%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP002-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:08:50 AM UTC 2026
% Result : Unsatisfiable 51.07s 6.97s
% Output : Proof 52.32s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : GRP002-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.38 % Computer : n009.cluster.edu
% 0.11/0.38 % Model : x86_64 x86_64
% 0.11/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.38 % Memory : 8046.5625MB
% 0.11/0.38 % OS : Linux 6.8.0-71-generic
% 0.11/0.38 % CPULimit : 300
% 0.11/0.38 % WCLimit : 300
% 0.11/0.38 % DateTime : Sun Sep 27 09:12:59 UTC 2026
% 0.11/0.38 % CPUTime :
% 0.11/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 51.07/6.97 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 51.07/6.97
% 51.07/6.97 % SZS status Unsatisfiable
% 51.07/6.97
% 52.32/7.09 % SZS output start Proof
% 52.32/7.09 Axiom 1 (right_identity): product(X, identity, X) = true.
% 52.32/7.09 Axiom 2 (right_inverse): product(X, inverse(X), identity) = true.
% 52.32/7.09 Axiom 3 (left_identity): product(identity, X, X) = true.
% 52.32/7.09 Axiom 4 (h_times_b_is_j): product(h, b, j) = true.
% 52.32/7.09 Axiom 5 (a_times_b_is_c): product(a, b, c) = true.
% 52.32/7.09 Axiom 6 (c_times_inverse_a_is_d): product(c, inverse(a), d) = true.
% 52.32/7.09 Axiom 7 (d_times_inverse_b_is_h): product(d, inverse(b), h) = true.
% 52.32/7.09 Axiom 8 (j_times_inverse_h_is_k): product(j, inverse(h), k) = true.
% 52.32/7.09 Axiom 9 (left_inverse): product(inverse(X), X, identity) = true.
% 52.32/7.09 Axiom 10 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 52.32/7.09 Axiom 11 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 52.32/7.09 Axiom 12 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 52.32/7.09 Axiom 13 (x_cubed_is_identity_2): ifeq(product(X, X, Y), true, product(Y, X, identity), true) = true.
% 52.32/7.09 Axiom 14 (total_function2): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 52.32/7.09 Axiom 15 (associativity1): ifeq(product(X, Y, Z), true, ifeq(product(W, Y, V), true, ifeq(product(U, W, X), true, product(U, V, Z), true), true), true) = true.
% 52.32/7.09 Axiom 16 (associativity2): ifeq(product(X, Y, Z), true, ifeq(product(W, Z, V), true, ifeq(product(W, X, U), true, product(U, Y, V), true), true), true) = true.
% 52.32/7.09
% 52.32/7.09 Lemma 17: j = d.
% 52.32/7.09 Proof:
% 52.32/7.09 j
% 52.32/7.09 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.09 ifeq2(true, true, j, d)
% 52.32/7.10 = { by axiom 16 (associativity2) R->L }
% 52.32/7.10 ifeq2(ifeq(product(inverse(b), b, identity), true, ifeq(product(d, identity, d), true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true), true, j, d)
% 52.32/7.10 = { by axiom 1 (right_identity) }
% 52.32/7.10 ifeq2(ifeq(product(inverse(b), b, identity), true, ifeq(true, true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true), true, j, d)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(inverse(b), b, identity), true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true, j, d)
% 52.32/7.10 = { by axiom 9 (left_inverse) }
% 52.32/7.10 ifeq2(ifeq(true, true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true, j, d)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true, j, d)
% 52.32/7.10 = { by axiom 7 (d_times_inverse_b_is_h) }
% 52.32/7.10 ifeq2(ifeq(true, true, product(h, b, d), true), true, j, d)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(product(h, b, d), true, j, d)
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(product(h, b, d), true, ifeq2(true, true, j, d), d)
% 52.32/7.10 = { by axiom 4 (h_times_b_is_j) R->L }
% 52.32/7.10 ifeq2(product(h, b, d), true, ifeq2(product(h, b, j), true, j, d), d)
% 52.32/7.10 = { by axiom 14 (total_function2) }
% 52.32/7.10 d
% 52.32/7.10
% 52.32/7.10 Lemma 18: ifeq2(product(identity, X, Y), true, Y, X) = X.
% 52.32/7.10 Proof:
% 52.32/7.10 ifeq2(product(identity, X, Y), true, Y, X)
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(true, true, ifeq2(product(identity, X, Y), true, Y, X), X)
% 52.32/7.10 = { by axiom 3 (left_identity) R->L }
% 52.32/7.10 ifeq2(product(identity, X, X), true, ifeq2(product(identity, X, Y), true, Y, X), X)
% 52.32/7.10 = { by axiom 14 (total_function2) }
% 52.32/7.10 X
% 52.32/7.10
% 52.32/7.10 Lemma 19: inverse(inverse(X)) = X.
% 52.32/7.10 Proof:
% 52.32/7.10 inverse(inverse(X))
% 52.32/7.10 = { by lemma 18 R->L }
% 52.32/7.10 ifeq2(product(identity, inverse(inverse(X)), X), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(ifeq(true, true, product(identity, inverse(inverse(X)), X), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 2 (right_inverse) R->L }
% 52.32/7.10 ifeq2(ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(ifeq(true, true, ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 2 (right_inverse) R->L }
% 52.32/7.10 ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), identity), true, ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), identity), true, ifeq(true, true, ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 1 (right_identity) R->L }
% 52.32/7.10 ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), identity), true, ifeq(product(X, identity, X), true, ifeq(product(X, inverse(X), identity), true, product(identity, inverse(inverse(X)), X), true), true), true), true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 16 (associativity2) }
% 52.32/7.10 ifeq2(true, true, X, inverse(inverse(X)))
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) }
% 52.32/7.10 X
% 52.32/7.10
% 52.32/7.10 Lemma 20: multiply(X, X) = inverse(X).
% 52.32/7.10 Proof:
% 52.32/7.10 multiply(X, X)
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(true, true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 16 (associativity2) R->L }
% 52.32/7.10 ifeq2(ifeq(product(X, inverse(X), identity), true, ifeq(product(multiply(X, X), identity, multiply(X, X)), true, ifeq(product(multiply(X, X), X, identity), true, product(identity, inverse(X), multiply(X, X)), true), true), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 1 (right_identity) }
% 52.32/7.10 ifeq2(ifeq(product(X, inverse(X), identity), true, ifeq(true, true, ifeq(product(multiply(X, X), X, identity), true, product(identity, inverse(X), multiply(X, X)), true), true), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(X, inverse(X), identity), true, ifeq(product(multiply(X, X), X, identity), true, product(identity, inverse(X), multiply(X, X)), true), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 2 (right_inverse) }
% 52.32/7.10 ifeq2(ifeq(true, true, ifeq(product(multiply(X, X), X, identity), true, product(identity, inverse(X), multiply(X, X)), true), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(multiply(X, X), X, identity), true, product(identity, inverse(X), multiply(X, X)), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(ifeq(ifeq(true, true, product(multiply(X, X), X, identity), true), true, product(identity, inverse(X), multiply(X, X)), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 12 (total_function1) R->L }
% 52.32/7.10 ifeq2(ifeq(ifeq(product(X, X, multiply(X, X)), true, product(multiply(X, X), X, identity), true), true, product(identity, inverse(X), multiply(X, X)), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 13 (x_cubed_is_identity_2) }
% 52.32/7.10 ifeq2(ifeq(true, true, product(identity, inverse(X), multiply(X, X)), true), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(product(identity, inverse(X), multiply(X, X)), true, multiply(X, X), inverse(X))
% 52.32/7.10 = { by lemma 18 }
% 52.32/7.10 inverse(X)
% 52.32/7.10
% 52.32/7.10 Lemma 21: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 52.32/7.10 Proof:
% 52.32/7.10 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 52.32/7.10 = { by axiom 12 (total_function1) R->L }
% 52.32/7.10 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 52.32/7.10 = { by axiom 14 (total_function2) }
% 52.32/7.10 Z
% 52.32/7.10
% 52.32/7.10 Lemma 22: multiply(multiply(X, Y), inverse(Y)) = X.
% 52.32/7.10 Proof:
% 52.32/7.10 multiply(multiply(X, Y), inverse(Y))
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(true, true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 16 (associativity2) R->L }
% 52.32/7.10 ifeq2(ifeq(product(Y, inverse(Y), identity), true, ifeq(product(X, identity, X), true, ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true), true), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 1 (right_identity) }
% 52.32/7.10 ifeq2(ifeq(product(Y, inverse(Y), identity), true, ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true), true), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(Y, inverse(Y), identity), true, ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 2 (right_inverse) }
% 52.32/7.10 ifeq2(ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(multiply(X, Y), inverse(Y), X), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 12 (total_function1) }
% 52.32/7.10 ifeq2(ifeq(true, true, product(multiply(X, Y), inverse(Y), X), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(product(multiply(X, Y), inverse(Y), X), true, multiply(multiply(X, Y), inverse(Y)), X)
% 52.32/7.10 = { by lemma 21 }
% 52.32/7.10 X
% 52.32/7.10
% 52.32/7.10 Lemma 23: multiply(inverse(X), multiply(X, Y)) = Y.
% 52.32/7.10 Proof:
% 52.32/7.10 multiply(inverse(X), multiply(X, Y))
% 52.32/7.10 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.10 ifeq2(true, true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 15 (associativity1) R->L }
% 52.32/7.10 ifeq2(ifeq(product(identity, Y, Y), true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(product(inverse(X), X, identity), true, product(inverse(X), multiply(X, Y), Y), true), true), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 3 (left_identity) }
% 52.32/7.10 ifeq2(ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(product(inverse(X), X, identity), true, product(inverse(X), multiply(X, Y), Y), true), true), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(product(inverse(X), X, identity), true, product(inverse(X), multiply(X, Y), Y), true), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 9 (left_inverse) }
% 52.32/7.10 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(true, true, product(inverse(X), multiply(X, Y), Y), true), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(inverse(X), multiply(X, Y), Y), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 12 (total_function1) }
% 52.32/7.10 ifeq2(ifeq(true, true, product(inverse(X), multiply(X, Y), Y), true), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.10 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.10 ifeq2(product(inverse(X), multiply(X, Y), Y), true, multiply(inverse(X), multiply(X, Y)), Y)
% 52.32/7.11 = { by lemma 21 }
% 52.32/7.11 Y
% 52.32/7.11
% 52.32/7.11 Lemma 24: inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
% 52.32/7.11 Proof:
% 52.32/7.11 inverse(multiply(X, Y))
% 52.32/7.11 = { by lemma 22 R->L }
% 52.32/7.11 multiply(multiply(inverse(multiply(X, Y)), multiply(multiply(X, Y), inverse(Y))), inverse(multiply(multiply(X, Y), inverse(Y))))
% 52.32/7.11 = { by lemma 23 }
% 52.32/7.11 multiply(inverse(Y), inverse(multiply(multiply(X, Y), inverse(Y))))
% 52.32/7.11 = { by lemma 22 }
% 52.32/7.11 multiply(inverse(Y), inverse(X))
% 52.32/7.11
% 52.32/7.11 Lemma 25: multiply(a, inverse(c)) = inverse(d).
% 52.32/7.11 Proof:
% 52.32/7.11 multiply(a, inverse(c))
% 52.32/7.11 = { by lemma 21 R->L }
% 52.32/7.11 multiply(ifeq2(product(inverse(d), c, a), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(true, true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 16 (associativity2) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(product(inverse(a), inverse(inverse(a)), identity), true, ifeq(product(c, identity, c), true, ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 1 (right_identity) }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(product(inverse(a), inverse(inverse(a)), identity), true, ifeq(true, true, ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(product(inverse(a), inverse(inverse(a)), identity), true, ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 2 (right_inverse) }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(true, true, ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 6 (c_times_inverse_a_is_d) }
% 52.32/7.11 multiply(ifeq2(ifeq(ifeq(true, true, product(d, inverse(inverse(a)), c), true), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 multiply(ifeq2(ifeq(product(d, inverse(inverse(a)), c), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by lemma 19 }
% 52.32/7.11 multiply(ifeq2(ifeq(product(d, a, c), true, product(inverse(d), c, a), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(product(d, a, c), true, ifeq(true, true, product(inverse(d), c, a), true), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 9 (left_inverse) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(product(d, a, c), true, ifeq(product(inverse(d), d, identity), true, product(inverse(d), c, a), true), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(true, true, ifeq(product(d, a, c), true, ifeq(product(inverse(d), d, identity), true, product(inverse(d), c, a), true), true), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 3 (left_identity) R->L }
% 52.32/7.11 multiply(ifeq2(ifeq(product(identity, a, a), true, ifeq(product(d, a, c), true, ifeq(product(inverse(d), d, identity), true, product(inverse(d), c, a), true), true), true), true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 15 (associativity1) }
% 52.32/7.11 multiply(ifeq2(true, true, multiply(inverse(d), c), a), inverse(c))
% 52.32/7.11 = { by axiom 11 (ifeq_axiom) }
% 52.32/7.11 multiply(multiply(inverse(d), c), inverse(c))
% 52.32/7.11 = { by lemma 22 }
% 52.32/7.11 inverse(d)
% 52.32/7.11
% 52.32/7.11 Lemma 26: multiply(a, multiply(b, X)) = multiply(c, X).
% 52.32/7.11 Proof:
% 52.32/7.11 multiply(a, multiply(b, X))
% 52.32/7.11 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.11 ifeq2(true, true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 15 (associativity1) R->L }
% 52.32/7.11 ifeq2(ifeq(product(c, X, multiply(c, X)), true, ifeq(product(b, X, multiply(b, X)), true, ifeq(product(a, b, c), true, product(a, multiply(b, X), multiply(c, X)), true), true), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 5 (a_times_b_is_c) }
% 52.32/7.11 ifeq2(ifeq(product(c, X, multiply(c, X)), true, ifeq(product(b, X, multiply(b, X)), true, ifeq(true, true, product(a, multiply(b, X), multiply(c, X)), true), true), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 ifeq2(ifeq(product(c, X, multiply(c, X)), true, ifeq(product(b, X, multiply(b, X)), true, product(a, multiply(b, X), multiply(c, X)), true), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 12 (total_function1) }
% 52.32/7.11 ifeq2(ifeq(true, true, ifeq(product(b, X, multiply(b, X)), true, product(a, multiply(b, X), multiply(c, X)), true), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 ifeq2(ifeq(product(b, X, multiply(b, X)), true, product(a, multiply(b, X), multiply(c, X)), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 12 (total_function1) }
% 52.32/7.11 ifeq2(ifeq(true, true, product(a, multiply(b, X), multiply(c, X)), true), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.11 ifeq2(product(a, multiply(b, X), multiply(c, X)), true, multiply(a, multiply(b, X)), multiply(c, X))
% 52.32/7.11 = { by lemma 21 }
% 52.32/7.11 multiply(c, X)
% 52.32/7.11
% 52.32/7.11 Lemma 27: multiply(inverse(a), inverse(c)) = multiply(b, c).
% 52.32/7.11 Proof:
% 52.32/7.11 multiply(inverse(a), inverse(c))
% 52.32/7.11 = { by lemma 20 R->L }
% 52.32/7.11 multiply(inverse(a), multiply(c, c))
% 52.32/7.11 = { by lemma 26 R->L }
% 52.32/7.11 multiply(inverse(a), multiply(a, multiply(b, c)))
% 52.32/7.11 = { by lemma 23 }
% 52.32/7.11 multiply(b, c)
% 52.32/7.11
% 52.32/7.11 Lemma 28: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 52.32/7.11 Proof:
% 52.32/7.11 multiply(multiply(X, Y), Z)
% 52.32/7.11 = { by lemma 21 R->L }
% 52.32/7.11 ifeq2(product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 ifeq2(ifeq(true, true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 12 (total_function1) R->L }
% 52.32/7.11 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 ifeq2(ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 12 (total_function1) R->L }
% 52.32/7.11 ifeq2(ifeq(product(Y, Z, multiply(Y, Z)), true, ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.11 ifeq2(ifeq(true, true, ifeq(product(Y, Z, multiply(Y, Z)), true, ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.11 = { by axiom 12 (total_function1) R->L }
% 52.32/7.12 ifeq2(ifeq(product(multiply(X, Y), Z, multiply(multiply(X, Y), Z)), true, ifeq(product(Y, Z, multiply(Y, Z)), true, ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true), true), true), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.12 = { by axiom 15 (associativity1) }
% 52.32/7.12 ifeq2(true, true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 52.32/7.12 = { by axiom 11 (ifeq_axiom) }
% 52.32/7.12 multiply(X, multiply(Y, Z))
% 52.32/7.12
% 52.32/7.12 Goal 1 (prove_k_times_inverse_b_is_e): product(k, inverse(b), identity) = true.
% 52.32/7.12 Proof:
% 52.32/7.12 product(k, inverse(b), identity)
% 52.32/7.12 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.12 ifeq(true, true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 3 (left_identity) R->L }
% 52.32/7.12 ifeq(product(identity, inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by lemma 21 R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(inverse(k), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by lemma 22 R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(multiply(inverse(k), d), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 11 (ifeq_axiom) R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(true, true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 15 (associativity1) R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(identity, h, h), true, ifeq(product(k, h, d), true, ifeq(product(inverse(k), k, identity), true, product(inverse(k), d, h), true), true), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 3 (left_identity) }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(true, true, ifeq(product(k, h, d), true, ifeq(product(inverse(k), k, identity), true, product(inverse(k), d, h), true), true), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(k, h, d), true, ifeq(product(inverse(k), k, identity), true, product(inverse(k), d, h), true), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 9 (left_inverse) }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(k, h, d), true, ifeq(true, true, product(inverse(k), d, h), true), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(k, h, d), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by lemma 17 R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(k, h, j), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by lemma 19 R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(product(k, inverse(inverse(h)), j), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(true, true, product(k, inverse(inverse(h)), j), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 8 (j_times_inverse_h_is_k) R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.12 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.12 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(true, true, ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 2 (right_inverse) R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(product(inverse(h), inverse(inverse(h)), identity), true, ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(product(inverse(h), inverse(inverse(h)), identity), true, ifeq(true, true, ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 1 (right_identity) R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(ifeq(product(inverse(h), inverse(inverse(h)), identity), true, ifeq(product(j, identity, j), true, ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true), true), true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 16 (associativity2) }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(ifeq2(ifeq(true, true, product(inverse(k), d, h), true), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(ifeq2(product(inverse(k), d, h), true, multiply(inverse(k), d), h), inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 21 }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, inverse(d)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 25 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, multiply(a, inverse(c))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 19 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, multiply(inverse(inverse(a)), inverse(c))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 20 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, multiply(multiply(inverse(a), inverse(a)), inverse(c))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 28 }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, multiply(inverse(a), multiply(inverse(a), inverse(c)))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 27 }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(h, multiply(inverse(a), multiply(b, c))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 19 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(inverse(inverse(h)), multiply(inverse(a), multiply(b, c))), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 28 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(multiply(inverse(inverse(h)), inverse(a)), multiply(b, c)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by lemma 24 R->L }
% 52.32/7.13 ifeq(product(ifeq2(product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.13 ifeq(product(ifeq2(ifeq(true, true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 16 (associativity2) R->L }
% 52.32/7.13 ifeq(product(ifeq2(ifeq(ifeq(product(multiply(c, a), b, multiply(c, c)), true, ifeq(product(multiply(c, a), multiply(c, c), multiply(multiply(c, a), multiply(c, c))), true, ifeq(product(multiply(c, a), multiply(c, a), multiply(multiply(c, a), multiply(c, a))), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 12 (total_function1) }
% 52.32/7.13 ifeq(product(ifeq2(ifeq(ifeq(product(multiply(c, a), b, multiply(c, c)), true, ifeq(true, true, ifeq(product(multiply(c, a), multiply(c, a), multiply(multiply(c, a), multiply(c, a))), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.13 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.13 ifeq(product(ifeq2(ifeq(ifeq(product(multiply(c, a), b, multiply(c, c)), true, ifeq(product(multiply(c, a), multiply(c, a), multiply(multiply(c, a), multiply(c, a))), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 12 (total_function1) }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(product(multiply(c, a), b, multiply(c, c)), true, ifeq(true, true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(product(multiply(c, a), b, multiply(c, c)), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(true, true, product(multiply(c, a), b, multiply(c, c)), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 12 (total_function1) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(product(c, a, multiply(c, a)), true, product(multiply(c, a), b, multiply(c, c)), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(true, true, ifeq(product(c, a, multiply(c, a)), true, product(multiply(c, a), b, multiply(c, c)), true), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 12 (total_function1) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(product(c, c, multiply(c, c)), true, ifeq(product(c, a, multiply(c, a)), true, product(multiply(c, a), b, multiply(c, c)), true), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(true, true, ifeq(product(c, c, multiply(c, c)), true, ifeq(product(c, a, multiply(c, a)), true, product(multiply(c, a), b, multiply(c, c)), true), true), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 5 (a_times_b_is_c) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(ifeq(product(a, b, c), true, ifeq(product(c, c, multiply(c, c)), true, ifeq(product(c, a, multiply(c, a)), true, product(multiply(c, a), b, multiply(c, c)), true), true), true), true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 16 (associativity2) }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(ifeq(true, true, product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), multiply(c, c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 20 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(multiply(c, a), multiply(c, a)), b, multiply(multiply(c, a), inverse(c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 20 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(inverse(multiply(c, a)), b, multiply(multiply(c, a), inverse(c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 24 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(inverse(a), inverse(c)), b, multiply(multiply(c, a), inverse(c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 27 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(multiply(c, a), inverse(c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 28 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(c, multiply(a, inverse(c)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 25 }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(c, inverse(d))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 26 R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(b, inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 21 R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(product(inverse(h), d, b), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(true, true, product(inverse(h), d, b), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 9 (left_inverse) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(product(inverse(h), h, identity), true, product(inverse(h), d, b), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by lemma 17 R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(product(inverse(h), h, identity), true, product(inverse(h), j, b), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(true, true, ifeq(product(inverse(h), h, identity), true, product(inverse(h), j, b), true), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 4 (h_times_b_is_j) R->L }
% 52.32/7.14 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(product(h, b, j), true, ifeq(product(inverse(h), h, identity), true, product(inverse(h), j, b), true), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.14 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(true, true, ifeq(product(h, b, j), true, ifeq(product(inverse(h), h, identity), true, product(inverse(h), j, b), true), true), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 3 (left_identity) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(ifeq(product(identity, b, b), true, ifeq(product(h, b, j), true, ifeq(product(inverse(h), h, identity), true, product(inverse(h), j, b), true), true), true), true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 15 (associativity1) }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(ifeq2(true, true, multiply(inverse(h), d), b), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 11 (ifeq_axiom) }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, multiply(multiply(inverse(h), d), inverse(d)))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by lemma 22 }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, inverse(h))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, inverse(h))), true, ifeq(true, true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 12 (total_function1) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, inverse(h))), true, ifeq(product(inverse(multiply(a, inverse(h))), multiply(b, c), multiply(inverse(multiply(a, inverse(h))), multiply(b, c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, inverse(h))), true, ifeq(true, true, ifeq(product(inverse(multiply(a, inverse(h))), multiply(b, c), multiply(inverse(multiply(a, inverse(h))), multiply(b, c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 9 (left_inverse) R->L }
% 52.32/7.15 ifeq(product(ifeq2(ifeq(product(multiply(b, c), b, multiply(a, inverse(h))), true, ifeq(product(inverse(multiply(a, inverse(h))), multiply(a, inverse(h)), identity), true, ifeq(product(inverse(multiply(a, inverse(h))), multiply(b, c), multiply(inverse(multiply(a, inverse(h))), multiply(b, c))), true, product(multiply(inverse(multiply(a, inverse(h))), multiply(b, c)), b, identity), true), true), true), true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 16 (associativity2) }
% 52.32/7.15 ifeq(product(ifeq2(true, true, multiply(inverse(k), b), identity), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 11 (ifeq_axiom) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, product(k, inverse(b), identity), true)
% 52.32/7.15 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(true, true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 15 (associativity1) R->L }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(product(identity, b, b), true, ifeq(product(inverse(k), b, multiply(inverse(k), b)), true, ifeq(product(k, inverse(k), identity), true, product(k, multiply(inverse(k), b), b), true), true), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 3 (left_identity) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(true, true, ifeq(product(inverse(k), b, multiply(inverse(k), b)), true, ifeq(product(k, inverse(k), identity), true, product(k, multiply(inverse(k), b), b), true), true), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(product(inverse(k), b, multiply(inverse(k), b)), true, ifeq(product(k, inverse(k), identity), true, product(k, multiply(inverse(k), b), b), true), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 2 (right_inverse) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(product(inverse(k), b, multiply(inverse(k), b)), true, ifeq(true, true, product(k, multiply(inverse(k), b), b), true), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(product(inverse(k), b, multiply(inverse(k), b)), true, product(k, multiply(inverse(k), b), b), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.15 = { by axiom 12 (total_function1) }
% 52.32/7.15 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(ifeq(true, true, product(k, multiply(inverse(k), b), b), true), true, product(k, inverse(b), identity), true), true)
% 52.32/7.16 = { by axiom 10 (ifeq_axiom) }
% 52.32/7.16 ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(product(k, multiply(inverse(k), b), b), true, product(k, inverse(b), identity), true), true)
% 52.32/7.16 = { by axiom 10 (ifeq_axiom) R->L }
% 52.32/7.16 ifeq(true, true, ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(product(k, multiply(inverse(k), b), b), true, product(k, inverse(b), identity), true), true), true)
% 52.32/7.16 = { by axiom 2 (right_inverse) R->L }
% 52.32/7.16 ifeq(product(b, inverse(b), identity), true, ifeq(product(multiply(inverse(k), b), inverse(b), inverse(b)), true, ifeq(product(k, multiply(inverse(k), b), b), true, product(k, inverse(b), identity), true), true), true)
% 52.32/7.16 = { by axiom 15 (associativity1) }
% 52.32/7.16 true
% 52.32/7.16 % SZS output end Proof
% 52.32/7.16
% 52.32/7.16 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------