TSTP Solution File: GRP039-3 by Z3---4.8.9.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP039-3 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 22:25:36 EDT 2022
% Result : Unsatisfiable 2.54s 1.85s
% Output : Proof 2.54s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP039-3 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.33 % Computer : n027.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Wed Aug 31 14:31:47 EDT 2022
% 0.13/0.34 % CPUTime :
% 0.13/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34 Usage: tptp [options] [-file:]file
% 0.13/0.34 -h, -? prints this message.
% 0.13/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.13/0.34 -m, -model generate model.
% 0.13/0.34 -p, -proof generate proof.
% 0.13/0.34 -c, -core generate unsat core of named formulas.
% 0.13/0.34 -st, -statistics display statistics.
% 0.13/0.34 -t:timeout set timeout (in second).
% 0.13/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.13/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34 -<param>:<value> configuration parameter and value.
% 0.13/0.34 -o:<output-file> file to place output in.
% 2.54/1.85 % SZS status Unsatisfiable
% 2.54/1.85 % SZS output start Proof
% 2.54/1.85 tff(product_type, type, (
% 2.54/1.85 product: ( $i * $i * $i ) > $o)).
% 2.54/1.85 tff(inverse_type, type, (
% 2.54/1.85 inverse: $i > $i)).
% 2.54/1.85 tff(a_type, type, (
% 2.54/1.85 a: $i)).
% 2.54/1.85 tff(c_type, type, (
% 2.54/1.85 c: $i)).
% 2.54/1.85 tff(multiply_type, type, (
% 2.54/1.85 multiply: ( $i * $i ) > $i)).
% 2.54/1.85 tff(b_type, type, (
% 2.54/1.85 b: $i)).
% 2.54/1.85 tff(identity_type, type, (
% 2.54/1.85 identity: $i)).
% 2.54/1.85 tff(subgroup_member_type, type, (
% 2.54/1.85 subgroup_member: $i > $o)).
% 2.54/1.85 tff(element_in_O2_type, type, (
% 2.54/1.85 element_in_O2: ( $i * $i ) > $i)).
% 2.54/1.85 tff(d_type, type, (
% 2.54/1.85 d: $i)).
% 2.54/1.85 tff(1,plain,
% 2.54/1.85 (![A: $i] : (inverse(inverse(A)) = A) <=> ![A: $i] : (inverse(inverse(A)) = A)),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(2,plain,
% 2.54/1.85 (![A: $i] : (inverse(inverse(A)) = A) <=> ![A: $i] : (inverse(inverse(A)) = A)),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(3,axiom,(![A: $i] : (inverse(inverse(A)) = A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','inverse_is_self_cancelling')).
% 2.54/1.85 tff(4,plain,
% 2.54/1.85 (![A: $i] : (inverse(inverse(A)) = A)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[3, 2])).
% 2.54/1.85 tff(5,plain,(
% 2.54/1.85 ![A: $i] : (inverse(inverse(A)) = A)),
% 2.54/1.85 inference(skolemize,[status(sab)],[4])).
% 2.54/1.85 tff(6,plain,
% 2.54/1.85 (![A: $i] : (inverse(inverse(A)) = A)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[5, 1])).
% 2.54/1.85 tff(7,plain,
% 2.54/1.85 ((~![A: $i] : (inverse(inverse(A)) = A)) | (inverse(inverse(c)) = c)),
% 2.54/1.85 inference(quant_inst,[status(thm)],[])).
% 2.54/1.85 tff(8,plain,
% 2.54/1.85 (inverse(inverse(c)) = c),
% 2.54/1.85 inference(unit_resolution,[status(thm)],[7, 6])).
% 2.54/1.85 tff(9,plain,
% 2.54/1.85 (product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a)) <=> product(inverse(multiply(c, a)), c, inverse(a))),
% 2.54/1.85 inference(monotonicity,[status(thm)],[8])).
% 2.54/1.85 tff(10,plain,
% 2.54/1.85 (product(inverse(multiply(c, a)), c, inverse(a)) <=> product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))),
% 2.54/1.85 inference(symmetry,[status(thm)],[9])).
% 2.54/1.85 tff(11,plain,
% 2.54/1.85 (^[Y: $i, X: $i] : refl(product(X, Y, multiply(X, Y)) <=> product(X, Y, multiply(X, Y)))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(12,plain,
% 2.54/1.85 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.54/1.85 inference(quant_intro,[status(thm)],[11])).
% 2.54/1.85 tff(13,plain,
% 2.54/1.85 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y)) <=> ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(14,axiom,(![Y: $i, X: $i] : product(X, Y, multiply(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','total_function1')).
% 2.54/1.85 tff(15,plain,
% 2.54/1.85 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[14, 13])).
% 2.54/1.85 tff(16,plain,(
% 2.54/1.85 ![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.54/1.85 inference(skolemize,[status(sab)],[15])).
% 2.54/1.85 tff(17,plain,
% 2.54/1.85 (![Y: $i, X: $i] : product(X, Y, multiply(X, Y))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[16, 12])).
% 2.54/1.85 tff(18,plain,
% 2.54/1.85 ((~![Y: $i, X: $i] : product(X, Y, multiply(X, Y))) | product(c, a, multiply(c, a))),
% 2.54/1.85 inference(quant_inst,[status(thm)],[])).
% 2.54/1.85 tff(19,plain,
% 2.54/1.85 (product(c, a, multiply(c, a))),
% 2.54/1.85 inference(unit_resolution,[status(thm)],[18, 17])).
% 2.54/1.85 tff(20,plain,
% 2.54/1.85 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(21,plain,
% 2.54/1.85 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.54/1.85 inference(quant_intro,[status(thm)],[20])).
% 2.54/1.85 tff(22,plain,
% 2.54/1.85 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(23,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_inverse')).
% 2.54/1.85 tff(24,plain,
% 2.54/1.85 (![X: $i] : product(inverse(X), X, identity)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[23, 22])).
% 2.54/1.85 tff(25,plain,(
% 2.54/1.85 ![X: $i] : product(inverse(X), X, identity)),
% 2.54/1.85 inference(skolemize,[status(sab)],[24])).
% 2.54/1.85 tff(26,plain,
% 2.54/1.85 (![X: $i] : product(inverse(X), X, identity)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[25, 21])).
% 2.54/1.85 tff(27,plain,
% 2.54/1.85 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(a), a, identity)),
% 2.54/1.85 inference(quant_inst,[status(thm)],[])).
% 2.54/1.85 tff(28,plain,
% 2.54/1.85 (product(inverse(a), a, identity)),
% 2.54/1.85 inference(unit_resolution,[status(thm)],[27, 26])).
% 2.54/1.85 tff(29,plain,
% 2.54/1.85 (product(b, inverse(a), c) <=> product(b, inverse(a), c)),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(30,axiom,(product(b, inverse(a), c)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_times_a_inverse_is_c')).
% 2.54/1.85 tff(31,plain,
% 2.54/1.85 (product(b, inverse(a), c)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[30, 29])).
% 2.54/1.85 tff(32,plain,
% 2.54/1.85 (^[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))))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(33,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(quant_intro,[status(thm)],[32])).
% 2.54/1.85 tff(34,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(35,plain,
% 2.54/1.85 (^[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)))))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(36,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(quant_intro,[status(thm)],[35])).
% 2.54/1.85 tff(37,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/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity1')).
% 2.54/1.85 tff(38,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[37, 36])).
% 2.54/1.85 tff(39,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[38, 34])).
% 2.54/1.85 tff(40,plain,(
% 2.54/1.85 ![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)))),
% 2.54/1.85 inference(skolemize,[status(sab)],[39])).
% 2.54/1.85 tff(41,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[40, 33])).
% 2.54/1.85 tff(42,plain,
% 2.54/1.85 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(43,plain,
% 2.54/1.85 ((product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c))) <=> ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(44,plain,
% 2.54/1.85 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a))))),
% 2.54/1.85 inference(monotonicity,[status(thm)],[43])).
% 2.54/1.85 tff(45,plain,
% 2.54/1.85 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a)))),
% 2.54/1.85 inference(transitivity,[status(thm)],[44, 42])).
% 2.54/1.85 tff(46,plain,
% 2.54/1.85 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(b, identity, multiply(c, a)) | (~product(c, a, multiply(c, a))) | (~product(inverse(a), a, identity)) | (~product(b, inverse(a), c)))),
% 2.54/1.85 inference(quant_inst,[status(thm)],[])).
% 2.54/1.85 tff(47,plain,
% 2.54/1.85 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(a), a, identity)) | (~product(c, a, multiply(c, a))) | product(b, identity, multiply(c, a))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[46, 45])).
% 2.54/1.85 tff(48,plain,
% 2.54/1.85 (product(b, identity, multiply(c, a))),
% 2.54/1.85 inference(unit_resolution,[status(thm)],[47, 41, 31, 28, 19])).
% 2.54/1.85 tff(49,plain,
% 2.54/1.85 (^[X: $i] : refl(product(X, identity, X) <=> product(X, identity, X))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(50,plain,
% 2.54/1.85 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.54/1.85 inference(quant_intro,[status(thm)],[49])).
% 2.54/1.85 tff(51,plain,
% 2.54/1.85 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(52,axiom,(![X: $i] : product(X, identity, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','right_identity')).
% 2.54/1.85 tff(53,plain,
% 2.54/1.85 (![X: $i] : product(X, identity, X)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[52, 51])).
% 2.54/1.85 tff(54,plain,(
% 2.54/1.85 ![X: $i] : product(X, identity, X)),
% 2.54/1.85 inference(skolemize,[status(sab)],[53])).
% 2.54/1.85 tff(55,plain,
% 2.54/1.85 (![X: $i] : product(X, identity, X)),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[54, 50])).
% 2.54/1.85 tff(56,plain,
% 2.54/1.85 ((~![X: $i] : product(X, identity, X)) | product(a, identity, a)),
% 2.54/1.85 inference(quant_inst,[status(thm)],[])).
% 2.54/1.85 tff(57,plain,
% 2.54/1.85 (product(a, identity, a)),
% 2.54/1.85 inference(unit_resolution,[status(thm)],[56, 55])).
% 2.54/1.85 tff(58,plain,
% 2.54/1.85 (^[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))))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(59,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(quant_intro,[status(thm)],[58])).
% 2.54/1.85 tff(60,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(61,plain,
% 2.54/1.85 (^[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)))))),
% 2.54/1.85 inference(bind,[status(th)],[])).
% 2.54/1.85 tff(62,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(quant_intro,[status(thm)],[61])).
% 2.54/1.85 tff(63,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/sandbox2/benchmark/Axioms/GRP003-0.ax','associativity2')).
% 2.54/1.85 tff(64,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[63, 62])).
% 2.54/1.85 tff(65,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[64, 60])).
% 2.54/1.85 tff(66,plain,(
% 2.54/1.85 ![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)))),
% 2.54/1.85 inference(skolemize,[status(sab)],[65])).
% 2.54/1.85 tff(67,plain,
% 2.54/1.85 (![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)))),
% 2.54/1.85 inference(modus_ponens,[status(thm)],[66, 59])).
% 2.54/1.85 tff(68,plain,
% 2.54/1.85 (((~![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, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a))))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(69,plain,
% 2.54/1.85 ((product(multiply(c, a), identity, multiply(c, a)) | (~product(a, identity, a)) | (~product(c, a, multiply(c, a))) | (~product(c, a, multiply(c, a)))) <=> ((~product(a, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a))))),
% 2.54/1.85 inference(rewrite,[status(thm)],[])).
% 2.54/1.85 tff(70,plain,
% 2.54/1.85 (((~![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(c, a), identity, multiply(c, a)) | (~product(a, identity, a)) | (~product(c, a, multiply(c, a))) | (~product(c, a, multiply(c, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | ((~product(a, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a)))))),
% 2.54/1.86 inference(monotonicity,[status(thm)],[69])).
% 2.54/1.86 tff(71,plain,
% 2.54/1.86 (((~![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(c, a), identity, multiply(c, a)) | (~product(a, identity, a)) | (~product(c, a, multiply(c, a))) | (~product(c, a, multiply(c, a))))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (~product(a, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a))))),
% 2.54/1.86 inference(transitivity,[status(thm)],[70, 68])).
% 2.54/1.86 tff(72,plain,
% 2.54/1.86 ((~![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(c, a), identity, multiply(c, a)) | (~product(a, identity, a)) | (~product(c, a, multiply(c, a))) | (~product(c, a, multiply(c, a))))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(73,plain,
% 2.54/1.86 ((~![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, identity, a)) | product(multiply(c, a), identity, multiply(c, a)) | (~product(c, a, multiply(c, a)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[72, 71])).
% 2.54/1.86 tff(74,plain,
% 2.54/1.86 (product(multiply(c, a), identity, multiply(c, a))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[73, 67, 57, 19])).
% 2.54/1.86 tff(75,plain,
% 2.54/1.86 (^[B: $i, D: $i, A: $i, C: $i] : refl(((~product(A, B, C)) | (D = A) | (~product(D, B, C))) <=> ((~product(A, B, C)) | (D = A) | (~product(D, B, C))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(76,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[75])).
% 2.54/1.86 tff(77,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(78,plain,
% 2.54/1.86 (^[B: $i, D: $i, A: $i, C: $i] : rewrite((((~product(A, B, C)) | (~product(D, B, C))) | (D = A)) <=> ((~product(A, B, C)) | (D = A) | (~product(D, B, C))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(79,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(D, B, C))) | (D = A)) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[78])).
% 2.54/1.86 tff(80,axiom,(![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(D, B, C))) | (D = A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_left_cancellation')).
% 2.54/1.86 tff(81,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[80, 79])).
% 2.54/1.86 tff(82,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[81, 77])).
% 2.54/1.86 tff(83,plain,(
% 2.54/1.86 ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(skolemize,[status(sab)],[82])).
% 2.54/1.86 tff(84,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[83, 76])).
% 2.54/1.86 tff(85,plain,
% 2.54/1.86 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))) | ((~product(multiply(c, a), identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, multiply(c, a))))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))) | (~product(multiply(c, a), identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, multiply(c, a))))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(86,plain,
% 2.54/1.86 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))) | ((~product(multiply(c, a), identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, multiply(c, a))))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(87,plain,
% 2.54/1.86 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, B, C)) | (D = A) | (~product(D, B, C)))) | (~product(multiply(c, a), identity, multiply(c, a))) | (b = multiply(c, a)) | (~product(b, identity, multiply(c, a)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[86, 85])).
% 2.54/1.86 tff(88,plain,
% 2.54/1.86 (b = multiply(c, a)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[87, 84, 74, 48])).
% 2.54/1.86 tff(89,plain,
% 2.54/1.86 (multiply(c, a) = b),
% 2.54/1.86 inference(symmetry,[status(thm)],[88])).
% 2.54/1.86 tff(90,plain,
% 2.54/1.86 (product(multiply(c, a), inverse(a), c) <=> product(b, inverse(a), c)),
% 2.54/1.86 inference(monotonicity,[status(thm)],[89])).
% 2.54/1.86 tff(91,plain,
% 2.54/1.86 (product(b, inverse(a), c) <=> product(multiply(c, a), inverse(a), c)),
% 2.54/1.86 inference(symmetry,[status(thm)],[90])).
% 2.54/1.86 tff(92,plain,
% 2.54/1.86 (product(multiply(c, a), inverse(a), c)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[31, 91])).
% 2.54/1.86 tff(93,plain,
% 2.54/1.86 (^[X: $i] : refl(product(identity, X, X) <=> product(identity, X, X))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(94,plain,
% 2.54/1.86 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 2.54/1.86 inference(quant_intro,[status(thm)],[93])).
% 2.54/1.86 tff(95,plain,
% 2.54/1.86 (![X: $i] : product(identity, X, X) <=> ![X: $i] : product(identity, X, X)),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(96,axiom,(![X: $i] : product(identity, X, X)), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-0.ax','left_identity')).
% 2.54/1.86 tff(97,plain,
% 2.54/1.86 (![X: $i] : product(identity, X, X)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[96, 95])).
% 2.54/1.86 tff(98,plain,(
% 2.54/1.86 ![X: $i] : product(identity, X, X)),
% 2.54/1.86 inference(skolemize,[status(sab)],[97])).
% 2.54/1.86 tff(99,plain,
% 2.54/1.86 (![X: $i] : product(identity, X, X)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[98, 94])).
% 2.54/1.86 tff(100,plain,
% 2.54/1.86 ((~![X: $i] : product(identity, X, X)) | product(identity, inverse(a), inverse(a))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(101,plain,
% 2.54/1.86 (product(identity, inverse(a), inverse(a))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[100, 99])).
% 2.54/1.86 tff(102,plain,
% 2.54/1.86 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(multiply(c, a)), multiply(c, a), identity)),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(103,plain,
% 2.54/1.86 (product(inverse(multiply(c, a)), multiply(c, a), identity)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[102, 26])).
% 2.54/1.86 tff(104,plain,
% 2.54/1.86 (((~![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(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(multiply(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(105,plain,
% 2.54/1.86 ((product(inverse(multiply(c, a)), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(multiply(c, a)), multiply(c, a), identity))) <=> ((~product(inverse(multiply(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(106,plain,
% 2.54/1.86 (((~![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(c, a)), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(multiply(c, a)), multiply(c, a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(inverse(multiply(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c))))),
% 2.54/1.86 inference(monotonicity,[status(thm)],[105])).
% 2.54/1.86 tff(107,plain,
% 2.54/1.86 (((~![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(c, a)), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(multiply(c, a)), multiply(c, a), identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(inverse(multiply(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c)))),
% 2.54/1.86 inference(transitivity,[status(thm)],[106, 104])).
% 2.54/1.86 tff(108,plain,
% 2.54/1.86 ((~![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(c, a)), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(multiply(c, a), inverse(a), c)) | (~product(inverse(multiply(c, a)), multiply(c, a), identity)))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(109,plain,
% 2.54/1.86 ((~![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(c, a)), multiply(c, a), identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[108, 107])).
% 2.54/1.86 tff(110,plain,
% 2.54/1.86 (product(inverse(multiply(c, a)), c, inverse(a)) | (~product(multiply(c, a), inverse(a), c))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[109, 41, 103, 101])).
% 2.54/1.86 tff(111,plain,
% 2.54/1.86 (product(inverse(multiply(c, a)), c, inverse(a))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[110, 92])).
% 2.54/1.86 tff(112,plain,
% 2.54/1.86 (product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[111, 10])).
% 2.54/1.86 tff(113,plain,
% 2.54/1.86 (inverse(b) = inverse(multiply(c, a))),
% 2.54/1.86 inference(monotonicity,[status(thm)],[88])).
% 2.54/1.86 tff(114,plain,
% 2.54/1.86 (subgroup_member(inverse(b)) <=> subgroup_member(inverse(multiply(c, a)))),
% 2.54/1.86 inference(monotonicity,[status(thm)],[113])).
% 2.54/1.86 tff(115,plain,
% 2.54/1.86 (subgroup_member(b) <=> subgroup_member(b)),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(116,axiom,(subgroup_member(b)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','b_is_in_subgroup')).
% 2.54/1.86 tff(117,plain,
% 2.54/1.86 (subgroup_member(b)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[116, 115])).
% 2.54/1.86 tff(118,plain,
% 2.54/1.86 (^[A: $i] : refl(((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ((~subgroup_member(A)) | subgroup_member(inverse(A))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(119,plain,
% 2.54/1.86 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[118])).
% 2.54/1.86 tff(120,plain,
% 2.54/1.86 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A))) <=> ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(121,axiom,(![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','subgroup_member_inverse_are_in_subgroup')).
% 2.54/1.86 tff(122,plain,
% 2.54/1.86 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[121, 120])).
% 2.54/1.86 tff(123,plain,(
% 2.54/1.86 ![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.54/1.86 inference(skolemize,[status(sab)],[122])).
% 2.54/1.86 tff(124,plain,
% 2.54/1.86 (![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[123, 119])).
% 2.54/1.86 tff(125,plain,
% 2.54/1.86 (((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))) <=> ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(126,plain,
% 2.54/1.86 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(127,plain,
% 2.54/1.86 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(b)) | subgroup_member(inverse(b))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[126, 125])).
% 2.54/1.86 tff(128,plain,
% 2.54/1.86 (subgroup_member(inverse(b))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[127, 124, 117])).
% 2.54/1.86 tff(129,plain,
% 2.54/1.86 (subgroup_member(inverse(multiply(c, a)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[128, 114])).
% 2.54/1.86 tff(130,assumption,(~subgroup_member(c)), introduced(assumption)).
% 2.54/1.86 tff(131,plain,
% 2.54/1.86 (^[B: $i, A: $i, C: $i] : refl((subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(132,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[131])).
% 2.54/1.86 tff(133,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(134,plain,
% 2.54/1.86 (^[B: $i, A: $i, C: $i] : trans(monotonicity(rewrite((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) <=> ((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> (((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)))), rewrite((((~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(135,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C)) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[134])).
% 2.54/1.86 tff(136,axiom,(![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, inverse(B), C))) | subgroup_member(C))), file('/export/starexec/sandbox2/benchmark/Axioms/GRP003-2.ax','closure_of_product_and_inverse')).
% 2.54/1.86 tff(137,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[136, 135])).
% 2.54/1.86 tff(138,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[137, 133])).
% 2.54/1.86 tff(139,plain,(
% 2.54/1.86 ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(skolemize,[status(sab)],[138])).
% 2.54/1.86 tff(140,plain,
% 2.54/1.86 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[139, 132])).
% 2.54/1.86 tff(141,plain,
% 2.54/1.86 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(142,plain,
% 2.54/1.86 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b)))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(143,plain,
% 2.54/1.86 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(a)) | (~subgroup_member(b))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[142, 141])).
% 2.54/1.86 tff(144,plain,
% 2.54/1.86 (subgroup_member(c) | (~subgroup_member(a))),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[143, 140, 117, 31])).
% 2.54/1.86 tff(145,plain,
% 2.54/1.86 (~subgroup_member(a)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[144, 130])).
% 2.54/1.86 tff(146,plain,
% 2.54/1.86 ((~subgroup_member(d)) <=> (~subgroup_member(d))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(147,axiom,(~subgroup_member(d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_d_is_in_subgroup')).
% 2.54/1.86 tff(148,plain,
% 2.54/1.86 (~subgroup_member(d)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[147, 146])).
% 2.54/1.86 tff(149,plain,
% 2.54/1.86 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(150,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[149])).
% 2.54/1.86 tff(151,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(152,plain,
% 2.54/1.86 (^[B: $i, A: $i] : trans(monotonicity(rewrite((product(A, element_in_O2(A, B), B) | subgroup_member(B)) <=> (subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)))), rewrite(((subgroup_member(B) | product(A, element_in_O2(A, B), B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))), (((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(153,plain,
% 2.54/1.86 (![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[152])).
% 2.54/1.86 tff(154,axiom,(![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','property_of_O2')).
% 2.54/1.86 tff(155,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[154, 153])).
% 2.54/1.86 tff(156,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[155, 151])).
% 2.54/1.86 tff(157,plain,(
% 2.54/1.86 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(skolemize,[status(sab)],[156])).
% 2.54/1.86 tff(158,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[157, 150])).
% 2.54/1.86 tff(159,plain,
% 2.54/1.86 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(160,plain,
% 2.54/1.86 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | (subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(161,plain,
% 2.54/1.86 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | product(A, element_in_O2(A, B), B))) | subgroup_member(a) | subgroup_member(d) | product(a, element_in_O2(a, d), d)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[160, 159])).
% 2.54/1.86 tff(162,plain,
% 2.54/1.86 (subgroup_member(a) | product(a, element_in_O2(a, d), d)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[161, 158, 148])).
% 2.54/1.86 tff(163,plain,
% 2.54/1.86 (product(a, element_in_O2(a, d), d)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[162, 145])).
% 2.54/1.86 tff(164,plain,
% 2.54/1.86 (product(a, c, d) <=> product(a, c, d)),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(165,axiom,(product(a, c, d)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','a_times_c_is_d')).
% 2.54/1.86 tff(166,plain,
% 2.54/1.86 (product(a, c, d)),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[165, 164])).
% 2.54/1.86 tff(167,plain,
% 2.54/1.86 (^[B: $i, D: $i, A: $i, C: $i] : refl(((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(168,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[167])).
% 2.54/1.86 tff(169,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(170,plain,
% 2.54/1.86 (^[B: $i, D: $i, A: $i, C: $i] : trans(monotonicity(rewrite(((~product(A, B, C)) | (~product(A, D, C))) <=> ((~product(A, D, C)) | (~product(A, B, C)))), ((((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> (((~product(A, D, C)) | (~product(A, B, C))) | (D = B)))), rewrite((((~product(A, D, C)) | (~product(A, B, C))) | (D = B)) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))), ((((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(171,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | (D = B)) <=> ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[170])).
% 2.54/1.86 tff(172,axiom,(![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | (D = B))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','product_right_cancellation')).
% 2.54/1.86 tff(173,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[172, 171])).
% 2.54/1.86 tff(174,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[173, 169])).
% 2.54/1.86 tff(175,plain,(
% 2.54/1.86 ![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(skolemize,[status(sab)],[174])).
% 2.54/1.86 tff(176,plain,
% 2.54/1.86 (![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[175, 168])).
% 2.54/1.86 tff(177,plain,
% 2.54/1.86 (((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(178,plain,
% 2.54/1.86 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | ((c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)))),
% 2.54/1.86 inference(quant_inst,[status(thm)],[])).
% 2.54/1.86 tff(179,plain,
% 2.54/1.86 ((~![B: $i, D: $i, A: $i, C: $i] : ((D = B) | (~product(A, D, C)) | (~product(A, B, C)))) | (c = element_in_O2(a, d)) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[178, 177])).
% 2.54/1.86 tff(180,plain,
% 2.54/1.86 (c = element_in_O2(a, d)),
% 2.54/1.86 inference(unit_resolution,[status(thm)],[179, 176, 166, 163])).
% 2.54/1.86 tff(181,plain,
% 2.54/1.86 (element_in_O2(a, d) = c),
% 2.54/1.86 inference(symmetry,[status(thm)],[180])).
% 2.54/1.86 tff(182,plain,
% 2.54/1.86 (subgroup_member(element_in_O2(a, d)) <=> subgroup_member(c)),
% 2.54/1.86 inference(monotonicity,[status(thm)],[181])).
% 2.54/1.86 tff(183,plain,
% 2.54/1.86 (subgroup_member(c) <=> subgroup_member(element_in_O2(a, d))),
% 2.54/1.86 inference(symmetry,[status(thm)],[182])).
% 2.54/1.86 tff(184,plain,
% 2.54/1.86 ((~subgroup_member(c)) <=> (~subgroup_member(element_in_O2(a, d)))),
% 2.54/1.86 inference(monotonicity,[status(thm)],[183])).
% 2.54/1.86 tff(185,plain,
% 2.54/1.86 (~subgroup_member(element_in_O2(a, d))),
% 2.54/1.86 inference(modus_ponens,[status(thm)],[130, 184])).
% 2.54/1.86 tff(186,plain,
% 2.54/1.86 (^[B: $i, A: $i] : refl((subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(187,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[186])).
% 2.54/1.86 tff(188,plain,
% 2.54/1.86 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.86 inference(rewrite,[status(thm)],[])).
% 2.54/1.86 tff(189,plain,
% 2.54/1.86 (^[B: $i, A: $i] : trans(monotonicity(rewrite((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) <=> (subgroup_member(B) | subgroup_member(element_in_O2(A, B)))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ((subgroup_member(B) | subgroup_member(element_in_O2(A, B))) | subgroup_member(A)))), rewrite(((subgroup_member(B) | subgroup_member(element_in_O2(A, B))) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))), (((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))))),
% 2.54/1.86 inference(bind,[status(th)],[])).
% 2.54/1.86 tff(190,plain,
% 2.54/1.86 (![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.86 inference(quant_intro,[status(thm)],[189])).
% 2.54/1.86 tff(191,axiom,(![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','an_element_in_O2')).
% 2.54/1.87 tff(192,plain,
% 2.54/1.87 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[191, 190])).
% 2.54/1.87 tff(193,plain,
% 2.54/1.87 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[192, 188])).
% 2.54/1.87 tff(194,plain,(
% 2.54/1.87 ![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.87 inference(skolemize,[status(sab)],[193])).
% 2.54/1.87 tff(195,plain,
% 2.54/1.87 (![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[194, 187])).
% 2.54/1.87 tff(196,plain,
% 2.54/1.87 (((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))) <=> ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(197,plain,
% 2.54/1.87 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d)))),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(198,plain,
% 2.54/1.87 ((~![B: $i, A: $i] : (subgroup_member(A) | subgroup_member(B) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(a) | subgroup_member(d) | subgroup_member(element_in_O2(a, d))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[197, 196])).
% 2.54/1.87 tff(199,plain,
% 2.54/1.87 (subgroup_member(element_in_O2(a, d))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[198, 195, 148, 145])).
% 2.54/1.87 tff(200,plain,
% 2.54/1.87 ($false),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[199, 185])).
% 2.54/1.87 tff(201,plain,(subgroup_member(c)), inference(lemma,lemma(discharge,[]))).
% 2.54/1.87 tff(202,plain,
% 2.54/1.87 (((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(c)) | subgroup_member(inverse(c)))) <=> ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(c)) | subgroup_member(inverse(c)))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(203,plain,
% 2.54/1.87 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(c)) | subgroup_member(inverse(c)))),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(204,plain,
% 2.54/1.87 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(c)) | subgroup_member(inverse(c))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[203, 202])).
% 2.54/1.87 tff(205,plain,
% 2.54/1.87 ((~subgroup_member(c)) | subgroup_member(inverse(c))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[204, 124])).
% 2.54/1.87 tff(206,plain,
% 2.54/1.87 (subgroup_member(inverse(c))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[205, 201])).
% 2.54/1.87 tff(207,plain,
% 2.54/1.87 ((~![A: $i] : (inverse(inverse(A)) = A)) | (inverse(inverse(a)) = a)),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(208,plain,
% 2.54/1.87 (inverse(inverse(a)) = a),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[207, 6])).
% 2.54/1.87 tff(209,plain,
% 2.54/1.87 (subgroup_member(inverse(inverse(a))) <=> subgroup_member(a)),
% 2.54/1.87 inference(monotonicity,[status(thm)],[208])).
% 2.54/1.87 tff(210,plain,
% 2.54/1.87 (subgroup_member(a) <=> subgroup_member(inverse(inverse(a)))),
% 2.54/1.87 inference(symmetry,[status(thm)],[209])).
% 2.54/1.87 tff(211,plain,
% 2.54/1.87 ((~subgroup_member(a)) <=> (~subgroup_member(inverse(inverse(a))))),
% 2.54/1.87 inference(monotonicity,[status(thm)],[210])).
% 2.54/1.87 tff(212,plain,
% 2.54/1.87 (product(a, inverse(inverse(c)), d) <=> product(a, c, d)),
% 2.54/1.87 inference(monotonicity,[status(thm)],[8])).
% 2.54/1.87 tff(213,plain,
% 2.54/1.87 (product(a, c, d) <=> product(a, inverse(inverse(c)), d)),
% 2.54/1.87 inference(symmetry,[status(thm)],[212])).
% 2.54/1.87 tff(214,plain,
% 2.54/1.87 (product(a, inverse(inverse(c)), d)),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[166, 213])).
% 2.54/1.87 tff(215,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d)))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(216,plain,
% 2.54/1.87 ((subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a))) <=> (subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d)))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(217,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d))))),
% 2.54/1.87 inference(monotonicity,[status(thm)],[216])).
% 2.54/1.87 tff(218,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d)))),
% 2.54/1.87 inference(transitivity,[status(thm)],[217, 215])).
% 2.54/1.87 tff(219,plain,
% 2.54/1.87 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, inverse(inverse(c)), d)) | (~subgroup_member(inverse(c))) | (~subgroup_member(a)))),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(220,plain,
% 2.54/1.87 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[219, 218])).
% 2.54/1.87 tff(221,plain,
% 2.54/1.87 ((~subgroup_member(a)) | (~subgroup_member(inverse(c))) | (~product(a, inverse(inverse(c)), d))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[220, 140, 148])).
% 2.54/1.87 tff(222,plain,
% 2.54/1.87 ((~subgroup_member(a)) | (~subgroup_member(inverse(c)))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[221, 214])).
% 2.54/1.87 tff(223,plain,
% 2.54/1.87 (~subgroup_member(a)),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[222, 206])).
% 2.54/1.87 tff(224,plain,
% 2.54/1.87 (~subgroup_member(inverse(inverse(a)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[223, 211])).
% 2.54/1.87 tff(225,plain,
% 2.54/1.87 (((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))) <=> ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(226,plain,
% 2.54/1.87 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(227,plain,
% 2.54/1.87 ((~![A: $i] : ((~subgroup_member(A)) | subgroup_member(inverse(A)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[226, 225])).
% 2.54/1.87 tff(228,plain,
% 2.54/1.87 ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[227, 124])).
% 2.54/1.87 tff(229,plain,
% 2.54/1.87 (~subgroup_member(inverse(a))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[228, 224])).
% 2.54/1.87 tff(230,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(231,plain,
% 2.54/1.87 ((subgroup_member(inverse(a)) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a))))) <=> (subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))))),
% 2.54/1.87 inference(rewrite,[status(thm)],[])).
% 2.54/1.87 tff(232,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a)))))),
% 2.54/1.87 inference(monotonicity,[status(thm)],[231])).
% 2.54/1.87 tff(233,plain,
% 2.54/1.87 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))))),
% 2.54/1.87 inference(transitivity,[status(thm)],[232, 230])).
% 2.54/1.87 tff(234,plain,
% 2.54/1.87 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))))),
% 2.54/1.87 inference(quant_inst,[status(thm)],[])).
% 2.54/1.87 tff(235,plain,
% 2.54/1.87 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, inverse(B), C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(inverse(c))) | (~subgroup_member(inverse(multiply(c, a)))) | (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a)))),
% 2.54/1.87 inference(modus_ponens,[status(thm)],[234, 233])).
% 2.54/1.87 tff(236,plain,
% 2.54/1.87 (~product(inverse(multiply(c, a)), inverse(inverse(c)), inverse(a))),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[235, 140, 229, 206, 129])).
% 2.54/1.87 tff(237,plain,
% 2.54/1.87 ($false),
% 2.54/1.87 inference(unit_resolution,[status(thm)],[236, 112])).
% 2.54/1.87 % SZS output end Proof
%------------------------------------------------------------------------------