TSTP Solution File: GRP040-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP040-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n024.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 22:25:37 EDT 2022
% Result : Unsatisfiable 2.32s 1.71s
% Output : Proof 2.32s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 125
% Syntax : Number of formulae : 296 ( 94 unt; 10 typ; 0 def)
% Number of atoms : 1621 ( 91 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 2304 (1016 ~;1147 |; 0 &)
% ( 141 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 47 ( 47 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 9 ( 5 >; 4 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 8 ( 8 usr; 5 con; 0-2 aty)
% Number of variables : 892 ( 840 !; 0 ?; 892 :)
% Comments :
%------------------------------------------------------------------------------
tff(subgroup_member_type,type,
subgroup_member: $i > $o ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(c_type,type,
c: $i ).
tff(element_in_O2_type,type,
element_in_O2: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(d_type,type,
d: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(identity_type,type,
identity: $i ).
tff(b_type,type,
b: $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(1,plain,
^ [X: $i] :
refl(
( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],]) ).
tff(2,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],]) ).
tff(4,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(5,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(c),identity,inverse(c)) ),
inference(quant_inst,[status(thm)],]) ).
tff(9,plain,
product(inverse(c),identity,inverse(c)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( ~ subgroup_member(d)
<=> ~ subgroup_member(d) ),
inference(rewrite,[status(thm)],]) ).
tff(11,axiom,
~ subgroup_member(d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d_in_subgroup) ).
tff(12,plain,
~ subgroup_member(d),
inference(modus_ponens,[status(thm)],[11,10]) ).
tff(13,plain,
( ~ subgroup_member(a)
<=> ~ subgroup_member(a) ),
inference(rewrite,[status(thm)],]) ).
tff(14,axiom,
~ subgroup_member(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_in_subgroup) ).
tff(15,plain,
~ subgroup_member(a),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
^ [B: $i,A: $i] :
refl(
( ( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
<=> ( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) )),
inference(bind,[status(th)],]) ).
tff(17,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) ),
inference(rewrite,[status(thm)],]) ).
tff(19,plain,
^ [B: $i,A: $i] :
trans(
monotonicity(
rewrite(
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
<=> ( product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) )),
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) ) )),
rewrite(
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) )),
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) )),
inference(bind,[status(th)],]) ).
tff(20,plain,
( ! [B: $i,A: $i] :
( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,axiom,
! [B: $i,A: $i] :
( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',property_of_O2) ).
tff(22,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[21,20]) ).
tff(23,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[22,18]) ).
tff(24,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ),
inference(skolemize,[status(sab)],[23]) ).
tff(25,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[24,17]) ).
tff(26,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(27,plain,
( ( subgroup_member(d)
| product(d,element_in_O2(d,a),a)
| subgroup_member(a) )
<=> ( subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(28,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(d)
| product(d,element_in_O2(d,a),a)
| subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) ) ),
inference(monotonicity,[status(thm)],[27]) ).
tff(29,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(d)
| product(d,element_in_O2(d,a),a)
| subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) ) ),
inference(transitivity,[status(thm)],[28,26]) ).
tff(30,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(d)
| product(d,element_in_O2(d,a),a)
| subgroup_member(a) ),
inference(quant_inst,[status(thm)],]) ).
tff(31,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| product(d,element_in_O2(d,a),a) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
product(d,element_in_O2(d,a),a),
inference(unit_resolution,[status(thm)],[31,25,15,12]) ).
tff(33,plain,
^ [X: $i] :
refl(
( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],]) ).
tff(34,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[33]) ).
tff(35,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],]) ).
tff(36,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(37,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[37]) ).
tff(39,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[38,34]) ).
tff(40,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,c,c) ),
inference(quant_inst,[status(thm)],]) ).
tff(41,plain,
product(identity,c,c),
inference(unit_resolution,[status(thm)],[40,39]) ).
tff(42,plain,
^ [X: $i] :
refl(
( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],]) ).
tff(43,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[42]) ).
tff(44,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],]) ).
tff(45,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(46,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[46]) ).
tff(48,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[47,43]) ).
tff(49,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(50,plain,
product(inverse(a),a,identity),
inference(unit_resolution,[status(thm)],[49,48]) ).
tff(51,plain,
( product(a,c,d)
<=> product(a,c,d) ),
inference(rewrite,[status(thm)],]) ).
tff(52,axiom,
product(a,c,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_times_c_is_d) ).
tff(53,plain,
product(a,c,d),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,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(55,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)],[54]) ).
tff(56,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(57,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(58,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)],[57]) ).
tff(59,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(60,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)],[59,58]) ).
tff(61,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)],[60,56]) ).
tff(62,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)],[61]) ).
tff(63,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)],[62,55]) ).
tff(64,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),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) )
<=> ( ~ ! [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),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(65,plain,
( ( product(inverse(a),d,c)
| ~ product(identity,c,c)
| ~ product(a,c,d)
| ~ product(inverse(a),a,identity) )
<=> ( ~ product(inverse(a),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(66,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),d,c)
| ~ product(identity,c,c)
| ~ product(a,c,d)
| ~ 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(inverse(a),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) ) ),
inference(monotonicity,[status(thm)],[65]) ).
tff(67,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),d,c)
| ~ product(identity,c,c)
| ~ product(a,c,d)
| ~ 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(inverse(a),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) ) ),
inference(transitivity,[status(thm)],[66,64]) ).
tff(68,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),d,c)
| ~ product(identity,c,c)
| ~ product(a,c,d)
| ~ product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(69,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),a,identity)
| product(inverse(a),d,c)
| ~ product(a,c,d)
| ~ product(identity,c,c) ),
inference(modus_ponens,[status(thm)],[68,67]) ).
tff(70,plain,
product(inverse(a),d,c),
inference(unit_resolution,[status(thm)],[69,63,53,50,41]) ).
tff(71,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(72,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)],[71]) ).
tff(73,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(74,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(75,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)],[74]) ).
tff(76,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(77,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)],[76,75]) ).
tff(78,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)],[77,73]) ).
tff(79,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)],[78]) ).
tff(80,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)],[79,72]) ).
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(inverse(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) )
<=> ( ~ ! [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(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) ) ),
inference(rewrite,[status(thm)],]) ).
tff(82,plain,
( ( product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c)
| ~ product(inverse(a),a,identity) )
<=> ( ~ product(inverse(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) ) ),
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(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c)
| ~ product(inverse(a),a,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(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) ) ),
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(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c)
| ~ product(inverse(a),a,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(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) ) ),
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(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c)
| ~ product(inverse(a),a,identity) ),
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(inverse(a),a,identity)
| product(c,element_in_O2(d,a),identity)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(inverse(a),d,c) ),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
product(c,element_in_O2(d,a),identity),
inference(unit_resolution,[status(thm)],[86,80,50,70,32]) ).
tff(88,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,element_in_O2(d,a),element_in_O2(d,a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(89,plain,
product(identity,element_in_O2(d,a),element_in_O2(d,a)),
inference(unit_resolution,[status(thm)],[88,39]) ).
tff(90,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(c),c,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(91,plain,
product(inverse(c),c,identity),
inference(unit_resolution,[status(thm)],[90,48]) ).
tff(92,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(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) )
<=> ( ~ ! [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(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(93,plain,
( ( product(inverse(c),identity,element_in_O2(d,a))
| ~ product(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity) )
<=> ( ~ product(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(94,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(c),identity,element_in_O2(d,a))
| ~ product(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,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(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) ) ),
inference(monotonicity,[status(thm)],[93]) ).
tff(95,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(c),identity,element_in_O2(d,a))
| ~ product(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,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(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) ) ),
inference(transitivity,[status(thm)],[94,92]) ).
tff(96,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(c),identity,element_in_O2(d,a))
| ~ product(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(97,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(identity,element_in_O2(d,a),element_in_O2(d,a))
| ~ product(c,element_in_O2(d,a),identity)
| ~ product(inverse(c),c,identity)
| product(inverse(c),identity,element_in_O2(d,a)) ),
inference(modus_ponens,[status(thm)],[96,95]) ).
tff(98,plain,
product(inverse(c),identity,element_in_O2(d,a)),
inference(unit_resolution,[status(thm)],[97,63,91,89,87]) ).
tff(99,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(100,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)],[99]) ).
tff(101,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(102,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(103,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)],[102]) ).
tff(104,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(105,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[104,103]) ).
tff(106,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[105,101]) ).
tff(107,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[106]) ).
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)],[107,100]) ).
tff(109,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(110,plain,
( ( ( inverse(c) = element_in_O2(d,a) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c)) )
<=> ( ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) ) ),
inference(rewrite,[status(thm)],]) ).
tff(111,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(c) = element_in_O2(d,a) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) ) ),
inference(monotonicity,[status(thm)],[110]) ).
tff(112,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(c) = element_in_O2(d,a) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) ) ),
inference(transitivity,[status(thm)],[111,109]) ).
tff(113,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(c) = element_in_O2(d,a) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c)) ),
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) )
| ~ product(inverse(c),identity,element_in_O2(d,a))
| ~ product(inverse(c),identity,inverse(c))
| ( inverse(c) = element_in_O2(d,a) ) ),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
inverse(c) = element_in_O2(d,a),
inference(unit_resolution,[status(thm)],[114,108,98,9]) ).
tff(116,plain,
( subgroup_member(inverse(c))
<=> subgroup_member(element_in_O2(d,a)) ),
inference(monotonicity,[status(thm)],[115]) ).
tff(117,plain,
( subgroup_member(element_in_O2(d,a))
<=> subgroup_member(inverse(c)) ),
inference(symmetry,[status(thm)],[116]) ).
tff(118,plain,
^ [B: $i,A: $i] :
refl(
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
<=> ( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) )),
inference(bind,[status(th)],]) ).
tff(119,plain,
( ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) ),
inference(quant_intro,[status(thm)],[118]) ).
tff(120,plain,
( ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) ),
inference(rewrite,[status(thm)],]) ).
tff(121,plain,
^ [B: $i,A: $i] :
trans(
monotonicity(
rewrite(
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B) )
<=> ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B) ) )),
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) ) )),
rewrite(
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) )),
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) )),
inference(bind,[status(th)],]) ).
tff(122,plain,
( ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ) ),
inference(quant_intro,[status(thm)],[121]) ).
tff(123,axiom,
! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',an_element_in_O2) ).
tff(124,plain,
! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[123,122]) ).
tff(125,plain,
! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[124,120]) ).
tff(126,plain,
! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ),
inference(skolemize,[status(sab)],[125]) ).
tff(127,plain,
! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) ),
inference(modus_ponens,[status(thm)],[126,119]) ).
tff(128,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(129,plain,
( ( subgroup_member(element_in_O2(d,a))
| subgroup_member(d)
| subgroup_member(a) )
<=> ( subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(130,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(element_in_O2(d,a))
| subgroup_member(d)
| subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) ) ),
inference(monotonicity,[status(thm)],[129]) ).
tff(131,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(element_in_O2(d,a))
| subgroup_member(d)
| subgroup_member(a) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) ) ),
inference(transitivity,[status(thm)],[130,128]) ).
tff(132,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(element_in_O2(d,a))
| subgroup_member(d)
| subgroup_member(a) ),
inference(quant_inst,[status(thm)],]) ).
tff(133,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(A)
| subgroup_member(B) )
| subgroup_member(a)
| subgroup_member(d)
| subgroup_member(element_in_O2(d,a)) ),
inference(modus_ponens,[status(thm)],[132,131]) ).
tff(134,plain,
subgroup_member(element_in_O2(d,a)),
inference(unit_resolution,[status(thm)],[133,127,15,12]) ).
tff(135,plain,
subgroup_member(inverse(c)),
inference(modus_ponens,[status(thm)],[134,117]) ).
tff(136,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(b),inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(137,plain,
product(identity,inverse(b),inverse(b)),
inference(unit_resolution,[status(thm)],[136,39]) ).
tff(138,plain,
^ [X: $i] :
refl(
( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],]) ).
tff(139,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[138]) ).
tff(140,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],]) ).
tff(141,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(142,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[141,140]) ).
tff(143,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[142]) ).
tff(144,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[143,139]) ).
tff(145,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(b,inverse(b),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(146,plain,
product(b,inverse(b),identity),
inference(unit_resolution,[status(thm)],[145,144]) ).
tff(147,plain,
( subgroup_member(b)
<=> subgroup_member(b) ),
inference(rewrite,[status(thm)],]) ).
tff(148,axiom,
subgroup_member(b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_is_in_subgroup) ).
tff(149,plain,
subgroup_member(b),
inference(modus_ponens,[status(thm)],[148,147]) ).
tff(150,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(151,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)],[150]) ).
tff(152,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(153,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(154,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)],[153]) ).
tff(155,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(156,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)],[155,154]) ).
tff(157,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)],[156,152]) ).
tff(158,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)],[157]) ).
tff(159,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)],[158,151]) ).
tff(160,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(b,inverse(b),identity)
| ~ subgroup_member(b) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(b,inverse(b),identity)
| ~ subgroup_member(b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(161,plain,
( ( subgroup_member(identity)
| ~ product(b,inverse(b),identity)
| ~ subgroup_member(b)
| ~ subgroup_member(b) )
<=> ( subgroup_member(identity)
| ~ product(b,inverse(b),identity)
| ~ subgroup_member(b) ) ),
inference(rewrite,[status(thm)],]) ).
tff(162,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(b,inverse(b),identity)
| ~ subgroup_member(b)
| ~ subgroup_member(b) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(b,inverse(b),identity)
| ~ subgroup_member(b) ) ),
inference(monotonicity,[status(thm)],[161]) ).
tff(163,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(b,inverse(b),identity)
| ~ subgroup_member(b)
| ~ subgroup_member(b) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(identity)
| ~ product(b,inverse(b),identity)
| ~ subgroup_member(b) ) ),
inference(transitivity,[status(thm)],[162,160]) ).
tff(164,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(b,inverse(b),identity)
| ~ subgroup_member(b)
| ~ subgroup_member(b) ),
inference(quant_inst,[status(thm)],]) ).
tff(165,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(b,inverse(b),identity)
| ~ subgroup_member(b) ),
inference(modus_ponens,[status(thm)],[164,163]) ).
tff(166,plain,
subgroup_member(identity),
inference(unit_resolution,[status(thm)],[165,159,149,146]) ).
tff(167,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(168,plain,
( ( subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b))
| ~ subgroup_member(b)
| ~ subgroup_member(identity) )
<=> ( ~ subgroup_member(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(169,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b))
| ~ subgroup_member(b)
| ~ 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(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) ) ),
inference(monotonicity,[status(thm)],[168]) ).
tff(170,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b))
| ~ subgroup_member(b)
| ~ 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(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) ) ),
inference(transitivity,[status(thm)],[169,167]) ).
tff(171,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b))
| ~ subgroup_member(b)
| ~ subgroup_member(identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(172,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| ~ subgroup_member(identity)
| ~ subgroup_member(b)
| subgroup_member(inverse(b))
| ~ product(identity,inverse(b),inverse(b)) ),
inference(modus_ponens,[status(thm)],[171,170]) ).
tff(173,plain,
subgroup_member(inverse(b)),
inference(unit_resolution,[status(thm)],[172,159,149,166,137]) ).
tff(174,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(identity),inverse(identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(175,plain,
product(identity,inverse(identity),inverse(identity)),
inference(unit_resolution,[status(thm)],[174,39]) ).
tff(176,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(identity,inverse(identity),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(177,plain,
product(identity,inverse(identity),identity),
inference(unit_resolution,[status(thm)],[176,144]) ).
tff(178,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(179,plain,
( ( ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),identity)
| ~ product(identity,inverse(identity),inverse(identity)) )
<=> ( ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(180,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),identity)
| ~ product(identity,inverse(identity),inverse(identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) ) ),
inference(monotonicity,[status(thm)],[179]) ).
tff(181,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),identity)
| ~ product(identity,inverse(identity),inverse(identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) ) ),
inference(transitivity,[status(thm)],[180,178]) ).
tff(182,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),identity)
| ~ product(identity,inverse(identity),inverse(identity)) ),
inference(quant_inst,[status(thm)],]) ).
tff(183,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(identity),identity)
| ( inverse(identity) = identity )
| ~ product(identity,inverse(identity),inverse(identity)) ),
inference(modus_ponens,[status(thm)],[182,181]) ).
tff(184,plain,
inverse(identity) = identity,
inference(unit_resolution,[status(thm)],[183,108,177,175]) ).
tff(185,plain,
identity = inverse(identity),
inference(symmetry,[status(thm)],[184]) ).
tff(186,plain,
( product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
<=> product(inverse(identity),inverse(a),multiply(inverse(identity),inverse(a))) ),
inference(monotonicity,[status(thm)],[185]) ).
tff(187,plain,
( product(inverse(identity),inverse(a),multiply(inverse(identity),inverse(a)))
<=> product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ),
inference(symmetry,[status(thm)],[186]) ).
tff(188,plain,
^ [Y: $i,X: $i] :
refl(
( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],]) ).
tff(189,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)],[188]) ).
tff(190,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(191,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(192,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[191,190]) ).
tff(193,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[192]) ).
tff(194,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[193,189]) ).
tff(195,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(inverse(identity),inverse(a),multiply(inverse(identity),inverse(a))) ),
inference(quant_inst,[status(thm)],]) ).
tff(196,plain,
product(inverse(identity),inverse(a),multiply(inverse(identity),inverse(a))),
inference(unit_resolution,[status(thm)],[195,194]) ).
tff(197,plain,
product(identity,inverse(a),multiply(inverse(identity),inverse(a))),
inference(modus_ponens,[status(thm)],[196,187]) ).
tff(198,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(199,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[198,39]) ).
tff(200,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(201,plain,
( ( ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ) ),
inference(rewrite,[status(thm)],]) ).
tff(202,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ) ),
inference(monotonicity,[status(thm)],[201]) ).
tff(203,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ) ),
inference(transitivity,[status(thm)],[202,200]) ).
tff(204,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],]) ).
tff(205,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(a),inverse(a))
| ( inverse(a) = multiply(inverse(identity),inverse(a)) )
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a))) ),
inference(modus_ponens,[status(thm)],[204,203]) ).
tff(206,plain,
inverse(a) = multiply(inverse(identity),inverse(a)),
inference(unit_resolution,[status(thm)],[205,108,199,197]) ).
tff(207,plain,
( subgroup_member(inverse(a))
<=> subgroup_member(multiply(inverse(identity),inverse(a))) ),
inference(monotonicity,[status(thm)],[206]) ).
tff(208,plain,
( ~ subgroup_member(inverse(a))
<=> ~ subgroup_member(multiply(inverse(identity),inverse(a))) ),
inference(monotonicity,[status(thm)],[207]) ).
tff(209,plain,
( ! [A: $i] : ( inverse(inverse(A)) = A )
<=> ! [A: $i] : ( inverse(inverse(A)) = A ) ),
inference(rewrite,[status(thm)],]) ).
tff(210,plain,
( ! [A: $i] : ( inverse(inverse(A)) = A )
<=> ! [A: $i] : ( inverse(inverse(A)) = A ) ),
inference(rewrite,[status(thm)],]) ).
tff(211,axiom,
! [A: $i] : ( inverse(inverse(A)) = A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_inverse_is_self_cancelling) ).
tff(212,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(modus_ponens,[status(thm)],[211,210]) ).
tff(213,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(skolemize,[status(sab)],[212]) ).
tff(214,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(modus_ponens,[status(thm)],[213,209]) ).
tff(215,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(a)) = a ) ),
inference(quant_inst,[status(thm)],]) ).
tff(216,plain,
inverse(inverse(a)) = a,
inference(unit_resolution,[status(thm)],[215,214]) ).
tff(217,plain,
( product(identity,inverse(inverse(a)),a)
<=> product(identity,a,a) ),
inference(monotonicity,[status(thm)],[216]) ).
tff(218,plain,
( product(identity,a,a)
<=> product(identity,inverse(inverse(a)),a) ),
inference(symmetry,[status(thm)],[217]) ).
tff(219,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(d)) = d ) ),
inference(quant_inst,[status(thm)],]) ).
tff(220,plain,
inverse(inverse(d)) = d,
inference(unit_resolution,[status(thm)],[219,214]) ).
tff(221,plain,
d = inverse(inverse(d)),
inference(symmetry,[status(thm)],[220]) ).
tff(222,plain,
( product(inverse(d),d,identity)
<=> product(inverse(d),inverse(inverse(d)),identity) ),
inference(monotonicity,[status(thm)],[221]) ).
tff(223,plain,
( product(inverse(d),inverse(inverse(d)),identity)
<=> product(inverse(d),d,identity) ),
inference(symmetry,[status(thm)],[222]) ).
tff(224,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(inverse(d),inverse(inverse(d)),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(225,plain,
product(inverse(d),inverse(inverse(d)),identity),
inference(unit_resolution,[status(thm)],[224,144]) ).
tff(226,plain,
product(inverse(d),d,identity),
inference(modus_ponens,[status(thm)],[225,223]) ).
tff(227,plain,
( product(d,inverse(d),identity)
<=> product(inverse(inverse(d)),inverse(d),identity) ),
inference(monotonicity,[status(thm)],[221]) ).
tff(228,plain,
( product(inverse(inverse(d)),inverse(d),identity)
<=> product(d,inverse(d),identity) ),
inference(symmetry,[status(thm)],[227]) ).
tff(229,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(inverse(d)),inverse(d),identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(230,plain,
product(inverse(inverse(d)),inverse(d),identity),
inference(unit_resolution,[status(thm)],[229,48]) ).
tff(231,plain,
product(d,inverse(d),identity),
inference(modus_ponens,[status(thm)],[230,228]) ).
tff(232,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(d,identity,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(233,plain,
product(d,identity,d),
inference(unit_resolution,[status(thm)],[232,7]) ).
tff(234,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,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) )
<=> ( ~ ! [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,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(235,plain,
( ( product(identity,d,d)
| ~ product(inverse(d),d,identity)
| ~ product(d,inverse(d),identity)
| ~ product(d,identity,d) )
<=> ( product(identity,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) ) ),
inference(rewrite,[status(thm)],]) ).
tff(236,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,d,d)
| ~ product(inverse(d),d,identity)
| ~ product(d,inverse(d),identity)
| ~ product(d,identity,d) )
<=> ( ~ ! [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,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) ) ),
inference(monotonicity,[status(thm)],[235]) ).
tff(237,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,d,d)
| ~ product(inverse(d),d,identity)
| ~ product(d,inverse(d),identity)
| ~ product(d,identity,d) )
<=> ( ~ ! [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,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) ) ),
inference(transitivity,[status(thm)],[236,234]) ).
tff(238,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,d,d)
| ~ product(inverse(d),d,identity)
| ~ product(d,inverse(d),identity)
| ~ product(d,identity,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(239,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,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity)
| ~ product(d,identity,d) ),
inference(modus_ponens,[status(thm)],[238,237]) ).
tff(240,plain,
( product(identity,d,d)
| ~ product(d,inverse(d),identity)
| ~ product(inverse(d),d,identity) ),
inference(unit_resolution,[status(thm)],[239,80,233]) ).
tff(241,plain,
product(identity,d,d),
inference(unit_resolution,[status(thm)],[240,231,226]) ).
tff(242,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(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) )
<=> ( ~ ! [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(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(243,plain,
( ( product(identity,a,a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(identity,d,d) )
<=> ( ~ product(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(244,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(identity,a,a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(identity,d,d) )
<=> ( ~ ! [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(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) ) ),
inference(monotonicity,[status(thm)],[243]) ).
tff(245,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(identity,a,a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(identity,d,d) )
<=> ( ~ ! [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(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) ) ),
inference(transitivity,[status(thm)],[244,242]) ).
tff(246,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(identity,a,a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(d,element_in_O2(d,a),a)
| ~ product(identity,d,d) ),
inference(quant_inst,[status(thm)],]) ).
tff(247,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(d,element_in_O2(d,a),a)
| ~ product(identity,d,d)
| product(identity,a,a) ),
inference(modus_ponens,[status(thm)],[246,245]) ).
tff(248,plain,
product(identity,a,a),
inference(unit_resolution,[status(thm)],[247,63,241,32]) ).
tff(249,plain,
product(identity,inverse(inverse(a)),a),
inference(modus_ponens,[status(thm)],[248,218]) ).
tff(250,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(251,plain,
( ( subgroup_member(a)
| ~ product(identity,inverse(inverse(a)),a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity) )
<=> ( subgroup_member(a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) ) ),
inference(rewrite,[status(thm)],]) ).
tff(252,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(inverse(a)),a)
| ~ subgroup_member(inverse(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)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) ) ),
inference(monotonicity,[status(thm)],[251]) ).
tff(253,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(inverse(a)),a)
| ~ subgroup_member(inverse(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)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) ) ),
inference(transitivity,[status(thm)],[252,250]) ).
tff(254,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(inverse(a)),a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(255,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(a)
| ~ subgroup_member(inverse(a))
| ~ subgroup_member(identity)
| ~ product(identity,inverse(inverse(a)),a) ),
inference(modus_ponens,[status(thm)],[254,253]) ).
tff(256,plain,
( ~ subgroup_member(inverse(a))
| ~ product(identity,inverse(inverse(a)),a) ),
inference(unit_resolution,[status(thm)],[255,159,15,166]) ).
tff(257,plain,
~ subgroup_member(inverse(a)),
inference(unit_resolution,[status(thm)],[256,249]) ).
tff(258,plain,
~ subgroup_member(multiply(inverse(identity),inverse(a))),
inference(modus_ponens,[status(thm)],[257,208]) ).
tff(259,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(c)) = c ) ),
inference(quant_inst,[status(thm)],]) ).
tff(260,plain,
inverse(inverse(c)) = c,
inference(unit_resolution,[status(thm)],[259,214]) ).
tff(261,plain,
( product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
<=> product(inverse(b),c,multiply(inverse(identity),inverse(a))) ),
inference(monotonicity,[status(thm)],[260]) ).
tff(262,plain,
( product(inverse(b),c,multiply(inverse(identity),inverse(a)))
<=> product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a))) ),
inference(symmetry,[status(thm)],[261]) ).
tff(263,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(264,plain,
product(inverse(b),b,identity),
inference(unit_resolution,[status(thm)],[263,48]) ).
tff(265,plain,
( product(b,inverse(a),c)
<=> product(b,inverse(a),c) ),
inference(rewrite,[status(thm)],]) ).
tff(266,axiom,
product(b,inverse(a),c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_times_a_inverse_is_c) ).
tff(267,plain,
product(b,inverse(a),c),
inference(modus_ponens,[status(thm)],[266,265]) ).
tff(268,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(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,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(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(269,plain,
( ( product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) )
<=> ( ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ) ),
inference(rewrite,[status(thm)],]) ).
tff(270,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(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,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(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ) ),
inference(monotonicity,[status(thm)],[269]) ).
tff(271,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(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,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(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ) ),
inference(transitivity,[status(thm)],[270,268]) ).
tff(272,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(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],]) ).
tff(273,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(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a)))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ),
inference(modus_ponens,[status(thm)],[272,271]) ).
tff(274,plain,
( ~ product(identity,inverse(a),multiply(inverse(identity),inverse(a)))
| product(inverse(b),c,multiply(inverse(identity),inverse(a))) ),
inference(unit_resolution,[status(thm)],[273,63,267,264]) ).
tff(275,plain,
product(inverse(b),c,multiply(inverse(identity),inverse(a))),
inference(unit_resolution,[status(thm)],[274,197]) ).
tff(276,plain,
product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a))),
inference(modus_ponens,[status(thm)],[275,262]) ).
tff(277,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(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(278,plain,
( ( subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) )
<=> ( subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ) ),
inference(rewrite,[status(thm)],]) ).
tff(279,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(identity),inverse(a)))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ) ),
inference(monotonicity,[status(thm)],[278]) ).
tff(280,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(identity),inverse(a)))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ) ),
inference(transitivity,[status(thm)],[279,277]) ).
tff(281,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(identity),inverse(a)))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) ),
inference(quant_inst,[status(thm)],]) ).
tff(282,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(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ),
inference(modus_ponens,[status(thm)],[281,280]) ).
tff(283,plain,
( subgroup_member(multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),multiply(inverse(identity),inverse(a)))
| ~ subgroup_member(inverse(b)) ),
inference(unit_resolution,[status(thm)],[282,159]) ).
tff(284,plain,
( ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) ),
inference(unit_resolution,[status(thm)],[283,276,258]) ).
tff(285,plain,
~ subgroup_member(inverse(c)),
inference(unit_resolution,[status(thm)],[284,173]) ).
tff(286,plain,
$false,
inference(unit_resolution,[status(thm)],[285,135]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.11 % Problem : GRP040-3 : TPTP v8.1.0. Released v1.0.0.
% 0.11/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n024.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 14:20:04 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.33 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.33 Usage: tptp [options] [-file:]file
% 0.12/0.33 -h, -? prints this message.
% 0.12/0.33 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.33 -m, -model generate model.
% 0.12/0.33 -p, -proof generate proof.
% 0.12/0.33 -c, -core generate unsat core of named formulas.
% 0.12/0.33 -st, -statistics display statistics.
% 0.12/0.33 -t:timeout set timeout (in second).
% 0.12/0.33 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.33 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.33 -<param>:<value> configuration parameter and value.
% 0.12/0.33 -o:<output-file> file to place output in.
% 2.32/1.71 % SZS status Unsatisfiable
% 2.32/1.71 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------