TSTP Solution File: GRP012-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP012-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n029.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 : Fri Sep 16 22:25:26 EDT 2022
% Result : Unsatisfiable 0.85s 0.81s
% Output : Proof 0.85s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 57
% Syntax : Number of formulae : 121 ( 47 unt; 6 typ; 0 def)
% Number of atoms : 584 ( 27 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 849 ( 403 ~; 395 |; 0 &)
% ( 51 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 23 ( 23 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 6 ( 3 >; 3 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 457 ( 419 !; 0 ?; 457 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(b_type,type,
b: $i ).
tff(identity_type,type,
identity: $i ).
tff(1,plain,
^ [X: $i] :
refl(
( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(5,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,b,b) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
product(identity,b,b),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [Y: $i,X: $i] :
refl(
( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(14,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(a,b,multiply(a,b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
product(a,b,multiply(a,b)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [X: $i] :
refl(
( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],]) ).
tff(22,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(23,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[22,21]) ).
tff(24,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[24,20]) ).
tff(26,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(27,plain,
product(inverse(a),a,identity),
inference(unit_resolution,[status(thm)],[26,25]) ).
tff(28,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
<=> ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
inference(bind,[status(th)],]) ).
tff(29,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[28]) ).
tff(30,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(rewrite,[status(thm)],]) ).
tff(31,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) ) )),
rewrite(
( ( ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| product(X,V,W) )
<=> ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) )),
inference(bind,[status(th)],]) ).
tff(32,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[31]) ).
tff(33,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(U,Z,W)
| product(X,V,W) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',associativity1) ).
tff(34,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[34,30]) ).
tff(36,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(skolemize,[status(sab)],[35]) ).
tff(37,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) ),
inference(modus_ponens,[status(thm)],[36,29]) ).
tff(38,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(39,plain,
( ( product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b)
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity) )
<=> ( ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(40,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b)
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) ) ),
inference(monotonicity,[status(thm)],[39]) ).
tff(41,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b)
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) ) ),
inference(transitivity,[status(thm)],[40,38]) ).
tff(42,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b)
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(43,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(a,b,multiply(a,b))
| ~ product(inverse(a),a,identity)
| product(inverse(a),multiply(a,b),b)
| ~ product(identity,b,b) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
product(inverse(a),multiply(a,b),b),
inference(unit_resolution,[status(thm)],[43,37,27,18,9]) ).
tff(45,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(inverse(b),inverse(a),multiply(inverse(b),inverse(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(46,plain,
product(inverse(b),inverse(a),multiply(inverse(b),inverse(a))),
inference(unit_resolution,[status(thm)],[45,16]) ).
tff(47,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(48,plain,
product(inverse(b),b,identity),
inference(unit_resolution,[status(thm)],[47,25]) ).
tff(49,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(
( ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(50,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[49]) ).
tff(51,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(rewrite,[status(thm)],]) ).
tff(52,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(
monotonicity(
rewrite(
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W)
| product(U,Z,W) ) )),
rewrite(
( ( ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
( ( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) )),
inference(bind,[status(th)],]) ).
tff(53,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ) ),
inference(quant_intro,[status(thm)],[52]) ).
tff(54,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(Y,Z,V)
| ~ product(X,V,W)
| product(U,Z,W) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',associativity2) ).
tff(55,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[55,51]) ).
tff(57,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(skolemize,[status(sab)],[56]) ).
tff(58,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) ),
inference(modus_ponens,[status(thm)],[57,50]) ).
tff(59,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(inverse(a),multiply(a,b),b)
| ~ product(inverse(b),inverse(a),multiply(inverse(b),inverse(a)))
| ~ product(inverse(b),b,identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(inverse(a),multiply(a,b),b)
| ~ product(inverse(b),inverse(a),multiply(inverse(b),inverse(a)))
| ~ product(inverse(b),b,identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(60,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(inverse(a),multiply(a,b),b)
| ~ product(inverse(b),inverse(a),multiply(inverse(b),inverse(a)))
| ~ product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(61,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(inverse(a),multiply(a,b),b)
| ~ product(inverse(b),inverse(a),multiply(inverse(b),inverse(a)))
| ~ product(inverse(b),b,identity) ),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
product(multiply(inverse(b),inverse(a)),multiply(a,b),identity),
inference(unit_resolution,[status(thm)],[61,58,48,46,44]) ).
tff(63,plain,
^ [X: $i] :
refl(
( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],]) ).
tff(64,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[63]) ).
tff(65,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],]) ).
tff(66,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(67,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[67]) ).
tff(69,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[68,64]) ).
tff(70,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(71,plain,
product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))),
inference(unit_resolution,[status(thm)],[70,69]) ).
tff(72,plain,
^ [X: $i] :
refl(
( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],]) ).
tff(73,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[72]) ).
tff(74,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],]) ).
tff(75,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(76,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[75,74]) ).
tff(77,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[76]) ).
tff(78,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[77,73]) ).
tff(79,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(multiply(a,b),inverse(multiply(a,b)),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(80,plain,
product(multiply(a,b),inverse(multiply(a,b)),identity),
inference(unit_resolution,[status(thm)],[79,78]) ).
tff(81,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(82,plain,
( ( product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))) )
<=> ( ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(83,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) ) ),
inference(monotonicity,[status(thm)],[82]) ).
tff(84,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) ) ),
inference(transitivity,[status(thm)],[83,81]) ).
tff(85,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(86,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(b),inverse(a)),multiply(a,b),identity)
| product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(inverse(b),inverse(a)),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity) ),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a))),
inference(unit_resolution,[status(thm)],[86,58,80,71,62]) ).
tff(88,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))) ),
inference(quant_inst,[status(thm)],]) ).
tff(89,plain,
product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))),
inference(unit_resolution,[status(thm)],[88,69]) ).
tff(90,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(multiply(a,b)),multiply(a,b),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(91,plain,
product(inverse(multiply(a,b)),multiply(a,b),identity),
inference(unit_resolution,[status(thm)],[90,25]) ).
tff(92,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(93,plain,
( ( product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))) )
<=> ( ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(94,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ) ),
inference(monotonicity,[status(thm)],[93]) ).
tff(95,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ) ),
inference(transitivity,[status(thm)],[94,92]) ).
tff(96,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b))) ),
inference(quant_inst,[status(thm)],]) ).
tff(97,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(inverse(multiply(a,b)),identity,inverse(multiply(a,b)))
| product(identity,inverse(multiply(a,b)),inverse(multiply(a,b)))
| ~ product(multiply(a,b),inverse(multiply(a,b)),identity)
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ),
inference(modus_ponens,[status(thm)],[96,95]) ).
tff(98,plain,
product(identity,inverse(multiply(a,b)),inverse(multiply(a,b))),
inference(unit_resolution,[status(thm)],[97,58,91,80,89]) ).
tff(99,plain,
( ( inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)) )
<=> ( inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(100,axiom,
inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_inverse_of_product_is_product_of_inverses) ).
tff(101,plain,
inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)),
inference(modus_ponens,[status(thm)],[100,99]) ).
tff(102,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(
( ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],]) ).
tff(103,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[102]) ).
tff(104,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(rewrite,[status(thm)],]) ).
tff(105,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(
( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],]) ).
tff(106,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] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[105]) ).
tff(107,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/GRP003-0.ax',total_function2) ).
tff(108,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[107,106]) ).
tff(109,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[108,104]) ).
tff(110,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[109]) ).
tff(111,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[110,103]) ).
tff(112,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)) )
| ~ product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(identity,inverse(multiply(a,b)),inverse(multiply(a,b))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)) )
| ~ product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(identity,inverse(multiply(a,b)),inverse(multiply(a,b))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(113,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)) )
| ~ product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(identity,inverse(multiply(a,b)),inverse(multiply(a,b))) ),
inference(quant_inst,[status(thm)],]) ).
tff(114,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)) )
| ~ product(identity,inverse(multiply(a,b)),multiply(inverse(b),inverse(a)))
| ~ product(identity,inverse(multiply(a,b)),inverse(multiply(a,b))) ),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
$false,
inference(unit_resolution,[status(thm)],[114,111,101,98,87]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP012-3 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34 % Computer : n029.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.35 % DateTime : Wed Aug 31 14:14:58 EDT 2022
% 0.13/0.35 % CPUTime :
% 0.13/0.35 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.35 Usage: tptp [options] [-file:]file
% 0.13/0.35 -h, -? prints this message.
% 0.13/0.35 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.35 -m, -model generate model.
% 0.13/0.35 -p, -proof generate proof.
% 0.13/0.35 -c, -core generate unsat core of named formulas.
% 0.13/0.35 -st, -statistics display statistics.
% 0.13/0.35 -t:timeout set timeout (in second).
% 0.13/0.35 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.35 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.35 -<param>:<value> configuration parameter and value.
% 0.13/0.35 -o:<output-file> file to place output in.
% 0.85/0.81 % SZS status Unsatisfiable
% 0.85/0.81 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------