%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP040-4 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:06:56 AM UTC 2025
% Result : Unsatisfiable 19.01s 19.19s
% Output : Proof 19.01s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 119
% Syntax : Number of formulae : 306 ( 105 unt; 0 typ; 0 def)
% Number of atoms : 1551 ( 69 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 2146 ( 949 ~;1054 |; 0 &)
% ( 143 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 48 ( 48 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 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 : 861 ( 808 !; 0 ?; 861 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(a_type,type,
a: $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(element_in_O2_type,type,
element_in_O2: ( $i * $i ) > $i ).
tff(c_type,type,
c: $i ).
tff(d_type,type,
d: $i ).
tff(identity_type,type,
identity: $i ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(subgroup_member_type,type,
subgroup_member: $i > $o ).
tff(1,plain,
( ! [A: $i] : ( inverse(inverse(A)) = A )
<=> ! [A: $i] : ( inverse(inverse(A)) = A ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,plain,
( ! [A: $i] : ( inverse(inverse(A)) = A )
<=> ! [A: $i] : ( inverse(inverse(A)) = A ) ),
inference(rewrite,[status(thm)],[]) ).
tff(3,axiom,
! [A: $i] : ( inverse(inverse(A)) = A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_inverse_is_self_cancelling) ).
tff(4,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(modus_ponens,[status(thm)],[3,2]) ).
tff(5,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(skolemize,[status(sab)],[4]) ).
tff(6,plain,
! [A: $i] : ( inverse(inverse(A)) = A ),
inference(modus_ponens,[status(thm)],[5,1]) ).
tff(7,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(element_in_O2(inverse(d),inverse(c)))) = element_in_O2(inverse(d),inverse(c)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(8,plain,
inverse(inverse(element_in_O2(inverse(d),inverse(c)))) = element_in_O2(inverse(d),inverse(c)),
inference(unit_resolution,[status(thm)],[7,6]) ).
tff(9,plain,
( product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a)
<=> product(identity,element_in_O2(inverse(d),inverse(c)),a) ),
inference(monotonicity,[status(thm)],[8]) ).
tff(10,plain,
( product(identity,element_in_O2(inverse(d),inverse(c)),a)
<=> product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ),
inference(symmetry,[status(thm)],[9]) ).
tff(11,plain,
^ [Y: $i,X: $i] :
refl(( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(12,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)],[11]) ).
tff(13,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(14,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(15,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[14,13]) ).
tff(16,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[15]) ).
tff(17,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[16,12]) ).
tff(18,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(identity,inverse(c),multiply(identity,inverse(c))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(19,plain,
product(identity,inverse(c),multiply(identity,inverse(c))),
inference(unit_resolution,[status(thm)],[18,17]) ).
tff(20,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(21,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[20]) ).
tff(22,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(24,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[24]) ).
tff(26,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(28,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[27,26]) ).
tff(29,plain,
^ [X: $i] :
refl(( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],[]) ).
tff(30,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[29]) ).
tff(31,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(32,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(33,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[32,31]) ).
tff(34,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[33]) ).
tff(35,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[34,30]) ).
tff(36,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(37,plain,
product(a,inverse(a),identity),
inference(unit_resolution,[status(thm)],[36,35]) ).
tff(38,plain,
^ [X: $i] :
refl(( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],[]) ).
tff(39,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[38]) ).
tff(40,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(42,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[42]) ).
tff(44,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(46,plain,
product(inverse(a),a,identity),
inference(unit_resolution,[status(thm)],[45,44]) ).
tff(47,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(48,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)],[47]) ).
tff(49,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(50,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(51,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)],[50]) ).
tff(52,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(53,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)],[52,51]) ).
tff(54,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)],[53,49]) ).
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) ),
inference(skolemize,[status(sab)],[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) ),
inference(modus_ponens,[status(thm)],[55,48]) ).
tff(57,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(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(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(inverse(a),a,identity)
| ~ product(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,plain,
( ( product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity)
| ~ product(inverse(a),a,identity) )
<=> ( ~ product(inverse(a),a,identity)
| ~ product(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(59,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),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity)
| ~ 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(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(a)) ) ),
inference(monotonicity,[status(thm)],[58]) ).
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) )
| product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity)
| ~ 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(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(a)) ) ),
inference(transitivity,[status(thm)],[59,57]) ).
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) )
| product(inverse(a),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity)
| ~ product(inverse(a),a,identity) ),
inference(quant_inst,[status(thm)],[]) ).
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) )
| ~ product(inverse(a),a,identity)
| ~ product(a,inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| product(inverse(a),identity,inverse(a)) ),
inference(modus_ponens,[status(thm)],[61,60]) ).
tff(63,plain,
product(inverse(a),identity,inverse(a)),
inference(unit_resolution,[status(thm)],[62,56,46,37,28]) ).
tff(64,plain,
( product(b,inverse(a),c)
<=> product(b,inverse(a),c) ),
inference(rewrite,[status(thm)],[]) ).
tff(65,axiom,
product(b,inverse(a),c),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_times_a_inverse_is_c) ).
tff(66,plain,
product(b,inverse(a),c),
inference(modus_ponens,[status(thm)],[65,64]) ).
tff(67,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(68,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)],[67]) ).
tff(69,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(70,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(71,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)],[70]) ).
tff(72,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(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) ),
inference(modus_ponens,[status(thm)],[72,71]) ).
tff(74,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)],[73,69]) ).
tff(75,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)],[74]) ).
tff(76,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)],[75,68]) ).
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) )
| ~ product(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,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(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,c) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,plain,
( ( product(c,identity,c)
| ~ product(inverse(a),identity,inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(b,inverse(a),c) )
<=> ( ~ product(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,c) ) ),
inference(rewrite,[status(thm)],[]) ).
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) )
| product(c,identity,c)
| ~ product(inverse(a),identity,inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(b,inverse(a),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(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,c) ) ),
inference(monotonicity,[status(thm)],[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) )
| product(c,identity,c)
| ~ product(inverse(a),identity,inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(b,inverse(a),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(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,c) ) ),
inference(transitivity,[status(thm)],[79,77]) ).
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(c,identity,c)
| ~ product(inverse(a),identity,inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(b,inverse(a),c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(82,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(b,inverse(a),c)
| ~ product(inverse(a),identity,inverse(a))
| product(c,identity,c) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
product(c,identity,c),
inference(unit_resolution,[status(thm)],[82,76,66,63]) ).
tff(84,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(c,inverse(c),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
product(c,inverse(c),identity),
inference(unit_resolution,[status(thm)],[84,35]) ).
tff(86,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(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(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(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(87,plain,
( ( product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,inverse(c),identity)
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(c,identity,c) )
<=> ( ~ product(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,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(c,multiply(identity,inverse(c)),identity)
| ~ product(c,inverse(c),identity)
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(c,identity,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(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ) ),
inference(monotonicity,[status(thm)],[87]) ).
tff(89,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(c,multiply(identity,inverse(c)),identity)
| ~ product(c,inverse(c),identity)
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(c,identity,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(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ) ),
inference(transitivity,[status(thm)],[88,86]) ).
tff(90,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(c,multiply(identity,inverse(c)),identity)
| ~ product(c,inverse(c),identity)
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(c,identity,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(91,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(c,inverse(c),identity)
| product(c,multiply(identity,inverse(c)),identity)
| ~ product(c,identity,c)
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ),
inference(modus_ponens,[status(thm)],[90,89]) ).
tff(92,plain,
product(c,multiply(identity,inverse(c)),identity),
inference(unit_resolution,[status(thm)],[91,56,85,83,19]) ).
tff(93,plain,
^ [X: $i] :
refl(( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],[]) ).
tff(94,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[93]) ).
tff(95,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(96,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(97,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[96,95]) ).
tff(98,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[97]) ).
tff(99,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[98,94]) ).
tff(100,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(identity),identity,inverse(identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(101,plain,
product(inverse(identity),identity,inverse(identity)),
inference(unit_resolution,[status(thm)],[100,99]) ).
tff(102,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(identity),identity,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(103,plain,
product(inverse(identity),identity,identity),
inference(unit_resolution,[status(thm)],[102,44]) ).
tff(104,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(105,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)],[104]) ).
tff(106,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(107,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(108,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)],[107]) ).
tff(109,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(110,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[109,108]) ).
tff(111,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[110,106]) ).
tff(112,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[111]) ).
tff(113,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[112,105]) ).
tff(114,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(inverse(identity),identity,identity)
| ~ product(inverse(identity),identity,inverse(identity)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(inverse(identity),identity,identity)
| ~ product(inverse(identity),identity,inverse(identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(115,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(inverse(identity),identity,identity)
| ~ product(inverse(identity),identity,inverse(identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(116,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(identity) = identity )
| ~ product(inverse(identity),identity,identity)
| ~ product(inverse(identity),identity,inverse(identity)) ),
inference(modus_ponens,[status(thm)],[115,114]) ).
tff(117,plain,
inverse(identity) = identity,
inference(unit_resolution,[status(thm)],[116,113,103,101]) ).
tff(118,plain,
identity = inverse(identity),
inference(symmetry,[status(thm)],[117]) ).
tff(119,plain,
( product(identity,d,multiply(inverse(identity),d))
<=> product(inverse(identity),d,multiply(inverse(identity),d)) ),
inference(monotonicity,[status(thm)],[118]) ).
tff(120,plain,
( product(inverse(identity),d,multiply(inverse(identity),d))
<=> product(identity,d,multiply(inverse(identity),d)) ),
inference(symmetry,[status(thm)],[119]) ).
tff(121,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(inverse(identity),d,multiply(inverse(identity),d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(122,plain,
product(inverse(identity),d,multiply(inverse(identity),d)),
inference(unit_resolution,[status(thm)],[121,17]) ).
tff(123,plain,
product(identity,d,multiply(inverse(identity),d)),
inference(modus_ponens,[status(thm)],[122,120]) ).
tff(124,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,d,d) ),
inference(quant_inst,[status(thm)],[]) ).
tff(125,plain,
product(identity,d,d),
inference(unit_resolution,[status(thm)],[124,26]) ).
tff(126,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(127,plain,
( ( ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d))
| ~ product(identity,d,d) )
<=> ( ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d))
| ~ product(identity,d,d) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) ) ),
inference(monotonicity,[status(thm)],[127]) ).
tff(129,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d))
| ~ product(identity,d,d) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) ) ),
inference(transitivity,[status(thm)],[128,126]) ).
tff(130,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d))
| ~ product(identity,d,d) ),
inference(quant_inst,[status(thm)],[]) ).
tff(131,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,d,d)
| ( d = multiply(inverse(identity),d) )
| ~ product(identity,d,multiply(inverse(identity),d)) ),
inference(modus_ponens,[status(thm)],[130,129]) ).
tff(132,plain,
d = multiply(inverse(identity),d),
inference(unit_resolution,[status(thm)],[131,113,125,123]) ).
tff(133,plain,
multiply(inverse(identity),d) = d,
inference(symmetry,[status(thm)],[132]) ).
tff(134,plain,
( product(a,c,multiply(inverse(identity),d))
<=> product(a,c,d) ),
inference(monotonicity,[status(thm)],[133]) ).
tff(135,plain,
( product(a,c,d)
<=> product(a,c,multiply(inverse(identity),d)) ),
inference(symmetry,[status(thm)],[134]) ).
tff(136,plain,
( product(a,c,d)
<=> product(a,c,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(137,axiom,
product(a,c,d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_times_c_is_d) ).
tff(138,plain,
product(a,c,d),
inference(modus_ponens,[status(thm)],[137,136]) ).
tff(139,plain,
product(a,c,multiply(inverse(identity),d)),
inference(modus_ponens,[status(thm)],[138,135]) ).
tff(140,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(a,identity,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(141,plain,
product(a,identity,a),
inference(unit_resolution,[status(thm)],[140,99]) ).
tff(142,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(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(143,plain,
( ( product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a) )
<=> ( ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(144,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ) ),
inference(monotonicity,[status(thm)],[143]) ).
tff(145,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ) ),
inference(transitivity,[status(thm)],[144,142]) ).
tff(146,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| ~ product(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(147,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(a,c,multiply(inverse(identity),d))
| ~ product(a,identity,a)
| ~ product(c,multiply(identity,inverse(c)),identity)
| product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ),
inference(modus_ponens,[status(thm)],[146,145]) ).
tff(148,plain,
product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a),
inference(unit_resolution,[status(thm)],[147,76,141,139,92]) ).
tff(149,plain,
inverse(d) = inverse(multiply(inverse(identity),d)),
inference(monotonicity,[status(thm)],[132]) ).
tff(150,plain,
inverse(multiply(inverse(identity),d)) = inverse(d),
inference(symmetry,[status(thm)],[149]) ).
tff(151,plain,
( product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c))
<=> product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(monotonicity,[status(thm)],[150]) ).
tff(152,plain,
( product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c))
<=> product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(symmetry,[status(thm)],[151]) ).
tff(153,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(c)) = c ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(154,plain,
inverse(inverse(c)) = c,
inference(unit_resolution,[status(thm)],[153,6]) ).
tff(155,plain,
( product(inverse(b),inverse(inverse(c)),inverse(a))
<=> product(inverse(b),c,inverse(a)) ),
inference(monotonicity,[status(thm)],[154]) ).
tff(156,plain,
( product(inverse(b),c,inverse(a))
<=> product(inverse(b),inverse(inverse(c)),inverse(a)) ),
inference(symmetry,[status(thm)],[155]) ).
tff(157,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(158,plain,
product(inverse(b),b,identity),
inference(unit_resolution,[status(thm)],[157,44]) ).
tff(159,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(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(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(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(160,plain,
( ( product(inverse(b),c,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) )
<=> ( ~ product(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(161,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,inverse(a))
| ~ product(identity,inverse(a),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(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(a)) ) ),
inference(monotonicity,[status(thm)],[160]) ).
tff(162,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,inverse(a))
| ~ product(identity,inverse(a),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(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(a)) ) ),
inference(transitivity,[status(thm)],[161,159]) ).
tff(163,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,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(b,inverse(a),c)
| ~ product(inverse(b),b,identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(164,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(b,inverse(a),c)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(inverse(b),b,identity)
| product(inverse(b),c,inverse(a)) ),
inference(modus_ponens,[status(thm)],[163,162]) ).
tff(165,plain,
product(inverse(b),c,inverse(a)),
inference(unit_resolution,[status(thm)],[164,56,66,158,28]) ).
tff(166,plain,
product(inverse(b),inverse(inverse(c)),inverse(a)),
inference(modus_ponens,[status(thm)],[165,156]) ).
tff(167,plain,
( subgroup_member(b)
<=> subgroup_member(b) ),
inference(rewrite,[status(thm)],[]) ).
tff(168,axiom,
subgroup_member(b),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',b_is_in_subgroup) ).
tff(169,plain,
subgroup_member(b),
inference(modus_ponens,[status(thm)],[168,167]) ).
tff(170,plain,
^ [A: $i] :
refl(( ( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
<=> ( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ) )),
inference(bind,[status(th)],[]) ).
tff(171,plain,
( ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
<=> ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ) ),
inference(quant_intro,[status(thm)],[170]) ).
tff(172,plain,
( ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
<=> ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(173,axiom,
! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',closure_of_inverse) ).
tff(174,plain,
! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ),
inference(modus_ponens,[status(thm)],[173,172]) ).
tff(175,plain,
! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ),
inference(skolemize,[status(sab)],[174]) ).
tff(176,plain,
! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) ),
inference(modus_ponens,[status(thm)],[175,171]) ).
tff(177,plain,
( ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(b)
| subgroup_member(inverse(b)) )
<=> ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(b)
| subgroup_member(inverse(b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(178,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(b)
| subgroup_member(inverse(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(179,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(b)
| subgroup_member(inverse(b)) ),
inference(modus_ponens,[status(thm)],[178,177]) ).
tff(180,plain,
subgroup_member(inverse(b)),
inference(unit_resolution,[status(thm)],[179,176,169]) ).
tff(181,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(a)) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(182,plain,
inverse(inverse(a)) = a,
inference(unit_resolution,[status(thm)],[181,6]) ).
tff(183,plain,
( subgroup_member(inverse(inverse(a)))
<=> subgroup_member(a) ),
inference(monotonicity,[status(thm)],[182]) ).
tff(184,plain,
( subgroup_member(a)
<=> subgroup_member(inverse(inverse(a))) ),
inference(symmetry,[status(thm)],[183]) ).
tff(185,plain,
( ~ subgroup_member(a)
<=> ~ subgroup_member(inverse(inverse(a))) ),
inference(monotonicity,[status(thm)],[184]) ).
tff(186,plain,
( ~ subgroup_member(a)
<=> ~ subgroup_member(a) ),
inference(rewrite,[status(thm)],[]) ).
tff(187,axiom,
~ subgroup_member(a),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',a_in_subgroup) ).
tff(188,plain,
~ subgroup_member(a),
inference(modus_ponens,[status(thm)],[187,186]) ).
tff(189,plain,
~ subgroup_member(inverse(inverse(a))),
inference(modus_ponens,[status(thm)],[188,185]) ).
tff(190,plain,
( ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(a))
| subgroup_member(inverse(inverse(a))) )
<=> ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(a))
| subgroup_member(inverse(inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(191,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(a))
| subgroup_member(inverse(inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(192,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(a))
| subgroup_member(inverse(inverse(a))) ),
inference(modus_ponens,[status(thm)],[191,190]) ).
tff(193,plain,
( ~ subgroup_member(inverse(a))
| subgroup_member(inverse(inverse(a))) ),
inference(unit_resolution,[status(thm)],[192,176]) ).
tff(194,plain,
~ subgroup_member(inverse(a)),
inference(unit_resolution,[status(thm)],[193,189]) ).
tff(195,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(196,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)],[195]) ).
tff(197,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(198,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(199,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)],[198]) ).
tff(200,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(201,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)],[200,199]) ).
tff(202,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)],[201,197]) ).
tff(203,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)],[202]) ).
tff(204,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)],[203,196]) ).
tff(205,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) )
<=> ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(206,plain,
( ( subgroup_member(inverse(a))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) )
<=> ( subgroup_member(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(207,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(inverse(b),inverse(inverse(c)),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(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ) ),
inference(monotonicity,[status(thm)],[206]) ).
tff(208,plain,
( ( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(inverse(b),inverse(inverse(c)),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(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ) ),
inference(transitivity,[status(thm)],[207,205]) ).
tff(209,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(210,plain,
( ~ ! [B: $i,A: $i,C: $i] :
( subgroup_member(C)
| ~ product(A,inverse(B),C)
| ~ subgroup_member(B)
| ~ subgroup_member(A) )
| subgroup_member(inverse(a))
| ~ subgroup_member(inverse(c))
| ~ subgroup_member(inverse(b))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ),
inference(modus_ponens,[status(thm)],[209,208]) ).
tff(211,plain,
( ~ subgroup_member(inverse(c))
| ~ product(inverse(b),inverse(inverse(c)),inverse(a)) ),
inference(unit_resolution,[status(thm)],[210,204,194,180]) ).
tff(212,plain,
~ subgroup_member(inverse(c)),
inference(unit_resolution,[status(thm)],[211,166]) ).
tff(213,plain,
( ~ ! [A: $i] : ( inverse(inverse(A)) = A )
| ( inverse(inverse(d)) = d ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(214,plain,
inverse(inverse(d)) = d,
inference(unit_resolution,[status(thm)],[213,6]) ).
tff(215,plain,
( subgroup_member(inverse(inverse(d)))
<=> subgroup_member(d) ),
inference(monotonicity,[status(thm)],[214]) ).
tff(216,plain,
( subgroup_member(d)
<=> subgroup_member(inverse(inverse(d))) ),
inference(symmetry,[status(thm)],[215]) ).
tff(217,plain,
( ~ subgroup_member(d)
<=> ~ subgroup_member(inverse(inverse(d))) ),
inference(monotonicity,[status(thm)],[216]) ).
tff(218,plain,
( ~ subgroup_member(d)
<=> ~ subgroup_member(d) ),
inference(rewrite,[status(thm)],[]) ).
tff(219,axiom,
~ subgroup_member(d),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',d_in_subgroup) ).
tff(220,plain,
~ subgroup_member(d),
inference(modus_ponens,[status(thm)],[219,218]) ).
tff(221,plain,
~ subgroup_member(inverse(inverse(d))),
inference(modus_ponens,[status(thm)],[220,217]) ).
tff(222,plain,
( ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(d))
| subgroup_member(inverse(inverse(d))) )
<=> ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(d))
| subgroup_member(inverse(inverse(d))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(223,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(d))
| subgroup_member(inverse(inverse(d))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(224,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(inverse(d))
| subgroup_member(inverse(inverse(d))) ),
inference(modus_ponens,[status(thm)],[223,222]) ).
tff(225,plain,
( ~ subgroup_member(inverse(d))
| subgroup_member(inverse(inverse(d))) ),
inference(unit_resolution,[status(thm)],[224,176]) ).
tff(226,plain,
~ subgroup_member(inverse(d)),
inference(unit_resolution,[status(thm)],[225,221]) ).
tff(227,plain,
^ [B: $i,A: $i] :
refl(( ( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) )),
inference(bind,[status(th)],[]) ).
tff(228,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) ),
inference(quant_intro,[status(thm)],[227]) ).
tff(229,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(230,plain,
^ [B: $i,A: $i] :
trans(monotonicity(rewrite(( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B) )
<=> ( subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) )),
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(B)
| product(A,element_in_O2(A,B),B)
| subgroup_member(A) ) )),
rewrite(( ( subgroup_member(B)
| product(A,element_in_O2(A,B),B)
| subgroup_member(A) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) )),
( ( product(A,element_in_O2(A,B),B)
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) )),
inference(bind,[status(th)],[]) ).
tff(231,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)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ) ),
inference(quant_intro,[status(thm)],[230]) ).
tff(232,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(233,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ),
inference(modus_ponens,[status(thm)],[232,231]) ).
tff(234,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ),
inference(modus_ponens,[status(thm)],[233,229]) ).
tff(235,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ),
inference(skolemize,[status(sab)],[234]) ).
tff(236,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) ),
inference(modus_ponens,[status(thm)],[235,228]) ).
tff(237,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
| subgroup_member(inverse(d))
| subgroup_member(inverse(c))
| product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
| subgroup_member(inverse(d))
| subgroup_member(inverse(c))
| product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(238,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
| subgroup_member(inverse(d))
| subgroup_member(inverse(c))
| product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(239,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| product(A,element_in_O2(A,B),B) )
| subgroup_member(inverse(d))
| subgroup_member(inverse(c))
| product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(modus_ponens,[status(thm)],[238,237]) ).
tff(240,plain,
( subgroup_member(inverse(c))
| product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(unit_resolution,[status(thm)],[239,236,226]) ).
tff(241,plain,
product(inverse(d),element_in_O2(inverse(d),inverse(c)),inverse(c)),
inference(unit_resolution,[status(thm)],[240,212]) ).
tff(242,plain,
product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)),
inference(modus_ponens,[status(thm)],[241,152]) ).
tff(243,plain,
( product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
<=> product(identity,inverse(d),inverse(d)) ),
inference(monotonicity,[status(thm)],[150,150]) ).
tff(244,plain,
( product(identity,inverse(d),inverse(d))
<=> product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d))) ),
inference(symmetry,[status(thm)],[243]) ).
tff(245,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(d),inverse(d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(246,plain,
product(identity,inverse(d),inverse(d)),
inference(unit_resolution,[status(thm)],[245,26]) ).
tff(247,plain,
product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d))),
inference(modus_ponens,[status(thm)],[246,244]) ).
tff(248,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(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(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(249,plain,
( ( product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c))
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) )
<=> ( ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(250,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c))
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| ~ product(identity,inverse(c),multiply(identity,inverse(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(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ) ),
inference(monotonicity,[status(thm)],[249]) ).
tff(251,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c))
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| ~ product(identity,inverse(c),multiply(identity,inverse(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(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ) ),
inference(transitivity,[status(thm)],[250,248]) ).
tff(252,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c))
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(253,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(identity,inverse(multiply(inverse(identity),d)),inverse(multiply(inverse(identity),d)))
| product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(identity,inverse(c),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(modus_ponens,[status(thm)],[252,251]) ).
tff(254,plain,
( product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),inverse(c)) ),
inference(unit_resolution,[status(thm)],[253,76,247,19]) ).
tff(255,plain,
product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c))),
inference(unit_resolution,[status(thm)],[254,242]) ).
tff(256,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(257,plain,
product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity),
inference(unit_resolution,[status(thm)],[256,35]) ).
tff(258,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(259,plain,
( ( product(identity,element_in_O2(inverse(d),inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) )
<=> ( ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(260,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,element_in_O2(inverse(d),inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) ) ),
inference(monotonicity,[status(thm)],[259]) ).
tff(261,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,element_in_O2(inverse(d),inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) ) ),
inference(transitivity,[status(thm)],[260,258]) ).
tff(262,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,element_in_O2(inverse(d),inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(263,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(multiply(inverse(identity),d),inverse(multiply(inverse(identity),d)),identity)
| ~ product(multiply(inverse(identity),d),multiply(identity,inverse(c)),a)
| ~ product(inverse(multiply(inverse(identity),d)),element_in_O2(inverse(d),inverse(c)),multiply(identity,inverse(c)))
| product(identity,element_in_O2(inverse(d),inverse(c)),a) ),
inference(modus_ponens,[status(thm)],[262,261]) ).
tff(264,plain,
product(identity,element_in_O2(inverse(d),inverse(c)),a),
inference(unit_resolution,[status(thm)],[263,76,257,255,148]) ).
tff(265,plain,
product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a),
inference(modus_ponens,[status(thm)],[264,10]) ).
tff(266,plain,
element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c)) = element_in_O2(inverse(d),inverse(c)),
inference(monotonicity,[status(thm)],[150]) ).
tff(267,plain,
element_in_O2(inverse(d),inverse(c)) = element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c)),
inference(symmetry,[status(thm)],[266]) ).
tff(268,plain,
( subgroup_member(element_in_O2(inverse(d),inverse(c)))
<=> subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ),
inference(monotonicity,[status(thm)],[267]) ).
tff(269,plain,
( subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c)))
<=> subgroup_member(element_in_O2(inverse(d),inverse(c))) ),
inference(symmetry,[status(thm)],[268]) ).
tff(270,plain,
( subgroup_member(inverse(d))
<=> subgroup_member(inverse(multiply(inverse(identity),d))) ),
inference(monotonicity,[status(thm)],[149]) ).
tff(271,plain,
( ~ subgroup_member(inverse(d))
<=> ~ subgroup_member(inverse(multiply(inverse(identity),d))) ),
inference(monotonicity,[status(thm)],[270]) ).
tff(272,plain,
~ subgroup_member(inverse(multiply(inverse(identity),d))),
inference(modus_ponens,[status(thm)],[226,271]) ).
tff(273,plain,
^ [B: $i,A: $i] :
refl(( ( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) )),
inference(bind,[status(th)],[]) ).
tff(274,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) ),
inference(quant_intro,[status(thm)],[273]) ).
tff(275,plain,
( ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(276,plain,
^ [B: $i,A: $i] :
trans(monotonicity(rewrite(( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B) )
<=> ( subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) )),
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(B)
| subgroup_member(element_in_O2(A,B))
| subgroup_member(A) ) )),
rewrite(( ( subgroup_member(B)
| subgroup_member(element_in_O2(A,B))
| subgroup_member(A) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) )),
( ( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) )),
inference(bind,[status(th)],[]) ).
tff(277,plain,
( ! [B: $i,A: $i] :
( subgroup_member(element_in_O2(A,B))
| subgroup_member(B)
| subgroup_member(A) )
<=> ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ) ),
inference(quant_intro,[status(thm)],[276]) ).
tff(278,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(279,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ),
inference(modus_ponens,[status(thm)],[278,277]) ).
tff(280,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ),
inference(modus_ponens,[status(thm)],[279,275]) ).
tff(281,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ),
inference(skolemize,[status(sab)],[280]) ).
tff(282,plain,
! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) ),
inference(modus_ponens,[status(thm)],[281,274]) ).
tff(283,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(284,plain,
( ( subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(inverse(c))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) )
<=> ( subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(285,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(inverse(c))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ) ),
inference(monotonicity,[status(thm)],[284]) ).
tff(286,plain,
( ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(inverse(c))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) )
<=> ( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ) ),
inference(transitivity,[status(thm)],[285,283]) ).
tff(287,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(inverse(c))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(288,plain,
( ~ ! [B: $i,A: $i] :
( subgroup_member(A)
| subgroup_member(B)
| subgroup_member(element_in_O2(A,B)) )
| subgroup_member(inverse(c))
| subgroup_member(inverse(multiply(inverse(identity),d)))
| subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))) ),
inference(modus_ponens,[status(thm)],[287,286]) ).
tff(289,plain,
subgroup_member(element_in_O2(inverse(multiply(inverse(identity),d)),inverse(c))),
inference(unit_resolution,[status(thm)],[288,282,212,272]) ).
tff(290,plain,
subgroup_member(element_in_O2(inverse(d),inverse(c))),
inference(modus_ponens,[status(thm)],[289,269]) ).
tff(291,plain,
( ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(element_in_O2(inverse(d),inverse(c)))
| subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))) )
<=> ( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(element_in_O2(inverse(d),inverse(c)))
| subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(292,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(element_in_O2(inverse(d),inverse(c)))
| subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(293,plain,
( ~ ! [A: $i] :
( ~ subgroup_member(A)
| subgroup_member(inverse(A)) )
| ~ subgroup_member(element_in_O2(inverse(d),inverse(c)))
| subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))) ),
inference(modus_ponens,[status(thm)],[292,291]) ).
tff(294,plain,
( ~ subgroup_member(element_in_O2(inverse(d),inverse(c)))
| subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))) ),
inference(unit_resolution,[status(thm)],[293,176]) ).
tff(295,plain,
subgroup_member(inverse(element_in_O2(inverse(d),inverse(c)))),
inference(unit_resolution,[status(thm)],[294,290]) ).
tff(296,plain,
( subgroup_member(identity)
<=> subgroup_member(identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(297,axiom,
subgroup_member(identity),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',identity_is_in_subgroup) ).
tff(298,plain,
subgroup_member(identity),
inference(modus_ponens,[status(thm)],[297,296]) ).
tff(299,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(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),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(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(300,plain,
( ( subgroup_member(a)
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ subgroup_member(identity) )
<=> ( subgroup_member(a)
| ~ subgroup_member(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(301,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(element_in_O2(inverse(d),inverse(c)))),a)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ 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(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ) ),
inference(monotonicity,[status(thm)],[300]) ).
tff(302,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(element_in_O2(inverse(d),inverse(c)))),a)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ 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(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ) ),
inference(transitivity,[status(thm)],[301,299]) ).
tff(303,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(element_in_O2(inverse(d),inverse(c)))),a)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ subgroup_member(identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(304,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(identity)
| ~ subgroup_member(inverse(element_in_O2(inverse(d),inverse(c))))
| ~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a) ),
inference(modus_ponens,[status(thm)],[303,302]) ).
tff(305,plain,
~ product(identity,inverse(inverse(element_in_O2(inverse(d),inverse(c)))),a),
inference(unit_resolution,[status(thm)],[304,204,298,188,295]) ).
tff(306,plain,
$false,
inference(unit_resolution,[status(thm)],[305,265]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP040-4 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n008.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 : Thu Jun 19 15:30:48 EDT 2025
% 0.12/0.33 % CPUTime :
% 19.01/19.19 % SZS status Unsatisfiable
% 19.01/19.19 % SZS output start Proof
% See solution above
% 19.01/19.22 % E exiting
%------------------------------------------------------------------------------