TSTP Solution File: CAT002-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : CAT002-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% 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 : Tue Sep 6 17:29:36 EDT 2022
% Result : Unsatisfiable 1.49s 1.23s
% Output : Proof 1.56s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 92
% Syntax : Number of formulae : 218 ( 59 unt; 10 typ; 0 def)
% Number of atoms : 1003 ( 124 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 1452 ( 683 ~; 672 |; 0 &)
% ( 97 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 26 ( 26 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 8 ( 4 >; 4 *; 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(g_type,type,
g: $i ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(domain_type,type,
domain: $i > $i ).
tff(h_type,type,
h: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(defined_type,type,
defined: ( $i * $i ) > $o ).
tff(b_type,type,
b: $i ).
tff(c_type,type,
c: $i ).
tff(a_type,type,
a: $i ).
tff(d_type,type,
d: $i ).
tff(1,plain,
^ [X: $i] :
refl(
( defined(X,domain(X))
<=> defined(X,domain(X)) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : defined(X,domain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',mapping_from_x_to_its_domain) ).
tff(5,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : defined(X,domain(X)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : defined(X,domain(X))
| defined(h,domain(h)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
defined(h,domain(h)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,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(11,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)],[10]) ).
tff(12,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(13,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(14,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,domain(h))
| product(h,domain(h),compose(h,domain(h))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,domain(h))
| product(h,domain(h),compose(h,domain(h))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(18,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,domain(h))
| product(h,domain(h),compose(h,domain(h))) ),
inference(quant_inst,[status(thm)],]) ).
tff(19,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(h,domain(h))
| product(h,domain(h),compose(h,domain(h))) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
product(h,domain(h),compose(h,domain(h))),
inference(unit_resolution,[status(thm)],[19,16,9]) ).
tff(21,plain,
^ [X: $i] :
refl(
( product(X,domain(X),X)
<=> product(X,domain(X),X) )),
inference(bind,[status(th)],]) ).
tff(22,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(quant_intro,[status(thm)],[21]) ).
tff(23,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(rewrite,[status(thm)],]) ).
tff(24,axiom,
! [X: $i] : product(X,domain(X),X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',product_on_domain) ).
tff(25,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [X: $i] : product(X,domain(X),X),
inference(skolemize,[status(sab)],[25]) ).
tff(27,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[26,22]) ).
tff(28,plain,
( ~ ! [X: $i] : product(X,domain(X),X)
| product(h,domain(h),h) ),
inference(quant_inst,[status(thm)],]) ).
tff(29,plain,
product(h,domain(h),h),
inference(unit_resolution,[status(thm)],[28,27]) ).
tff(30,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(31,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)],[30]) ).
tff(32,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(33,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(34,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)],[33]) ).
tff(35,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(36,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[35,34]) ).
tff(37,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[36,32]) ).
tff(38,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[38,31]) ).
tff(40,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,domain(h),h)
| ~ product(h,domain(h),compose(h,domain(h)))
| ( h = compose(h,domain(h)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,domain(h),h)
| ~ product(h,domain(h),compose(h,domain(h)))
| ( h = compose(h,domain(h)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(41,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,domain(h),h)
| ~ product(h,domain(h),compose(h,domain(h)))
| ( h = compose(h,domain(h)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(42,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(h,domain(h),h)
| ~ product(h,domain(h),compose(h,domain(h)))
| ( h = compose(h,domain(h)) ) ),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
h = compose(h,domain(h)),
inference(unit_resolution,[status(thm)],[42,39,29,20]) ).
tff(44,plain,
compose(h,domain(h)) = h,
inference(symmetry,[status(thm)],[43]) ).
tff(45,plain,
( ( compose(h,domain(h)) = g )
<=> ( h = g ) ),
inference(monotonicity,[status(thm)],[44]) ).
tff(46,plain,
( ( h = g )
<=> ( compose(h,domain(h)) = g ) ),
inference(symmetry,[status(thm)],[45]) ).
tff(47,plain,
( ( h != g )
<=> ( compose(h,domain(h)) != g ) ),
inference(monotonicity,[status(thm)],[46]) ).
tff(48,plain,
( ( h != g )
<=> ( h != g ) ),
inference(rewrite,[status(thm)],]) ).
tff(49,axiom,
h != g,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_h_equals_g) ).
tff(50,plain,
h != g,
inference(modus_ponens,[status(thm)],[49,48]) ).
tff(51,plain,
compose(h,domain(h)) != g,
inference(modus_ponens,[status(thm)],[50,47]) ).
tff(52,plain,
( product(a,b,c)
<=> product(a,b,c) ),
inference(rewrite,[status(thm)],]) ).
tff(53,axiom,
product(a,b,c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ab_equals_c) ).
tff(54,plain,
product(a,b,c),
inference(modus_ponens,[status(thm)],[53,52]) ).
tff(55,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(56,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)],[55]) ).
tff(57,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(58,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(59,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(skolemize,[status(sab)],[59]) ).
tff(61,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) ),
inference(modus_ponens,[status(thm)],[60,56]) ).
tff(62,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(63,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(64,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)],[63,62]) ).
tff(65,plain,
defined(a,b),
inference(unit_resolution,[status(thm)],[64,61,54]) ).
tff(66,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(67,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(68,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)],[67,66]) ).
tff(69,plain,
product(a,b,compose(a,b)),
inference(unit_resolution,[status(thm)],[68,16,65]) ).
tff(70,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(71,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(72,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)],[71,70]) ).
tff(73,plain,
c = compose(a,b),
inference(unit_resolution,[status(thm)],[72,39,54,69]) ).
tff(74,plain,
compose(a,b) = c,
inference(symmetry,[status(thm)],[73]) ).
tff(75,plain,
( defined(compose(a,b),g)
<=> defined(c,g) ),
inference(monotonicity,[status(thm)],[74]) ).
tff(76,plain,
( defined(c,g)
<=> defined(compose(a,b),g) ),
inference(symmetry,[status(thm)],[75]) ).
tff(77,plain,
( product(c,g,d)
<=> product(c,g,d) ),
inference(rewrite,[status(thm)],]) ).
tff(78,axiom,
product(c,g,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cg_equals_d) ).
tff(79,plain,
product(c,g,d),
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,g,d)
| defined(c,g) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,g,d)
| defined(c,g) ) ),
inference(rewrite,[status(thm)],]) ).
tff(81,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,g,d)
| defined(c,g) ),
inference(quant_inst,[status(thm)],]) ).
tff(82,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,g,d)
| defined(c,g) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
defined(c,g),
inference(unit_resolution,[status(thm)],[82,61,79]) ).
tff(84,plain,
defined(compose(a,b),g),
inference(modus_ponens,[status(thm)],[83,76]) ).
tff(85,plain,
^ [Xy: $i,Z: $i,Y: $i,X: $i] :
refl(
( ( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
<=> ( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],]) ).
tff(86,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[85]) ).
tff(87,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) ),
inference(rewrite,[status(thm)],]) ).
tff(88,plain,
^ [Xy: $i,Z: $i,Y: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ product(X,Y,Xy)
| ~ defined(Xy,Z) )
<=> ( ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ defined(Xy,Z)
| defined(Y,Z) )
<=> ( ~ defined(Xy,Z)
| ~ product(X,Y,Xy)
| defined(Y,Z) ) )),
rewrite(
( ( ~ defined(Xy,Z)
| ~ product(X,Y,Xy)
| defined(Y,Z) )
<=> ( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ defined(Xy,Z)
| defined(Y,Z) )
<=> ( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],]) ).
tff(89,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Xy)
| ~ defined(Xy,Z)
| defined(Y,Z) )
<=> ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[88]) ).
tff(90,axiom,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Xy)
| ~ defined(Xy,Z)
| defined(Y,Z) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',associative_property2) ).
tff(91,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[91,87]) ).
tff(93,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(skolemize,[status(sab)],[92]) ).
tff(94,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[93,86]) ).
tff(95,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) ) ),
inference(rewrite,[status(thm)],]) ).
tff(96,plain,
( ( defined(b,g)
| ~ defined(compose(a,b),g)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) ) ),
inference(rewrite,[status(thm)],]) ).
tff(97,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,g)
| ~ defined(compose(a,b),g)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) ) ),
inference(monotonicity,[status(thm)],[96]) ).
tff(98,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,g)
| ~ defined(compose(a,b),g)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) ) ),
inference(transitivity,[status(thm)],[97,95]) ).
tff(99,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,g)
| ~ defined(compose(a,b),g)
| ~ product(a,b,compose(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(100,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ defined(compose(a,b),g)
| defined(b,g) ),
inference(modus_ponens,[status(thm)],[99,98]) ).
tff(101,plain,
defined(b,g),
inference(unit_resolution,[status(thm)],[100,94,69,84]) ).
tff(102,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,g)
| product(b,g,compose(b,g)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,g)
| product(b,g,compose(b,g)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(103,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,g)
| product(b,g,compose(b,g)) ),
inference(quant_inst,[status(thm)],]) ).
tff(104,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,g)
| product(b,g,compose(b,g)) ),
inference(modus_ponens,[status(thm)],[103,102]) ).
tff(105,plain,
product(b,g,compose(b,g)),
inference(unit_resolution,[status(thm)],[104,16,101]) ).
tff(106,plain,
( product(c,h,d)
<=> product(c,h,d) ),
inference(rewrite,[status(thm)],]) ).
tff(107,axiom,
product(c,h,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ch_equals_d) ).
tff(108,plain,
product(c,h,d),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,h,d)
| defined(c,h) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,h,d)
| defined(c,h) ) ),
inference(rewrite,[status(thm)],]) ).
tff(110,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,h,d)
| defined(c,h) ),
inference(quant_inst,[status(thm)],]) ).
tff(111,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| defined(X,Y) )
| ~ product(c,h,d)
| defined(c,h) ),
inference(modus_ponens,[status(thm)],[110,109]) ).
tff(112,plain,
defined(c,h),
inference(unit_resolution,[status(thm)],[111,61,108]) ).
tff(113,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(c,h)
| product(c,h,compose(c,h)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(c,h)
| product(c,h,compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(114,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(c,h)
| product(c,h,compose(c,h)) ),
inference(quant_inst,[status(thm)],]) ).
tff(115,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(c,h)
| product(c,h,compose(c,h)) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
product(c,h,compose(c,h)),
inference(unit_resolution,[status(thm)],[115,16,112]) ).
tff(117,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(c,h,d)
| ~ product(c,h,compose(c,h))
| ( d = compose(c,h) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(c,h,d)
| ~ product(c,h,compose(c,h))
| ( d = compose(c,h) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(118,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(c,h,d)
| ~ product(c,h,compose(c,h))
| ( d = compose(c,h) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(119,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(c,h,d)
| ~ product(c,h,compose(c,h))
| ( d = compose(c,h) ) ),
inference(modus_ponens,[status(thm)],[118,117]) ).
tff(120,plain,
d = compose(c,h),
inference(unit_resolution,[status(thm)],[119,39,108,116]) ).
tff(121,plain,
compose(c,h) = d,
inference(symmetry,[status(thm)],[120]) ).
tff(122,plain,
( product(compose(a,b),g,compose(c,h))
<=> product(c,g,d) ),
inference(monotonicity,[status(thm)],[74,121]) ).
tff(123,plain,
( product(c,g,d)
<=> product(compose(a,b),g,compose(c,h)) ),
inference(symmetry,[status(thm)],[122]) ).
tff(124,plain,
product(compose(a,b),g,compose(c,h)),
inference(modus_ponens,[status(thm)],[79,123]) ).
tff(125,plain,
^ [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
refl(
( ( product(X,Yz,Xyz)
| ~ 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) ) )),
inference(bind,[status(th)],]) ).
tff(126,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[125]) ).
tff(127,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) ),
inference(rewrite,[status(thm)],]) ).
tff(128,plain,
^ [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
trans(
monotonicity(
rewrite(
( ( ~ product(X,Y,Xy)
| ~ product(Xy,Z,Xyz)
| ~ product(Y,Z,Yz) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ product(Xy,Z,Xyz)
| ~ product(Y,Z,Yz)
| product(X,Yz,Xyz) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| product(X,Yz,Xyz) ) )),
rewrite(
( ( ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy)
| product(X,Yz,Xyz) )
<=> ( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(X,Y,Xy)
| ~ product(Xy,Z,Xyz)
| ~ product(Y,Z,Yz)
| product(X,Yz,Xyz) )
<=> ( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],]) ).
tff(129,plain,
( ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(X,Y,Xy)
| ~ product(Xy,Z,Xyz)
| ~ product(Y,Z,Yz)
| product(X,Yz,Xyz) )
<=> ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ) ),
inference(quant_intro,[status(thm)],[128]) ).
tff(130,axiom,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( ~ product(X,Y,Xy)
| ~ product(Xy,Z,Xyz)
| ~ product(Y,Z,Yz)
| product(X,Yz,Xyz) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',category_theory_axiom2) ).
tff(131,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[131,127]) ).
tff(133,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ),
inference(skolemize,[status(sab)],[132]) ).
tff(134,plain,
! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[133,126]) ).
tff(135,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(136,plain,
( ( product(a,compose(b,g),compose(c,h))
| ~ product(b,g,compose(b,g))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(137,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,g),compose(c,h))
| ~ product(b,g,compose(b,g))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) ) ),
inference(monotonicity,[status(thm)],[136]) ).
tff(138,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,g),compose(c,h))
| ~ product(b,g,compose(b,g))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) ) ),
inference(transitivity,[status(thm)],[137,135]) ).
tff(139,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,g),compose(c,h))
| ~ product(b,g,compose(b,g))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(a,b,compose(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(140,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),g,compose(c,h))
| ~ product(b,g,compose(b,g))
| product(a,compose(b,g),compose(c,h)) ),
inference(modus_ponens,[status(thm)],[139,138]) ).
tff(141,plain,
product(a,compose(b,g),compose(c,h)),
inference(unit_resolution,[status(thm)],[140,134,69,124,105]) ).
tff(142,plain,
( product(b,compose(h,domain(h)),compose(b,h))
<=> product(b,h,compose(b,h)) ),
inference(monotonicity,[status(thm)],[44]) ).
tff(143,plain,
( product(b,h,compose(b,h))
<=> product(b,compose(h,domain(h)),compose(b,h)) ),
inference(symmetry,[status(thm)],[142]) ).
tff(144,plain,
( defined(compose(a,b),h)
<=> defined(c,h) ),
inference(monotonicity,[status(thm)],[74]) ).
tff(145,plain,
( defined(c,h)
<=> defined(compose(a,b),h) ),
inference(symmetry,[status(thm)],[144]) ).
tff(146,plain,
defined(compose(a,b),h),
inference(modus_ponens,[status(thm)],[112,145]) ).
tff(147,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) ) ),
inference(rewrite,[status(thm)],]) ).
tff(148,plain,
( ( defined(b,h)
| ~ defined(compose(a,b),h)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) ) ),
inference(rewrite,[status(thm)],]) ).
tff(149,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,h)
| ~ defined(compose(a,b),h)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) ) ),
inference(monotonicity,[status(thm)],[148]) ).
tff(150,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,h)
| ~ defined(compose(a,b),h)
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) ) ),
inference(transitivity,[status(thm)],[149,147]) ).
tff(151,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(b,h)
| ~ defined(compose(a,b),h)
| ~ product(a,b,compose(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(152,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| defined(b,h)
| ~ defined(compose(a,b),h) ),
inference(modus_ponens,[status(thm)],[151,150]) ).
tff(153,plain,
( defined(b,h)
| ~ defined(compose(a,b),h) ),
inference(unit_resolution,[status(thm)],[152,94,69]) ).
tff(154,plain,
defined(b,h),
inference(unit_resolution,[status(thm)],[153,146]) ).
tff(155,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,h)
| product(b,h,compose(b,h)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,h)
| product(b,h,compose(b,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(156,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,h)
| product(b,h,compose(b,h)) ),
inference(quant_inst,[status(thm)],]) ).
tff(157,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(b,h)
| product(b,h,compose(b,h)) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
product(b,h,compose(b,h)),
inference(unit_resolution,[status(thm)],[157,16,154]) ).
tff(159,plain,
product(b,compose(h,domain(h)),compose(b,h)),
inference(modus_ponens,[status(thm)],[158,143]) ).
tff(160,plain,
( product(compose(a,b),compose(h,domain(h)),compose(c,h))
<=> product(c,h,compose(c,h)) ),
inference(monotonicity,[status(thm)],[74,44]) ).
tff(161,plain,
( product(c,h,compose(c,h))
<=> product(compose(a,b),compose(h,domain(h)),compose(c,h)) ),
inference(symmetry,[status(thm)],[160]) ).
tff(162,plain,
product(compose(a,b),compose(h,domain(h)),compose(c,h)),
inference(modus_ponens,[status(thm)],[116,161]) ).
tff(163,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(164,plain,
( ( product(a,compose(b,h),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(165,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,h),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ) ),
inference(monotonicity,[status(thm)],[164]) ).
tff(166,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,h),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(a,b,compose(a,b)) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ) ),
inference(transitivity,[status(thm)],[165,163]) ).
tff(167,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| product(a,compose(b,h),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(a,b,compose(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(168,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,Xyz: $i,X: $i,Yz: $i] :
( product(X,Yz,Xyz)
| ~ product(Y,Z,Yz)
| ~ product(Xy,Z,Xyz)
| ~ product(X,Y,Xy) )
| ~ product(a,b,compose(a,b))
| ~ product(compose(a,b),compose(h,domain(h)),compose(c,h))
| ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ),
inference(modus_ponens,[status(thm)],[167,166]) ).
tff(169,plain,
( ~ product(b,compose(h,domain(h)),compose(b,h))
| product(a,compose(b,h),compose(c,h)) ),
inference(unit_resolution,[status(thm)],[168,134,69,162]) ).
tff(170,plain,
product(a,compose(b,h),compose(c,h)),
inference(unit_resolution,[status(thm)],[169,159]) ).
tff(171,plain,
^ [W: $i,Y: $i,X: $i] :
refl(
( ( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
<=> ( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ) )),
inference(bind,[status(th)],]) ).
tff(172,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ) ),
inference(quant_intro,[status(thm)],[171]) ).
tff(173,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(174,plain,
^ [W: $i,Y: $i,X: $i] :
rewrite(
( ( ~ product(a,X,W)
| ~ product(a,Y,W)
| ( X = Y ) )
<=> ( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ) )),
inference(bind,[status(th)],]) ).
tff(175,plain,
( ! [W: $i,Y: $i,X: $i] :
( ~ product(a,X,W)
| ~ product(a,Y,W)
| ( X = Y ) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ) ),
inference(quant_intro,[status(thm)],[174]) ).
tff(176,axiom,
! [W: $i,Y: $i,X: $i] :
( ~ product(a,X,W)
| ~ product(a,Y,W)
| ( X = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cancellation_for_product1) ).
tff(177,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ),
inference(modus_ponens,[status(thm)],[176,175]) ).
tff(178,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ),
inference(modus_ponens,[status(thm)],[177,173]) ).
tff(179,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ),
inference(skolemize,[status(sab)],[178]) ).
tff(180,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) ),
inference(modus_ponens,[status(thm)],[179,172]) ).
tff(181,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(182,plain,
( ( ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h))
| ~ product(a,compose(b,h),compose(c,h)) )
<=> ( ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(183,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h))
| ~ product(a,compose(b,h),compose(c,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) ) ),
inference(monotonicity,[status(thm)],[182]) ).
tff(184,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h))
| ~ product(a,compose(b,h),compose(c,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) ) ),
inference(transitivity,[status(thm)],[183,181]) ).
tff(185,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h))
| ~ product(a,compose(b,h),compose(c,h)) ),
inference(quant_inst,[status(thm)],]) ).
tff(186,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(a,Y,W)
| ~ product(a,X,W) )
| ~ product(a,compose(b,h),compose(c,h))
| ( compose(b,h) = compose(b,g) )
| ~ product(a,compose(b,g),compose(c,h)) ),
inference(modus_ponens,[status(thm)],[185,184]) ).
tff(187,plain,
compose(b,h) = compose(b,g),
inference(unit_resolution,[status(thm)],[186,180,170,141]) ).
tff(188,plain,
( product(b,g,compose(b,h))
<=> product(b,g,compose(b,g)) ),
inference(monotonicity,[status(thm)],[187]) ).
tff(189,plain,
( product(b,g,compose(b,g))
<=> product(b,g,compose(b,h)) ),
inference(symmetry,[status(thm)],[188]) ).
tff(190,plain,
product(b,g,compose(b,h)),
inference(modus_ponens,[status(thm)],[105,189]) ).
tff(191,plain,
^ [W: $i,Y: $i,X: $i] :
refl(
( ( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
<=> ( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ) )),
inference(bind,[status(th)],]) ).
tff(192,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ) ),
inference(quant_intro,[status(thm)],[191]) ).
tff(193,plain,
( ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(194,plain,
^ [W: $i,Y: $i,X: $i] :
rewrite(
( ( ~ product(b,X,W)
| ~ product(b,Y,W)
| ( X = Y ) )
<=> ( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ) )),
inference(bind,[status(th)],]) ).
tff(195,plain,
( ! [W: $i,Y: $i,X: $i] :
( ~ product(b,X,W)
| ~ product(b,Y,W)
| ( X = Y ) )
<=> ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ) ),
inference(quant_intro,[status(thm)],[194]) ).
tff(196,axiom,
! [W: $i,Y: $i,X: $i] :
( ~ product(b,X,W)
| ~ product(b,Y,W)
| ( X = Y ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',cancellation_for_product2) ).
tff(197,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ),
inference(modus_ponens,[status(thm)],[196,195]) ).
tff(198,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ),
inference(modus_ponens,[status(thm)],[197,193]) ).
tff(199,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ),
inference(skolemize,[status(sab)],[198]) ).
tff(200,plain,
! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) ),
inference(modus_ponens,[status(thm)],[199,192]) ).
tff(201,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(202,plain,
( ( ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h))
| ~ product(b,compose(h,domain(h)),compose(b,h)) )
<=> ( ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(203,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h))
| ~ product(b,compose(h,domain(h)),compose(b,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ) ),
inference(monotonicity,[status(thm)],[202]) ).
tff(204,plain,
( ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h))
| ~ product(b,compose(h,domain(h)),compose(b,h)) )
<=> ( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ) ),
inference(transitivity,[status(thm)],[203,201]) ).
tff(205,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h))
| ~ product(b,compose(h,domain(h)),compose(b,h)) ),
inference(quant_inst,[status(thm)],]) ).
tff(206,plain,
( ~ ! [W: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ product(b,Y,W)
| ~ product(b,X,W) )
| ~ product(b,compose(h,domain(h)),compose(b,h))
| ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ),
inference(modus_ponens,[status(thm)],[205,204]) ).
tff(207,plain,
( ( compose(h,domain(h)) = g )
| ~ product(b,g,compose(b,h)) ),
inference(unit_resolution,[status(thm)],[206,200,159]) ).
tff(208,plain,
$false,
inference(unit_resolution,[status(thm)],[207,190,51]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : CAT002-1 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.35 % Computer : n011.cluster.edu
% 0.12/0.35 % Model : x86_64 x86_64
% 0.12/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.35 % Memory : 8042.1875MB
% 0.12/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.35 % CPULimit : 300
% 0.12/0.35 % WCLimit : 300
% 0.12/0.35 % DateTime : Tue Aug 30 05:45:55 EDT 2022
% 0.12/0.35 % CPUTime :
% 0.12/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.35 Usage: tptp [options] [-file:]file
% 0.12/0.35 -h, -? prints this message.
% 0.12/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.35 -m, -model generate model.
% 0.12/0.35 -p, -proof generate proof.
% 0.12/0.35 -c, -core generate unsat core of named formulas.
% 0.12/0.35 -st, -statistics display statistics.
% 0.12/0.35 -t:timeout set timeout (in second).
% 0.12/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.35 -<param>:<value> configuration parameter and value.
% 0.12/0.35 -o:<output-file> file to place output in.
% 1.49/1.23 % SZS status Unsatisfiable
% 1.49/1.23 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------