%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT004-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:00:19 AM UTC 2025
% Result : Unsatisfiable 0.21s 0.55s
% Output : Proof 0.42s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 83
% Syntax : Number of formulae : 204 ( 59 unt; 0 typ; 0 def)
% Number of atoms : 1015 ( 128 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 1478 ( 699 ~; 681 |; 0 &)
% ( 98 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 32 ( 32 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 6 con; 0-2 aty)
% Number of variables : 657 ( 610 !; 0 ?; 657 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(g_type,type,
g: $i ).
tff(h_type,type,
h: $i ).
tff(c_type,type,
c: $i ).
tff(b_type,type,
b: $i ).
tff(defined_type,type,
defined: ( $i * $i ) > $o ).
tff(d_type,type,
d: $i ).
tff(domain_type,type,
domain: $i > $i ).
tff(1,plain,
( product(a,b,c)
<=> product(a,b,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
product(a,b,c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ab_equals_c) ).
tff(3,plain,
product(a,b,c),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| defined(X,Y) )
<=> ( ~ product(X,Y,Z)
| defined(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',associative_property1) ).
tff(8,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(a,b,c)
| defined(a,b) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(a,b,c)
| defined(a,b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(a,b,c)
| defined(a,b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(a,b,c)
| defined(a,b) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
defined(a,b),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,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(16,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)],[15]) ).
tff(17,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(18,axiom,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',closure_of_composition) ).
tff(19,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(skolemize,[status(sab)],[19]) ).
tff(21,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[20,16]) ).
tff(22,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,b)
| product(a,b,compose(a,b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,b)
| product(a,b,compose(a,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,b)
| product(a,b,compose(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,b)
| product(a,b,compose(a,b)) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
product(a,b,compose(a,b)),
inference(unit_resolution,[status(thm)],[24,21,14]) ).
tff(26,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ) )),
inference(bind,[status(th)],[]) ).
tff(27,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ) ),
inference(quant_intro,[status(thm)],[26]) ).
tff(28,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ) )),
inference(bind,[status(th)],[]) ).
tff(30,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ) ),
inference(quant_intro,[status(thm)],[29]) ).
tff(31,axiom,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',composition_is_well_defined) ).
tff(32,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[32,28]) ).
tff(34,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(skolemize,[status(sab)],[33]) ).
tff(35,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[34,27]) ).
tff(36,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,b,c)
| ~ product(a,b,compose(a,b))
| ( c = compose(a,b) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,b,c)
| ~ product(a,b,compose(a,b))
| ( c = compose(a,b) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(37,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,b,c)
| ~ product(a,b,compose(a,b))
| ( c = compose(a,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,b,c)
| ~ product(a,b,compose(a,b))
| ( c = compose(a,b) ) ),
inference(modus_ponens,[status(thm)],[37,36]) ).
tff(39,plain,
c = compose(a,b),
inference(unit_resolution,[status(thm)],[38,35,3,25]) ).
tff(40,plain,
compose(a,b) = c,
inference(symmetry,[status(thm)],[39]) ).
tff(41,plain,
( product(h,compose(a,b),compose(h,c))
<=> product(h,c,compose(h,c)) ),
inference(monotonicity,[status(thm)],[40]) ).
tff(42,plain,
( product(h,c,compose(h,c))
<=> product(h,compose(a,b),compose(h,c)) ),
inference(symmetry,[status(thm)],[41]) ).
tff(43,plain,
( product(h,c,d)
<=> product(h,c,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(44,axiom,
product(h,c,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',hc_equals_d) ).
tff(45,plain,
product(h,c,d),
inference(modus_ponens,[status(thm)],[44,43]) ).
tff(46,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(h,c,d)
| defined(h,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(h,c,d)
| defined(h,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(h,c,d)
| defined(h,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(48,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(h,c,d)
| defined(h,c) ),
inference(modus_ponens,[status(thm)],[47,46]) ).
tff(49,plain,
defined(h,c),
inference(unit_resolution,[status(thm)],[48,10,45]) ).
tff(50,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,c)
| product(h,c,compose(h,c)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,c)
| product(h,c,compose(h,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,c)
| product(h,c,compose(h,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(52,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,c)
| product(h,c,compose(h,c)) ),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
product(h,c,compose(h,c)),
inference(unit_resolution,[status(thm)],[52,21,49]) ).
tff(54,plain,
product(h,compose(a,b),compose(h,c)),
inference(modus_ponens,[status(thm)],[53,42]) ).
tff(55,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(56,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)],[55]) ).
tff(57,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(58,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(59,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)],[58]) ).
tff(60,axiom,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ defined(X,Yz)
| defined(X,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',category_theory_axiom3) ).
tff(61,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(skolemize,[status(sab)],[62]) ).
tff(64,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[63,56]) ).
tff(65,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(66,plain,
( ( ~ defined(h,c)
| defined(h,a)
| ~ product(a,b,c) )
<=> ( ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(67,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(h,c)
| defined(h,a)
| ~ product(a,b,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) ) ),
inference(monotonicity,[status(thm)],[66]) ).
tff(68,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(h,c)
| defined(h,a)
| ~ product(a,b,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) ) ),
inference(transitivity,[status(thm)],[67,65]) ).
tff(69,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(h,c)
| defined(h,a)
| ~ product(a,b,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(70,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(h,c)
| defined(h,a) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
defined(h,a),
inference(unit_resolution,[status(thm)],[70,64,3,49]) ).
tff(72,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,a)
| product(h,a,compose(h,a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,a)
| product(h,a,compose(h,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(73,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,a)
| product(h,a,compose(h,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,a)
| product(h,a,compose(h,a)) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
product(h,a,compose(h,a)),
inference(unit_resolution,[status(thm)],[74,21,71]) ).
tff(76,plain,
^ [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
refl(( ( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) )),
inference(bind,[status(th)],[]) ).
tff(77,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) ),
inference(quant_intro,[status(thm)],[76]) ).
tff(78,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(79,plain,
^ [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy) ) )),
rewrite(( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy)
| product(Xy,Z,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz)
| product(Xy,Z,Xyz) ) )),
rewrite(( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz)
| product(Xy,Z,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy)
| product(Xy,Z,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) )),
inference(bind,[status(th)],[]) ).
tff(80,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy)
| product(Xy,Z,Xyz) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ) ),
inference(quant_intro,[status(thm)],[79]) ).
tff(81,axiom,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ product(X,Yz,Xyz)
| ~ product(X,Y,Xy)
| product(Xy,Z,Xyz) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',category_theory_axiom5) ).
tff(82,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ),
inference(modus_ponens,[status(thm)],[82,78]) ).
tff(84,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ),
inference(skolemize,[status(sab)],[83]) ).
tff(85,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) ),
inference(modus_ponens,[status(thm)],[84,77]) ).
tff(86,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ( ~ product(a,b,compose(a,b))
| product(compose(h,a),b,compose(h,c))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c)) )
<=> ( ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(h,a),b,compose(h,c))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) ) ),
inference(monotonicity,[status(thm)],[87]) ).
tff(89,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(h,a),b,compose(h,c))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) ) ),
inference(transitivity,[status(thm)],[88,86]) ).
tff(90,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(h,a),b,compose(h,c))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(91,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(h,a,compose(h,a))
| ~ product(h,compose(a,b),compose(h,c))
| product(compose(h,a),b,compose(h,c)) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
product(compose(h,a),b,compose(h,c)),
inference(unit_resolution,[status(thm)],[91,85,25,75,54]) ).
tff(93,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,c,d)
| ~ product(h,c,compose(h,c))
| ( d = compose(h,c) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,c,d)
| ~ product(h,c,compose(h,c))
| ( d = compose(h,c) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(94,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,c,d)
| ~ product(h,c,compose(h,c))
| ( d = compose(h,c) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(95,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,c,d)
| ~ product(h,c,compose(h,c))
| ( d = compose(h,c) ) ),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
d = compose(h,c),
inference(unit_resolution,[status(thm)],[95,35,45,53]) ).
tff(97,plain,
compose(h,c) = d,
inference(symmetry,[status(thm)],[96]) ).
tff(98,plain,
^ [X: $i] :
refl(( defined(X,domain(X))
<=> defined(X,domain(X)) )),
inference(bind,[status(th)],[]) ).
tff(99,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(quant_intro,[status(thm)],[98]) ).
tff(100,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,axiom,
! [X: $i] : defined(X,domain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',mapping_from_x_to_its_domain) ).
tff(102,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [X: $i] : defined(X,domain(X)),
inference(skolemize,[status(sab)],[102]) ).
tff(104,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[103,99]) ).
tff(105,plain,
( ~ ! [X: $i] : defined(X,domain(X))
| defined(g,domain(g)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(106,plain,
defined(g,domain(g)),
inference(unit_resolution,[status(thm)],[105,104]) ).
tff(107,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,domain(g))
| product(g,domain(g),compose(g,domain(g))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,domain(g))
| product(g,domain(g),compose(g,domain(g))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(108,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,domain(g))
| product(g,domain(g),compose(g,domain(g))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(109,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,domain(g))
| product(g,domain(g),compose(g,domain(g))) ),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,plain,
product(g,domain(g),compose(g,domain(g))),
inference(unit_resolution,[status(thm)],[109,21,106]) ).
tff(111,plain,
^ [X: $i] :
refl(( product(X,domain(X),X)
<=> product(X,domain(X),X) )),
inference(bind,[status(th)],[]) ).
tff(112,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(quant_intro,[status(thm)],[111]) ).
tff(113,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(rewrite,[status(thm)],[]) ).
tff(114,axiom,
! [X: $i] : product(X,domain(X),X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',product_on_domain) ).
tff(115,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
! [X: $i] : product(X,domain(X),X),
inference(skolemize,[status(sab)],[115]) ).
tff(117,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[116,112]) ).
tff(118,plain,
( ~ ! [X: $i] : product(X,domain(X),X)
| product(g,domain(g),g) ),
inference(quant_inst,[status(thm)],[]) ).
tff(119,plain,
product(g,domain(g),g),
inference(unit_resolution,[status(thm)],[118,117]) ).
tff(120,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(g,domain(g),g)
| ~ product(g,domain(g),compose(g,domain(g)))
| ( g = compose(g,domain(g)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(g,domain(g),g)
| ~ product(g,domain(g),compose(g,domain(g)))
| ( g = compose(g,domain(g)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(121,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(g,domain(g),g)
| ~ product(g,domain(g),compose(g,domain(g)))
| ( g = compose(g,domain(g)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(122,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(g,domain(g),g)
| ~ product(g,domain(g),compose(g,domain(g)))
| ( g = compose(g,domain(g)) ) ),
inference(modus_ponens,[status(thm)],[121,120]) ).
tff(123,plain,
g = compose(g,domain(g)),
inference(unit_resolution,[status(thm)],[122,35,119,110]) ).
tff(124,plain,
compose(g,domain(g)) = g,
inference(symmetry,[status(thm)],[123]) ).
tff(125,plain,
( product(compose(g,domain(g)),compose(a,b),compose(h,c))
<=> product(g,c,d) ),
inference(monotonicity,[status(thm)],[124,40,97]) ).
tff(126,plain,
( product(g,c,d)
<=> product(compose(g,domain(g)),compose(a,b),compose(h,c)) ),
inference(symmetry,[status(thm)],[125]) ).
tff(127,plain,
( product(g,c,d)
<=> product(g,c,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,axiom,
product(g,c,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',gc_equals_d) ).
tff(129,plain,
product(g,c,d),
inference(modus_ponens,[status(thm)],[128,127]) ).
tff(130,plain,
product(compose(g,domain(g)),compose(a,b),compose(h,c)),
inference(modus_ponens,[status(thm)],[129,126]) ).
tff(131,plain,
( product(compose(g,domain(g)),a,compose(g,a))
<=> product(g,a,compose(g,a)) ),
inference(monotonicity,[status(thm)],[124]) ).
tff(132,plain,
( product(g,a,compose(g,a))
<=> product(compose(g,domain(g)),a,compose(g,a)) ),
inference(symmetry,[status(thm)],[131]) ).
tff(133,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(g,c,d)
| defined(g,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(g,c,d)
| defined(g,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(134,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(g,c,d)
| defined(g,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(135,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(g,c,d)
| defined(g,c) ),
inference(modus_ponens,[status(thm)],[134,133]) ).
tff(136,plain,
defined(g,c),
inference(unit_resolution,[status(thm)],[135,10,129]) ).
tff(137,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(138,plain,
( ( ~ defined(g,c)
| defined(g,a)
| ~ product(a,b,c) )
<=> ( ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(139,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(g,c)
| defined(g,a)
| ~ product(a,b,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) ) ),
inference(monotonicity,[status(thm)],[138]) ).
tff(140,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(g,c)
| defined(g,a)
| ~ product(a,b,c) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) ) ),
inference(transitivity,[status(thm)],[139,137]) ).
tff(141,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ defined(g,c)
| defined(g,a)
| ~ product(a,b,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(142,plain,
( ~ ! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) )
| ~ product(a,b,c)
| ~ defined(g,c)
| defined(g,a) ),
inference(modus_ponens,[status(thm)],[141,140]) ).
tff(143,plain,
defined(g,a),
inference(unit_resolution,[status(thm)],[142,64,3,136]) ).
tff(144,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,a)
| product(g,a,compose(g,a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,a)
| product(g,a,compose(g,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(145,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,a)
| product(g,a,compose(g,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(146,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(g,a)
| product(g,a,compose(g,a)) ),
inference(modus_ponens,[status(thm)],[145,144]) ).
tff(147,plain,
product(g,a,compose(g,a)),
inference(unit_resolution,[status(thm)],[146,21,143]) ).
tff(148,plain,
product(compose(g,domain(g)),a,compose(g,a)),
inference(modus_ponens,[status(thm)],[147,132]) ).
tff(149,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(150,plain,
( ( ~ product(a,b,compose(a,b))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) )
<=> ( ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(151,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ) ),
inference(monotonicity,[status(thm)],[150]) ).
tff(152,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ) ),
inference(transitivity,[status(thm)],[151,149]) ).
tff(153,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(154,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| ~ product(X,Yz,Xyz) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ),
inference(modus_ponens,[status(thm)],[153,152]) ).
tff(155,plain,
( ~ product(compose(g,domain(g)),a,compose(g,a))
| product(compose(g,a),b,compose(h,c))
| ~ product(compose(g,domain(g)),compose(a,b),compose(h,c)) ),
inference(unit_resolution,[status(thm)],[154,85,25]) ).
tff(156,plain,
product(compose(g,a),b,compose(h,c)),
inference(unit_resolution,[status(thm)],[155,148,130]) ).
tff(157,plain,
^ [W: $i,Y: $i,X: $i] :
refl(( ( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
<=> ( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ) )),
inference(bind,[status(th)],[]) ).
tff(158,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ) ),
inference(quant_intro,[status(thm)],[157]) ).
tff(159,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(160,plain,
^ [W: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,b,W)
| ~ product(Y,b,W)
| ( X = Y ) )
<=> ( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ) )),
inference(bind,[status(th)],[]) ).
tff(161,plain,
( ! [W: $i,Y: $i,X: $i] :
( ~ product(X,b,W)
| ~ product(Y,b,W)
| ( X = Y ) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ) ),
inference(quant_intro,[status(thm)],[160]) ).
tff(162,axiom,
! [W: $i,Y: $i,X: $i] :
( ~ product(X,b,W)
| ~ product(Y,b,W)
| ( X = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cancellation_for_product2) ).
tff(163,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ),
inference(modus_ponens,[status(thm)],[162,161]) ).
tff(164,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ),
inference(modus_ponens,[status(thm)],[163,159]) ).
tff(165,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ),
inference(skolemize,[status(sab)],[164]) ).
tff(166,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) ),
inference(modus_ponens,[status(thm)],[165,158]) ).
tff(167,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(168,plain,
( ( ( compose(h,a) = compose(g,a) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c)) )
<=> ( ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(169,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ( compose(h,a) = compose(g,a) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) ) ),
inference(monotonicity,[status(thm)],[168]) ).
tff(170,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ( compose(h,a) = compose(g,a) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) ) ),
inference(transitivity,[status(thm)],[169,167]) ).
tff(171,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ( compose(h,a) = compose(g,a) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(172,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,b,W)
| ~ product(X,b,W) )
| ~ product(compose(g,a),b,compose(h,c))
| ~ product(compose(h,a),b,compose(h,c))
| ( compose(h,a) = compose(g,a) ) ),
inference(modus_ponens,[status(thm)],[171,170]) ).
tff(173,plain,
compose(h,a) = compose(g,a),
inference(unit_resolution,[status(thm)],[172,166,156,92]) ).
tff(174,plain,
compose(g,a) = compose(h,a),
inference(symmetry,[status(thm)],[173]) ).
tff(175,plain,
( product(h,a,compose(g,a))
<=> product(h,a,compose(h,a)) ),
inference(monotonicity,[status(thm)],[174]) ).
tff(176,plain,
( product(h,a,compose(h,a))
<=> product(h,a,compose(g,a)) ),
inference(symmetry,[status(thm)],[175]) ).
tff(177,plain,
product(h,a,compose(g,a)),
inference(modus_ponens,[status(thm)],[75,176]) ).
tff(178,plain,
( ( g = h )
<=> ( h = g ) ),
inference(commutativity,[status(thm)],[]) ).
tff(179,plain,
( ( compose(g,domain(g)) = h )
<=> ( g = h ) ),
inference(monotonicity,[status(thm)],[124]) ).
tff(180,plain,
( ( compose(g,domain(g)) = h )
<=> ( h = g ) ),
inference(transitivity,[status(thm)],[179,178]) ).
tff(181,plain,
( ( h = g )
<=> ( compose(g,domain(g)) = h ) ),
inference(symmetry,[status(thm)],[180]) ).
tff(182,plain,
( ( h != g )
<=> ( compose(g,domain(g)) != h ) ),
inference(monotonicity,[status(thm)],[181]) ).
tff(183,plain,
( ( h != g )
<=> ( h != g ) ),
inference(rewrite,[status(thm)],[]) ).
tff(184,axiom,
h != g,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_h_equals_g) ).
tff(185,plain,
h != g,
inference(modus_ponens,[status(thm)],[184,183]) ).
tff(186,plain,
compose(g,domain(g)) != h,
inference(modus_ponens,[status(thm)],[185,182]) ).
tff(187,plain,
^ [W: $i,Y: $i,X: $i] :
refl(( ( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
<=> ( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ) )),
inference(bind,[status(th)],[]) ).
tff(188,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ) ),
inference(quant_intro,[status(thm)],[187]) ).
tff(189,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(190,plain,
^ [W: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,a,W)
| ~ product(Y,a,W)
| ( X = Y ) )
<=> ( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ) )),
inference(bind,[status(th)],[]) ).
tff(191,plain,
( ! [W: $i,Y: $i,X: $i] :
( ~ product(X,a,W)
| ~ product(Y,a,W)
| ( X = Y ) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ) ),
inference(quant_intro,[status(thm)],[190]) ).
tff(192,axiom,
! [W: $i,Y: $i,X: $i] :
( ~ product(X,a,W)
| ~ product(Y,a,W)
| ( X = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cancellation_for_product1) ).
tff(193,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ),
inference(modus_ponens,[status(thm)],[192,191]) ).
tff(194,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ),
inference(modus_ponens,[status(thm)],[193,189]) ).
tff(195,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ),
inference(skolemize,[status(sab)],[194]) ).
tff(196,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) ),
inference(modus_ponens,[status(thm)],[195,188]) ).
tff(197,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(198,plain,
( ( ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a))
| ~ product(compose(g,domain(g)),a,compose(g,a)) )
<=> ( ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(199,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a))
| ~ product(compose(g,domain(g)),a,compose(g,a)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) ) ),
inference(monotonicity,[status(thm)],[198]) ).
tff(200,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a))
| ~ product(compose(g,domain(g)),a,compose(g,a)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) ) ),
inference(transitivity,[status(thm)],[199,197]) ).
tff(201,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a))
| ~ product(compose(g,domain(g)),a,compose(g,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(202,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(Y,a,W)
| ~ product(X,a,W) )
| ~ product(compose(g,domain(g)),a,compose(g,a))
| ( compose(g,domain(g)) = h )
| ~ product(h,a,compose(g,a)) ),
inference(modus_ponens,[status(thm)],[201,200]) ).
tff(203,plain,
~ product(h,a,compose(g,a)),
inference(unit_resolution,[status(thm)],[202,196,186,148]) ).
tff(204,plain,
$false,
inference(unit_resolution,[status(thm)],[203,177]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : CAT004-1 : TPTP v9.0.0. Released v1.0.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n011.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 11:49:26 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.55 % SZS status Unsatisfiable
% 0.21/0.55 % SZS output start Proof
% See solution above
% 0.42/0.57 % E exiting
%------------------------------------------------------------------------------