),$product(-1,count(X15))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[45]) ).
tff(47,plain,
( ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) )
<=> ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(48,axiom,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax6) ).
tff(49,plain,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( count(remove(X14,X15)) = count(X15) ) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(remove(X14,X15)),$product(-1,count(X15))) = 0 ) ),
inference(modus_ponens,[status(thm)],[49,46]) ).
tff(51,plain,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[50,44]) ).
tff(52,plain,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ),
inference(skolemize,[status(sab)],[51]) ).
tff(53,plain,
! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[52,42]) ).
tff(54,plain,
( ~ ! [X14: $int,X15: collection] :
( ~ in(X14,X15)
<=> ( $sum(count(X15),$product(-1,count(remove(X14,X15)))) = 0 ) )
| ( ~ in(3,add(2,empty))
<=> ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(55,plain,
( ~ in(3,add(2,empty))
<=> ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) = 0 ) ),
inference(unit_resolution,[status(thm)],[54,53]) ).
tff(56,plain,
^ [Z: $int,X1: collection,X2: $int] :
refl(
( ( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
<=> ( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ) )),
inference(bind,[status(th)],]) ).
tff(57,plain,
( ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
<=> ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ) ),
inference(quant_intro,[status(thm)],[56]) ).
tff(58,plain,
^ [Z: $int,X1: collection,X2: $int] :
rewrite(
( ( ( in(Z,X1)
| ( $sum(Z,$product(-1,X2)) = 0 ) )
<=> in(Z,add(X2,X1)) )
<=> ( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ) )),
inference(bind,[status(th)],]) ).
tff(59,plain,
( ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(Z,$product(-1,X2)) = 0 ) )
<=> in(Z,add(X2,X1)) )
<=> ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,plain,
^ [Z: $int,X1: collection,X2: $int] :
rewrite(
( ( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) )
<=> ( ( in(Z,X1)
| ( $sum(Z,$product(-1,X2)) = 0 ) )
<=> in(Z,add(X2,X1)) ) )),
inference(bind,[status(th)],]) ).
tff(61,plain,
( ! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) )
<=> ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(Z,$product(-1,X2)) = 0 ) )
<=> in(Z,add(X2,X1)) ) ),
inference(quant_intro,[status(thm)],[60]) ).
tff(62,plain,
( ! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) )
<=> ! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(63,plain,
^ [Z: $int,X1: collection,X2: $int] :
rewrite(
( ( ( in(Z,X1)
| ( Z = X2 ) )
<=> in(Z,add(X2,X1)) )
<=> ( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) ) )),
inference(bind,[status(th)],]) ).
tff(64,plain,
( ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( Z = X2 ) )
<=> in(Z,add(X2,X1)) )
<=> ! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) ) ),
inference(quant_intro,[status(thm)],[63]) ).
tff(65,axiom,
! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( Z = X2 ) )
<=> in(Z,add(X2,X1)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=0.ax',ax4) ).
tff(66,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) ),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( ( Z = X2 )
| in(Z,X1) )
<=> in(Z,add(X2,X1)) ),
inference(modus_ponens,[status(thm)],[66,62]) ).
tff(68,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(Z,$product(-1,X2)) = 0 ) )
<=> in(Z,add(X2,X1)) ),
inference(modus_ponens,[status(thm)],[67,61]) ).
tff(69,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ),
inference(modus_ponens,[status(thm)],[68,59]) ).
tff(70,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ),
inference(skolemize,[status(sab)],[69]) ).
tff(71,plain,
! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) ),
inference(modus_ponens,[status(thm)],[70,57]) ).
tff(72,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(3,empty)
<=> in(3,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(3,empty)
<=> in(3,add(2,empty)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(73,plain,
( ( ( in(3,empty)
| ( $sum(2,$product(-1,3)) = 0 ) )
<=> in(3,add(2,empty)) )
<=> ( in(3,empty)
<=> in(3,add(2,empty)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(74,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(3,empty)
| ( $sum(2,$product(-1,3)) = 0 ) )
<=> in(3,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(3,empty)
<=> in(3,add(2,empty)) ) ) ),
inference(monotonicity,[status(thm)],[73]) ).
tff(75,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(3,empty)
| ( $sum(2,$product(-1,3)) = 0 ) )
<=> in(3,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(3,empty)
<=> in(3,add(2,empty)) ) ) ),
inference(transitivity,[status(thm)],[74,72]) ).
tff(76,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(3,empty)
| ( $sum(2,$product(-1,3)) = 0 ) )
<=> in(3,add(2,empty)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(77,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(3,empty)
<=> in(3,add(2,empty)) ) ),
inference(modus_ponens,[status(thm)],[76,75]) ).
tff(78,plain,
( in(3,empty)
<=> in(3,add(2,empty)) ),
inference(unit_resolution,[status(thm)],[77,71]) ).
tff(79,plain,
( ~ ! [U: $int] : ~ in(U,empty)
| ~ in(3,empty) ),
inference(quant_inst,[status(thm)],]) ).
tff(80,plain,
~ in(3,empty),
inference(unit_resolution,[status(thm)],[79,34]) ).
tff(81,plain,
( ~ ( in(3,empty)
<=> in(3,add(2,empty)) )
| in(3,empty)
| ~ in(3,add(2,empty)) ),
inference(tautology,[status(thm)],]) ).
tff(82,plain,
~ in(3,add(2,empty)),
inference(unit_resolution,[status(thm)],[81,80,78]) ).
tff(83,plain,
( ~ ( ~ in(3,add(2,empty))
<=> ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) = 0 ) )
| in(3,add(2,empty))
| ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) = 0 ) ),
inference(tautology,[status(thm)],]) ).
tff(84,plain,
$sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) = 0,
inference(unit_resolution,[status(thm)],[83,82,55]) ).
tff(85,plain,
( ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) != 0 )
| $lesseq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(86,plain,
$lesseq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0),
inference(unit_resolution,[status(thm)],[85,84]) ).
tff(87,plain,
( ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) )
<=> ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(88,plain,
( ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(add(1,add(5,remove(3,add(2,empty))))))) = -1 ) )
<=> ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(89,plain,
( ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(add(1,add(5,remove(3,add(2,empty))))))) = -1 ) ) )
<=> ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) ) ),
inference(monotonicity,[status(thm)],[88]) ).
tff(90,plain,
( ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(add(1,add(5,remove(3,add(2,empty))))))) = -1 ) ) )
<=> ( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) ) ),
inference(transitivity,[status(thm)],[89,87]) ).
tff(91,plain,
( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(add(1,add(5,remove(3,add(2,empty))))))) = -1 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(92,plain,
( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ) ),
inference(modus_ponens,[status(thm)],[91,90]) ).
tff(93,plain,
( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ),
inference(unit_resolution,[status(thm)],[92,25]) ).
tff(94,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(95,plain,
( ( ( in(1,remove(3,add(2,empty)))
| ( $sum(5,$product(-1,1)) = 0 ) )
<=> in(1,add(5,remove(3,add(2,empty)))) )
<=> ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(96,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,remove(3,add(2,empty)))
| ( $sum(5,$product(-1,1)) = 0 ) )
<=> in(1,add(5,remove(3,add(2,empty)))) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) ) ),
inference(monotonicity,[status(thm)],[95]) ).
tff(97,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,remove(3,add(2,empty)))
| ( $sum(5,$product(-1,1)) = 0 ) )
<=> in(1,add(5,remove(3,add(2,empty)))) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) ) ),
inference(transitivity,[status(thm)],[96,94]) ).
tff(98,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,remove(3,add(2,empty)))
| ( $sum(5,$product(-1,1)) = 0 ) )
<=> in(1,add(5,remove(3,add(2,empty)))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(99,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ) ),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) ),
inference(unit_resolution,[status(thm)],[99,71]) ).
tff(101,plain,
^ [V: $int,W: collection] :
refl(
( in(V,add(V,W))
<=> in(V,add(V,W)) )),
inference(bind,[status(th)],]) ).
tff(102,plain,
( ! [V: $int,W: collection] : in(V,add(V,W))
<=> ! [V: $int,W: collection] : in(V,add(V,W)) ),
inference(quant_intro,[status(thm)],[101]) ).
tff(103,plain,
( ! [V: $int,W: collection] : in(V,add(V,W))
<=> ! [V: $int,W: collection] : in(V,add(V,W)) ),
inference(rewrite,[status(thm)],]) ).
tff(104,axiom,
! [V: $int,W: collection] : in(V,add(V,W)),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=0.ax',ax2) ).
tff(105,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(modus_ponens,[status(thm)],[104,103]) ).
tff(106,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(skolemize,[status(sab)],[105]) ).
tff(107,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(modus_ponens,[status(thm)],[106,102]) ).
tff(108,plain,
( ~ ! [V: $int,W: collection] : in(V,add(V,W))
| in(2,add(2,empty)) ),
inference(quant_inst,[status(thm)],]) ).
tff(109,plain,
in(2,add(2,empty)),
inference(unit_resolution,[status(thm)],[108,107]) ).
tff(110,plain,
^ [X16: $int,X17: collection] :
refl(
( ( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
<=> ( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ) )),
inference(bind,[status(th)],]) ).
tff(111,plain,
( ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
<=> ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,plain,
( ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
<=> ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(113,plain,
^ [X16: $int,X17: collection] :
rewrite(
( ( in(X16,X17)
=> ( X17 = add(X16,remove(X16,X17)) ) )
<=> ( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ) )),
inference(bind,[status(th)],]) ).
tff(114,plain,
( ! [X16: $int,X17: collection] :
( in(X16,X17)
=> ( X17 = add(X16,remove(X16,X17)) ) )
<=> ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ) ),
inference(quant_intro,[status(thm)],[113]) ).
tff(115,axiom,
! [X16: $int,X17: collection] :
( in(X16,X17)
=> ( X17 = add(X16,remove(X16,X17)) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax7) ).
tff(116,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[116,112]) ).
tff(118,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(skolemize,[status(sab)],[117]) ).
tff(119,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[118,111]) ).
tff(120,plain,
( ( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(2,add(2,empty))
| ( add(2,empty) = add(2,remove(2,add(2,empty))) ) )
<=> ( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(2,add(2,empty))
| ( add(2,empty) = add(2,remove(2,add(2,empty))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(121,plain,
( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(2,add(2,empty))
| ( add(2,empty) = add(2,remove(2,add(2,empty))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(122,plain,
( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(2,add(2,empty))
| ( add(2,empty) = add(2,remove(2,add(2,empty))) ) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
add(2,empty) = add(2,remove(2,add(2,empty))),
inference(unit_resolution,[status(thm)],[122,119,109]) ).
tff(124,plain,
add(2,remove(2,add(2,empty))) = add(2,empty),
inference(symmetry,[status(thm)],[123]) ).
tff(125,plain,
remove(3,add(2,remove(2,add(2,empty)))) = remove(3,add(2,empty)),
inference(monotonicity,[status(thm)],[124]) ).
tff(126,plain,
( in(1,remove(3,add(2,remove(2,add(2,empty)))))
<=> in(1,remove(3,add(2,empty))) ),
inference(monotonicity,[status(thm)],[125]) ).
tff(127,plain,
( ~ in(1,remove(3,add(2,remove(2,add(2,empty)))))
<=> ~ in(1,remove(3,add(2,empty))) ),
inference(monotonicity,[status(thm)],[126]) ).
tff(128,plain,
^ [X3: $int,X4: collection,X5: $int] :
refl(
( ( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ) )),
inference(bind,[status(th)],]) ).
tff(129,plain,
( ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ) ),
inference(quant_intro,[status(thm)],[128]) ).
tff(130,plain,
^ [X3: $int,X4: collection,X5: $int] :
rewrite(
( ( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ) )),
inference(bind,[status(th)],]) ).
tff(131,plain,
( ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ) ),
inference(quant_intro,[status(thm)],[130]) ).
tff(132,plain,
^ [X3: $int,X4: collection,X5: $int] :
rewrite(
( ( ( in(X3,X4)
& ( $sum(X3,$product(-1,X5)) != 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) ) )),
inference(bind,[status(th)],]) ).
tff(133,plain,
( ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X3,$product(-1,X5)) != 0 ) )
<=> in(X3,remove(X5,X4)) )
<=> ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) ) ),
inference(quant_intro,[status(thm)],[132]) ).
tff(134,plain,
^ [X3: $int,X4: collection,X5: $int] :
rewrite(
( ( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) )
<=> ( ( in(X3,X4)
& ( $sum(X3,$product(-1,X5)) != 0 ) )
<=> in(X3,remove(X5,X4)) ) )),
inference(bind,[status(th)],]) ).
tff(135,plain,
( ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) )
<=> ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X3,$product(-1,X5)) != 0 ) )
<=> in(X3,remove(X5,X4)) ) ),
inference(quant_intro,[status(thm)],[134]) ).
tff(136,plain,
( ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) )
<=> ! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(137,axiom,
! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=0.ax',ax5) ).
tff(138,plain,
! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( X3 != X5 ) )
<=> in(X3,remove(X5,X4)) ),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X3,$product(-1,X5)) != 0 ) )
<=> in(X3,remove(X5,X4)) ),
inference(modus_ponens,[status(thm)],[138,135]) ).
tff(140,plain,
! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) ),
inference(modus_ponens,[status(thm)],[139,133]) ).
tff(141,plain,
! [X3: $int,X4: collection,X5: $int] :
( ( in(X3,X4)
& ( $sum(X5,$product(-1,X3)) != 0 ) )
<=> in(X3,remove(X5,X4)) ),
inference(skolemize,[status(sab)],[140]) ).
tff(142,plain,
! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ),
inference(modus_ponens,[status(thm)],[141,131]) ).
tff(143,plain,
! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) ),
inference(modus_ponens,[status(thm)],[142,129]) ).
tff(144,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(145,plain,
( ( ~ ( ~ in(1,add(2,remove(2,add(2,empty))))
| ( $sum(3,$product(-1,1)) = 0 ) )
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) )
<=> ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(146,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(1,add(2,remove(2,add(2,empty))))
| ( $sum(3,$product(-1,1)) = 0 ) )
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ) ),
inference(monotonicity,[status(thm)],[145]) ).
tff(147,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(1,add(2,remove(2,add(2,empty))))
| ( $sum(3,$product(-1,1)) = 0 ) )
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ) ),
inference(transitivity,[status(thm)],[146,144]) ).
tff(148,plain,
( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(1,add(2,remove(2,add(2,empty))))
| ( $sum(3,$product(-1,1)) = 0 ) )
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(149,plain,
( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ) ),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,plain,
( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) ),
inference(unit_resolution,[status(thm)],[149,143]) ).
tff(151,plain,
( in(1,add(2,empty))
<=> in(1,add(2,remove(2,add(2,empty)))) ),
inference(monotonicity,[status(thm)],[123]) ).
tff(152,plain,
( ~ in(1,add(2,empty))
<=> ~ in(1,add(2,remove(2,add(2,empty)))) ),
inference(monotonicity,[status(thm)],[151]) ).
tff(153,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,empty)
<=> in(1,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,empty)
<=> in(1,add(2,empty)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(154,plain,
( ( ( in(1,empty)
| ( $sum(2,$product(-1,1)) = 0 ) )
<=> in(1,add(2,empty)) )
<=> ( in(1,empty)
<=> in(1,add(2,empty)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(155,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,empty)
| ( $sum(2,$product(-1,1)) = 0 ) )
<=> in(1,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,empty)
<=> in(1,add(2,empty)) ) ) ),
inference(monotonicity,[status(thm)],[154]) ).
tff(156,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,empty)
| ( $sum(2,$product(-1,1)) = 0 ) )
<=> in(1,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,empty)
<=> in(1,add(2,empty)) ) ) ),
inference(transitivity,[status(thm)],[155,153]) ).
tff(157,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(1,empty)
| ( $sum(2,$product(-1,1)) = 0 ) )
<=> in(1,add(2,empty)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(158,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(1,empty)
<=> in(1,add(2,empty)) ) ),
inference(modus_ponens,[status(thm)],[157,156]) ).
tff(159,plain,
( in(1,empty)
<=> in(1,add(2,empty)) ),
inference(unit_resolution,[status(thm)],[158,71]) ).
tff(160,plain,
( ~ ! [U: $int] : ~ in(U,empty)
| ~ in(1,empty) ),
inference(quant_inst,[status(thm)],]) ).
tff(161,plain,
~ in(1,empty),
inference(unit_resolution,[status(thm)],[160,34]) ).
tff(162,plain,
( ~ ( in(1,empty)
<=> in(1,add(2,empty)) )
| in(1,empty)
| ~ in(1,add(2,empty)) ),
inference(tautology,[status(thm)],]) ).
tff(163,plain,
~ in(1,add(2,empty)),
inference(unit_resolution,[status(thm)],[162,161,159]) ).
tff(164,plain,
~ in(1,add(2,remove(2,add(2,empty)))),
inference(modus_ponens,[status(thm)],[163,152]) ).
tff(165,plain,
( ~ ( in(1,add(2,remove(2,add(2,empty))))
<=> in(1,remove(3,add(2,remove(2,add(2,empty))))) )
| in(1,add(2,remove(2,add(2,empty))))
| ~ in(1,remove(3,add(2,remove(2,add(2,empty))))) ),
inference(tautology,[status(thm)],]) ).
tff(166,plain,
~ in(1,remove(3,add(2,remove(2,add(2,empty))))),
inference(unit_resolution,[status(thm)],[165,164,150]) ).
tff(167,plain,
~ in(1,remove(3,add(2,empty))),
inference(modus_ponens,[status(thm)],[166,127]) ).
tff(168,plain,
( ~ ( in(1,remove(3,add(2,empty)))
<=> in(1,add(5,remove(3,add(2,empty)))) )
| in(1,remove(3,add(2,empty)))
| ~ in(1,add(5,remove(3,add(2,empty)))) ),
inference(tautology,[status(thm)],]) ).
tff(169,plain,
~ in(1,add(5,remove(3,add(2,empty)))),
inference(unit_resolution,[status(thm)],[168,167,100]) ).
tff(170,plain,
( ~ ( ~ in(1,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) )
| in(1,add(5,remove(3,add(2,empty))))
| ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1 ) ),
inference(tautology,[status(thm)],]) ).
tff(171,plain,
$sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 1,
inference(unit_resolution,[status(thm)],[170,169,93]) ).
tff(172,plain,
( ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) != 1 )
| $greatereq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(173,plain,
$greatereq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1),
inference(unit_resolution,[status(thm)],[172,171]) ).
tff(174,plain,
( ~ ! [X6: collection] : $greatereq(count(X6),0)
| $greatereq(count(remove(5,add(5,remove(3,add(2,empty))))),0) ),
inference(quant_inst,[status(thm)],]) ).
tff(175,plain,
$greatereq(count(remove(5,add(5,remove(3,add(2,empty))))),0),
inference(unit_resolution,[status(thm)],[174,7]) ).
tff(176,plain,
^ [X7: collection] :
refl(
( ( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ( ( X7 = empty )
<=> ( count(X7) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(177,plain,
( ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ) ),
inference(quant_intro,[status(thm)],[176]) ).
tff(178,plain,
( ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(179,axiom,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax2) ).
tff(180,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(modus_ponens,[status(thm)],[179,178]) ).
tff(181,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(skolemize,[status(sab)],[180]) ).
tff(182,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(modus_ponens,[status(thm)],[181,177]) ).
tff(183,plain,
( ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) )
<=> ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(184,plain,
( ( $true
<=> ( count(empty) = 0 ) )
<=> ( count(empty) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(185,plain,
( ( empty = empty )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(186,plain,
( ( ( empty = empty )
<=> ( count(empty) = 0 ) )
<=> ( $true
<=> ( count(empty) = 0 ) ) ),
inference(monotonicity,[status(thm)],[185]) ).
tff(187,plain,
( ( ( empty = empty )
<=> ( count(empty) = 0 ) )
<=> ( count(empty) = 0 ) ),
inference(transitivity,[status(thm)],[186,184]) ).
tff(188,plain,
( ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( ( empty = empty )
<=> ( count(empty) = 0 ) ) )
<=> ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) ) ),
inference(monotonicity,[status(thm)],[187]) ).
tff(189,plain,
( ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( ( empty = empty )
<=> ( count(empty) = 0 ) ) )
<=> ( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) ) ),
inference(transitivity,[status(thm)],[188,183]) ).
tff(190,plain,
( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( ( empty = empty )
<=> ( count(empty) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(191,plain,
( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) ),
inference(modus_ponens,[status(thm)],[190,189]) ).
tff(192,plain,
count(empty) = 0,
inference(unit_resolution,[status(thm)],[191,182]) ).
tff(193,plain,
( ( count(empty) != 0 )
| $lesseq(count(empty),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(194,plain,
$lesseq(count(empty),0),
inference(unit_resolution,[status(thm)],[193,192]) ).
tff(195,plain,
^ [X12: $int,X13: collection] :
refl(
( ( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) )
<=> ( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(196,plain,
( ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) )
<=> ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[195]) ).
tff(197,plain,
^ [X12: $int,X13: collection] :
rewrite(
( ( in(X12,X13)
<=> ( $sum(count(remove(X12,X13)),$product(-1,count(X13))) = -1 ) )
<=> ( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ) )),
inference(bind,[status(th)],]) ).
tff(198,plain,
( ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(remove(X12,X13)),$product(-1,count(X13))) = -1 ) )
<=> ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ) ),
inference(quant_intro,[status(thm)],[197]) ).
tff(199,plain,
^ [X12: $int,X13: collection] :
rewrite(
( ( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) )
<=> ( in(X12,X13)
<=> ( $sum(count(remove(X12,X13)),$product(-1,count(X13))) = -1 ) ) )),
inference(bind,[status(th)],]) ).
tff(200,plain,
( ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) )
<=> ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(remove(X12,X13)),$product(-1,count(X13))) = -1 ) ) ),
inference(quant_intro,[status(thm)],[199]) ).
tff(201,plain,
( ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) )
<=> ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(202,plain,
^ [X12: $int,X13: collection] :
rewrite(
( ( in(X12,X13)
<=> ( count(remove(X12,X13)) = $difference(count(X13),1) ) )
<=> ( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) ) )),
inference(bind,[status(th)],]) ).
tff(203,plain,
( ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $difference(count(X13),1) ) )
<=> ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) ) ),
inference(quant_intro,[status(thm)],[202]) ).
tff(204,axiom,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $difference(count(X13),1) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax5) ).
tff(205,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) ),
inference(modus_ponens,[status(thm)],[204,203]) ).
tff(206,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( count(remove(X12,X13)) = $sum(-1,count(X13)) ) ),
inference(modus_ponens,[status(thm)],[205,201]) ).
tff(207,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(remove(X12,X13)),$product(-1,count(X13))) = -1 ) ),
inference(modus_ponens,[status(thm)],[206,200]) ).
tff(208,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ),
inference(modus_ponens,[status(thm)],[207,198]) ).
tff(209,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ),
inference(skolemize,[status(sab)],[208]) ).
tff(210,plain,
! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) ),
inference(modus_ponens,[status(thm)],[209,196]) ).
tff(211,plain,
( ~ ! [X12: $int,X13: collection] :
( in(X12,X13)
<=> ( $sum(count(X13),$product(-1,count(remove(X12,X13)))) = 1 ) )
| ( in(5,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(212,plain,
( in(5,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) ),
inference(unit_resolution,[status(thm)],[211,210]) ).
tff(213,plain,
( ~ ! [V: $int,W: collection] : in(V,add(V,W))
| in(5,add(5,remove(3,add(2,empty)))) ),
inference(quant_inst,[status(thm)],]) ).
tff(214,plain,
in(5,add(5,remove(3,add(2,empty)))),
inference(unit_resolution,[status(thm)],[213,107]) ).
tff(215,plain,
( ~ ( in(5,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) )
| ~ in(5,add(5,remove(3,add(2,empty))))
| ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) ),
inference(tautology,[status(thm)],]) ).
tff(216,plain,
( ~ ( in(5,add(5,remove(3,add(2,empty))))
<=> ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) )
| ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1 ) ),
inference(unit_resolution,[status(thm)],[215,214]) ).
tff(217,plain,
$sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) = 1,
inference(unit_resolution,[status(thm)],[216,212]) ).
tff(218,plain,
( ( $sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))) != 1 )
| $greatereq($sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(219,plain,
$greatereq($sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))),1),
inference(unit_resolution,[status(thm)],[218,217]) ).
tff(220,plain,
( ( $sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))) != 0 )
| $greatereq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(221,plain,
$greatereq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0),
inference(unit_resolution,[status(thm)],[220,84]) ).
tff(222,plain,
( ( $sum(count(empty),$product(-1,count(add(2,empty)))) != -1 )
| $greatereq($sum(count(empty),$product(-1,count(add(2,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(223,plain,
$greatereq($sum(count(empty),$product(-1,count(add(2,empty)))),-1),
inference(unit_resolution,[status(thm)],[222,38]) ).
tff(224,plain,
( $lesseq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0)
| ~ $greatereq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0)
| ~ $lesseq(count(empty),0)
| ~ $greatereq($sum(count(empty),$product(-1,count(add(2,empty)))),-1)
| ~ $greatereq(count(remove(5,add(5,remove(3,add(2,empty))))),0)
| ~ $greatereq($sum(count(add(5,remove(3,add(2,empty)))),$product(-1,count(remove(5,add(5,remove(3,add(2,empty))))))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(225,plain,
$lesseq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0),
inference(unit_resolution,[status(thm)],[224,223,221,219,194,175]) ).
tff(226,plain,
^ [X10: $int,X11: collection] :
refl(
( ( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) )
<=> ( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(227,plain,
( ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) )
<=> ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[226]) ).
tff(228,plain,
^ [X10: $int,X11: collection] :
rewrite(
( ( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) )
<=> ( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(229,plain,
( ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) )
<=> ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[228]) ).
tff(230,plain,
^ [X10: $int,X11: collection] :
rewrite(
( ( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) )
<=> ( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(231,plain,
( ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) )
<=> ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) ) ),
inference(quant_intro,[status(thm)],[230]) ).
tff(232,plain,
( ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) )
<=> ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(233,axiom,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax4) ).
tff(234,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) ),
inference(modus_ponens,[status(thm)],[233,232]) ).
tff(235,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) ),
inference(modus_ponens,[status(thm)],[234,231]) ).
tff(236,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[235,229]) ).
tff(237,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(skolemize,[status(sab)],[236]) ).
tff(238,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[237,227]) ).
tff(239,plain,
( ~ ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) )
| ( in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(240,plain,
( in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 0 ) ),
inference(unit_resolution,[status(thm)],[239,238]) ).
tff(241,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(242,plain,
( ( ~ ( ~ in(5,add(2,empty))
| ( $sum(3,$product(-1,5)) = 0 ) )
<=> in(5,remove(3,add(2,empty))) )
<=> ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(243,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(5,add(2,empty))
| ( $sum(3,$product(-1,5)) = 0 ) )
<=> in(5,remove(3,add(2,empty))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) ) ),
inference(monotonicity,[status(thm)],[242]) ).
tff(244,plain,
( ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(5,add(2,empty))
| ( $sum(3,$product(-1,5)) = 0 ) )
<=> in(5,remove(3,add(2,empty))) ) )
<=> ( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) ) ),
inference(transitivity,[status(thm)],[243,241]) ).
tff(245,plain,
( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( ~ ( ~ in(5,add(2,empty))
| ( $sum(3,$product(-1,5)) = 0 ) )
<=> in(5,remove(3,add(2,empty))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(246,plain,
( ~ ! [X3: $int,X4: collection,X5: $int] :
( ~ ( ~ in(X3,X4)
| ( $sum(X5,$product(-1,X3)) = 0 ) )
<=> in(X3,remove(X5,X4)) )
| ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ) ),
inference(modus_ponens,[status(thm)],[245,244]) ).
tff(247,plain,
( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) ),
inference(unit_resolution,[status(thm)],[246,143]) ).
tff(248,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(5,empty)
<=> in(5,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(5,empty)
<=> in(5,add(2,empty)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(249,plain,
( ( ( in(5,empty)
| ( $sum(2,$product(-1,5)) = 0 ) )
<=> in(5,add(2,empty)) )
<=> ( in(5,empty)
<=> in(5,add(2,empty)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(250,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(5,empty)
| ( $sum(2,$product(-1,5)) = 0 ) )
<=> in(5,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(5,empty)
<=> in(5,add(2,empty)) ) ) ),
inference(monotonicity,[status(thm)],[249]) ).
tff(251,plain,
( ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(5,empty)
| ( $sum(2,$product(-1,5)) = 0 ) )
<=> in(5,add(2,empty)) ) )
<=> ( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(5,empty)
<=> in(5,add(2,empty)) ) ) ),
inference(transitivity,[status(thm)],[250,248]) ).
tff(252,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( ( in(5,empty)
| ( $sum(2,$product(-1,5)) = 0 ) )
<=> in(5,add(2,empty)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(253,plain,
( ~ ! [Z: $int,X1: collection,X2: $int] :
( ( in(Z,X1)
| ( $sum(X2,$product(-1,Z)) = 0 ) )
<=> in(Z,add(X2,X1)) )
| ( in(5,empty)
<=> in(5,add(2,empty)) ) ),
inference(modus_ponens,[status(thm)],[252,251]) ).
tff(254,plain,
( in(5,empty)
<=> in(5,add(2,empty)) ),
inference(unit_resolution,[status(thm)],[253,71]) ).
tff(255,plain,
( ~ ! [U: $int] : ~ in(U,empty)
| ~ in(5,empty) ),
inference(quant_inst,[status(thm)],]) ).
tff(256,plain,
~ in(5,empty),
inference(unit_resolution,[status(thm)],[255,34]) ).
tff(257,plain,
( ~ ( in(5,empty)
<=> in(5,add(2,empty)) )
| in(5,empty)
| ~ in(5,add(2,empty)) ),
inference(tautology,[status(thm)],]) ).
tff(258,plain,
~ in(5,add(2,empty)),
inference(unit_resolution,[status(thm)],[257,256,254]) ).
tff(259,plain,
( ~ ( in(5,add(2,empty))
<=> in(5,remove(3,add(2,empty))) )
| in(5,add(2,empty))
| ~ in(5,remove(3,add(2,empty))) ),
inference(tautology,[status(thm)],]) ).
tff(260,plain,
~ in(5,remove(3,add(2,empty))),
inference(unit_resolution,[status(thm)],[259,258,247]) ).
tff(261,plain,
( ~ ( in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 0 ) )
| in(5,remove(3,add(2,empty)))
| ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) != 0 ) ),
inference(tautology,[status(thm)],]) ).
tff(262,plain,
$sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) != 0,
inference(unit_resolution,[status(thm)],[261,260,240]) ).
tff(263,plain,
( ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = 0 )
| ~ $lesseq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0)
| ~ $greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(264,plain,
~ $greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0),
inference(unit_resolution,[status(thm)],[263,262,225]) ).
tff(265,plain,
( $greatereq(count(add(1,add(5,remove(3,add(2,empty))))),3)
| ~ $greatereq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1)
| $greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),0)
| ~ $lesseq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0)
| ~ $greatereq(count(empty),0)
| ~ $lesseq($sum(count(empty),$product(-1,count(add(2,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(266,plain,
$greatereq(count(add(1,add(5,remove(3,add(2,empty))))),3),
inference(unit_resolution,[status(thm)],[265,264,173,86,40,9]) ).
tff(267,plain,
( ( $sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))) != 1 )
| $lesseq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(268,plain,
$lesseq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1),
inference(unit_resolution,[status(thm)],[267,171]) ).
tff(269,plain,
( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = -1 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(270,plain,
( ~ in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = -1 ) ),
inference(unit_resolution,[status(thm)],[269,25]) ).
tff(271,plain,
( ~ ( ~ in(5,remove(3,add(2,empty)))
<=> ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = -1 ) )
| in(5,remove(3,add(2,empty)))
| ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = -1 ) ),
inference(tautology,[status(thm)],]) ).
tff(272,plain,
$sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) = -1,
inference(unit_resolution,[status(thm)],[271,260,270]) ).
tff(273,plain,
( ( $sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))) != -1 )
| $greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(274,plain,
$greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),-1),
inference(unit_resolution,[status(thm)],[273,272]) ).
tff(275,plain,
( $lesseq(count(add(1,add(5,remove(3,add(2,empty))))),3)
| ~ $lesseq($sum(count(add(1,add(5,remove(3,add(2,empty))))),$product(-1,count(add(5,remove(3,add(2,empty)))))),1)
| ~ $greatereq($sum(count(remove(3,add(2,empty))),$product(-1,count(add(5,remove(3,add(2,empty)))))),-1)
| ~ $greatereq($sum(count(add(2,empty)),$product(-1,count(remove(3,add(2,empty))))),0)
| ~ $lesseq(count(empty),0)
| ~ $greatereq($sum(count(empty),$product(-1,count(add(2,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(276,plain,
$lesseq(count(add(1,add(5,remove(3,add(2,empty))))),3),
inference(unit_resolution,[status(thm)],[275,274,268,221,223,194]) ).
tff(277,plain,
( ( count(add(1,add(5,remove(3,add(2,empty))))) != 3 )
<=> ( count(add(1,add(5,remove(3,add(2,empty))))) != 3 ) ),
inference(rewrite,[status(thm)],]) ).
tff(278,axiom,
count(add(1,add(5,remove(3,add(2,empty))))) != 3,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',co1) ).
tff(279,plain,
count(add(1,add(5,remove(3,add(2,empty))))) != 3,
inference(modus_ponens,[status(thm)],[278,277]) ).
tff(280,plain,
( ( count(add(1,add(5,remove(3,add(2,empty))))) = 3 )
| ~ $lesseq(count(add(1,add(5,remove(3,add(2,empty))))),3)
| ~ $greatereq(count(add(1,add(5,remove(3,add(2,empty))))),3) ),
inference(theory_lemma,[status(thm)],]) ).
tff(281,plain,
( ~ $lesseq(count(add(1,add(5,remove(3,add(2,empty))))),3)
| ~ $greatereq(count(add(1,add(5,remove(3,add(2,empty))))),3) ),
inference(unit_resolution,[status(thm)],[280,279]) ).
tff(282,plain,
$false,
inference(unit_resolution,[status(thm)],[281,276,266]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : DAT048_1 : TPTP v8.1.0. Released v5.0.0.
% 0.07/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n021.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 31 01:42:11 EDT 2022
% 0.12/0.34 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.20/0.44 % SZS status Theorem
% 0.20/0.44 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------