TSTP Solution File: GRP039-6 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : GRP039-6 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 16 22:25:37 EDT 2022
% Result : Unsatisfiable 2.27s 1.72s
% Output : Proof 2.27s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP039-6 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.12 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33 % Computer : n001.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Aug 31 14:38:49 EDT 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.12/0.34 Usage: tptp [options] [-file:]file
% 0.12/0.34 -h, -? prints this message.
% 0.12/0.34 -smt2 print SMT-LIB2 benchmark.
% 0.12/0.34 -m, -model generate model.
% 0.12/0.34 -p, -proof generate proof.
% 0.12/0.34 -c, -core generate unsat core of named formulas.
% 0.12/0.34 -st, -statistics display statistics.
% 0.12/0.34 -t:timeout set timeout (in second).
% 0.12/0.34 -smt2status display status in smt2 format instead of SZS.
% 0.12/0.34 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.12/0.34 -<param>:<value> configuration parameter and value.
% 0.12/0.34 -o:<output-file> file to place output in.
% 2.27/1.72 % SZS status Unsatisfiable
% 2.27/1.72 % SZS output start Proof
% 2.27/1.72 tff(subgroup_member_type, type, (
% 2.27/1.72 subgroup_member: $i > $o)).
% 2.27/1.72 tff(element_in_O2_type, type, (
% 2.27/1.72 element_in_O2: ( $i * $i ) > $i)).
% 2.27/1.72 tff(d_type, type, (
% 2.27/1.72 d: $i)).
% 2.27/1.72 tff(a_type, type, (
% 2.27/1.72 a: $i)).
% 2.27/1.72 tff(c_type, type, (
% 2.27/1.72 c: $i)).
% 2.27/1.72 tff(product_type, type, (
% 2.27/1.72 product: ( $i * $i * $i ) > $o)).
% 2.27/1.72 tff(inverse_type, type, (
% 2.27/1.72 inverse: $i > $i)).
% 2.27/1.72 tff(b_type, type, (
% 2.27/1.72 b: $i)).
% 2.27/1.72 tff(equalish_type, type, (
% 2.27/1.72 equalish: ( $i * $i ) > $o)).
% 2.27/1.72 tff(identity_type, type, (
% 2.27/1.72 identity: $i)).
% 2.27/1.72 tff(1,assumption,(subgroup_member(a)), introduced(assumption)).
% 2.27/1.72 tff(2,plain,
% 2.27/1.72 ((~subgroup_member(d)) <=> (~subgroup_member(d))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(3,axiom,(~subgroup_member(d)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_d_is_in_subgroup')).
% 2.27/1.72 tff(4,plain,
% 2.27/1.72 (~subgroup_member(d)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[3, 2])).
% 2.27/1.72 tff(5,plain,
% 2.27/1.72 (product(a, c, d) <=> product(a, c, d)),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(6,axiom,(product(a, c, d)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_times_c_is_d')).
% 2.27/1.72 tff(7,plain,
% 2.27/1.72 (product(a, c, d)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[6, 5])).
% 2.27/1.72 tff(8,plain,
% 2.27/1.72 (^[B: $i, A: $i, C: $i] : refl((subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(9,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[8])).
% 2.27/1.72 tff(10,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(11,plain,
% 2.27/1.72 (^[B: $i, A: $i, C: $i] : trans(monotonicity(rewrite((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) <=> ((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> (((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)))), rewrite((((~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))), (((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(12,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C)) <=> ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[11])).
% 2.27/1.72 tff(13,axiom,(![B: $i, A: $i, C: $i] : ((((~subgroup_member(A)) | (~subgroup_member(B))) | (~product(A, B, C))) | subgroup_member(C))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','closure_of_product')).
% 2.27/1.72 tff(14,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[13, 12])).
% 2.27/1.72 tff(15,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[14, 10])).
% 2.27/1.72 tff(16,plain,(
% 2.27/1.72 ![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(skolemize,[status(sab)],[15])).
% 2.27/1.72 tff(17,plain,
% 2.27/1.72 (![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[16, 9])).
% 2.27/1.72 tff(18,plain,
% 2.27/1.72 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, c, d)) | (~subgroup_member(c)) | (~subgroup_member(a)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~product(a, c, d)) | (~subgroup_member(c)) | (~subgroup_member(a)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(19,plain,
% 2.27/1.72 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(d) | (~product(a, c, d)) | (~subgroup_member(c)) | (~subgroup_member(a)))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(20,plain,
% 2.27/1.72 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(d) | (~product(a, c, d)) | (~subgroup_member(c)) | (~subgroup_member(a))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[19, 18])).
% 2.27/1.72 tff(21,plain,
% 2.27/1.72 ((~subgroup_member(c)) | (~subgroup_member(a))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[20, 17, 7, 4])).
% 2.27/1.72 tff(22,plain,
% 2.27/1.72 (~subgroup_member(c)),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[21, 1])).
% 2.27/1.72 tff(23,plain,
% 2.27/1.72 (^[X: $i] : refl(((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ((~subgroup_member(X)) | subgroup_member(inverse(X))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(24,plain,
% 2.27/1.72 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[23])).
% 2.27/1.72 tff(25,plain,
% 2.27/1.72 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X))) <=> ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(26,axiom,(![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','closure_of_inverse')).
% 2.27/1.72 tff(27,plain,
% 2.27/1.72 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[26, 25])).
% 2.27/1.72 tff(28,plain,(
% 2.27/1.72 ![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 2.27/1.72 inference(skolemize,[status(sab)],[27])).
% 2.27/1.72 tff(29,plain,
% 2.27/1.72 (![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[28, 24])).
% 2.27/1.72 tff(30,plain,
% 2.27/1.72 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(a)) | subgroup_member(inverse(a)))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(a)) | subgroup_member(inverse(a)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(31,plain,
% 2.27/1.72 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(a)) | subgroup_member(inverse(a)))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(32,plain,
% 2.27/1.72 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(a)) | subgroup_member(inverse(a))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[31, 30])).
% 2.27/1.72 tff(33,plain,
% 2.27/1.72 ((~subgroup_member(a)) | subgroup_member(inverse(a))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[32, 29])).
% 2.27/1.72 tff(34,plain,
% 2.27/1.72 (subgroup_member(inverse(a))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[33, 1])).
% 2.27/1.72 tff(35,plain,
% 2.27/1.72 (product(b, inverse(a), c) <=> product(b, inverse(a), c)),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(36,axiom,(product(b, inverse(a), c)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_times_a_inverse_is_c')).
% 2.27/1.72 tff(37,plain,
% 2.27/1.72 (product(b, inverse(a), c)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[36, 35])).
% 2.27/1.72 tff(38,plain,
% 2.27/1.72 (subgroup_member(b) <=> subgroup_member(b)),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(39,axiom,(subgroup_member(b)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_is_in_subgroup')).
% 2.27/1.72 tff(40,plain,
% 2.27/1.72 (subgroup_member(b)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[39, 38])).
% 2.27/1.72 tff(41,plain,
% 2.27/1.72 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(42,plain,
% 2.27/1.72 ((subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b))) <=> ((~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(43,plain,
% 2.27/1.72 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | ((~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b))))),
% 2.27/1.72 inference(monotonicity,[status(thm)],[42])).
% 2.27/1.72 tff(44,plain,
% 2.27/1.72 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 2.27/1.72 inference(transitivity,[status(thm)],[43, 41])).
% 2.27/1.72 tff(45,plain,
% 2.27/1.72 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(c) | (~product(b, inverse(a), c)) | (~subgroup_member(inverse(a))) | (~subgroup_member(b)))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(46,plain,
% 2.27/1.72 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (~product(b, inverse(a), c)) | subgroup_member(c) | (~subgroup_member(inverse(a))) | (~subgroup_member(b))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[45, 44])).
% 2.27/1.72 tff(47,plain,
% 2.27/1.72 (subgroup_member(c) | (~subgroup_member(inverse(a)))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[46, 17, 40, 37])).
% 2.27/1.72 tff(48,plain,
% 2.27/1.72 ($false),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[47, 34, 22])).
% 2.27/1.72 tff(49,plain,(~subgroup_member(a)), inference(lemma,lemma(discharge,[]))).
% 2.27/1.72 tff(50,plain,
% 2.27/1.72 (^[B: $i, A: $i] : refl((subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B))) <=> (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(51,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[50])).
% 2.27/1.72 tff(52,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(53,plain,
% 2.27/1.72 (^[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(B) | subgroup_member(A) | 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(A) | subgroup_member(element_in_O2(A, B)))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(54,plain,
% 2.27/1.72 (![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[53])).
% 2.27/1.72 tff(55,axiom,(![B: $i, A: $i] : ((subgroup_member(element_in_O2(A, B)) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','an_element_in_O2')).
% 2.27/1.72 tff(56,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[55, 54])).
% 2.27/1.72 tff(57,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[56, 52])).
% 2.27/1.72 tff(58,plain,(
% 2.27/1.72 ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(skolemize,[status(sab)],[57])).
% 2.27/1.72 tff(59,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[58, 51])).
% 2.27/1.72 tff(60,plain,
% 2.27/1.72 (((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(a, d)))) <=> ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(a, d)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(61,plain,
% 2.27/1.72 ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))) | (subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(a, d)))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(62,plain,
% 2.27/1.72 ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | subgroup_member(element_in_O2(A, B)))) | subgroup_member(d) | subgroup_member(a) | subgroup_member(element_in_O2(a, d))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[61, 60])).
% 2.27/1.72 tff(63,plain,
% 2.27/1.72 (subgroup_member(element_in_O2(a, d))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[62, 59, 4, 49])).
% 2.27/1.72 tff(64,plain,
% 2.27/1.72 (^[B: $i, A: $i] : refl((subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B)) <=> (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B)))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(65,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[64])).
% 2.27/1.72 tff(66,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B)) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(67,plain,
% 2.27/1.72 (^[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(B) | subgroup_member(A) | 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) | subgroup_member(A) | product(A, element_in_O2(A, B), B))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(68,plain,
% 2.27/1.72 (![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A)) <=> ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[67])).
% 2.27/1.72 tff(69,axiom,(![B: $i, A: $i] : ((product(A, element_in_O2(A, B), B) | subgroup_member(B)) | subgroup_member(A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','property_of_O2')).
% 2.27/1.72 tff(70,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[69, 68])).
% 2.27/1.72 tff(71,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[70, 66])).
% 2.27/1.72 tff(72,plain,(
% 2.27/1.72 ![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(skolemize,[status(sab)],[71])).
% 2.27/1.72 tff(73,plain,
% 2.27/1.72 (![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[72, 65])).
% 2.27/1.72 tff(74,plain,
% 2.27/1.72 (((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))) | (subgroup_member(d) | subgroup_member(a) | product(a, element_in_O2(a, d), d))) <=> ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))) | subgroup_member(d) | subgroup_member(a) | product(a, element_in_O2(a, d), d))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(75,plain,
% 2.27/1.72 ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))) | (subgroup_member(d) | subgroup_member(a) | product(a, element_in_O2(a, d), d))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(76,plain,
% 2.27/1.72 ((~![B: $i, A: $i] : (subgroup_member(B) | subgroup_member(A) | product(A, element_in_O2(A, B), B))) | subgroup_member(d) | subgroup_member(a) | product(a, element_in_O2(a, d), d)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[75, 74])).
% 2.27/1.72 tff(77,plain,
% 2.27/1.72 (product(a, element_in_O2(a, d), d)),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[76, 73, 4, 49])).
% 2.27/1.72 tff(78,plain,
% 2.27/1.72 (^[B: $i, D: $i, A: $i, C: $i] : refl(((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C))) <=> ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(79,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[78])).
% 2.27/1.72 tff(80,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C))) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(81,plain,
% 2.27/1.72 (^[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))) | equalish(D, B)) <=> (((~product(A, D, C)) | (~product(A, B, C))) | equalish(D, B)))), rewrite((((~product(A, D, C)) | (~product(A, B, C))) | equalish(D, B)) <=> ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))), ((((~product(A, B, C)) | (~product(A, D, C))) | equalish(D, B)) <=> ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(82,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | equalish(D, B)) <=> ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(quant_intro,[status(thm)],[81])).
% 2.27/1.72 tff(83,axiom,(![B: $i, D: $i, A: $i, C: $i] : (((~product(A, B, C)) | (~product(A, D, C))) | equalish(D, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','product_right_cancellation')).
% 2.27/1.72 tff(84,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[83, 82])).
% 2.27/1.72 tff(85,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[84, 80])).
% 2.27/1.72 tff(86,plain,(
% 2.27/1.72 ![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(skolemize,[status(sab)],[85])).
% 2.27/1.72 tff(87,plain,
% 2.27/1.72 (![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[86, 79])).
% 2.27/1.72 tff(88,plain,
% 2.27/1.72 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(89,plain,
% 2.27/1.72 (((~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c) | (~product(a, c, d))) <=> ((~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c))),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(90,plain,
% 2.27/1.72 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c) | (~product(a, c, d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c)))),
% 2.27/1.72 inference(monotonicity,[status(thm)],[89])).
% 2.27/1.72 tff(91,plain,
% 2.27/1.72 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c) | (~product(a, c, d)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c))),
% 2.27/1.72 inference(transitivity,[status(thm)],[90, 88])).
% 2.27/1.72 tff(92,plain,
% 2.27/1.72 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c) | (~product(a, c, d)))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(93,plain,
% 2.27/1.72 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(a, c, d)) | (~product(a, element_in_O2(a, d), d)) | equalish(element_in_O2(a, d), c)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[92, 91])).
% 2.27/1.72 tff(94,plain,
% 2.27/1.72 (equalish(element_in_O2(a, d), c)),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[93, 87, 7, 77])).
% 2.27/1.72 tff(95,plain,
% 2.27/1.72 (^[X: $i] : refl(product(X, identity, X) <=> product(X, identity, X))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(96,plain,
% 2.27/1.72 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.27/1.72 inference(quant_intro,[status(thm)],[95])).
% 2.27/1.72 tff(97,plain,
% 2.27/1.72 (![X: $i] : product(X, identity, X) <=> ![X: $i] : product(X, identity, X)),
% 2.27/1.72 inference(rewrite,[status(thm)],[])).
% 2.27/1.72 tff(98,axiom,(![X: $i] : product(X, identity, X)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','right_identity')).
% 2.27/1.72 tff(99,plain,
% 2.27/1.72 (![X: $i] : product(X, identity, X)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[98, 97])).
% 2.27/1.72 tff(100,plain,(
% 2.27/1.72 ![X: $i] : product(X, identity, X)),
% 2.27/1.72 inference(skolemize,[status(sab)],[99])).
% 2.27/1.72 tff(101,plain,
% 2.27/1.72 (![X: $i] : product(X, identity, X)),
% 2.27/1.72 inference(modus_ponens,[status(thm)],[100, 96])).
% 2.27/1.72 tff(102,plain,
% 2.27/1.72 ((~![X: $i] : product(X, identity, X)) | product(inverse(a), identity, inverse(a))),
% 2.27/1.72 inference(quant_inst,[status(thm)],[])).
% 2.27/1.72 tff(103,plain,
% 2.27/1.72 (product(inverse(a), identity, inverse(a))),
% 2.27/1.72 inference(unit_resolution,[status(thm)],[102, 101])).
% 2.27/1.72 tff(104,plain,
% 2.27/1.72 (^[X: $i] : refl(product(inverse(X), X, identity) <=> product(inverse(X), X, identity))),
% 2.27/1.72 inference(bind,[status(th)],[])).
% 2.27/1.72 tff(105,plain,
% 2.27/1.72 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.27/1.73 inference(quant_intro,[status(thm)],[104])).
% 2.27/1.73 tff(106,plain,
% 2.27/1.73 (![X: $i] : product(inverse(X), X, identity) <=> ![X: $i] : product(inverse(X), X, identity)),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(107,axiom,(![X: $i] : product(inverse(X), X, identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','left_inverse')).
% 2.27/1.73 tff(108,plain,
% 2.27/1.73 (![X: $i] : product(inverse(X), X, identity)),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[107, 106])).
% 2.27/1.73 tff(109,plain,(
% 2.27/1.73 ![X: $i] : product(inverse(X), X, identity)),
% 2.27/1.73 inference(skolemize,[status(sab)],[108])).
% 2.27/1.73 tff(110,plain,
% 2.27/1.73 (![X: $i] : product(inverse(X), X, identity)),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[109, 105])).
% 2.27/1.73 tff(111,plain,
% 2.27/1.73 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(a), a, identity)),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(112,plain,
% 2.27/1.73 (product(inverse(a), a, identity)),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[111, 110])).
% 2.27/1.73 tff(113,plain,
% 2.27/1.73 (^[X: $i] : refl(product(X, inverse(X), identity) <=> product(X, inverse(X), identity))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(114,plain,
% 2.27/1.73 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 2.27/1.73 inference(quant_intro,[status(thm)],[113])).
% 2.27/1.73 tff(115,plain,
% 2.27/1.73 (![X: $i] : product(X, inverse(X), identity) <=> ![X: $i] : product(X, inverse(X), identity)),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(116,axiom,(![X: $i] : product(X, inverse(X), identity)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','right_inverse')).
% 2.27/1.73 tff(117,plain,
% 2.27/1.73 (![X: $i] : product(X, inverse(X), identity)),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[116, 115])).
% 2.27/1.73 tff(118,plain,(
% 2.27/1.73 ![X: $i] : product(X, inverse(X), identity)),
% 2.27/1.73 inference(skolemize,[status(sab)],[117])).
% 2.27/1.73 tff(119,plain,
% 2.27/1.73 (![X: $i] : product(X, inverse(X), identity)),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[118, 114])).
% 2.27/1.73 tff(120,plain,
% 2.27/1.73 ((~![X: $i] : product(X, inverse(X), identity)) | product(a, inverse(a), identity)),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(121,plain,
% 2.27/1.73 (product(a, inverse(a), identity)),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[120, 119])).
% 2.27/1.73 tff(122,plain,
% 2.27/1.73 (^[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))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(123,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[122])).
% 2.27/1.73 tff(124,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(125,plain,
% 2.27/1.73 (^[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)))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(126,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[125])).
% 2.27/1.73 tff(127,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(X, V, W))) | product(U, Z, W))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','associativity2')).
% 2.27/1.73 tff(128,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[127, 126])).
% 2.27/1.73 tff(129,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[128, 124])).
% 2.27/1.73 tff(130,plain,(
% 2.27/1.73 ![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)))),
% 2.27/1.73 inference(skolemize,[status(sab)],[129])).
% 2.27/1.73 tff(131,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[130, 123])).
% 2.27/1.73 tff(132,plain,
% 2.27/1.73 (((~![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, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(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, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(a)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(133,plain,
% 2.27/1.73 ((product(identity, inverse(a), inverse(a)) | (~product(a, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a)))) <=> ((~product(a, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(a)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(134,plain,
% 2.27/1.73 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(a)) | (~product(a, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(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, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(a))))),
% 2.27/1.73 inference(monotonicity,[status(thm)],[133])).
% 2.27/1.73 tff(135,plain,
% 2.27/1.73 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(a)) | (~product(a, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(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, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(a)))),
% 2.27/1.73 inference(transitivity,[status(thm)],[134, 132])).
% 2.27/1.73 tff(136,plain,
% 2.27/1.73 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(U, Z, W) | (~product(Y, Z, V)) | (~product(X, Y, U)) | (~product(X, V, W)))) | (product(identity, inverse(a), inverse(a)) | (~product(a, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(137,plain,
% 2.27/1.73 ((~![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, inverse(a), identity)) | (~product(inverse(a), a, identity)) | (~product(inverse(a), identity, inverse(a))) | product(identity, inverse(a), inverse(a))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[136, 135])).
% 2.27/1.73 tff(138,plain,
% 2.27/1.73 (product(identity, inverse(a), inverse(a))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[137, 131, 121, 112, 103])).
% 2.27/1.73 tff(139,plain,
% 2.27/1.73 ((~![X: $i] : product(inverse(X), X, identity)) | product(inverse(b), b, identity)),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(140,plain,
% 2.27/1.73 (product(inverse(b), b, identity)),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[139, 110])).
% 2.27/1.73 tff(141,plain,
% 2.27/1.73 (^[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))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(142,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[141])).
% 2.27/1.73 tff(143,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(144,plain,
% 2.27/1.73 (^[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)))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(145,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[144])).
% 2.27/1.73 tff(146,axiom,(![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : ((((~product(X, Y, U)) | (~product(Y, Z, V))) | (~product(U, Z, W))) | product(X, V, W))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','associativity1')).
% 2.27/1.73 tff(147,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[146, 145])).
% 2.27/1.73 tff(148,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[147, 143])).
% 2.27/1.73 tff(149,plain,(
% 2.27/1.73 ![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)))),
% 2.27/1.73 inference(skolemize,[status(sab)],[148])).
% 2.27/1.73 tff(150,plain,
% 2.27/1.73 (![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)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[149, 142])).
% 2.27/1.73 tff(151,plain,
% 2.27/1.73 (((~![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(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(152,plain,
% 2.27/1.73 ((product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity))) <=> ((~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(153,plain,
% 2.27/1.73 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | ((~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a))))),
% 2.27/1.73 inference(monotonicity,[status(thm)],[152])).
% 2.27/1.73 tff(154,plain,
% 2.27/1.73 (((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)))) <=> ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a)))),
% 2.27/1.73 inference(transitivity,[status(thm)],[153, 151])).
% 2.27/1.73 tff(155,plain,
% 2.27/1.73 ((~![W: $i, V: $i, Z: $i, Y: $i, U: $i, X: $i] : (product(X, V, W) | (~product(U, Z, W)) | (~product(Y, Z, V)) | (~product(X, Y, U)))) | (product(inverse(b), c, inverse(a)) | (~product(identity, inverse(a), inverse(a))) | (~product(b, inverse(a), c)) | (~product(inverse(b), b, identity)))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(156,plain,
% 2.27/1.73 ((~![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(b), b, identity)) | (~product(identity, inverse(a), inverse(a))) | product(inverse(b), c, inverse(a))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[155, 154])).
% 2.27/1.73 tff(157,plain,
% 2.27/1.73 (product(inverse(b), c, inverse(a))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[156, 150, 37, 140, 138])).
% 2.27/1.73 tff(158,plain,
% 2.27/1.73 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(159,plain,
% 2.27/1.73 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(b)) | subgroup_member(inverse(b)))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(160,plain,
% 2.27/1.73 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(b)) | subgroup_member(inverse(b))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[159, 158])).
% 2.27/1.73 tff(161,plain,
% 2.27/1.73 (subgroup_member(inverse(b))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[160, 29, 40])).
% 2.27/1.73 tff(162,plain,
% 2.27/1.73 ((~![X: $i] : product(X, inverse(X), identity)) | product(inverse(a), inverse(inverse(a)), identity)),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(163,plain,
% 2.27/1.73 (product(inverse(a), inverse(inverse(a)), identity)),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[162, 119])).
% 2.27/1.73 tff(164,plain,
% 2.27/1.73 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(165,plain,
% 2.27/1.73 (((~product(inverse(a), inverse(inverse(a)), identity)) | equalish(inverse(inverse(a)), a) | (~product(inverse(a), a, identity))) <=> ((~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(166,plain,
% 2.27/1.73 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(inverse(a), inverse(inverse(a)), identity)) | equalish(inverse(inverse(a)), a) | (~product(inverse(a), a, identity)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a)))),
% 2.27/1.73 inference(monotonicity,[status(thm)],[165])).
% 2.27/1.73 tff(167,plain,
% 2.27/1.73 (((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(inverse(a), inverse(inverse(a)), identity)) | equalish(inverse(inverse(a)), a) | (~product(inverse(a), a, identity)))) <=> ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a))),
% 2.27/1.73 inference(transitivity,[status(thm)],[166, 164])).
% 2.27/1.73 tff(168,plain,
% 2.27/1.73 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | ((~product(inverse(a), inverse(inverse(a)), identity)) | equalish(inverse(inverse(a)), a) | (~product(inverse(a), a, identity)))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(169,plain,
% 2.27/1.73 ((~![B: $i, D: $i, A: $i, C: $i] : ((~product(A, D, C)) | equalish(D, B) | (~product(A, B, C)))) | (~product(inverse(a), inverse(inverse(a)), identity)) | (~product(inverse(a), a, identity)) | equalish(inverse(inverse(a)), a)),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[168, 167])).
% 2.27/1.73 tff(170,plain,
% 2.27/1.73 (equalish(inverse(inverse(a)), a)),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[169, 87, 112, 163])).
% 2.27/1.73 tff(171,plain,
% 2.27/1.73 (^[B: $i, A: $i] : refl((subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B))) <=> (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(172,plain,
% 2.27/1.73 (![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[171])).
% 2.27/1.73 tff(173,plain,
% 2.27/1.73 (![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B))) <=> ![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(174,plain,
% 2.27/1.73 (^[B: $i, A: $i] : trans(monotonicity(rewrite(((~equalish(A, B)) | (~subgroup_member(A))) <=> ((~subgroup_member(A)) | (~equalish(A, B)))), ((((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> (((~subgroup_member(A)) | (~equalish(A, B))) | subgroup_member(B)))), rewrite((((~subgroup_member(A)) | (~equalish(A, B))) | subgroup_member(B)) <=> (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))), ((((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))))),
% 2.27/1.73 inference(bind,[status(th)],[])).
% 2.27/1.73 tff(175,plain,
% 2.27/1.73 (![B: $i, A: $i] : (((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B)) <=> ![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(quant_intro,[status(thm)],[174])).
% 2.27/1.73 tff(176,axiom,(![B: $i, A: $i] : (((~equalish(A, B)) | (~subgroup_member(A))) | subgroup_member(B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','subgroup_member_substitution')).
% 2.27/1.73 tff(177,plain,
% 2.27/1.73 (![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[176, 175])).
% 2.27/1.73 tff(178,plain,
% 2.27/1.73 (![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[177, 173])).
% 2.27/1.73 tff(179,plain,(
% 2.27/1.73 ![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(skolemize,[status(sab)],[178])).
% 2.27/1.73 tff(180,plain,
% 2.27/1.73 (![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[179, 172])).
% 2.27/1.73 tff(181,plain,
% 2.27/1.73 (((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | (subgroup_member(a) | (~subgroup_member(inverse(inverse(a)))) | (~equalish(inverse(inverse(a)), a)))) <=> ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | subgroup_member(a) | (~subgroup_member(inverse(inverse(a)))) | (~equalish(inverse(inverse(a)), a)))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(182,plain,
% 2.27/1.73 ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | (subgroup_member(a) | (~subgroup_member(inverse(inverse(a)))) | (~equalish(inverse(inverse(a)), a)))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(183,plain,
% 2.27/1.73 ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | subgroup_member(a) | (~subgroup_member(inverse(inverse(a)))) | (~equalish(inverse(inverse(a)), a))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[182, 181])).
% 2.27/1.73 tff(184,plain,
% 2.27/1.73 (subgroup_member(a) | (~subgroup_member(inverse(inverse(a))))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[183, 180, 170])).
% 2.27/1.73 tff(185,plain,
% 2.27/1.73 (~subgroup_member(inverse(inverse(a)))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[184, 49])).
% 2.27/1.73 tff(186,plain,
% 2.27/1.73 (((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))) <=> ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(187,plain,
% 2.27/1.73 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a))))),
% 2.27/1.73 inference(quant_inst,[status(thm)],[])).
% 2.27/1.73 tff(188,plain,
% 2.27/1.73 ((~![X: $i] : ((~subgroup_member(X)) | subgroup_member(inverse(X)))) | (~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.27/1.73 inference(modus_ponens,[status(thm)],[187, 186])).
% 2.27/1.73 tff(189,plain,
% 2.27/1.73 ((~subgroup_member(inverse(a))) | subgroup_member(inverse(inverse(a)))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[188, 29])).
% 2.27/1.73 tff(190,plain,
% 2.27/1.73 (~subgroup_member(inverse(a))),
% 2.27/1.73 inference(unit_resolution,[status(thm)],[189, 185])).
% 2.27/1.73 tff(191,plain,
% 2.27/1.73 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(192,plain,
% 2.27/1.73 ((subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b)))) <=> (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 2.27/1.73 inference(rewrite,[status(thm)],[])).
% 2.27/1.73 tff(193,plain,
% 2.27/1.73 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a)))))),
% 2.27/1.73 inference(monotonicity,[status(thm)],[192])).
% 2.27/1.73 tff(194,plain,
% 2.27/1.73 (((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))) <=> ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a))))),
% 2.27/1.74 inference(transitivity,[status(thm)],[193, 191])).
% 2.27/1.74 tff(195,plain,
% 2.27/1.74 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | (subgroup_member(inverse(a)) | (~product(inverse(b), c, inverse(a))) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))))),
% 2.27/1.74 inference(quant_inst,[status(thm)],[])).
% 2.27/1.74 tff(196,plain,
% 2.27/1.74 ((~![B: $i, A: $i, C: $i] : (subgroup_member(C) | (~product(A, B, C)) | (~subgroup_member(B)) | (~subgroup_member(A)))) | subgroup_member(inverse(a)) | (~subgroup_member(c)) | (~subgroup_member(inverse(b))) | (~product(inverse(b), c, inverse(a)))),
% 2.27/1.74 inference(modus_ponens,[status(thm)],[195, 194])).
% 2.27/1.74 tff(197,plain,
% 2.27/1.74 (~subgroup_member(c)),
% 2.27/1.74 inference(unit_resolution,[status(thm)],[196, 17, 190, 161, 157])).
% 2.27/1.74 tff(198,plain,
% 2.27/1.74 (((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | (subgroup_member(c) | (~subgroup_member(element_in_O2(a, d))) | (~equalish(element_in_O2(a, d), c)))) <=> ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | subgroup_member(c) | (~subgroup_member(element_in_O2(a, d))) | (~equalish(element_in_O2(a, d), c)))),
% 2.27/1.74 inference(rewrite,[status(thm)],[])).
% 2.27/1.74 tff(199,plain,
% 2.27/1.74 ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | (subgroup_member(c) | (~subgroup_member(element_in_O2(a, d))) | (~equalish(element_in_O2(a, d), c)))),
% 2.27/1.74 inference(quant_inst,[status(thm)],[])).
% 2.27/1.74 tff(200,plain,
% 2.27/1.74 ((~![B: $i, A: $i] : (subgroup_member(B) | (~subgroup_member(A)) | (~equalish(A, B)))) | subgroup_member(c) | (~subgroup_member(element_in_O2(a, d))) | (~equalish(element_in_O2(a, d), c))),
% 2.27/1.74 inference(modus_ponens,[status(thm)],[199, 198])).
% 2.27/1.74 tff(201,plain,
% 2.27/1.74 ($false),
% 2.27/1.74 inference(unit_resolution,[status(thm)],[200, 180, 197, 94, 63])).
% 2.27/1.74 % SZS output end Proof
%------------------------------------------------------------------------------