%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : GRP012-3 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:06:50 AM UTC 2025
% Result : Unsatisfiable 11.77s 11.95s
% Output : Proof 11.77s
% Verified :
% SZS Type : Refutation
% Derivation depth : 24
% Number of leaves : 80
% Syntax : Number of formulae : 207 ( 75 unt; 0 typ; 0 def)
% Number of atoms : 1104 ( 102 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 1665 ( 791 ~; 783 |; 0 &)
% ( 91 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 23 ( 23 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 3 con; 0-2 aty)
% Number of variables : 748 ( 710 !; 0 ?; 748 :)
% Comments :
%------------------------------------------------------------------------------
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(inverse_type,type,
inverse: $i > $i ).
tff(a_type,type,
a: $i ).
tff(b_type,type,
b: $i ).
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(identity_type,type,
identity: $i ).
tff(1,plain,
^ [Y: $i,X: $i] :
refl(( product(X,Y,multiply(X,Y))
<=> product(X,Y,multiply(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
<=> ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function1) ).
tff(5,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [Y: $i,X: $i] : product(X,Y,multiply(X,Y)),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(identity,inverse(b),multiply(identity,inverse(b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
product(identity,inverse(b),multiply(identity,inverse(b))),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
^ [X: $i] :
refl(( product(identity,X,X)
<=> product(identity,X,X) )),
inference(bind,[status(th)],[]) ).
tff(11,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(quant_intro,[status(thm)],[10]) ).
tff(12,plain,
( ! [X: $i] : product(identity,X,X)
<=> ! [X: $i] : product(identity,X,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(13,axiom,
! [X: $i] : product(identity,X,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_identity) ).
tff(14,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [X: $i] : product(identity,X,X),
inference(skolemize,[status(sab)],[14]) ).
tff(16,plain,
! [X: $i] : product(identity,X,X),
inference(modus_ponens,[status(thm)],[15,11]) ).
tff(17,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(b),inverse(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(18,plain,
product(identity,inverse(b),inverse(b)),
inference(unit_resolution,[status(thm)],[17,16]) ).
tff(19,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
refl(( ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(20,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[19]) ).
tff(21,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(22,plain,
^ [W: $i,Z: $i,Y: $i,X: $i] :
rewrite(( ( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(23,plain,
( ! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) )
<=> ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ) ),
inference(quant_intro,[status(thm)],[22]) ).
tff(24,axiom,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ~ product(X,Y,Z)
| ~ product(X,Y,W)
| ( Z = W ) ),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',total_function2) ).
tff(25,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[25,21]) ).
tff(27,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(skolemize,[status(sab)],[26]) ).
tff(28,plain,
! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) ),
inference(modus_ponens,[status(thm)],[27,20]) ).
tff(29,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ( ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b)))
| ~ product(identity,inverse(b),inverse(b)) )
<=> ( ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b)))
| ~ product(identity,inverse(b),inverse(b)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) ) ),
inference(monotonicity,[status(thm)],[30]) ).
tff(32,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b)))
| ~ product(identity,inverse(b),inverse(b)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) ) ),
inference(transitivity,[status(thm)],[31,29]) ).
tff(33,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b)))
| ~ product(identity,inverse(b),inverse(b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(34,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(b),inverse(b))
| ( inverse(b) = multiply(identity,inverse(b)) )
| ~ product(identity,inverse(b),multiply(identity,inverse(b))) ),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
inverse(b) = multiply(identity,inverse(b)),
inference(unit_resolution,[status(thm)],[34,28,18,9]) ).
tff(36,plain,
multiply(identity,inverse(b)) = inverse(b),
inference(symmetry,[status(thm)],[35]) ).
tff(37,plain,
( product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
<=> product(inverse(b),inverse(inverse(b)),identity) ),
inference(monotonicity,[status(thm)],[36]) ).
tff(38,plain,
( product(inverse(b),inverse(inverse(b)),identity)
<=> product(multiply(identity,inverse(b)),inverse(inverse(b)),identity) ),
inference(symmetry,[status(thm)],[37]) ).
tff(39,plain,
^ [X: $i] :
refl(( product(X,inverse(X),identity)
<=> product(X,inverse(X),identity) )),
inference(bind,[status(th)],[]) ).
tff(40,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(quant_intro,[status(thm)],[39]) ).
tff(41,plain,
( ! [X: $i] : product(X,inverse(X),identity)
<=> ! [X: $i] : product(X,inverse(X),identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(42,axiom,
! [X: $i] : product(X,inverse(X),identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_inverse) ).
tff(43,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(skolemize,[status(sab)],[43]) ).
tff(45,plain,
! [X: $i] : product(X,inverse(X),identity),
inference(modus_ponens,[status(thm)],[44,40]) ).
tff(46,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(inverse(b),inverse(inverse(b)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(47,plain,
product(inverse(b),inverse(inverse(b)),identity),
inference(unit_resolution,[status(thm)],[46,45]) ).
tff(48,plain,
product(multiply(identity,inverse(b)),inverse(inverse(b)),identity),
inference(modus_ponens,[status(thm)],[47,38]) ).
tff(49,plain,
^ [X: $i] :
refl(( product(X,identity,X)
<=> product(X,identity,X) )),
inference(bind,[status(th)],[]) ).
tff(50,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(quant_intro,[status(thm)],[49]) ).
tff(51,plain,
( ! [X: $i] : product(X,identity,X)
<=> ! [X: $i] : product(X,identity,X) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,axiom,
! [X: $i] : product(X,identity,X),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',right_identity) ).
tff(53,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
! [X: $i] : product(X,identity,X),
inference(skolemize,[status(sab)],[53]) ).
tff(55,plain,
! [X: $i] : product(X,identity,X),
inference(modus_ponens,[status(thm)],[54,50]) ).
tff(56,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(b,identity,b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(57,plain,
product(b,identity,b),
inference(unit_resolution,[status(thm)],[56,55]) ).
tff(58,plain,
( product(b,multiply(identity,inverse(b)),identity)
<=> product(b,inverse(b),identity) ),
inference(monotonicity,[status(thm)],[36]) ).
tff(59,plain,
( product(b,inverse(b),identity)
<=> product(b,multiply(identity,inverse(b)),identity) ),
inference(symmetry,[status(thm)],[58]) ).
tff(60,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(b,inverse(b),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(61,plain,
product(b,inverse(b),identity),
inference(unit_resolution,[status(thm)],[60,45]) ).
tff(62,plain,
product(b,multiply(identity,inverse(b)),identity),
inference(modus_ponens,[status(thm)],[61,59]) ).
tff(63,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(64,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)],[63]) ).
tff(65,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(66,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(67,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)],[66]) ).
tff(68,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(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) ),
inference(modus_ponens,[status(thm)],[68,67]) ).
tff(70,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)],[69,65]) ).
tff(71,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)],[70]) ).
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) ),
inference(modus_ponens,[status(thm)],[71,64]) ).
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) )
| ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(74,plain,
( ( product(identity,inverse(inverse(b)),b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(b,identity,b) )
<=> ( ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ) ),
inference(rewrite,[status(thm)],[]) ).
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) )
| product(identity,inverse(inverse(b)),b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(b,identity,b) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ) ),
inference(monotonicity,[status(thm)],[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) )
| product(identity,inverse(inverse(b)),b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(b,identity,b) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U)
| ~ product(X,V,W) )
| ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ) ),
inference(transitivity,[status(thm)],[75,73]) ).
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(identity,inverse(inverse(b)),b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(b,identity,b) ),
inference(quant_inst,[status(thm)],[]) ).
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) )
| ~ product(b,identity,b)
| ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
( ~ product(multiply(identity,inverse(b)),inverse(inverse(b)),identity)
| product(identity,inverse(inverse(b)),b) ),
inference(unit_resolution,[status(thm)],[78,72,62,57]) ).
tff(80,plain,
product(identity,inverse(inverse(b)),b),
inference(unit_resolution,[status(thm)],[79,48]) ).
tff(81,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(inverse(b)),inverse(inverse(b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(82,plain,
product(identity,inverse(inverse(b)),inverse(inverse(b))),
inference(unit_resolution,[status(thm)],[81,16]) ).
tff(83,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(84,plain,
( ( ( b = inverse(inverse(b)) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b) )
<=> ( ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(85,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( b = inverse(inverse(b)) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) ) ),
inference(monotonicity,[status(thm)],[84]) ).
tff(86,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( b = inverse(inverse(b)) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) ) ),
inference(transitivity,[status(thm)],[85,83]) ).
tff(87,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( b = inverse(inverse(b)) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b) ),
inference(quant_inst,[status(thm)],[]) ).
tff(88,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(identity,inverse(inverse(b)),inverse(inverse(b)))
| ~ product(identity,inverse(inverse(b)),b)
| ( b = inverse(inverse(b)) ) ),
inference(modus_ponens,[status(thm)],[87,86]) ).
tff(89,plain,
b = inverse(inverse(b)),
inference(unit_resolution,[status(thm)],[88,28,82,80]) ).
tff(90,plain,
multiply(a,b) = multiply(a,inverse(inverse(b))),
inference(monotonicity,[status(thm)],[89]) ).
tff(91,plain,
inverse(multiply(a,b)) = inverse(multiply(a,inverse(inverse(b)))),
inference(monotonicity,[status(thm)],[90]) ).
tff(92,plain,
inverse(multiply(a,inverse(inverse(b)))) = inverse(multiply(a,b)),
inference(symmetry,[status(thm)],[91]) ).
tff(93,plain,
( product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
<=> product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b))) ),
inference(monotonicity,[status(thm)],[92]) ).
tff(94,plain,
( product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
<=> product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))) ),
inference(symmetry,[status(thm)],[93]) ).
tff(95,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(a,identity,multiply(a,identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(96,plain,
product(a,identity,multiply(a,identity)),
inference(unit_resolution,[status(thm)],[95,7]) ).
tff(97,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(a,b,multiply(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(98,plain,
product(a,b,multiply(a,b)),
inference(unit_resolution,[status(thm)],[97,7]) ).
tff(99,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,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(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(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(100,plain,
( ( product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity)) )
<=> ( ~ product(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,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(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(a,b,multiply(a,b))
| ~ product(a,identity,multiply(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(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(monotonicity,[status(thm)],[100]) ).
tff(102,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(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(a,b,multiply(a,b))
| ~ product(a,identity,multiply(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(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(transitivity,[status(thm)],[101,99]) ).
tff(103,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(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| ~ product(a,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(104,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,b,multiply(a,b))
| ~ product(a,identity,multiply(a,identity))
| ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ),
inference(modus_ponens,[status(thm)],[103,102]) ).
tff(105,plain,
( ~ product(b,multiply(identity,inverse(b)),identity)
| product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ),
inference(unit_resolution,[status(thm)],[104,72,98,96]) ).
tff(106,plain,
product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)),
inference(unit_resolution,[status(thm)],[105,62]) ).
tff(107,plain,
( product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
<=> product(identity,inverse(b),inverse(b)) ),
inference(monotonicity,[status(thm)],[36,36]) ).
tff(108,plain,
( product(identity,inverse(b),inverse(b))
<=> product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b))) ),
inference(symmetry,[status(thm)],[107]) ).
tff(109,plain,
product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b))),
inference(modus_ponens,[status(thm)],[18,108]) ).
tff(110,plain,
^ [X: $i] :
refl(( product(inverse(X),X,identity)
<=> product(inverse(X),X,identity) )),
inference(bind,[status(th)],[]) ).
tff(111,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(quant_intro,[status(thm)],[110]) ).
tff(112,plain,
( ! [X: $i] : product(inverse(X),X,identity)
<=> ! [X: $i] : product(inverse(X),X,identity) ),
inference(rewrite,[status(thm)],[]) ).
tff(113,axiom,
! [X: $i] : product(inverse(X),X,identity),
file('/export/starexec/sandbox/benchmark/Axioms/GRP003-0.ax',left_inverse) ).
tff(114,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[113,112]) ).
tff(115,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(skolemize,[status(sab)],[114]) ).
tff(116,plain,
! [X: $i] : product(inverse(X),X,identity),
inference(modus_ponens,[status(thm)],[115,111]) ).
tff(117,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(multiply(a,b)),multiply(a,b),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(118,plain,
product(inverse(multiply(a,b)),multiply(a,b),identity),
inference(unit_resolution,[status(thm)],[117,116]) ).
tff(119,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(120,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)],[119]) ).
tff(121,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(122,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(123,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)],[122]) ).
tff(124,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(125,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)],[124,123]) ).
tff(126,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)],[125,121]) ).
tff(127,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)],[126]) ).
tff(128,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)],[127,120]) ).
tff(129,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,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(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(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(130,plain,
( ( product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) )
<=> ( ~ product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(131,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(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(inverse(multiply(a,b)),multiply(a,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,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(monotonicity,[status(thm)],[130]) ).
tff(132,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(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(inverse(multiply(a,b)),multiply(a,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,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ) ),
inference(transitivity,[status(thm)],[131,129]) ).
tff(133,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(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(identity,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(134,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,multiply(identity,inverse(b)),multiply(identity,inverse(b)))
| ~ product(inverse(multiply(a,b)),multiply(a,b),identity)
| product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,b),multiply(identity,inverse(b)),multiply(a,identity)) ),
inference(modus_ponens,[status(thm)],[133,132]) ).
tff(135,plain,
product(inverse(multiply(a,b)),multiply(a,identity),multiply(identity,inverse(b))),
inference(unit_resolution,[status(thm)],[134,128,118,109,106]) ).
tff(136,plain,
product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))),
inference(modus_ponens,[status(thm)],[135,94]) ).
tff(137,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(a),inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(138,plain,
product(identity,inverse(a),inverse(a)),
inference(unit_resolution,[status(thm)],[137,16]) ).
tff(139,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(140,plain,
product(a,inverse(a),identity),
inference(unit_resolution,[status(thm)],[139,45]) ).
tff(141,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(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(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(multiply(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(142,plain,
( ( product(multiply(a,identity),inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,identity,multiply(a,identity))
| ~ product(a,inverse(a),identity) )
<=> ( product(multiply(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(143,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(a,identity),inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,identity,multiply(a,identity))
| ~ product(a,inverse(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(multiply(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity) ) ),
inference(monotonicity,[status(thm)],[142]) ).
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(a,identity),inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,identity,multiply(a,identity))
| ~ product(a,inverse(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(multiply(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity) ) ),
inference(transitivity,[status(thm)],[143,141]) ).
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(a,identity),inverse(a),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,identity,multiply(a,identity))
| ~ product(a,inverse(a),identity) ),
inference(quant_inst,[status(thm)],[]) ).
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(a,identity),inverse(a),identity)
| ~ product(a,identity,multiply(a,identity))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(a,inverse(a),identity) ),
inference(modus_ponens,[status(thm)],[145,144]) ).
tff(147,plain,
product(multiply(a,identity),inverse(a),identity),
inference(unit_resolution,[status(thm)],[146,72,140,138,96]) ).
tff(148,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(multiply(a,identity)),multiply(a,identity),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(149,plain,
product(inverse(multiply(a,identity)),multiply(a,identity),identity),
inference(unit_resolution,[status(thm)],[148,116]) ).
tff(150,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(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),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(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(151,plain,
( ( product(inverse(multiply(a,identity)),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| ~ product(inverse(multiply(a,identity)),multiply(a,identity),identity) )
<=> ( ~ product(inverse(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),identity,inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(152,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(multiply(a,identity)),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| ~ product(inverse(multiply(a,identity)),multiply(a,identity),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(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),identity,inverse(a)) ) ),
inference(monotonicity,[status(thm)],[151]) ).
tff(153,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(multiply(a,identity)),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| ~ product(inverse(multiply(a,identity)),multiply(a,identity),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(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),identity,inverse(a)) ) ),
inference(transitivity,[status(thm)],[152,150]) ).
tff(154,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(multiply(a,identity)),identity,inverse(a))
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| ~ product(inverse(multiply(a,identity)),multiply(a,identity),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(155,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(multiply(a,identity)),multiply(a,identity),identity)
| ~ product(identity,inverse(a),inverse(a))
| ~ product(multiply(a,identity),inverse(a),identity)
| product(inverse(multiply(a,identity)),identity,inverse(a)) ),
inference(modus_ponens,[status(thm)],[154,153]) ).
tff(156,plain,
product(inverse(multiply(a,identity)),identity,inverse(a)),
inference(unit_resolution,[status(thm)],[155,128,138,149,147]) ).
tff(157,plain,
( ~ ! [X: $i] : product(X,identity,X)
| product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(158,plain,
product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))),
inference(unit_resolution,[status(thm)],[157,55]) ).
tff(159,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(160,plain,
( ( ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity)))
| ~ product(inverse(multiply(a,identity)),identity,inverse(a)) )
<=> ( ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(161,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity)))
| ~ product(inverse(multiply(a,identity)),identity,inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ) ),
inference(monotonicity,[status(thm)],[160]) ).
tff(162,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity)))
| ~ product(inverse(multiply(a,identity)),identity,inverse(a)) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ) ),
inference(transitivity,[status(thm)],[161,159]) ).
tff(163,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity)))
| ~ product(inverse(multiply(a,identity)),identity,inverse(a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(164,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(a))
| ( inverse(a) = inverse(multiply(a,identity)) )
| ~ product(inverse(multiply(a,identity)),identity,inverse(multiply(a,identity))) ),
inference(modus_ponens,[status(thm)],[163,162]) ).
tff(165,plain,
inverse(a) = inverse(multiply(a,identity)),
inference(unit_resolution,[status(thm)],[164,28,158,156]) ).
tff(166,plain,
inverse(multiply(a,identity)) = inverse(a),
inference(symmetry,[status(thm)],[165]) ).
tff(167,plain,
( product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
<=> product(inverse(b),inverse(a),multiply(inverse(b),inverse(a))) ),
inference(monotonicity,[status(thm)],[36,166]) ).
tff(168,plain,
( product(inverse(b),inverse(a),multiply(inverse(b),inverse(a)))
<=> product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a))) ),
inference(symmetry,[status(thm)],[167]) ).
tff(169,plain,
( ~ ! [Y: $i,X: $i] : product(X,Y,multiply(X,Y))
| product(inverse(b),inverse(a),multiply(inverse(b),inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(170,plain,
product(inverse(b),inverse(a),multiply(inverse(b),inverse(a))),
inference(unit_resolution,[status(thm)],[169,7]) ).
tff(171,plain,
product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a))),
inference(modus_ponens,[status(thm)],[170,168]) ).
tff(172,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(multiply(a,identity),inverse(multiply(a,identity)),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(173,plain,
product(multiply(a,identity),inverse(multiply(a,identity)),identity),
inference(unit_resolution,[status(thm)],[172,45]) ).
tff(174,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(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),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(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(175,plain,
( ( product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))) )
<=> ( ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(176,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(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ) ),
inference(monotonicity,[status(thm)],[175]) ).
tff(177,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(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( product(X,V,W)
| ~ product(U,Z,W)
| ~ product(Y,Z,V)
| ~ product(X,Y,U) )
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ) ),
inference(transitivity,[status(thm)],[176,174]) ).
tff(178,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(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(179,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(multiply(identity,inverse(b)),inverse(multiply(a,identity)),multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| ~ product(multiply(a,identity),inverse(multiply(a,identity)),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ),
inference(modus_ponens,[status(thm)],[178,177]) ).
tff(180,plain,
( ~ product(inverse(multiply(a,inverse(inverse(b)))),multiply(a,identity),multiply(identity,inverse(b)))
| product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ),
inference(unit_resolution,[status(thm)],[179,128,173,171]) ).
tff(181,plain,
product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))),
inference(unit_resolution,[status(thm)],[180,136]) ).
tff(182,plain,
( ~ ! [X: $i] : product(X,inverse(X),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(183,plain,
product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity),
inference(unit_resolution,[status(thm)],[182,45]) ).
tff(184,plain,
( ~ ! [X: $i] : product(inverse(X),X,identity)
| product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(185,plain,
product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity),
inference(unit_resolution,[status(thm)],[184,116]) ).
tff(186,plain,
( ~ ! [X: $i] : product(identity,X,X)
| product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b))))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(187,plain,
product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b))))),
inference(unit_resolution,[status(thm)],[186,16]) ).
tff(188,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(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(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(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(189,plain,
( ( product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity) )
<=> ( ~ product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(190,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(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(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(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ) ),
inference(monotonicity,[status(thm)],[189]) ).
tff(191,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(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(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(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ) ),
inference(transitivity,[status(thm)],[190,188]) ).
tff(192,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(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity)
| ~ product(inverse(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity) ),
inference(quant_inst,[status(thm)],[]) ).
tff(193,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(multiply(a,inverse(inverse(b)))),inverse(inverse(multiply(a,inverse(inverse(b))))),identity)
| product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(identity,inverse(multiply(a,inverse(inverse(b)))),inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(inverse(multiply(a,inverse(inverse(b))))),inverse(multiply(a,inverse(inverse(b)))),identity) ),
inference(modus_ponens,[status(thm)],[192,191]) ).
tff(194,plain,
product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))),
inference(unit_resolution,[status(thm)],[193,128,187,185,183]) ).
tff(195,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(196,plain,
( ( ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) )
<=> ( ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(197,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) ) ),
inference(monotonicity,[status(thm)],[196]) ).
tff(198,plain,
( ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) )
<=> ( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) ) ),
inference(transitivity,[status(thm)],[197,195]) ).
tff(199,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b)))))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(200,plain,
( ~ ! [W: $i,Z: $i,Y: $i,X: $i] :
( ( Z = W )
| ~ product(X,Y,W)
| ~ product(X,Y,Z) )
| ( multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))) )
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,multiply(inverse(b),inverse(a)))
| ~ product(inverse(multiply(a,inverse(inverse(b)))),identity,inverse(multiply(a,inverse(inverse(b))))) ),
inference(modus_ponens,[status(thm)],[199,198]) ).
tff(201,plain,
multiply(inverse(b),inverse(a)) = inverse(multiply(a,inverse(inverse(b)))),
inference(unit_resolution,[status(thm)],[200,28,194,181]) ).
tff(202,plain,
inverse(multiply(a,inverse(inverse(b)))) = multiply(inverse(b),inverse(a)),
inference(symmetry,[status(thm)],[201]) ).
tff(203,plain,
inverse(multiply(a,b)) = multiply(inverse(b),inverse(a)),
inference(transitivity,[status(thm)],[91,202]) ).
tff(204,plain,
( ( inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)) )
<=> ( inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(205,axiom,
inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_inverse_of_product_is_product_of_inverses) ).
tff(206,plain,
inverse(multiply(a,b)) != multiply(inverse(b),inverse(a)),
inference(modus_ponens,[status(thm)],[205,204]) ).
tff(207,plain,
$false,
inference(unit_resolution,[status(thm)],[206,203]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : GRP012-3 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n015.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Jun 19 15:27:55 EDT 2025
% 0.13/0.34 % CPUTime :
% 11.77/11.95 % SZS status Unsatisfiable
% 11.77/11.95 % SZS output start Proof
% See solution above
% 11.77/11.98 % E exiting
%------------------------------------------------------------------------------