%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : RNG039-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n011.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 05:23:58 AM UTC 2025
% Result : Unsatisfiable 0.19s 0.46s
% Output : Proof 0.19s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 108
% Syntax : Number of formulae : 297 ( 135 unt; 0 typ; 0 def)
% Number of atoms : 1075 ( 195 equ)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 1390 ( 647 ~; 626 |; 0 &)
% ( 117 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 35 ( 35 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 : 6 ( 6 usr; 4 con; 0-2 aty)
% Number of variables : 735 ( 677 !; 0 ?; 735 :)
% Comments :
%------------------------------------------------------------------------------
tff(product_type,type,
product: ( $i * $i * $i ) > $o ).
tff(add_type,type,
add: ( $i * $i ) > $i ).
tff(a_type,type,
a: $i ).
tff(c_type,type,
c: $i ).
tff(sum_type,type,
sum: ( $i * $i * $i ) > $o ).
tff(multiply_type,type,
multiply: ( $i * $i ) > $i ).
tff(b_type,type,
b: $i ).
tff(d_type,type,
d: $i ).
tff(1,plain,
( ( d = c )
<=> ( c = d ) ),
inference(commutativity,[status(thm)],[]) ).
tff(2,plain,
( product(a,b,c)
<=> product(a,b,c) ),
inference(rewrite,[status(thm)],[]) ).
tff(3,axiom,
product(a,b,c),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',a_times_b) ).
tff(4,plain,
product(a,b,c),
inference(modus_ponens,[status(thm)],[3,2]) ).
tff(5,plain,
( product(a,b,multiply(c,b))
<=> product(a,b,multiply(c,b)) ),
inference(rewrite,[status(thm)],[]) ).
tff(6,axiom,
product(a,b,multiply(c,b)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause45) ).
tff(7,plain,
product(a,b,multiply(c,b)),
inference(modus_ponens,[status(thm)],[6,5]) ).
tff(8,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
<=> ( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ) )),
inference(bind,[status(th)],[]) ).
tff(9,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[8]) ).
tff(10,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(11,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
rewrite(( ( ~ product(X,Y,U)
| ~ product(X,Y,V)
| ( U = V ) )
<=> ( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ) )),
inference(bind,[status(th)],[]) ).
tff(12,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[11]) ).
tff(13,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,U)
| ~ product(X,Y,V)
| ( U = V ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',multiplication_is_well_defined) ).
tff(14,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[13,12]) ).
tff(15,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[14,10]) ).
tff(16,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ),
inference(skolemize,[status(sab)],[15]) ).
tff(17,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[16,9]) ).
tff(18,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,c)
| ( c = multiply(c,b) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,c)
| ( c = multiply(c,b) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,c)
| ( c = multiply(c,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(20,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,c)
| ( c = multiply(c,b) ) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
c = multiply(c,b),
inference(unit_resolution,[status(thm)],[20,17,7,4]) ).
tff(22,plain,
multiply(c,b) = c,
inference(symmetry,[status(thm)],[21]) ).
tff(23,plain,
( product(b,a,d)
<=> product(b,a,d) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,axiom,
product(b,a,d),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',b_times_a) ).
tff(25,plain,
product(b,a,d),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
( product(b,a,multiply(d,a))
<=> product(b,a,multiply(d,a)) ),
inference(rewrite,[status(thm)],[]) ).
tff(27,axiom,
product(b,a,multiply(d,a)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause48) ).
tff(28,plain,
product(b,a,multiply(d,a)),
inference(modus_ponens,[status(thm)],[27,26]) ).
tff(29,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,d)
| ( d = multiply(d,a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,d)
| ( d = multiply(d,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(30,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,d)
| ( d = multiply(d,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(31,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,d)
| ( d = multiply(d,a) ) ),
inference(modus_ponens,[status(thm)],[30,29]) ).
tff(32,plain,
d = multiply(d,a),
inference(unit_resolution,[status(thm)],[31,17,28,25]) ).
tff(33,plain,
multiply(d,a) = d,
inference(symmetry,[status(thm)],[32]) ).
tff(34,plain,
( ( multiply(d,a) = multiply(c,b) )
<=> ( d = c ) ),
inference(monotonicity,[status(thm)],[33,22]) ).
tff(35,plain,
( ( multiply(d,a) = multiply(c,b) )
<=> ( c = d ) ),
inference(transitivity,[status(thm)],[34,1]) ).
tff(36,plain,
( ( c = d )
<=> ( multiply(d,a) = multiply(c,b) ) ),
inference(symmetry,[status(thm)],[35]) ).
tff(37,plain,
( ( c != d )
<=> ( multiply(d,a) != multiply(c,b) ) ),
inference(monotonicity,[status(thm)],[36]) ).
tff(38,plain,
( ( c != d )
<=> ( c != d ) ),
inference(rewrite,[status(thm)],[]) ).
tff(39,axiom,
c != d,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_c_equals_d) ).
tff(40,plain,
c != d,
inference(modus_ponens,[status(thm)],[39,38]) ).
tff(41,plain,
multiply(d,a) != multiply(c,b),
inference(modus_ponens,[status(thm)],[40,37]) ).
tff(42,plain,
multiply(multiply(c,b),multiply(d,a)) = multiply(c,d),
inference(monotonicity,[status(thm)],[22,33]) ).
tff(43,plain,
multiply(c,d) = multiply(multiply(c,b),multiply(d,a)),
inference(symmetry,[status(thm)],[42]) ).
tff(44,plain,
multiply(b,multiply(d,a)) = multiply(b,d),
inference(monotonicity,[status(thm)],[33]) ).
tff(45,plain,
multiply(b,d) = multiply(b,multiply(d,a)),
inference(symmetry,[status(thm)],[44]) ).
tff(46,plain,
( product(b,a,multiply(b,d))
<=> product(b,a,multiply(b,d)) ),
inference(rewrite,[status(thm)],[]) ).
tff(47,axiom,
product(b,a,multiply(b,d)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause58) ).
tff(48,plain,
product(b,a,multiply(b,d)),
inference(modus_ponens,[status(thm)],[47,46]) ).
tff(49,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,multiply(b,d))
| ( multiply(b,d) = multiply(d,a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,multiply(b,d))
| ( multiply(b,d) = multiply(d,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(50,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,multiply(b,d))
| ( multiply(b,d) = multiply(d,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(b,a,multiply(d,a))
| ~ product(b,a,multiply(b,d))
| ( multiply(b,d) = multiply(d,a) ) ),
inference(modus_ponens,[status(thm)],[50,49]) ).
tff(52,plain,
multiply(b,d) = multiply(d,a),
inference(unit_resolution,[status(thm)],[51,17,28,48]) ).
tff(53,plain,
multiply(d,a) = multiply(b,d),
inference(symmetry,[status(thm)],[52]) ).
tff(54,plain,
multiply(d,a) = multiply(b,multiply(d,a)),
inference(transitivity,[status(thm)],[53,45]) ).
tff(55,plain,
( product(a,multiply(d,a),multiply(c,d))
<=> product(a,multiply(b,multiply(d,a)),multiply(multiply(c,b),multiply(d,a))) ),
inference(monotonicity,[status(thm)],[54,43]) ).
tff(56,plain,
( product(a,multiply(b,multiply(d,a)),multiply(multiply(c,b),multiply(d,a)))
<=> product(a,multiply(d,a),multiply(c,d)) ),
inference(symmetry,[status(thm)],[55]) ).
tff(57,plain,
^ [B: $i,A: $i] :
refl(( product(a,multiply(b,A),multiply(B,A))
<=> product(a,multiply(b,A),multiply(B,A)) )),
inference(bind,[status(th)],[]) ).
tff(58,plain,
( ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
<=> ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)) ),
inference(quant_intro,[status(thm)],[57]) ).
tff(59,plain,
( ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
<=> ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)) ),
inference(rewrite,[status(thm)],[]) ).
tff(60,axiom,
! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause44) ).
tff(61,plain,
! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)),
inference(modus_ponens,[status(thm)],[60,59]) ).
tff(62,plain,
! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)),
inference(skolemize,[status(sab)],[61]) ).
tff(63,plain,
! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A)),
inference(modus_ponens,[status(thm)],[62,58]) ).
tff(64,plain,
( ~ ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
| product(a,multiply(b,multiply(d,a)),multiply(multiply(c,b),multiply(d,a))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(65,plain,
product(a,multiply(b,multiply(d,a)),multiply(multiply(c,b),multiply(d,a))),
inference(unit_resolution,[status(thm)],[64,63]) ).
tff(66,plain,
product(a,multiply(d,a),multiply(c,d)),
inference(modus_ponens,[status(thm)],[65,56]) ).
tff(67,plain,
^ [A: $i] :
refl(( ( multiply(A,A) = A )
<=> ( multiply(A,A) = A ) )),
inference(bind,[status(th)],[]) ).
tff(68,plain,
( ! [A: $i] : ( multiply(A,A) = A )
<=> ! [A: $i] : ( multiply(A,A) = A ) ),
inference(quant_intro,[status(thm)],[67]) ).
tff(69,plain,
( ! [A: $i] : ( multiply(A,A) = A )
<=> ! [A: $i] : ( multiply(A,A) = A ) ),
inference(rewrite,[status(thm)],[]) ).
tff(70,axiom,
! [A: $i] : ( multiply(A,A) = A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause35) ).
tff(71,plain,
! [A: $i] : ( multiply(A,A) = A ),
inference(modus_ponens,[status(thm)],[70,69]) ).
tff(72,plain,
! [A: $i] : ( multiply(A,A) = A ),
inference(skolemize,[status(sab)],[71]) ).
tff(73,plain,
! [A: $i] : ( multiply(A,A) = A ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
( ~ ! [A: $i] : ( multiply(A,A) = A )
| ( multiply(a,a) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(75,plain,
multiply(a,a) = a,
inference(unit_resolution,[status(thm)],[74,73]) ).
tff(76,plain,
a = multiply(a,a),
inference(symmetry,[status(thm)],[75]) ).
tff(77,plain,
( ( multiply(b,a) = d )
<=> ( multiply(b,a) = d ) ),
inference(rewrite,[status(thm)],[]) ).
tff(78,axiom,
multiply(b,a) = d,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause37) ).
tff(79,plain,
multiply(b,a) = d,
inference(modus_ponens,[status(thm)],[78,77]) ).
tff(80,plain,
d = multiply(b,a),
inference(symmetry,[status(thm)],[79]) ).
tff(81,plain,
multiply(d,a) = multiply(b,a),
inference(transitivity,[status(thm)],[33,80]) ).
tff(82,plain,
( product(a,multiply(d,a),a)
<=> product(a,multiply(b,a),multiply(a,a)) ),
inference(monotonicity,[status(thm)],[81,76]) ).
tff(83,plain,
( product(a,multiply(b,a),multiply(a,a))
<=> product(a,multiply(d,a),a) ),
inference(symmetry,[status(thm)],[82]) ).
tff(84,plain,
( ~ ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
| product(a,multiply(b,a),multiply(a,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(85,plain,
product(a,multiply(b,a),multiply(a,a)),
inference(unit_resolution,[status(thm)],[84,63]) ).
tff(86,plain,
product(a,multiply(d,a),a),
inference(modus_ponens,[status(thm)],[85,83]) ).
tff(87,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),a)
| ~ product(a,multiply(d,a),multiply(c,d))
| ( multiply(c,d) = a ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),a)
| ~ product(a,multiply(d,a),multiply(c,d))
| ( multiply(c,d) = a ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),a)
| ~ product(a,multiply(d,a),multiply(c,d))
| ( multiply(c,d) = a ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(89,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),a)
| ~ product(a,multiply(d,a),multiply(c,d))
| ( multiply(c,d) = a ) ),
inference(modus_ponens,[status(thm)],[88,87]) ).
tff(90,plain,
( ~ product(a,multiply(d,a),a)
| ~ product(a,multiply(d,a),multiply(c,d))
| ( multiply(c,d) = a ) ),
inference(unit_resolution,[status(thm)],[89,17]) ).
tff(91,plain,
multiply(c,d) = a,
inference(unit_resolution,[status(thm)],[90,86,66]) ).
tff(92,plain,
( product(multiply(c,b),a,multiply(a,d))
<=> product(c,a,multiply(a,d)) ),
inference(monotonicity,[status(thm)],[22]) ).
tff(93,plain,
( product(c,a,multiply(a,d))
<=> product(multiply(c,b),a,multiply(a,d)) ),
inference(symmetry,[status(thm)],[92]) ).
tff(94,plain,
( product(c,a,multiply(a,d))
<=> product(c,a,multiply(a,d)) ),
inference(rewrite,[status(thm)],[]) ).
tff(95,axiom,
product(c,a,multiply(a,d)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause60) ).
tff(96,plain,
product(c,a,multiply(a,d)),
inference(modus_ponens,[status(thm)],[95,94]) ).
tff(97,plain,
product(multiply(c,b),a,multiply(a,d)),
inference(modus_ponens,[status(thm)],[96,93]) ).
tff(98,plain,
^ [A: $i] :
refl(( product(multiply(A,b),a,multiply(A,d))
<=> product(multiply(A,b),a,multiply(A,d)) )),
inference(bind,[status(th)],[]) ).
tff(99,plain,
( ! [A: $i] : product(multiply(A,b),a,multiply(A,d))
<=> ! [A: $i] : product(multiply(A,b),a,multiply(A,d)) ),
inference(quant_intro,[status(thm)],[98]) ).
tff(100,plain,
( ! [A: $i] : product(multiply(A,b),a,multiply(A,d))
<=> ! [A: $i] : product(multiply(A,b),a,multiply(A,d)) ),
inference(rewrite,[status(thm)],[]) ).
tff(101,axiom,
! [A: $i] : product(multiply(A,b),a,multiply(A,d)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause57) ).
tff(102,plain,
! [A: $i] : product(multiply(A,b),a,multiply(A,d)),
inference(modus_ponens,[status(thm)],[101,100]) ).
tff(103,plain,
! [A: $i] : product(multiply(A,b),a,multiply(A,d)),
inference(skolemize,[status(sab)],[102]) ).
tff(104,plain,
! [A: $i] : product(multiply(A,b),a,multiply(A,d)),
inference(modus_ponens,[status(thm)],[103,99]) ).
tff(105,plain,
( ~ ! [A: $i] : product(multiply(A,b),a,multiply(A,d))
| product(multiply(c,b),a,multiply(c,d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(106,plain,
product(multiply(c,b),a,multiply(c,d)),
inference(unit_resolution,[status(thm)],[105,104]) ).
tff(107,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(multiply(c,b),a,multiply(a,d))
| ~ product(multiply(c,b),a,multiply(c,d))
| ( multiply(c,d) = multiply(a,d) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(multiply(c,b),a,multiply(a,d))
| ~ product(multiply(c,b),a,multiply(c,d))
| ( multiply(c,d) = multiply(a,d) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(108,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(multiply(c,b),a,multiply(a,d))
| ~ product(multiply(c,b),a,multiply(c,d))
| ( multiply(c,d) = multiply(a,d) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(109,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(multiply(c,b),a,multiply(a,d))
| ~ product(multiply(c,b),a,multiply(c,d))
| ( multiply(c,d) = multiply(a,d) ) ),
inference(modus_ponens,[status(thm)],[108,107]) ).
tff(110,plain,
( ~ product(multiply(c,b),a,multiply(a,d))
| ( multiply(c,d) = multiply(a,d) ) ),
inference(unit_resolution,[status(thm)],[109,17,106]) ).
tff(111,plain,
multiply(c,d) = multiply(a,d),
inference(unit_resolution,[status(thm)],[110,97]) ).
tff(112,plain,
multiply(a,d) = multiply(c,d),
inference(symmetry,[status(thm)],[111]) ).
tff(113,plain,
( product(a,multiply(d,a),multiply(d,a))
<=> product(a,multiply(b,d),multiply(b,d)) ),
inference(monotonicity,[status(thm)],[53,53]) ).
tff(114,plain,
( product(a,multiply(b,d),multiply(b,d))
<=> product(a,multiply(d,a),multiply(d,a)) ),
inference(symmetry,[status(thm)],[113]) ).
tff(115,plain,
( ~ ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
| product(a,multiply(b,d),multiply(b,d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(116,plain,
product(a,multiply(b,d),multiply(b,d)),
inference(unit_resolution,[status(thm)],[115,63]) ).
tff(117,plain,
product(a,multiply(d,a),multiply(d,a)),
inference(modus_ponens,[status(thm)],[116,114]) ).
tff(118,plain,
( product(a,multiply(d,a),multiply(a,d))
<=> product(a,multiply(b,d),multiply(a,d)) ),
inference(monotonicity,[status(thm)],[53]) ).
tff(119,plain,
( product(a,multiply(b,d),multiply(a,d))
<=> product(a,multiply(d,a),multiply(a,d)) ),
inference(symmetry,[status(thm)],[118]) ).
tff(120,plain,
( ~ ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
| product(a,multiply(b,d),multiply(a,d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(121,plain,
product(a,multiply(b,d),multiply(a,d)),
inference(unit_resolution,[status(thm)],[120,63]) ).
tff(122,plain,
product(a,multiply(d,a),multiply(a,d)),
inference(modus_ponens,[status(thm)],[121,119]) ).
tff(123,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),multiply(d,a))
| ~ product(a,multiply(d,a),multiply(a,d))
| ( multiply(a,d) = multiply(d,a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),multiply(d,a))
| ~ product(a,multiply(d,a),multiply(a,d))
| ( multiply(a,d) = multiply(d,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(124,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),multiply(d,a))
| ~ product(a,multiply(d,a),multiply(a,d))
| ( multiply(a,d) = multiply(d,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(125,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,multiply(d,a),multiply(d,a))
| ~ product(a,multiply(d,a),multiply(a,d))
| ( multiply(a,d) = multiply(d,a) ) ),
inference(modus_ponens,[status(thm)],[124,123]) ).
tff(126,plain,
( ~ product(a,multiply(d,a),multiply(d,a))
| ~ product(a,multiply(d,a),multiply(a,d))
| ( multiply(a,d) = multiply(d,a) ) ),
inference(unit_resolution,[status(thm)],[125,17]) ).
tff(127,plain,
multiply(a,d) = multiply(d,a),
inference(unit_resolution,[status(thm)],[126,122,117]) ).
tff(128,plain,
multiply(d,a) = multiply(a,d),
inference(symmetry,[status(thm)],[127]) ).
tff(129,plain,
multiply(d,a) = a,
inference(transitivity,[status(thm)],[128,112,91]) ).
tff(130,plain,
( sum(multiply(c,b),add(c,a),multiply(d,a))
<=> sum(c,add(c,a),a) ),
inference(monotonicity,[status(thm)],[22,129]) ).
tff(131,plain,
( sum(c,add(c,a),a)
<=> sum(multiply(c,b),add(c,a),multiply(d,a)) ),
inference(symmetry,[status(thm)],[130]) ).
tff(132,plain,
^ [B: $i,A: $i] :
refl(( sum(A,add(A,B),B)
<=> sum(A,add(A,B),B) )),
inference(bind,[status(th)],[]) ).
tff(133,plain,
( ! [B: $i,A: $i] : sum(A,add(A,B),B)
<=> ! [B: $i,A: $i] : sum(A,add(A,B),B) ),
inference(quant_intro,[status(thm)],[132]) ).
tff(134,plain,
( ! [B: $i,A: $i] : sum(A,add(A,B),B)
<=> ! [B: $i,A: $i] : sum(A,add(A,B),B) ),
inference(rewrite,[status(thm)],[]) ).
tff(135,axiom,
! [B: $i,A: $i] : sum(A,add(A,B),B),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',absorbtion1) ).
tff(136,plain,
! [B: $i,A: $i] : sum(A,add(A,B),B),
inference(modus_ponens,[status(thm)],[135,134]) ).
tff(137,plain,
! [B: $i,A: $i] : sum(A,add(A,B),B),
inference(skolemize,[status(sab)],[136]) ).
tff(138,plain,
! [B: $i,A: $i] : sum(A,add(A,B),B),
inference(modus_ponens,[status(thm)],[137,133]) ).
tff(139,plain,
( ~ ! [B: $i,A: $i] : sum(A,add(A,B),B)
| sum(c,add(c,a),a) ),
inference(quant_inst,[status(thm)],[]) ).
tff(140,plain,
sum(c,add(c,a),a),
inference(unit_resolution,[status(thm)],[139,138]) ).
tff(141,plain,
sum(multiply(c,b),add(c,a),multiply(d,a)),
inference(modus_ponens,[status(thm)],[140,131]) ).
tff(142,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
refl(( ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) )),
inference(bind,[status(th)],[]) ).
tff(143,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[142]) ).
tff(144,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(145,plain,
^ [V: $i,Y: $i,U: $i,X: $i] :
rewrite(( ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) )),
inference(bind,[status(th)],[]) ).
tff(146,plain,
( ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
<=> ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ) ),
inference(quant_intro,[status(thm)],[145]) ).
tff(147,axiom,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',addition_is_well_defined) ).
tff(148,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[147,146]) ).
tff(149,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[148,144]) ).
tff(150,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(skolemize,[status(sab)],[149]) ).
tff(151,plain,
! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) ),
inference(modus_ponens,[status(thm)],[150,143]) ).
tff(152,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(153,plain,
( ( ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b))
| ( multiply(d,a) = multiply(c,b) ) )
<=> ( ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(154,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b))
| ( multiply(d,a) = multiply(c,b) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ) ),
inference(monotonicity,[status(thm)],[153]) ).
tff(155,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b))
| ( multiply(d,a) = multiply(c,b) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ) ),
inference(transitivity,[status(thm)],[154,152]) ).
tff(156,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b))
| ( multiply(d,a) = multiply(c,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(157,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ),
inference(modus_ponens,[status(thm)],[156,155]) ).
tff(158,plain,
( ( multiply(d,a) = multiply(c,b) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ sum(multiply(c,b),add(c,a),multiply(c,b)) ),
inference(unit_resolution,[status(thm)],[157,151]) ).
tff(159,plain,
~ sum(multiply(c,b),add(c,a),multiply(c,b)),
inference(unit_resolution,[status(thm)],[158,141,41]) ).
tff(160,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) )),
inference(bind,[status(th)],[]) ).
tff(161,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(quant_intro,[status(thm)],[160]) ).
tff(162,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(163,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',commutativity_of_addition) ).
tff(164,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[163,162]) ).
tff(165,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(skolemize,[status(sab)],[164]) ).
tff(166,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) ),
inference(modus_ponens,[status(thm)],[165,161]) ).
tff(167,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(add(c,a),multiply(c,b),multiply(c,b))
| sum(multiply(c,b),add(c,a),multiply(c,b)) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(add(c,a),multiply(c,b),multiply(c,b))
| sum(multiply(c,b),add(c,a),multiply(c,b)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(168,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(add(c,a),multiply(c,b),multiply(c,b))
| sum(multiply(c,b),add(c,a),multiply(c,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(169,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ sum(X,Y,Z)
| sum(Y,X,Z) )
| ~ sum(add(c,a),multiply(c,b),multiply(c,b))
| sum(multiply(c,b),add(c,a),multiply(c,b)) ),
inference(modus_ponens,[status(thm)],[168,167]) ).
tff(170,plain,
~ sum(add(c,a),multiply(c,b),multiply(c,b)),
inference(unit_resolution,[status(thm)],[169,166,159]) ).
tff(171,plain,
( sum(add(c,a),multiply(d,a),multiply(c,b))
<=> sum(add(c,a),a,c) ),
inference(monotonicity,[status(thm)],[129,22]) ).
tff(172,plain,
( sum(add(c,a),a,c)
<=> sum(add(c,a),multiply(d,a),multiply(c,b)) ),
inference(symmetry,[status(thm)],[171]) ).
tff(173,plain,
^ [B: $i,A: $i] :
refl(( sum(add(A,B),B,A)
<=> sum(add(A,B),B,A) )),
inference(bind,[status(th)],[]) ).
tff(174,plain,
( ! [B: $i,A: $i] : sum(add(A,B),B,A)
<=> ! [B: $i,A: $i] : sum(add(A,B),B,A) ),
inference(quant_intro,[status(thm)],[173]) ).
tff(175,plain,
( ! [B: $i,A: $i] : sum(add(A,B),B,A)
<=> ! [B: $i,A: $i] : sum(add(A,B),B,A) ),
inference(rewrite,[status(thm)],[]) ).
tff(176,axiom,
! [B: $i,A: $i] : sum(add(A,B),B,A),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',absorbtion2) ).
tff(177,plain,
! [B: $i,A: $i] : sum(add(A,B),B,A),
inference(modus_ponens,[status(thm)],[176,175]) ).
tff(178,plain,
! [B: $i,A: $i] : sum(add(A,B),B,A),
inference(skolemize,[status(sab)],[177]) ).
tff(179,plain,
! [B: $i,A: $i] : sum(add(A,B),B,A),
inference(modus_ponens,[status(thm)],[178,174]) ).
tff(180,plain,
( ~ ! [B: $i,A: $i] : sum(add(A,B),B,A)
| sum(add(c,a),a,c) ),
inference(quant_inst,[status(thm)],[]) ).
tff(181,plain,
sum(add(c,a),a,c),
inference(unit_resolution,[status(thm)],[180,179]) ).
tff(182,plain,
sum(add(c,a),multiply(d,a),multiply(c,b)),
inference(modus_ponens,[status(thm)],[181,172]) ).
tff(183,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
refl(( ( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
<=> ( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
inference(bind,[status(th)],[]) ).
tff(184,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[183]) ).
tff(185,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(186,plain,
^ [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) )
<=> ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) ) )),
rewrite(( ( ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U)
| sum(X,V,W) )
<=> ( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
( ( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) )
<=> ( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) )),
inference(bind,[status(th)],[]) ).
tff(187,plain,
( ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) )
<=> ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ) ),
inference(quant_intro,[status(thm)],[186]) ).
tff(188,axiom,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(Y,Z,V)
| ~ sum(U,Z,W)
| sum(X,V,W) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',associativity_of_addition1) ).
tff(189,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[188,187]) ).
tff(190,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[189,185]) ).
tff(191,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(skolemize,[status(sab)],[190]) ).
tff(192,plain,
! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) ),
inference(modus_ponens,[status(thm)],[191,184]) ).
tff(193,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(194,plain,
( ( sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) )
<=> ( ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(195,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ) ),
inference(monotonicity,[status(thm)],[194]) ).
tff(196,plain,
( ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) )
<=> ( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ) ),
inference(transitivity,[status(thm)],[195,193]) ).
tff(197,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(198,plain,
( ~ ! [W: $i,V: $i,Z: $i,Y: $i,U: $i,X: $i] :
( sum(X,V,W)
| ~ sum(U,Z,W)
| ~ sum(Y,Z,V)
| ~ sum(X,Y,U) )
| ~ sum(add(c,a),multiply(d,a),multiply(c,b))
| sum(add(c,a),multiply(c,b),multiply(c,b))
| ~ sum(add(c,a),add(c,a),add(c,a)) ),
inference(modus_ponens,[status(thm)],[197,196]) ).
tff(199,plain,
~ sum(add(c,a),add(c,a),add(c,a)),
inference(unit_resolution,[status(thm)],[198,192,182,170]) ).
tff(200,plain,
( ~ ! [A: $i] : ( multiply(A,A) = A )
| ( multiply(b,b) = b ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(201,plain,
multiply(b,b) = b,
inference(unit_resolution,[status(thm)],[200,73]) ).
tff(202,plain,
b = multiply(b,b),
inference(symmetry,[status(thm)],[201]) ).
tff(203,plain,
( product(a,b,b)
<=> product(a,multiply(b,b),multiply(b,b)) ),
inference(monotonicity,[status(thm)],[202,202]) ).
tff(204,plain,
( product(a,multiply(b,b),multiply(b,b))
<=> product(a,b,b) ),
inference(symmetry,[status(thm)],[203]) ).
tff(205,plain,
( ~ ! [B: $i,A: $i] : product(a,multiply(b,A),multiply(B,A))
| product(a,multiply(b,b),multiply(b,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(206,plain,
product(a,multiply(b,b),multiply(b,b)),
inference(unit_resolution,[status(thm)],[205,63]) ).
tff(207,plain,
product(a,b,b),
inference(modus_ponens,[status(thm)],[206,204]) ).
tff(208,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,b)
| ( b = multiply(c,b) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,b)
| ( b = multiply(c,b) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(209,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,b)
| ( b = multiply(c,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(210,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ product(X,Y,V)
| ~ product(X,Y,U)
| ( U = V ) )
| ~ product(a,b,multiply(c,b))
| ~ product(a,b,b)
| ( b = multiply(c,b) ) ),
inference(modus_ponens,[status(thm)],[209,208]) ).
tff(211,plain,
( ~ product(a,b,b)
| ( b = multiply(c,b) ) ),
inference(unit_resolution,[status(thm)],[210,17,7]) ).
tff(212,plain,
b = multiply(c,b),
inference(unit_resolution,[status(thm)],[211,207]) ).
tff(213,plain,
b = c,
inference(transitivity,[status(thm)],[212,22]) ).
tff(214,plain,
add(b,a) = add(c,a),
inference(monotonicity,[status(thm)],[213]) ).
tff(215,plain,
add(c,a) = add(b,a),
inference(symmetry,[status(thm)],[214]) ).
tff(216,plain,
( product(multiply(d,a),add(c,a),add(c,a))
<=> product(a,add(b,a),add(c,a)) ),
inference(monotonicity,[status(thm)],[129,215]) ).
tff(217,plain,
( product(a,add(b,a),add(c,a))
<=> product(multiply(d,a),add(c,a),add(c,a)) ),
inference(symmetry,[status(thm)],[216]) ).
tff(218,plain,
( product(a,add(b,a),add(c,a))
<=> product(a,add(b,a),add(c,a)) ),
inference(rewrite,[status(thm)],[]) ).
tff(219,axiom,
product(a,add(b,a),add(c,a)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause63) ).
tff(220,plain,
product(a,add(b,a),add(c,a)),
inference(modus_ponens,[status(thm)],[219,218]) ).
tff(221,plain,
product(multiply(d,a),add(c,a),add(c,a)),
inference(modus_ponens,[status(thm)],[220,217]) ).
tff(222,plain,
( sum(b,multiply(d,a),add(b,d))
<=> sum(b,d,add(b,d)) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(223,plain,
( sum(b,d,add(b,d))
<=> sum(b,multiply(d,a),add(b,d)) ),
inference(symmetry,[status(thm)],[222]) ).
tff(224,plain,
^ [Y: $i,X: $i] :
refl(( sum(X,Y,add(X,Y))
<=> sum(X,Y,add(X,Y)) )),
inference(bind,[status(th)],[]) ).
tff(225,plain,
( ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
<=> ! [Y: $i,X: $i] : sum(X,Y,add(X,Y)) ),
inference(quant_intro,[status(thm)],[224]) ).
tff(226,plain,
( ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
<=> ! [Y: $i,X: $i] : sum(X,Y,add(X,Y)) ),
inference(rewrite,[status(thm)],[]) ).
tff(227,axiom,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',closure_of_addition) ).
tff(228,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[227,226]) ).
tff(229,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(skolemize,[status(sab)],[228]) ).
tff(230,plain,
! [Y: $i,X: $i] : sum(X,Y,add(X,Y)),
inference(modus_ponens,[status(thm)],[229,225]) ).
tff(231,plain,
( ~ ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
| sum(b,d,add(b,d)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(232,plain,
sum(b,d,add(b,d)),
inference(unit_resolution,[status(thm)],[231,230]) ).
tff(233,plain,
sum(b,multiply(d,a),add(b,d)),
inference(modus_ponens,[status(thm)],[232,223]) ).
tff(234,plain,
( sum(b,multiply(d,a),add(d,b))
<=> sum(b,d,add(d,b)) ),
inference(monotonicity,[status(thm)],[33]) ).
tff(235,plain,
( sum(b,d,add(d,b))
<=> sum(b,multiply(d,a),add(d,b)) ),
inference(symmetry,[status(thm)],[234]) ).
tff(236,plain,
^ [B: $i,A: $i] :
refl(( sum(A,B,add(B,A))
<=> sum(A,B,add(B,A)) )),
inference(bind,[status(th)],[]) ).
tff(237,plain,
( ! [B: $i,A: $i] : sum(A,B,add(B,A))
<=> ! [B: $i,A: $i] : sum(A,B,add(B,A)) ),
inference(quant_intro,[status(thm)],[236]) ).
tff(238,plain,
( ! [B: $i,A: $i] : sum(A,B,add(B,A))
<=> ! [B: $i,A: $i] : sum(A,B,add(B,A)) ),
inference(rewrite,[status(thm)],[]) ).
tff(239,axiom,
! [B: $i,A: $i] : sum(A,B,add(B,A)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause38) ).
tff(240,plain,
! [B: $i,A: $i] : sum(A,B,add(B,A)),
inference(modus_ponens,[status(thm)],[239,238]) ).
tff(241,plain,
! [B: $i,A: $i] : sum(A,B,add(B,A)),
inference(skolemize,[status(sab)],[240]) ).
tff(242,plain,
! [B: $i,A: $i] : sum(A,B,add(B,A)),
inference(modus_ponens,[status(thm)],[241,237]) ).
tff(243,plain,
( ~ ! [B: $i,A: $i] : sum(A,B,add(B,A))
| sum(b,d,add(d,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(244,plain,
sum(b,d,add(d,b)),
inference(unit_resolution,[status(thm)],[243,242]) ).
tff(245,plain,
sum(b,multiply(d,a),add(d,b)),
inference(modus_ponens,[status(thm)],[244,235]) ).
tff(246,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(b,multiply(d,a),add(b,d))
| ~ sum(b,multiply(d,a),add(d,b))
| ( add(b,d) = add(d,b) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(b,multiply(d,a),add(b,d))
| ~ sum(b,multiply(d,a),add(d,b))
| ( add(b,d) = add(d,b) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(247,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(b,multiply(d,a),add(b,d))
| ~ sum(b,multiply(d,a),add(d,b))
| ( add(b,d) = add(d,b) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(248,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(b,multiply(d,a),add(b,d))
| ~ sum(b,multiply(d,a),add(d,b))
| ( add(b,d) = add(d,b) ) ),
inference(modus_ponens,[status(thm)],[247,246]) ).
tff(249,plain,
( ~ sum(b,multiply(d,a),add(b,d))
| ~ sum(b,multiply(d,a),add(d,b))
| ( add(b,d) = add(d,b) ) ),
inference(unit_resolution,[status(thm)],[248,151]) ).
tff(250,plain,
add(b,d) = add(d,b),
inference(unit_resolution,[status(thm)],[249,245,233]) ).
tff(251,plain,
d = a,
inference(transitivity,[status(thm)],[32,128,112,91]) ).
tff(252,plain,
add(b,d) = add(c,a),
inference(monotonicity,[status(thm)],[213,251]) ).
tff(253,plain,
add(c,a) = add(b,d),
inference(symmetry,[status(thm)],[252]) ).
tff(254,plain,
add(c,a) = add(d,b),
inference(transitivity,[status(thm)],[253,250]) ).
tff(255,plain,
( ~ ! [Y: $i,X: $i] : sum(X,Y,add(X,Y))
| sum(a,b,add(a,b)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(256,plain,
sum(a,b,add(a,b)),
inference(unit_resolution,[status(thm)],[255,230]) ).
tff(257,plain,
( ~ ! [B: $i,A: $i] : sum(A,B,add(B,A))
| sum(a,b,add(b,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(258,plain,
sum(a,b,add(b,a)),
inference(unit_resolution,[status(thm)],[257,242]) ).
tff(259,plain,
( ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,b,add(a,b))
| ~ sum(a,b,add(b,a))
| ( add(a,b) = add(b,a) ) )
<=> ( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,b,add(a,b))
| ~ sum(a,b,add(b,a))
| ( add(a,b) = add(b,a) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(260,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,b,add(a,b))
| ~ sum(a,b,add(b,a))
| ( add(a,b) = add(b,a) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(261,plain,
( ~ ! [V: $i,Y: $i,U: $i,X: $i] :
( ~ sum(X,Y,U)
| ~ sum(X,Y,V)
| ( U = V ) )
| ~ sum(a,b,add(a,b))
| ~ sum(a,b,add(b,a))
| ( add(a,b) = add(b,a) ) ),
inference(modus_ponens,[status(thm)],[260,259]) ).
tff(262,plain,
add(a,b) = add(b,a),
inference(unit_resolution,[status(thm)],[261,151,258,256]) ).
tff(263,plain,
add(b,a) = add(a,b),
inference(symmetry,[status(thm)],[262]) ).
tff(264,plain,
add(c,a) = add(a,b),
inference(transitivity,[status(thm)],[215,263]) ).
tff(265,plain,
multiply(c,b) = b,
inference(symmetry,[status(thm)],[212]) ).
tff(266,plain,
( product(multiply(c,b),add(c,a),add(c,a))
<=> product(b,add(a,b),add(d,b)) ),
inference(monotonicity,[status(thm)],[265,264,254]) ).
tff(267,plain,
( product(b,add(a,b),add(d,b))
<=> product(multiply(c,b),add(c,a),add(c,a)) ),
inference(symmetry,[status(thm)],[266]) ).
tff(268,plain,
( product(b,add(a,b),add(d,b))
<=> product(b,add(a,b),add(d,b)) ),
inference(rewrite,[status(thm)],[]) ).
tff(269,axiom,
product(b,add(a,b),add(d,b)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause65) ).
tff(270,plain,
product(b,add(a,b),add(d,b)),
inference(modus_ponens,[status(thm)],[269,268]) ).
tff(271,plain,
product(multiply(c,b),add(c,a),add(c,a)),
inference(modus_ponens,[status(thm)],[270,267]) ).
tff(272,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
refl(( ( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
<=> ( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
inference(bind,[status(th)],[]) ).
tff(273,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) ),
inference(quant_intro,[status(thm)],[272]) ).
tff(274,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(275,plain,
^ [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
( ( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1)
| ~ product(V3,X,V4) ) )),
rewrite(( ( ~ sum(Y,Z,V3)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1)
| ~ product(V3,X,V4) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
( ( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
( ( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) )
<=> ( ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1)
| sum(V1,V2,V4) ) )),
rewrite(( ( ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1)
| sum(V1,V2,V4) )
<=> ( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
( ( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) )
<=> ( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) )),
inference(bind,[status(th)],[]) ).
tff(276,plain,
( ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) )
<=> ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ) ),
inference(quant_intro,[status(thm)],[275]) ).
tff(277,axiom,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( ~ product(Y,X,V1)
| ~ product(Z,X,V2)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| sum(V1,V2,V4) ),
file('/export/starexec/sandbox2/benchmark/Axioms/RNG001-0.ax',distributivity3) ).
tff(278,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ),
inference(modus_ponens,[status(thm)],[277,276]) ).
tff(279,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ),
inference(modus_ponens,[status(thm)],[278,274]) ).
tff(280,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ),
inference(skolemize,[status(sab)],[279]) ).
tff(281,plain,
! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) ),
inference(modus_ponens,[status(thm)],[280,273]) ).
tff(282,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(283,plain,
( ( sum(add(c,a),add(c,a),add(c,a))
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a))
| ~ product(multiply(c,b),add(c,a),add(c,a)) )
<=> ( ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(284,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| sum(add(c,a),add(c,a),add(c,a))
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a))
| ~ product(multiply(c,b),add(c,a),add(c,a)) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) ) ),
inference(monotonicity,[status(thm)],[283]) ).
tff(285,plain,
( ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| sum(add(c,a),add(c,a),add(c,a))
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a))
| ~ product(multiply(c,b),add(c,a),add(c,a)) )
<=> ( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) ) ),
inference(transitivity,[status(thm)],[284,282]) ).
tff(286,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| sum(add(c,a),add(c,a),add(c,a))
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a))
| ~ product(multiply(c,b),add(c,a),add(c,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(287,plain,
( ~ ! [Z: $i,Y: $i,V1: $i,X: $i,V2: $i,V3: $i,V4: $i] :
( sum(V1,V2,V4)
| ~ sum(Y,Z,V3)
| ~ product(V3,X,V4)
| ~ product(Z,X,V2)
| ~ product(Y,X,V1) )
| ~ sum(multiply(c,b),add(c,a),multiply(d,a))
| ~ product(multiply(c,b),add(c,a),add(c,a))
| ~ product(multiply(d,a),add(c,a),add(c,a))
| sum(add(c,a),add(c,a),add(c,a))
| ~ product(add(c,a),add(c,a),add(c,a)) ),
inference(modus_ponens,[status(thm)],[286,285]) ).
tff(288,plain,
~ product(add(c,a),add(c,a),add(c,a)),
inference(unit_resolution,[status(thm)],[287,281,141,271,221,199]) ).
tff(289,plain,
^ [A: $i] :
refl(( product(A,A,A)
<=> product(A,A,A) )),
inference(bind,[status(th)],[]) ).
tff(290,plain,
( ! [A: $i] : product(A,A,A)
<=> ! [A: $i] : product(A,A,A) ),
inference(quant_intro,[status(thm)],[289]) ).
tff(291,plain,
( ! [A: $i] : product(A,A,A)
<=> ! [A: $i] : product(A,A,A) ),
inference(rewrite,[status(thm)],[]) ).
tff(292,axiom,
! [A: $i] : product(A,A,A),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',clause71) ).
tff(293,plain,
! [A: $i] : product(A,A,A),
inference(modus_ponens,[status(thm)],[292,291]) ).
tff(294,plain,
! [A: $i] : product(A,A,A),
inference(skolemize,[status(sab)],[293]) ).
tff(295,plain,
! [A: $i] : product(A,A,A),
inference(modus_ponens,[status(thm)],[294,290]) ).
tff(296,plain,
( ~ ! [A: $i] : product(A,A,A)
| product(add(c,a),add(c,a),add(c,a)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(297,plain,
$false,
inference(unit_resolution,[status(thm)],[296,295,288]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : RNG039-1 : TPTP v9.0.0. Released v1.0.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n011.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 03:15:42 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.19/0.46 % SZS status Unsatisfiable
% 0.19/0.46 % SZS output start Proof
% See solution above
% 0.19/0.49 % E exiting
%------------------------------------------------------------------------------