%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : CAT014-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:00:23 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 27
% Syntax : Number of formulae : 61 ( 26 unt; 0 typ; 0 def)
% Number of atoms : 239 ( 37 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 308 ( 143 ~; 137 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 13 ( 13 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 9 ( 7 usr; 1 prp; 0-3 aty)
% Number of functors : 2 ( 2 usr; 1 con; 0-1 aty)
% Number of variables : 148 ( 133 !; 0 ?; 148 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(codomain_type,type,
codomain: $i > $i ).
tff(a_type,type,
a: $i ).
tff(defined_type,type,
defined: ( $i * $i ) > $o ).
tff(identity_map_type,type,
identity_map: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( defined(codomain(X),X)
<=> defined(codomain(X),X) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : defined(codomain(X),X)
<=> ! [X: $i] : defined(codomain(X),X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : defined(codomain(X),X)
<=> ! [X: $i] : defined(codomain(X),X) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : defined(codomain(X),X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',mapping_from_codomain_of_x_to_x) ).
tff(5,plain,
! [X: $i] : defined(codomain(X),X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : defined(codomain(X),X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : defined(codomain(X),X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : defined(codomain(X),X)
| defined(codomain(codomain(a)),codomain(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
defined(codomain(codomain(a)),codomain(a)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( identity_map(codomain(X))
<=> identity_map(codomain(X)) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : identity_map(codomain(X))
<=> ! [X: $i] : identity_map(codomain(X)) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : identity_map(codomain(X)),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',codomain_is_an_identity_map) ).
tff(14,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : identity_map(codomain(X)),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : identity_map(codomain(X)),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : identity_map(codomain(X))
| identity_map(codomain(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
identity_map(codomain(a)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,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)],[22]) ).
tff(24,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(25,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(codomain(a)),codomain(a))
| ~ identity_map(codomain(a))
| product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(codomain(a)),codomain(a))
| ~ identity_map(codomain(a))
| product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(codomain(a)),codomain(a))
| ~ identity_map(codomain(a))
| product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(31,plain,
( ~ ! [Y: $i,X: $i] :
( ~ defined(X,Y)
| ~ identity_map(Y)
| product(X,Y,X) )
| ~ defined(codomain(codomain(a)),codomain(a))
| ~ identity_map(codomain(a))
| product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
product(codomain(codomain(a)),codomain(a),codomain(codomain(a))),
inference(unit_resolution,[status(thm)],[31,28,18,9]) ).
tff(33,plain,
^ [X: $i] :
refl(( product(codomain(X),X,X)
<=> product(codomain(X),X,X) )),
inference(bind,[status(th)],[]) ).
tff(34,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,plain,
( ! [X: $i] : product(codomain(X),X,X)
<=> ! [X: $i] : product(codomain(X),X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,axiom,
! [X: $i] : product(codomain(X),X,X),
file('/export/starexec/sandbox/benchmark/Axioms/CAT001-0.ax',product_on_codomain) ).
tff(37,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
! [X: $i] : product(codomain(X),X,X),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [X: $i] : product(codomain(X),X,X),
inference(modus_ponens,[status(thm)],[38,34]) ).
tff(40,plain,
( ~ ! [X: $i] : product(codomain(X),X,X)
| product(codomain(codomain(a)),codomain(a),codomain(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(41,plain,
product(codomain(codomain(a)),codomain(a),codomain(a)),
inference(unit_resolution,[status(thm)],[40,39]) ).
tff(42,plain,
( ( codomain(codomain(a)) != codomain(a) )
<=> ( codomain(codomain(a)) != codomain(a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,axiom,
codomain(codomain(a)) != codomain(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_codomain_is_idempotent) ).
tff(44,plain,
codomain(codomain(a)) != codomain(a),
inference(modus_ponens,[status(thm)],[43,42]) ).
tff(45,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(46,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)],[45]) ).
tff(47,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(48,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(49,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)],[48]) ).
tff(50,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(51,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[51,47]) ).
tff(53,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(skolemize,[status(sab)],[52]) ).
tff(54,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
inference(modus_ponens,[status(thm)],[53,46]) ).
tff(55,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(56,plain,
( ( ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a)))
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ( codomain(codomain(a)) = codomain(a) ) )
<=> ( ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(57,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a)))
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ( codomain(codomain(a)) = codomain(a) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ) ),
inference(monotonicity,[status(thm)],[56]) ).
tff(58,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a)))
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ( codomain(codomain(a)) = codomain(a) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ) ),
inference(transitivity,[status(thm)],[57,55]) ).
tff(59,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a)))
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ( codomain(codomain(a)) = codomain(a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(60,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
| ( codomain(codomain(a)) = codomain(a) )
| ~ product(codomain(codomain(a)),codomain(a),codomain(a))
| ~ product(codomain(codomain(a)),codomain(a),codomain(codomain(a))) ),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
$false,
inference(unit_resolution,[status(thm)],[60,54,44,41,32]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : CAT014-1 : TPTP v9.0.0. Released v1.0.0.
% 0.00/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Jun 19 11:54:04 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.19/0.37 % SZS status Unsatisfiable
% 0.19/0.37 % SZS output start Proof
% See solution above
% 0.19/0.38 % E exiting
%------------------------------------------------------------------------------