TSTP Solution File: GRP034-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP034-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n017.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 22:25:34 EDT 2022
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 43
% Syntax : Number of formulae : 98 ( 38 unt; 6 typ; 0 def)
% Number of atoms : 408 ( 35 equ)
% Maximal formula atoms : 16 ( 4 avg)
% Number of connectives : 569 ( 267 ~; 258 |; 0 &)
% ( 44 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 14 ( 14 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 7 ( 4 >; 3 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-2 aty)
% Number of variables : 221 ( 202 !; 0 ?; 221 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(identity_type,type,
identity: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(subgroup_member_type,type,
subgroup_member: $i > $o ).
tff(1,plain,
^ [Y: $i,X: $i] :
refl(
( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(2,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)],[1]) ).
tff(3,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(4,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(5,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(inverse(a),identity,multiply(inverse(a),identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
product(inverse(a),identity,multiply(inverse(a),identity)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(
( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],]) ).
tff(11,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],]) ).
tff(13,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(14,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(a),identity,inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(18,plain,
product(inverse(a),identity,inverse(a)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,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(20,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)],[19]) ).
tff(21,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(22,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(23,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)],[22]) ).
tff(24,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(25,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(30,plain,
( ( ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,multiply(inverse(a),identity))
| ~ product(inverse(a),identity,inverse(a)) )
<=> ( ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(31,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,multiply(inverse(a),identity))
| ~ product(inverse(a),identity,inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) ) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,multiply(inverse(a),identity))
| ~ product(inverse(a),identity,inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) ) ),
inference(transitivity,[status(thm)],[31,29]) ).
tff(33,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,multiply(inverse(a),identity))
| ~ product(inverse(a),identity,inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(34,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(a),identity) )
| ~ product(inverse(a),identity,inverse(a))
| ~ product(inverse(a),identity,multiply(inverse(a),identity)) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
inverse(a) = multiply(inverse(a),identity),
inference(unit_resolution,[status(thm)],[34,28,18,9]) ).
tff(36,plain,
( product(identity,inverse(a),multiply(inverse(a),identity))
<=> product(identity,multiply(inverse(a),identity),multiply(inverse(a),identity)) ),
inference(monotonicity,[status(thm)],[35]) ).
tff(37,plain,
( product(identity,multiply(inverse(a),identity),multiply(inverse(a),identity))
<=> product(identity,inverse(a),multiply(inverse(a),identity)) ),
inference(symmetry,[status(thm)],[36]) ).
tff(38,plain,
^ [X: $i] :
refl(
( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],]) ).
tff(39,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(41,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(42,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,multiply(inverse(a),identity),multiply(inverse(a),identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(46,plain,
product(identity,multiply(inverse(a),identity),multiply(inverse(a),identity)),
inference(unit_resolution,[status(thm)],[45,44]) ).
tff(47,plain,
product(identity,inverse(a),multiply(inverse(a),identity)),
inference(modus_ponens,[status(thm)],[46,37]) ).
tff(48,plain,
multiply(inverse(a),identity) = inverse(a),
inference(symmetry,[status(thm)],[35]) ).
tff(49,plain,
( subgroup_member(multiply(inverse(a),identity))
<=> subgroup_member(inverse(a)) ),
inference(monotonicity,[status(thm)],[48]) ).
tff(50,plain,
( subgroup_member(inverse(a))
<=> subgroup_member(multiply(inverse(a),identity)) ),
inference(symmetry,[status(thm)],[49]) ).
tff(51,plain,
( ~ subgroup_member(inverse(a))
<=> ~ subgroup_member(multiply(inverse(a),identity)) ),
inference(monotonicity,[status(thm)],[50]) ).
tff(52,plain,
( ~ subgroup_member(inverse(a))
<=> ~ subgroup_member(inverse(a)) ),
inference(rewrite,[status(thm)],]) ).
tff(53,axiom,
~ subgroup_member(inverse(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_a_inverse_is_in_subgroup) ).
tff(54,plain,
~ subgroup_member(inverse(a)),
inference(modus_ponens,[status(thm)],[53,52]) ).
tff(55,plain,
~ subgroup_member(multiply(inverse(a),identity)),
inference(modus_ponens,[status(thm)],[54,51]) ).
tff(56,plain,
^ [X: $i] :
refl(
( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],]) ).
tff(57,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[56]) ).
tff(58,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],]) ).
tff(59,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(60,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[59,58]) ).
tff(61,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[60]) ).
tff(62,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[61,57]) ).
tff(63,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(64,plain,
product(a,inverse(a),identity),
inference(unit_resolution,[status(thm)],[63,62]) ).
tff(65,plain,
( subgroup_member(a)
<=> subgroup_member(a) ),
inference(rewrite,[status(thm)],]) ).
tff(66,axiom,
subgroup_member(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_is_in_subgroup) ).
tff(67,plain,
subgroup_member(a),
inference(modus_ponens,[status(thm)],[66,65]) ).
tff(68,plain,
^ [B: $i,A: $i,C: $i] :
refl(
( ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
inference(bind,[status(th)],]) ).
tff(69,plain,
( ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(quant_intro,[status(thm)],[68]) ).
tff(70,plain,
( ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(rewrite,[status(thm)],]) ).
tff(71,plain,
^ [B: $i,A: $i,C: $i] :
trans(
monotonicity(
rewrite(
( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C) )
<=> ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A)
| subgroup_member(C) ) )),
rewrite(
( ( ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A)
| subgroup_member(C) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
( ( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) )),
inference(bind,[status(th)],]) ).
tff(72,plain,
( ! [B: $i,A: $i,C: $i] :
( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) )
<=> ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ) ),
inference(quant_intro,[status(thm)],[71]) ).
tff(73,axiom,
! [B: $i,A: $i,C: $i] :
( ~ subgroup_member(A)
| ~ subgroup_member(B)
| ~ product(A,inverse(B),C)
| subgroup_member(C) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-2.ax',closure_of_product_and_inverse) ).
tff(74,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[74,70]) ).
tff(76,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(skolemize,[status(sab)],[75]) ).
tff(77,plain,
! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) ),
inference(modus_ponens,[status(thm)],[76,69]) ).
tff(78,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(79,plain,
( ( subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(80,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(monotonicity,[status(thm)],[79]) ).
tff(81,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ) ),
inference(transitivity,[status(thm)],[80,78]) ).
tff(82,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a)
| ~ subgroup_member(a) ),
inference(quant_inst,[status(thm)],]) ).
tff(83,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(a,inverse(a),identity)
| ~ subgroup_member(a) ),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
subgroup_member(identity),
inference(unit_resolution,[status(thm)],[83,77,67,64]) ).
tff(85,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(86,plain,
( ( subgroup_member(multiply(inverse(a),identity))
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(87,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(a),identity))
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) ) ),
inference(monotonicity,[status(thm)],[86]) ).
tff(88,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(a),identity))
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) ) ),
inference(transitivity,[status(thm)],[87,85]) ).
tff(89,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(a),identity))
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(a)
| ~ subgroup_member(identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(90,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(a)
| ~ product(identity,inverse(a),multiply(inverse(a),identity))
| ~ subgroup_member(identity)
| subgroup_member(multiply(inverse(a),identity)) ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
( ~ product(identity,inverse(a),multiply(inverse(a),identity))
| subgroup_member(multiply(inverse(a),identity)) ),
inference(unit_resolution,[status(thm)],[90,77,67,84]) ).
tff(92,plain,
$false,
inference(unit_resolution,[status(thm)],[91,55,47]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.12 % Problem : GRP034-3 : TPTP v8.1.0. Released v1.0.0.
% 0.08/0.13 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n017.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
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Aug 31 13:39:51 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------