TSTP Solution File: CAT008-1 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : CAT008-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n015.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 17:29:41 EDT 2022
% Result : Unsatisfiable 0.20s 0.49s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 89
% Syntax : Number of formulae : 203 ( 67 unt; 8 typ; 0 def)
% Number of atoms : 978 ( 48 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 1378 ( 649 ~; 629 |; 0 &)
% ( 100 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 54 ( 54 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 10 ( 6 >; 4 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 2 con; 0-2 aty)
% Number of variables : 544 ( 495 !; 0 ?; 544 :)
% Comments :
%------------------------------------------------------------------------------
tff(codomain_type,type,
codomain: $i > $i ).
tff(b_type,type,
b: $i ).
tff(domain_type,type,
domain: $i > $i ).
tff(a_type,type,
a: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(defined_type,type,
defined: ( $i * $i ) > $o ).
tff(compose_type,type,
compose: ( $i * $i ) > $i ).
tff(identity_map_type,type,
identity_map: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(
( product(X,domain(X),X)
<=> product(X,domain(X),X) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(X,domain(X),X)
<=> ! [X: $i] : product(X,domain(X),X) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : product(X,domain(X),X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',product_on_domain) ).
tff(5,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(X,domain(X),X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(X,domain(X),X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(X,domain(X),X)
| product(domain(a),domain(domain(a)),domain(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
product(domain(a),domain(domain(a)),domain(a)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(
( defined(X,domain(X))
<=> defined(X,domain(X)) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : defined(X,domain(X))
<=> ! [X: $i] : defined(X,domain(X)) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [X: $i] : defined(X,domain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',mapping_from_x_to_its_domain) ).
tff(14,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : defined(X,domain(X)),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : defined(X,domain(X)),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : defined(X,domain(X))
| defined(a,domain(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
defined(a,domain(a)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) ),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,domain(a))
| product(a,domain(a),compose(a,domain(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,domain(a))
| product(a,domain(a),compose(a,domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(27,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,domain(a))
| product(a,domain(a),compose(a,domain(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(28,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,compose(X,Y)) )
| ~ defined(a,domain(a))
| product(a,domain(a),compose(a,domain(a))) ),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
product(a,domain(a),compose(a,domain(a))),
inference(unit_resolution,[status(thm)],[28,25,18]) ).
tff(30,plain,
( ~ ! [X: $i] : product(X,domain(X),X)
| product(a,domain(a),a) ),
inference(quant_inst,[status(thm)],]) ).
tff(31,plain,
product(a,domain(a),a),
inference(unit_resolution,[status(thm)],[30,7]) ).
tff(32,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(33,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)],[32]) ).
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(rewrite,[status(thm)],]) ).
tff(35,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(36,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)],[35]) ).
tff(37,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(38,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[37,36]) ).
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)],[38,34]) ).
tff(40,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(skolemize,[status(sab)],[39]) ).
tff(41,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[40,33]) ).
tff(42,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,domain(a),a)
| ~ product(a,domain(a),compose(a,domain(a)))
| ( a = compose(a,domain(a)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,domain(a),a)
| ~ product(a,domain(a),compose(a,domain(a)))
| ( a = compose(a,domain(a)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(43,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,domain(a),a)
| ~ product(a,domain(a),compose(a,domain(a)))
| ( a = compose(a,domain(a)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(44,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(a,domain(a),a)
| ~ product(a,domain(a),compose(a,domain(a)))
| ( a = compose(a,domain(a)) ) ),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,plain,
a = compose(a,domain(a)),
inference(unit_resolution,[status(thm)],[44,41,31,29]) ).
tff(46,plain,
compose(a,domain(a)) = a,
inference(symmetry,[status(thm)],[45]) ).
tff(47,plain,
( product(compose(a,domain(a)),domain(a),compose(a,domain(a)))
<=> product(a,domain(a),a) ),
inference(monotonicity,[status(thm)],[46,46]) ).
tff(48,plain,
( product(a,domain(a),a)
<=> product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ),
inference(symmetry,[status(thm)],[47]) ).
tff(49,plain,
product(compose(a,domain(a)),domain(a),compose(a,domain(a))),
inference(modus_ponens,[status(thm)],[31,48]) ).
tff(50,plain,
( defined(compose(a,domain(a)),domain(a))
<=> defined(a,domain(a)) ),
inference(monotonicity,[status(thm)],[46]) ).
tff(51,plain,
( defined(a,domain(a))
<=> defined(compose(a,domain(a)),domain(a)) ),
inference(symmetry,[status(thm)],[50]) ).
tff(52,plain,
defined(compose(a,domain(a)),domain(a)),
inference(modus_ponens,[status(thm)],[18,51]) ).
tff(53,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(54,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)],[53]) ).
tff(55,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(56,plain,
^ [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
trans(
monotonicity(
trans(
monotonicity(
rewrite(
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz) )
<=> ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz) ) )),
rewrite(
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz) )
<=> ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz) )
<=> ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz)
| defined(Xy,Z) )
<=> ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| defined(Xy,Z) ) )),
rewrite(
( ( ~ defined(X,Yz)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| defined(Xy,Z) )
<=> ( ~ defined(X,Yz)
| defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
( ( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz)
| defined(Xy,Z) )
<=> ( ~ defined(X,Yz)
| defined(Xy,Z)
| ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy) ) )),
inference(bind,[status(th)],]) ).
tff(57,plain,
( ! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz)
| defined(Xy,Z) )
<=> ! [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)],[56]) ).
tff(58,axiom,
! [Xy: $i,Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ product(Y,Z,Yz)
| ~ product(X,Y,Xy)
| ~ defined(X,Yz)
| defined(Xy,Z) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',category_theory_axiom4) ).
tff(59,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)],[58,57]) ).
tff(60,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)],[59,55]) ).
tff(61,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)],[60]) ).
tff(62,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)],[61,54]) ).
tff(63,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(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) )
<=> ( ~ ! [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(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(64,plain,
( ( ~ defined(compose(a,domain(a)),domain(a))
| defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) )
<=> ( defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(65,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(compose(a,domain(a)),domain(a))
| defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) )
<=> ( ~ ! [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(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ) ),
inference(monotonicity,[status(thm)],[64]) ).
tff(66,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(compose(a,domain(a)),domain(a))
| defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) )
<=> ( ~ ! [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(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ) ),
inference(transitivity,[status(thm)],[65,63]) ).
tff(67,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(compose(a,domain(a)),domain(a))
| defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(68,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(compose(a,domain(a)),domain(domain(a)))
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ defined(compose(a,domain(a)),domain(a))
| ~ product(compose(a,domain(a)),domain(a),compose(a,domain(a))) ),
inference(modus_ponens,[status(thm)],[67,66]) ).
tff(69,plain,
defined(compose(a,domain(a)),domain(domain(a))),
inference(unit_resolution,[status(thm)],[68,62,52,49,9]) ).
tff(70,plain,
( product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
<=> product(a,codomain(b),a) ),
inference(monotonicity,[status(thm)],[46,46]) ).
tff(71,plain,
( product(a,codomain(b),a)
<=> product(compose(a,domain(a)),codomain(b),compose(a,domain(a))) ),
inference(symmetry,[status(thm)],[70]) ).
tff(72,plain,
^ [X: $i] :
refl(
( product(codomain(X),X,X)
<=> product(codomain(X),X,X) )),
inference(bind,[status(th)],]) ).
tff(73,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(75,axiom,
! [X: $i] : product(codomain(X),X,X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',product_on_codomain) ).
tff(76,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [X: $i] : product(codomain(X),X,X),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
( ~ ! [X: $i] : product(codomain(X),X,X)
| product(codomain(b),b,b) ),
inference(quant_inst,[status(thm)],]) ).
tff(80,plain,
product(codomain(b),b,b),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
( defined(a,b)
<=> defined(a,b) ),
inference(rewrite,[status(thm)],]) ).
tff(82,axiom,
defined(a,b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',ab_defined) ).
tff(83,plain,
defined(a,b),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,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(85,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)],[84]) ).
tff(86,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(87,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(88,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)],[87]) ).
tff(89,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(90,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[90,86]) ).
tff(92,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(skolemize,[status(sab)],[91]) ).
tff(93,plain,
! [Z: $i,Y: $i,X: $i,Yz: $i] :
( ~ defined(X,Yz)
| defined(X,Y)
| ~ product(Y,Z,Yz) ),
inference(modus_ponens,[status(thm)],[92,85]) ).
tff(94,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(95,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(96,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)],[95,94]) ).
tff(97,plain,
defined(a,codomain(b)),
inference(unit_resolution,[status(thm)],[96,93,83,80]) ).
tff(98,plain,
^ [X: $i] :
refl(
( identity_map(codomain(X))
<=> identity_map(codomain(X)) )),
inference(bind,[status(th)],]) ).
tff(99,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(quant_intro,[status(thm)],[98]) ).
tff(100,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(rewrite,[status(thm)],]) ).
tff(101,axiom,
! [X: $i] : identity_map(codomain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',codomain_is_an_identity_map) ).
tff(102,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [X: $i] : identity_map(codomain(X)),
inference(skolemize,[status(sab)],[102]) ).
tff(104,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[103,99]) ).
tff(105,plain,
( ~ ! [X: $i] : identity_map(codomain(X))
| identity_map(codomain(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(106,plain,
identity_map(codomain(b)),
inference(unit_resolution,[status(thm)],[105,104]) ).
tff(107,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) )),
inference(bind,[status(th)],]) ).
tff(108,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) ),
inference(quant_intro,[status(thm)],[107]) ).
tff(109,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(110,plain,
^ [Y: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ defined(X,Y)
| ~ identity_map(Y) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y) ) )),
( ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) )),
rewrite(
( ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) )),
( ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) )),
inference(bind,[status(th)],]) ).
tff(111,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,axiom,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',identity2) ).
tff(113,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[112,111]) ).
tff(114,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[113,109]) ).
tff(115,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(skolemize,[status(sab)],[114]) ).
tff(116,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[115,108]) ).
tff(117,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| product(a,codomain(b),a) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| product(a,codomain(b),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(118,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| product(a,codomain(b),a) ),
inference(quant_inst,[status(thm)],]) ).
tff(119,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(a,codomain(b))
| ~ identity_map(codomain(b))
| product(a,codomain(b),a) ),
inference(modus_ponens,[status(thm)],[118,117]) ).
tff(120,plain,
product(a,codomain(b),a),
inference(unit_resolution,[status(thm)],[119,116,106,97]) ).
tff(121,plain,
product(compose(a,domain(a)),codomain(b),compose(a,domain(a))),
inference(modus_ponens,[status(thm)],[120,71]) ).
tff(122,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(123,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)],[122]) ).
tff(124,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(125,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(126,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)],[125]) ).
tff(127,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(128,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[127,126]) ).
tff(129,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[128,124]) ).
tff(130,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(skolemize,[status(sab)],[129]) ).
tff(131,plain,
! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) ),
inference(modus_ponens,[status(thm)],[130,123]) ).
tff(132,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(133,plain,
( ( defined(codomain(b),domain(domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a))) )
<=> ( ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(134,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(codomain(b),domain(domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a))) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) ) ),
inference(monotonicity,[status(thm)],[133]) ).
tff(135,plain,
( ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(codomain(b),domain(domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a))) )
<=> ( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) ) ),
inference(transitivity,[status(thm)],[134,132]) ).
tff(136,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| defined(codomain(b),domain(domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(137,plain,
( ~ ! [Xy: $i,Z: $i,Y: $i,X: $i] :
( defined(Y,Z)
| ~ defined(Xy,Z)
| ~ product(X,Y,Xy) )
| ~ product(compose(a,domain(a)),codomain(b),compose(a,domain(a)))
| ~ defined(compose(a,domain(a)),domain(domain(a)))
| defined(codomain(b),domain(domain(a))) ),
inference(modus_ponens,[status(thm)],[136,135]) ).
tff(138,plain,
defined(codomain(b),domain(domain(a))),
inference(unit_resolution,[status(thm)],[137,131,121,69]) ).
tff(139,plain,
^ [Y: $i,X: $i] :
refl(
( ( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
<=> ( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) )),
inference(bind,[status(th)],]) ).
tff(140,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) ),
inference(quant_intro,[status(thm)],[139]) ).
tff(141,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) ),
inference(rewrite,[status(thm)],]) ).
tff(142,plain,
^ [Y: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ defined(X,Y)
| ~ identity_map(X) )
<=> ( ~ defined(X,Y)
| ~ identity_map(X) ) )),
( ( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) )
<=> ( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) ) )),
rewrite(
( ( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) )
<=> ( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) )),
( ( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) )
<=> ( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) )),
inference(bind,[status(th)],]) ).
tff(143,plain,
( ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) )
<=> ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ) ),
inference(quant_intro,[status(thm)],[142]) ).
tff(144,axiom,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(X)
| product(X,Y,Y) ),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',identity1) ).
tff(145,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ),
inference(modus_ponens,[status(thm)],[144,143]) ).
tff(146,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ),
inference(modus_ponens,[status(thm)],[145,141]) ).
tff(147,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ),
inference(skolemize,[status(sab)],[146]) ).
tff(148,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) ),
inference(modus_ponens,[status(thm)],[147,140]) ).
tff(149,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(150,plain,
( ( ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ identity_map(codomain(b)) )
<=> ( ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(151,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ identity_map(codomain(b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) ) ),
inference(monotonicity,[status(thm)],[150]) ).
tff(152,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ identity_map(codomain(b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) ) ),
inference(transitivity,[status(thm)],[151,149]) ).
tff(153,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ identity_map(codomain(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(154,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(codomain(b))
| product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a))) ),
inference(modus_ponens,[status(thm)],[153,152]) ).
tff(155,plain,
product(codomain(b),domain(domain(a)),domain(domain(a))),
inference(unit_resolution,[status(thm)],[154,148,106,138]) ).
tff(156,plain,
^ [X: $i] :
refl(
( identity_map(domain(X))
<=> identity_map(domain(X)) )),
inference(bind,[status(th)],]) ).
tff(157,plain,
( ! [X: $i] : identity_map(domain(X))
<=> ! [X: $i] : identity_map(domain(X)) ),
inference(quant_intro,[status(thm)],[156]) ).
tff(158,plain,
( ! [X: $i] : identity_map(domain(X))
<=> ! [X: $i] : identity_map(domain(X)) ),
inference(rewrite,[status(thm)],]) ).
tff(159,axiom,
! [X: $i] : identity_map(domain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',domain_is_an_identity_map) ).
tff(160,plain,
! [X: $i] : identity_map(domain(X)),
inference(modus_ponens,[status(thm)],[159,158]) ).
tff(161,plain,
! [X: $i] : identity_map(domain(X)),
inference(skolemize,[status(sab)],[160]) ).
tff(162,plain,
! [X: $i] : identity_map(domain(X)),
inference(modus_ponens,[status(thm)],[161,157]) ).
tff(163,plain,
( ~ ! [X: $i] : identity_map(domain(X))
| identity_map(domain(domain(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(164,plain,
identity_map(domain(domain(a))),
inference(unit_resolution,[status(thm)],[163,162]) ).
tff(165,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(166,plain,
( ( ~ defined(codomain(b),domain(domain(a)))
| ~ identity_map(domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) )
<=> ( ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(167,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(b),domain(domain(a)))
| ~ identity_map(domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ) ),
inference(monotonicity,[status(thm)],[166]) ).
tff(168,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(b),domain(domain(a)))
| ~ identity_map(domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ) ),
inference(transitivity,[status(thm)],[167,165]) ).
tff(169,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(b),domain(domain(a)))
| ~ identity_map(domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(170,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ identity_map(domain(domain(a)))
| ~ defined(codomain(b),domain(domain(a)))
| product(codomain(b),domain(domain(a)),codomain(b)) ),
inference(modus_ponens,[status(thm)],[169,168]) ).
tff(171,plain,
product(codomain(b),domain(domain(a)),codomain(b)),
inference(unit_resolution,[status(thm)],[170,116,164,138]) ).
tff(172,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ product(codomain(b),domain(domain(a)),codomain(b))
| ( domain(domain(a)) = codomain(b) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ product(codomain(b),domain(domain(a)),codomain(b))
| ( domain(domain(a)) = codomain(b) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(173,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ product(codomain(b),domain(domain(a)),codomain(b))
| ( domain(domain(a)) = codomain(b) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(174,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(b),domain(domain(a)),domain(domain(a)))
| ~ product(codomain(b),domain(domain(a)),codomain(b))
| ( domain(domain(a)) = codomain(b) ) ),
inference(modus_ponens,[status(thm)],[173,172]) ).
tff(175,plain,
domain(domain(a)) = codomain(b),
inference(unit_resolution,[status(thm)],[174,41,171,155]) ).
tff(176,plain,
( ~ ! [X: $i] : defined(X,domain(X))
| defined(domain(a),domain(domain(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(177,plain,
defined(domain(a),domain(domain(a))),
inference(unit_resolution,[status(thm)],[176,16]) ).
tff(178,plain,
( ~ ! [X: $i] : identity_map(domain(X))
| identity_map(domain(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(179,plain,
identity_map(domain(a)),
inference(unit_resolution,[status(thm)],[178,162]) ).
tff(180,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(181,plain,
( ( ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a)))
| ~ identity_map(domain(a)) )
<=> ( ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(182,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a)))
| ~ identity_map(domain(a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) ) ),
inference(monotonicity,[status(thm)],[181]) ).
tff(183,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a)))
| ~ identity_map(domain(a)) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) ) ),
inference(transitivity,[status(thm)],[182,180]) ).
tff(184,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a)))
| ~ identity_map(domain(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(185,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| product(X,Y,Y)
| ~ identity_map(X) )
| ~ identity_map(domain(a))
| ~ defined(domain(a),domain(domain(a)))
| product(domain(a),domain(domain(a)),domain(domain(a))) ),
inference(modus_ponens,[status(thm)],[184,183]) ).
tff(186,plain,
product(domain(a),domain(domain(a)),domain(domain(a))),
inference(unit_resolution,[status(thm)],[185,148,179,177]) ).
tff(187,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(domain(a),domain(domain(a)),domain(domain(a)))
| ( domain(a) = domain(domain(a)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(domain(a),domain(domain(a)),domain(domain(a)))
| ( domain(a) = domain(domain(a)) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(188,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(domain(a),domain(domain(a)),domain(domain(a)))
| ( domain(a) = domain(domain(a)) ) ),
inference(quant_inst,[status(thm)],]) ).
tff(189,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(domain(a),domain(domain(a)),domain(a))
| ~ product(domain(a),domain(domain(a)),domain(domain(a)))
| ( domain(a) = domain(domain(a)) ) ),
inference(modus_ponens,[status(thm)],[188,187]) ).
tff(190,plain,
domain(a) = domain(domain(a)),
inference(unit_resolution,[status(thm)],[189,41,9,186]) ).
tff(191,plain,
domain(a) = codomain(b),
inference(transitivity,[status(thm)],[190,175]) ).
tff(192,plain,
( ( domain(a) != codomain(b) )
<=> ( domain(a) != codomain(b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(193,axiom,
domain(a) != codomain(b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_domain_of_a_equals_codomain_of_b) ).
tff(194,plain,
domain(a) != codomain(b),
inference(modus_ponens,[status(thm)],[193,192]) ).
tff(195,plain,
$false,
inference(unit_resolution,[status(thm)],[194,191]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : CAT008-1 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 30 06:17:20 EDT 2022
% 0.12/0.34 % 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.
% 0.20/0.49 % SZS status Unsatisfiable
% 0.20/0.49 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------