%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT018-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n014.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 : Sat Jun 21 05:00:24 AM UTC 2025
% Result : Unsatisfiable 0.12s 0.38s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 39
% Syntax : Number of formulae : 86 ( 29 unt; 0 typ; 0 def)
% Number of atoms : 489 ( 0 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 716 ( 340 ~; 330 |; 0 &)
% ( 46 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 27 ( 27 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 282 ( 254 !; 0 ?; 282 :)
% Comments :
%------------------------------------------------------------------------------
tff(defined_type,type,
defined: ( $i * $i ) > $o ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(c_type,type,
c: $i ).
tff(b_type,type,
b: $i ).
tff(codomain_type,type,
codomain: $i > $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(a_type,type,
a: $i ).
tff(identity_map_type,type,
identity_map: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( product(codomain(X),X,X)
<=> product(codomain(X),X,X) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : product(codomain(X),X,X),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',product_on_codomain) ).
tff(5,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(codomain(X),X,X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(codomain(X),X,X)
| product(codomain(b),b,b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(codomain(b),b,b),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( defined(b,c)
<=> defined(b,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(11,axiom,
defined(b,c),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',assume_bc_exists) ).
tff(12,plain,
defined(b,c),
inference(modus_ponens,[status(thm)],[11,10]) ).
tff(13,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
<=> ( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,axiom,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',closure_of_composition) ).
tff(17,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[18,14]) ).
tff(20,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,c)
| product(b,c,compose(b,c)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,c)
| product(b,c,compose(b,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(21,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,c)
| product(b,c,compose(b,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(22,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,c)
| product(b,c,compose(b,c)) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
product(b,c,compose(b,c)),
inference(unit_resolution,[status(thm)],[22,19,12]) ).
tff(24,plain,
^ [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
refl(( ( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
<=> ( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],[]) ).
tff(25,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[24]) ).
tff(26,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,plain,
^ [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
trans(monotonicity(rewrite(( ( ~ product(X,Y,Xy)
| ~ product(Y,Z,Yz)
| ~ defined(Xy,Z) )
<=> ( ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ product(Y,Z,Yz)
| ~ defined(Xy,Z)
| defined(X,Yz) )
<=> ( ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| defined(X,Yz) ) )),
rewrite(( ( ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| defined(X,Yz) )
<=> ( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ product(Y,Z,Yz)
| ~ defined(Xy,Z)
| defined(X,Yz) )
<=> ( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],[]) ).
tff(28,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(X,Y,Xy)
| ~ product(Y,Z,Yz)
| ~ defined(Xy,Z)
| defined(X,Yz) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[27]) ).
tff(29,axiom,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(X,Y,Xy)
| ~ product(Y,Z,Yz)
| ~ defined(Xy,Z)
| defined(X,Yz) ),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',category_theory_axiom1) ).
tff(30,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[30,26]) ).
tff(32,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ),
inference(skolemize,[status(sab)],[31]) ).
tff(33,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[32,25]) ).
tff(34,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(35,plain,
( ( defined(codomain(b),compose(b,c))
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b) )
<=> ( ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| defined(codomain(b),compose(b,c))
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) ) ),
inference(monotonicity,[status(thm)],[35]) ).
tff(37,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| defined(codomain(b),compose(b,c))
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) ) ),
inference(transitivity,[status(thm)],[36,34]) ).
tff(38,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| defined(codomain(b),compose(b,c))
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(39,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( defined(X,Yz)
| ~ defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
| ~ defined(b,c)
| ~ product(b,c,compose(b,c))
| ~ product(codomain(b),b,b)
| defined(codomain(b),compose(b,c)) ),
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
defined(codomain(b),compose(b,c)),
inference(unit_resolution,[status(thm)],[39,33,12,23,9]) ).
tff(41,plain,
( defined(a,b)
<=> defined(a,b) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
defined(a,b),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',assume_ab_exists) ).
tff(43,plain,
defined(a,b),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
^ [Z: $i,Y: $i,X: $i,Yz: $i] :
refl(( ( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
<=> ( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) )),
inference(bind,[status(th)],[]) ).
tff(45,plain,
( ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
<=> ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) ),
inference(quant_intro,[status(thm)],[44]) ).
tff(46,plain,
( ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
<=> ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,plain,
^ [Z: $i,Y: $i,X: $i,Yz: $i] :
trans(monotonicity(rewrite(( ( ~ product(Y,Z,Yz)
| ~ defined(X,Yz) )
<=> ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ defined(X,Yz)
| defined(X,Y) )
<=> ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| defined(X,Y) ) )),
rewrite(( ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| defined(X,Y) )
<=> ( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ defined(X,Yz)
| defined(X,Y) )
<=> ( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) )),
inference(bind,[status(th)],[]) ).
tff(48,plain,
( ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ defined(X,Yz)
| defined(X,Y) )
<=> ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ) ),
inference(quant_intro,[status(thm)],[47]) ).
tff(49,axiom,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ defined(X,Yz)
| defined(X,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',category_theory_axiom3) ).
tff(50,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[50,46]) ).
tff(52,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(skolemize,[status(sab)],[51]) ).
tff(53,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[52,45]) ).
tff(54,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(a,b)
| defined(a,codomain(b))
| ~ product(codomain(b),b,b) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(a,b)
| defined(a,codomain(b))
| ~ product(codomain(b),b,b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(55,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(a,b)
| defined(a,codomain(b))
| ~ product(codomain(b),b,b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(56,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(a,b)
| defined(a,codomain(b))
| ~ product(codomain(b),b,b) ),
inference(modus_ponens,[status(thm)],[55,54]) ).
tff(57,plain,
defined(a,codomain(b)),
inference(unit_resolution,[status(thm)],[56,53,43,9]) ).
tff(58,plain,
^ [X: $i] :
refl(( identity_map(codomain(X))
<=> identity_map(codomain(X)) )),
inference(bind,[status(th)],[]) ).
tff(59,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(quant_intro,[status(thm)],[58]) ).
tff(60,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(rewrite,[status(thm)],[]) ).
tff(61,axiom,
! [X: $i] : identity_map(codomain(X)),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',codomain_is_an_identity_map) ).
tff(62,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[61,60]) ).
tff(63,plain,
! [X: $i] : identity_map(codomain(X)),
inference(skolemize,[status(sab)],[62]) ).
tff(64,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[63,59]) ).
tff(65,plain,
( ~ ! [X: $i] : identity_map(codomain(X))
| identity_map(codomain(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(66,plain,
identity_map(codomain(b)),
inference(unit_resolution,[status(thm)],[65,64]) ).
tff(67,plain,
( ~ defined(a,compose(b,c))
<=> ~ defined(a,compose(b,c)) ),
inference(rewrite,[status(thm)],[]) ).
tff(68,axiom,
~ defined(a,compose(b,c)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_a_bc_exists) ).
tff(69,plain,
~ defined(a,compose(b,c)),
inference(modus_ponens,[status(thm)],[68,67]) ).
tff(70,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
<=> ( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(71,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) ),
inference(quant_intro,[status(thm)],[70]) ).
tff(72,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ defined(X,Y)
| ~ defined(Y,Z)
| ~ identity_map(Y) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) )),
( ( ~ defined(X,Y)
| ~ defined(Y,Z)
| ~ identity_map(Y)
| defined(X,Z) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| ~ defined(Y,Z)
| defined(X,Z) ) )),
rewrite(( ( ~ defined(X,Y)
| ~ identity_map(Y)
| ~ defined(Y,Z)
| defined(X,Z) )
<=> ( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) )),
( ( ~ defined(X,Y)
| ~ defined(Y,Z)
| ~ identity_map(Y)
| defined(X,Z) )
<=> ( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(74,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ defined(Y,Z)
| ~ identity_map(Y)
| defined(X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ) ),
inference(quant_intro,[status(thm)],[73]) ).
tff(75,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ defined(Y,Z)
| ~ identity_map(Y)
| defined(X,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/CAT001-0.ax',category_theory_axiom6) ).
tff(76,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ),
inference(modus_ponens,[status(thm)],[76,72]) ).
tff(78,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ),
inference(skolemize,[status(sab)],[77]) ).
tff(79,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) ),
inference(modus_ponens,[status(thm)],[78,71]) ).
tff(80,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
( ( ~ defined(a,codomain(b))
| defined(a,compose(b,c))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) )
<=> ( defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(82,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| ~ defined(a,codomain(b))
| defined(a,compose(b,c))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ) ),
inference(monotonicity,[status(thm)],[81]) ).
tff(83,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| ~ defined(a,codomain(b))
| defined(a,compose(b,c))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ) ),
inference(transitivity,[status(thm)],[82,80]) ).
tff(84,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| ~ defined(a,codomain(b))
| defined(a,compose(b,c))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ defined(X,Y)
| defined(X,Z)
| ~ identity_map(Y)
| ~ defined(Y,Z) )
| defined(a,compose(b,c))
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| ~ defined(codomain(b),compose(b,c)) ),
inference(modus_ponens,[status(thm)],[84,83]) ).
tff(86,plain,
$false,
inference(unit_resolution,[status(thm)],[85,79,69,66,57,40]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : CAT018-1 : TPTP v9.0.0. Released v1.0.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n014.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Thu Jun 19 11:52:54 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.38 % SZS status Unsatisfiable
% 0.12/0.38 % SZS output start Proof
% See solution above
% 0.20/0.40 % E exiting
%------------------------------------------------------------------------------