TSTP Solution File: DAT041_1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : DAT041_1 : TPTP v8.1.0. Released v5.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n028.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 14:36:29 EDT 2022
% Result : Theorem 0.22s 0.42s
% Output : Proof 0.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 70
% Syntax : Number of formulae : 165 ( 59 unt; 5 typ; 0 def)
% Number of atoms : 402 ( 158 equ)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 377 ( 152 ~; 74 |; 0 &)
% ( 148 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of FOOLs : 17 ( 17 fml; 0 var)
% Number arithmetic : 1328 ( 126 atm; 337 fun; 744 num; 121 var)
% Number of types : 2 ( 0 usr; 1 ari)
% Number of type conns : 7 ( 4 >; 3 *; 0 +; 0 <<)
% Number of predicates : 11 ( 4 usr; 3 prp; 0-2 aty)
% Number of functors : 9 ( 4 usr; 4 con; 0-2 aty)
% Number of variables : 269 ( 203 !; 38 ?; 269 :)
% 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,
^ [V: $int,W: collection] :
refl(
( in(V,add(V,W))
<=> in(V,add(V,W)) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [V: $int,W: collection] : in(V,add(V,W))
<=> ! [V: $int,W: collection] : in(V,add(V,W)) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [V: $int,W: collection] : in(V,add(V,W))
<=> ! [V: $int,W: collection] : in(V,add(V,W)) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [V: $int,W: collection] : in(V,add(V,W)),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=0.ax',ax2) ).
tff(5,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [V: $int,W: collection] : in(V,add(V,W)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [V: $int,W: collection] : in(V,add(V,W))
| in(0,add(0,empty)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
in(0,add(0,empty)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,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(11,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)],[10]) ).
tff(12,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(13,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(14,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)],[13]) ).
tff(15,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(16,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[15,14]) ).
tff(17,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[16,12]) ).
tff(18,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) ),
inference(modus_ponens,[status(thm)],[18,11]) ).
tff(20,plain,
( ( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(0,add(0,empty))
| ( add(0,empty) = add(0,remove(0,add(0,empty))) ) )
<=> ( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(0,add(0,empty))
| ( add(0,empty) = add(0,remove(0,add(0,empty))) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(21,plain,
( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(0,add(0,empty))
| ( add(0,empty) = add(0,remove(0,add(0,empty))) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(22,plain,
( ~ ! [X16: $int,X17: collection] :
( ~ in(X16,X17)
| ( X17 = add(X16,remove(X16,X17)) ) )
| ~ in(0,add(0,empty))
| ( add(0,empty) = add(0,remove(0,add(0,empty))) ) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
add(0,empty) = add(0,remove(0,add(0,empty))),
inference(unit_resolution,[status(thm)],[22,19,9]) ).
tff(24,plain,
add(0,remove(0,add(0,empty))) = add(0,empty),
inference(symmetry,[status(thm)],[23]) ).
tff(25,plain,
add(0,add(0,remove(0,add(0,empty)))) = add(0,add(0,empty)),
inference(monotonicity,[status(thm)],[24]) ).
tff(26,plain,
count(add(0,add(0,remove(0,add(0,empty))))) = count(add(0,add(0,empty))),
inference(monotonicity,[status(thm)],[25]) ).
tff(27,plain,
count(add(0,add(0,empty))) = count(add(0,add(0,remove(0,add(0,empty))))),
inference(symmetry,[status(thm)],[26]) ).
tff(28,plain,
( ( count(add(0,add(0,empty))) != count(add(0,add(0,remove(0,add(0,empty))))) )
| $greatereq($sum(count(add(0,add(0,empty))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(29,plain,
$greatereq($sum(count(add(0,add(0,empty))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),0),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,plain,
^ [U: collection,V: $int] :
refl(
( $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
<=> $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) )),
inference(bind,[status(th)],]) ).
tff(31,plain,
( ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
<=> ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
^ [U: collection,V: $int] :
rewrite(
( ~ ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
<=> $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) )),
inference(bind,[status(th)],]) ).
tff(33,plain,
( ! [U: collection,V: $int] :
~ ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
<=> ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) ),
inference(quant_intro,[status(thm)],[32]) ).
tff(34,plain,
( ~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
<=> ~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) ),
inference(rewrite,[status(thm)],]) ).
tff(35,plain,
( ~ ? [U: collection,V: $int] : ~ $lesseq($sum(count(U),$product(-1,count(add(V,U)))),-1)
<=> ~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1) ),
inference(rewrite,[status(thm)],]) ).
tff(36,plain,
( ~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U))))
<=> ~ ? [U: collection,V: $int] : ~ $lesseq($sum(count(U),$product(-1,count(add(V,U)))),-1) ),
inference(rewrite,[status(thm)],]) ).
tff(37,plain,
( ~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U))))
<=> ~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))) ),
inference(rewrite,[status(thm)],]) ).
tff(38,plain,
( ~ ? [U: collection,V: $int] : $greater($sum(count(U),1),count(add(V,U)))
<=> ~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))) ),
inference(rewrite,[status(thm)],]) ).
tff(39,axiom,
~ ? [U: collection,V: $int] : $greater($sum(count(U),1),count(add(V,U))),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',co1) ).
tff(40,plain,
~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))),
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))),
inference(modus_ponens,[status(thm)],[40,37]) ).
tff(42,plain,
~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))),
inference(modus_ponens,[status(thm)],[41,37]) ).
tff(43,plain,
~ ? [U: collection,V: $int] : ~ $lesseq(count(U),$sum(-1,count(add(V,U)))),
inference(modus_ponens,[status(thm)],[42,37]) ).
tff(44,plain,
~ ? [U: collection,V: $int] : ~ $lesseq($sum(count(U),$product(-1,count(add(V,U)))),-1),
inference(modus_ponens,[status(thm)],[43,36]) ).
tff(45,plain,
~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(modus_ponens,[status(thm)],[44,35]) ).
tff(46,plain,
~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(modus_ponens,[status(thm)],[45,34]) ).
tff(47,plain,
~ ? [U: collection,V: $int] : ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(modus_ponens,[status(thm)],[46,34]) ).
tff(48,plain,
^ [U: collection,V: $int] :
refl(
$oeq(
~ ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
~ ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1))),
inference(bind,[status(th)],]) ).
tff(49,plain,
! [U: collection,V: $int] :
~ ~ $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(nnf-neg,[status(sab)],[47,48]) ).
tff(50,plain,
! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(modus_ponens,[status(thm)],[49,33]) ).
tff(51,plain,
! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1),
inference(modus_ponens,[status(thm)],[50,31]) ).
tff(52,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ) ),
inference(rewrite,[status(thm)],]) ).
tff(53,plain,
( $greatereq($sum($product(-1,count(remove(0,add(0,empty)))),count(add(0,remove(0,add(0,empty))))),1)
<=> $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ),
inference(rewrite,[status(thm)],]) ).
tff(54,plain,
$sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))) = $sum($product(-1,count(remove(0,add(0,empty)))),count(add(0,remove(0,add(0,empty))))),
inference(rewrite,[status(thm)],]) ).
tff(55,plain,
( $greatereq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))),1)
<=> $greatereq($sum($product(-1,count(remove(0,add(0,empty)))),count(add(0,remove(0,add(0,empty))))),1) ),
inference(monotonicity,[status(thm)],[54]) ).
tff(56,plain,
( $greatereq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))),1)
<=> $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ),
inference(transitivity,[status(thm)],[55,53]) ).
tff(57,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ) ),
inference(monotonicity,[status(thm)],[56]) ).
tff(58,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ) ),
inference(transitivity,[status(thm)],[57,52]) ).
tff(59,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(remove(0,add(0,empty))))),1) ),
inference(quant_inst,[status(thm)],]) ).
tff(60,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1) ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
$lesseq($sum(count(remove(0,add(0,empty))),$product(-1,count(add(0,remove(0,add(0,empty)))))),-1),
inference(unit_resolution,[status(thm)],[60,51]) ).
tff(62,plain,
^ [X6: collection] :
refl(
( $greatereq(count(X6),0)
<=> $greatereq(count(X6),0) )),
inference(bind,[status(th)],]) ).
tff(63,plain,
( ! [X6: collection] : $greatereq(count(X6),0)
<=> ! [X6: collection] : $greatereq(count(X6),0) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,plain,
( ! [X6: collection] : $greatereq(count(X6),0)
<=> ! [X6: collection] : $greatereq(count(X6),0) ),
inference(rewrite,[status(thm)],]) ).
tff(65,axiom,
! [X6: collection] : $greatereq(count(X6),0),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax1) ).
tff(66,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(skolemize,[status(sab)],[66]) ).
tff(68,plain,
! [X6: collection] : $greatereq(count(X6),0),
inference(modus_ponens,[status(thm)],[67,63]) ).
tff(69,plain,
( ~ ! [X6: collection] : $greatereq(count(X6),0)
| $greatereq(count(remove(0,add(0,empty))),0) ),
inference(quant_inst,[status(thm)],]) ).
tff(70,plain,
$greatereq(count(remove(0,add(0,empty))),0),
inference(unit_resolution,[status(thm)],[69,68]) ).
tff(71,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ) ),
inference(rewrite,[status(thm)],]) ).
tff(72,plain,
( $greatereq($sum($product(-1,count(add(0,remove(0,add(0,empty))))),count(add(0,add(0,remove(0,add(0,empty)))))),1)
<=> $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ),
inference(rewrite,[status(thm)],]) ).
tff(73,plain,
$sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))) = $sum($product(-1,count(add(0,remove(0,add(0,empty))))),count(add(0,add(0,remove(0,add(0,empty)))))),
inference(rewrite,[status(thm)],]) ).
tff(74,plain,
( $greatereq($sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))),1)
<=> $greatereq($sum($product(-1,count(add(0,remove(0,add(0,empty))))),count(add(0,add(0,remove(0,add(0,empty)))))),1) ),
inference(monotonicity,[status(thm)],[73]) ).
tff(75,plain,
( $greatereq($sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))),1)
<=> $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ),
inference(transitivity,[status(thm)],[74,72]) ).
tff(76,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ) ),
inference(monotonicity,[status(thm)],[75]) ).
tff(77,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ) ),
inference(transitivity,[status(thm)],[76,71]) ).
tff(78,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,add(0,remove(0,add(0,empty))))),$product(-1,count(add(0,remove(0,add(0,empty)))))),1) ),
inference(quant_inst,[status(thm)],]) ).
tff(79,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1) ),
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
$lesseq($sum(count(add(0,remove(0,add(0,empty)))),$product(-1,count(add(0,add(0,remove(0,add(0,empty))))))),-1),
inference(unit_resolution,[status(thm)],[79,51]) ).
tff(81,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(82,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)],[81]) ).
tff(83,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(84,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)],[83]) ).
tff(85,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(86,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)],[85]) ).
tff(87,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(88,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(89,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)],[88]) ).
tff(90,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(91,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( count(add(X8,X9)) = $sum(1,count(X9)) ) ),
inference(modus_ponens,[status(thm)],[91,87]) ).
tff(93,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(add(X8,X9)),$product(-1,count(X9))) = 1 ) ),
inference(modus_ponens,[status(thm)],[92,86]) ).
tff(94,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[93,84]) ).
tff(95,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(skolemize,[status(sab)],[94]) ).
tff(96,plain,
! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) ),
inference(modus_ponens,[status(thm)],[95,82]) ).
tff(97,plain,
( ~ ! [X8: $int,X9: collection] :
( ~ in(X8,X9)
<=> ( $sum(count(X9),$product(-1,count(add(X8,X9)))) = -1 ) )
| ( ~ in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = -1 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(98,plain,
( ~ in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = -1 ) ),
inference(unit_resolution,[status(thm)],[97,96]) ).
tff(99,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(100,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)],[99]) ).
tff(101,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(102,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)],[101]) ).
tff(103,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(104,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)],[103]) ).
tff(105,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(106,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(107,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( count(add(X10,X11)) = count(X11) ) ),
inference(modus_ponens,[status(thm)],[106,105]) ).
tff(108,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(add(X10,X11)),$product(-1,count(X11))) = 0 ) ),
inference(modus_ponens,[status(thm)],[107,104]) ).
tff(109,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[108,102]) ).
tff(110,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(skolemize,[status(sab)],[109]) ).
tff(111,plain,
! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) ),
inference(modus_ponens,[status(thm)],[110,100]) ).
tff(112,plain,
( ~ ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) )
| ( in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(113,plain,
( in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = 0 ) ),
inference(unit_resolution,[status(thm)],[112,111]) ).
tff(114,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ) ),
inference(rewrite,[status(thm)],]) ).
tff(115,plain,
( $greatereq($sum($product(-1,count(empty)),count(add(0,empty))),1)
<=> $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ),
inference(rewrite,[status(thm)],]) ).
tff(116,plain,
$sum(count(add(0,empty)),$product(-1,count(empty))) = $sum($product(-1,count(empty)),count(add(0,empty))),
inference(rewrite,[status(thm)],]) ).
tff(117,plain,
( $greatereq($sum(count(add(0,empty)),$product(-1,count(empty))),1)
<=> $greatereq($sum($product(-1,count(empty)),count(add(0,empty))),1) ),
inference(monotonicity,[status(thm)],[116]) ).
tff(118,plain,
( $greatereq($sum(count(add(0,empty)),$product(-1,count(empty))),1)
<=> $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ),
inference(transitivity,[status(thm)],[117,115]) ).
tff(119,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,empty)),$product(-1,count(empty))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ) ),
inference(monotonicity,[status(thm)],[118]) ).
tff(120,plain,
( ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,empty)),$product(-1,count(empty))),1) )
<=> ( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ) ),
inference(transitivity,[status(thm)],[119,114]) ).
tff(121,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $greatereq($sum(count(add(0,empty)),$product(-1,count(empty))),1) ),
inference(quant_inst,[status(thm)],]) ).
tff(122,plain,
( ~ ! [U: collection,V: $int] : $greatereq($sum(count(add(V,U)),$product(-1,count(U))),1)
| $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
$lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1),
inference(unit_resolution,[status(thm)],[122,51]) ).
tff(124,plain,
( ~ $greatereq($sum(count(empty),$product(-1,count(add(0,empty)))),0)
| ~ $lesseq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(125,plain,
~ $greatereq($sum(count(empty),$product(-1,count(add(0,empty)))),0),
inference(unit_resolution,[status(thm)],[124,123]) ).
tff(126,plain,
( ( $sum(count(empty),$product(-1,count(add(0,empty)))) != 0 )
| $greatereq($sum(count(empty),$product(-1,count(add(0,empty)))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(127,plain,
$sum(count(empty),$product(-1,count(add(0,empty)))) != 0,
inference(unit_resolution,[status(thm)],[126,125]) ).
tff(128,plain,
( ~ ( in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = 0 ) )
| ~ in(0,empty)
| ( $sum(count(empty),$product(-1,count(add(0,empty)))) = 0 ) ),
inference(tautology,[status(thm)],]) ).
tff(129,plain,
~ in(0,empty),
inference(unit_resolution,[status(thm)],[128,127,113]) ).
tff(130,plain,
( ~ ( ~ in(0,empty)
<=> ( $sum(count(empty),$product(-1,count(add(0,empty)))) = -1 ) )
| in(0,empty)
| ( $sum(count(empty),$product(-1,count(add(0,empty)))) = -1 ) ),
inference(tautology,[status(thm)],]) ).
tff(131,plain,
$sum(count(empty),$product(-1,count(add(0,empty)))) = -1,
inference(unit_resolution,[status(thm)],[130,129,98]) ).
tff(132,plain,
( ( $sum(count(empty),$product(-1,count(add(0,empty)))) != -1 )
| $greatereq($sum(count(empty),$product(-1,count(add(0,empty)))),-1) ),
inference(theory_lemma,[status(thm)],]) ).
tff(133,plain,
$greatereq($sum(count(empty),$product(-1,count(add(0,empty)))),-1),
inference(unit_resolution,[status(thm)],[132,131]) ).
tff(134,plain,
^ [X7: collection] :
refl(
( ( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ( ( X7 = empty )
<=> ( count(X7) = 0 ) ) )),
inference(bind,[status(th)],]) ).
tff(135,plain,
( ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ) ),
inference(quant_intro,[status(thm)],[134]) ).
tff(136,plain,
( ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
<=> ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(137,axiom,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/DAT002=1.ax',ax2) ).
tff(138,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(skolemize,[status(sab)],[138]) ).
tff(140,plain,
! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) ),
inference(modus_ponens,[status(thm)],[139,135]) ).
tff(141,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(142,plain,
( ( $true
<=> ( count(empty) = 0 ) )
<=> ( count(empty) = 0 ) ),
inference(rewrite,[status(thm)],]) ).
tff(143,plain,
( ( empty = empty )
<=> $true ),
inference(rewrite,[status(thm)],]) ).
tff(144,plain,
( ( ( empty = empty )
<=> ( count(empty) = 0 ) )
<=> ( $true
<=> ( count(empty) = 0 ) ) ),
inference(monotonicity,[status(thm)],[143]) ).
tff(145,plain,
( ( ( empty = empty )
<=> ( count(empty) = 0 ) )
<=> ( count(empty) = 0 ) ),
inference(transitivity,[status(thm)],[144,142]) ).
tff(146,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)],[145]) ).
tff(147,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)],[146,141]) ).
tff(148,plain,
( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( ( empty = empty )
<=> ( count(empty) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(149,plain,
( ~ ! [X7: collection] :
( ( X7 = empty )
<=> ( count(X7) = 0 ) )
| ( count(empty) = 0 ) ),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,plain,
count(empty) = 0,
inference(unit_resolution,[status(thm)],[149,140]) ).
tff(151,plain,
( ( count(empty) != 0 )
| $lesseq(count(empty),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(152,plain,
$lesseq(count(empty),0),
inference(unit_resolution,[status(thm)],[151,150]) ).
tff(153,plain,
( ~ ! [X10: $int,X11: collection] :
( in(X10,X11)
<=> ( $sum(count(X11),$product(-1,count(add(X10,X11)))) = 0 ) )
| ( in(0,add(0,empty))
<=> ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(154,plain,
( in(0,add(0,empty))
<=> ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) ),
inference(unit_resolution,[status(thm)],[153,111]) ).
tff(155,plain,
( ~ ( in(0,add(0,empty))
<=> ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) )
| ~ in(0,add(0,empty))
| ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) ),
inference(tautology,[status(thm)],]) ).
tff(156,plain,
( ~ ( in(0,add(0,empty))
<=> ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) )
| ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0 ) ),
inference(unit_resolution,[status(thm)],[155,9]) ).
tff(157,plain,
$sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) = 0,
inference(unit_resolution,[status(thm)],[156,154]) ).
tff(158,plain,
( ( $sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))) != 0 )
| $greatereq($sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))),0) ),
inference(theory_lemma,[status(thm)],]) ).
tff(159,plain,
$greatereq($sum(count(add(0,empty)),$product(-1,count(add(0,add(0,empty))))),0),
inference(unit_resolution,[status(thm)],[158,157]) ).
tff(160,plain,
$false,
inference(theory_lemma,[status(thm)],[159,152,133,80,70,61,29]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.13 % Problem : DAT041_1 : TPTP v8.1.0. Released v5.0.0.
% 0.05/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.35 % Computer : n028.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Wed Aug 31 01:54:06 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.36 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.36 Usage: tptp [options] [-file:]file
% 0.13/0.36 -h, -? prints this message.
% 0.13/0.36 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.36 -m, -model generate model.
% 0.13/0.36 -p, -proof generate proof.
% 0.13/0.36 -c, -core generate unsat core of named formulas.
% 0.13/0.36 -st, -statistics display statistics.
% 0.13/0.36 -t:timeout set timeout (in second).
% 0.13/0.36 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.36 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.36 -<param>:<value> configuration parameter and value.
% 0.13/0.36 -o:<output-file> file to place output in.
% 0.22/0.42 % SZS status Theorem
% 0.22/0.42 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------