TSTP Solution File: GRP002-10 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP002-10 : TPTP v8.1.2. Released v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n023.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 : Thu Aug 31 01:16:29 EDT 2023
% Result : Unsatisfiable 53.99s 7.47s
% Output : Proof 54.53s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.02/0.12 % Problem : GRP002-10 : TPTP v8.1.2. Released v7.3.0.
% 0.02/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n023.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Mon Aug 28 22:35:10 EDT 2023
% 0.13/0.33 % CPUTime :
% 53.99/7.47 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 53.99/7.47
% 53.99/7.47 % SZS status Unsatisfiable
% 53.99/7.47
% 53.99/7.52 % SZS output start Proof
% 53.99/7.52 Axiom 1 (right_identity): product(X, identity, X) = true.
% 53.99/7.52 Axiom 2 (left_identity): product(identity, X, X) = true.
% 53.99/7.52 Axiom 3 (h_times_b_is_j): product(h, b, j) = true.
% 53.99/7.52 Axiom 4 (a_times_b_is_c): product(a, b, c) = true.
% 53.99/7.52 Axiom 5 (right_inverse): product(X, inverse(X), identity) = true.
% 53.99/7.52 Axiom 6 (c_times_inverse_a_is_d): product(c, inverse(a), d) = true.
% 53.99/7.52 Axiom 7 (d_times_inverse_b_is_h): product(d, inverse(b), h) = true.
% 53.99/7.52 Axiom 8 (j_times_inverse_h_is_k): product(j, inverse(h), k) = true.
% 53.99/7.52 Axiom 9 (left_inverse): product(inverse(X), X, identity) = true.
% 53.99/7.52 Axiom 10 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y.
% 53.99/7.52 Axiom 11 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 53.99/7.52 Axiom 12 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 53.99/7.52 Axiom 13 (x_cubed_is_identity_1): ifeq(product(X, X, Y), true, product(X, Y, identity), true) = true.
% 53.99/7.52 Axiom 14 (total_function2): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 53.99/7.52 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.
% 53.99/7.52 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.
% 53.99/7.52
% 53.99/7.52 Lemma 17: ifeq(product(X, Y, identity), true, ifeq(product(Z, X, W), true, product(W, Y, Z), true), true) = true.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq(product(X, Y, identity), true, ifeq(product(Z, X, W), true, product(W, Y, Z), true), true)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq(product(X, Y, identity), true, ifeq(true, true, ifeq(product(Z, X, W), true, product(W, Y, Z), true), true), true)
% 53.99/7.52 = { by axiom 1 (right_identity) R->L }
% 53.99/7.52 ifeq(product(X, Y, identity), true, ifeq(product(Z, identity, Z), true, ifeq(product(Z, X, W), true, product(W, Y, Z), true), true), true)
% 53.99/7.52 = { by axiom 16 (associativity2) }
% 53.99/7.52 true
% 53.99/7.52
% 53.99/7.52 Lemma 18: j = d.
% 53.99/7.52 Proof:
% 53.99/7.52 j
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(true, true, j, d)
% 53.99/7.52 = { by lemma 17 R->L }
% 53.99/7.52 ifeq2(ifeq(product(inverse(b), b, identity), true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true, j, d)
% 53.99/7.52 = { by axiom 9 (left_inverse) }
% 53.99/7.52 ifeq2(ifeq(true, true, ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true), true, j, d)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) }
% 53.99/7.52 ifeq2(ifeq(product(d, inverse(b), h), true, product(h, b, d), true), true, j, d)
% 53.99/7.52 = { by axiom 7 (d_times_inverse_b_is_h) }
% 53.99/7.52 ifeq2(ifeq(true, true, product(h, b, d), true), true, j, d)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) }
% 53.99/7.52 ifeq2(product(h, b, d), true, j, d)
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(product(h, b, d), true, ifeq2(true, true, j, d), d)
% 53.99/7.52 = { by axiom 3 (h_times_b_is_j) R->L }
% 53.99/7.52 ifeq2(product(h, b, d), true, ifeq2(product(h, b, j), true, j, d), d)
% 53.99/7.52 = { by axiom 14 (total_function2) }
% 53.99/7.52 d
% 53.99/7.52
% 53.99/7.52 Lemma 19: ifeq(product(X, Y, Z), true, ifeq(product(W, X, identity), true, product(W, Z, Y), true), true) = true.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq(product(X, Y, Z), true, ifeq(product(W, X, identity), true, product(W, Z, Y), true), true)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq(true, true, ifeq(product(X, Y, Z), true, ifeq(product(W, X, identity), true, product(W, Z, Y), true), true), true)
% 53.99/7.52 = { by axiom 2 (left_identity) R->L }
% 53.99/7.52 ifeq(product(identity, Y, Y), true, ifeq(product(X, Y, Z), true, ifeq(product(W, X, identity), true, product(W, Z, Y), true), true), true)
% 53.99/7.52 = { by axiom 15 (associativity1) }
% 53.99/7.52 true
% 53.99/7.52
% 53.99/7.52 Lemma 20: ifeq(product(inverse(X), Y, Z), true, product(X, Z, Y), true) = true.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq(product(inverse(X), Y, Z), true, product(X, Z, Y), true)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq(product(inverse(X), Y, Z), true, ifeq(true, true, product(X, Z, Y), true), true)
% 53.99/7.52 = { by axiom 5 (right_inverse) R->L }
% 53.99/7.52 ifeq(product(inverse(X), Y, Z), true, ifeq(product(X, inverse(X), identity), true, product(X, Z, Y), true), true)
% 53.99/7.52 = { by lemma 19 }
% 53.99/7.52 true
% 53.99/7.52
% 53.99/7.52 Lemma 21: ifeq2(product(X, identity, Y), true, Y, X) = X.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq2(product(X, identity, Y), true, Y, X)
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(true, true, ifeq2(product(X, identity, Y), true, Y, X), X)
% 53.99/7.52 = { by axiom 1 (right_identity) R->L }
% 53.99/7.52 ifeq2(product(X, identity, X), true, ifeq2(product(X, identity, Y), true, Y, X), X)
% 53.99/7.52 = { by axiom 14 (total_function2) }
% 53.99/7.52 X
% 53.99/7.52
% 53.99/7.52 Lemma 22: inverse(inverse(X)) = X.
% 53.99/7.52 Proof:
% 53.99/7.52 inverse(inverse(X))
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(true, true, inverse(inverse(X)), X)
% 53.99/7.52 = { by lemma 20 R->L }
% 53.99/7.52 ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), identity), true, product(X, identity, inverse(inverse(X))), true), true, inverse(inverse(X)), X)
% 53.99/7.52 = { by axiom 5 (right_inverse) }
% 53.99/7.52 ifeq2(ifeq(true, true, product(X, identity, inverse(inverse(X))), true), true, inverse(inverse(X)), X)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) }
% 53.99/7.52 ifeq2(product(X, identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 53.99/7.52 = { by lemma 21 }
% 53.99/7.52 X
% 53.99/7.52
% 53.99/7.52 Lemma 23: multiply(X, X) = inverse(X).
% 53.99/7.52 Proof:
% 53.99/7.52 multiply(X, X)
% 53.99/7.52 = { by lemma 22 R->L }
% 53.99/7.52 multiply(X, inverse(inverse(X)))
% 53.99/7.52 = { by lemma 22 R->L }
% 53.99/7.52 multiply(inverse(inverse(X)), inverse(inverse(X)))
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(true, true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by lemma 20 R->L }
% 53.99/7.52 ifeq2(ifeq(product(inverse(inverse(X)), multiply(inverse(inverse(X)), inverse(inverse(X))), identity), true, product(inverse(X), identity, multiply(inverse(inverse(X)), inverse(inverse(X)))), true), true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq2(ifeq(ifeq(true, true, product(inverse(inverse(X)), multiply(inverse(inverse(X)), inverse(inverse(X))), identity), true), true, product(inverse(X), identity, multiply(inverse(inverse(X)), inverse(inverse(X)))), true), true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by axiom 12 (total_function1) R->L }
% 53.99/7.52 ifeq2(ifeq(ifeq(product(inverse(inverse(X)), inverse(inverse(X)), multiply(inverse(inverse(X)), inverse(inverse(X)))), true, product(inverse(inverse(X)), multiply(inverse(inverse(X)), inverse(inverse(X))), identity), true), true, product(inverse(X), identity, multiply(inverse(inverse(X)), inverse(inverse(X)))), true), true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by axiom 13 (x_cubed_is_identity_1) }
% 53.99/7.52 ifeq2(ifeq(true, true, product(inverse(X), identity, multiply(inverse(inverse(X)), inverse(inverse(X)))), true), true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) }
% 53.99/7.52 ifeq2(product(inverse(X), identity, multiply(inverse(inverse(X)), inverse(inverse(X)))), true, multiply(inverse(inverse(X)), inverse(inverse(X))), inverse(X))
% 53.99/7.52 = { by lemma 21 }
% 53.99/7.52 inverse(X)
% 53.99/7.52
% 53.99/7.52 Lemma 24: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.52 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 53.99/7.52 = { by axiom 12 (total_function1) R->L }
% 53.99/7.52 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 53.99/7.52 = { by axiom 14 (total_function2) }
% 53.99/7.52 Z
% 53.99/7.52
% 53.99/7.52 Lemma 25: ifeq(product(X, Y, Z), true, ifeq(product(W, X, V), true, product(W, Z, multiply(V, Y)), true), true) = true.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq(product(X, Y, Z), true, ifeq(product(W, X, V), true, product(W, Z, multiply(V, Y)), true), true)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq(true, true, ifeq(product(X, Y, Z), true, ifeq(product(W, X, V), true, product(W, Z, multiply(V, Y)), true), true), true)
% 53.99/7.52 = { by axiom 12 (total_function1) R->L }
% 53.99/7.52 ifeq(product(V, Y, multiply(V, Y)), true, ifeq(product(X, Y, Z), true, ifeq(product(W, X, V), true, product(W, Z, multiply(V, Y)), true), true), true)
% 53.99/7.52 = { by axiom 15 (associativity1) }
% 53.99/7.52 true
% 53.99/7.52
% 53.99/7.52 Lemma 26: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 53.99/7.52 Proof:
% 53.99/7.52 multiply(multiply(X, Y), Z)
% 53.99/7.52 = { by lemma 24 R->L }
% 53.99/7.52 ifeq2(product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 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))
% 53.99/7.52 = { by axiom 12 (total_function1) R->L }
% 53.99/7.52 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))
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 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))
% 53.99/7.52 = { by axiom 12 (total_function1) R->L }
% 53.99/7.52 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))
% 53.99/7.52 = { by lemma 25 }
% 53.99/7.52 ifeq2(true, true, multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), Z))
% 53.99/7.52 = { by axiom 11 (ifeq_axiom) }
% 53.99/7.52 multiply(X, multiply(Y, Z))
% 53.99/7.52
% 53.99/7.52 Lemma 27: ifeq(product(X, Y, Z), true, product(inverse(X), Z, Y), true) = true.
% 53.99/7.52 Proof:
% 53.99/7.52 ifeq(product(X, Y, Z), true, product(inverse(X), Z, Y), true)
% 53.99/7.52 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.52 ifeq(product(X, Y, Z), true, ifeq(true, true, product(inverse(X), Z, Y), true), true)
% 53.99/7.52 = { by axiom 9 (left_inverse) R->L }
% 53.99/7.52 ifeq(product(X, Y, Z), true, ifeq(product(inverse(X), X, identity), true, product(inverse(X), Z, Y), true), true)
% 53.99/7.53 = { by lemma 19 }
% 53.99/7.53 true
% 53.99/7.53
% 53.99/7.53 Lemma 28: product(inverse(X), multiply(X, Y), Y) = true.
% 53.99/7.53 Proof:
% 53.99/7.53 product(inverse(X), multiply(X, Y), Y)
% 53.99/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.53 ifeq(true, true, product(inverse(X), multiply(X, Y), Y), true)
% 53.99/7.53 = { by axiom 12 (total_function1) R->L }
% 53.99/7.53 ifeq(product(X, Y, multiply(X, Y)), true, product(inverse(X), multiply(X, Y), Y), true)
% 53.99/7.53 = { by lemma 27 }
% 53.99/7.53 true
% 53.99/7.53
% 53.99/7.53 Lemma 29: multiply(inverse(X), multiply(X, Y)) = Y.
% 53.99/7.53 Proof:
% 53.99/7.53 multiply(inverse(X), multiply(X, Y))
% 53.99/7.53 = { by axiom 11 (ifeq_axiom) R->L }
% 53.99/7.53 ifeq2(true, true, multiply(inverse(X), multiply(X, Y)), Y)
% 53.99/7.53 = { by lemma 28 R->L }
% 53.99/7.53 ifeq2(product(inverse(X), multiply(X, Y), Y), true, multiply(inverse(X), multiply(X, Y)), Y)
% 53.99/7.53 = { by lemma 24 }
% 53.99/7.53 Y
% 53.99/7.53
% 53.99/7.53 Lemma 30: multiply(inverse(a), multiply(c, X)) = multiply(b, X).
% 53.99/7.53 Proof:
% 53.99/7.53 multiply(inverse(a), multiply(c, X))
% 53.99/7.53 = { by lemma 24 R->L }
% 53.99/7.53 multiply(inverse(a), ifeq2(product(a, multiply(b, X), multiply(c, X)), true, multiply(a, multiply(b, X)), multiply(c, X)))
% 53.99/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.53 multiply(inverse(a), ifeq2(ifeq(true, true, product(a, multiply(b, X), multiply(c, X)), true), true, multiply(a, multiply(b, X)), multiply(c, X)))
% 53.99/7.53 = { by axiom 12 (total_function1) R->L }
% 53.99/7.53 multiply(inverse(a), 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)))
% 53.99/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 53.99/7.53 multiply(inverse(a), 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)))
% 53.99/7.53 = { by axiom 12 (total_function1) R->L }
% 54.53/7.53 multiply(inverse(a), 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)))
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 multiply(inverse(a), 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)))
% 54.53/7.53 = { by axiom 4 (a_times_b_is_c) R->L }
% 54.53/7.53 multiply(inverse(a), 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)))
% 54.53/7.53 = { by axiom 15 (associativity1) }
% 54.53/7.53 multiply(inverse(a), ifeq2(true, true, multiply(a, multiply(b, X)), multiply(c, X)))
% 54.53/7.53 = { by axiom 11 (ifeq_axiom) }
% 54.53/7.53 multiply(inverse(a), multiply(a, multiply(b, X)))
% 54.53/7.53 = { by lemma 29 }
% 54.53/7.53 multiply(b, X)
% 54.53/7.53
% 54.53/7.53 Lemma 31: ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true) = true.
% 54.53/7.53 Proof:
% 54.53/7.53 ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 ifeq(true, true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true)
% 54.53/7.53 = { by axiom 5 (right_inverse) R->L }
% 54.53/7.53 ifeq(product(Y, inverse(Y), identity), true, ifeq(product(X, Y, Z), true, product(Z, inverse(Y), X), true), true)
% 54.53/7.53 = { by lemma 17 }
% 54.53/7.53 true
% 54.53/7.53
% 54.53/7.53 Lemma 32: multiply(multiply(X, Y), inverse(Y)) = X.
% 54.53/7.53 Proof:
% 54.53/7.53 multiply(multiply(X, Y), inverse(Y))
% 54.53/7.53 = { by axiom 11 (ifeq_axiom) R->L }
% 54.53/7.53 ifeq2(true, true, multiply(multiply(X, Y), inverse(Y)), X)
% 54.53/7.53 = { by lemma 31 R->L }
% 54.53/7.53 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)
% 54.53/7.53 = { by axiom 12 (total_function1) }
% 54.53/7.53 ifeq2(ifeq(true, true, product(multiply(X, Y), inverse(Y), X), true), true, multiply(multiply(X, Y), inverse(Y)), X)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.53 ifeq2(product(multiply(X, Y), inverse(Y), X), true, multiply(multiply(X, Y), inverse(Y)), X)
% 54.53/7.53 = { by lemma 24 }
% 54.53/7.53 X
% 54.53/7.53
% 54.53/7.53 Goal 1 (prove_k_times_inverse_b_is_e): product(k, inverse(b), identity) = true.
% 54.53/7.53 Proof:
% 54.53/7.53 product(k, inverse(b), identity)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 ifeq(true, true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 31 R->L }
% 54.53/7.53 ifeq(ifeq(product(inverse(b), multiply(b, c), c), true, product(c, inverse(multiply(b, c)), inverse(b)), true), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 28 }
% 54.53/7.53 ifeq(ifeq(true, true, product(c, inverse(multiply(b, c)), inverse(b)), true), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.53 ifeq(product(c, inverse(multiply(b, c)), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 30 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(c, c))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 23 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), inverse(c))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 32 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(multiply(inverse(c), d), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 11 (ifeq_axiom) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(ifeq2(true, true, multiply(inverse(c), d), inverse(a)), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 27 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(ifeq2(ifeq(product(c, inverse(a), d), true, product(inverse(c), d, inverse(a)), true), true, multiply(inverse(c), d), inverse(a)), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 6 (c_times_inverse_a_is_d) }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(ifeq2(ifeq(true, true, product(inverse(c), d, inverse(a)), true), true, multiply(inverse(c), d), inverse(a)), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(ifeq2(product(inverse(c), d, inverse(a)), true, multiply(inverse(c), d), inverse(a)), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 24 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(a), multiply(inverse(a), inverse(d)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 26 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(multiply(inverse(a), inverse(a)), inverse(d))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 23 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(inverse(a)), inverse(d))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 32 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(inverse(a)), multiply(multiply(inverse(d), multiply(d, inverse(a))), inverse(multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 29 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(inverse(inverse(a)), multiply(inverse(a), inverse(multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 29 }
% 54.53/7.53 ifeq(product(c, inverse(inverse(multiply(d, inverse(a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 23 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(multiply(d, inverse(a)), multiply(d, inverse(a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 26 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), multiply(d, inverse(a))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 24 R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(true, true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 12 (total_function1) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(product(inverse(a), inverse(a), multiply(inverse(a), inverse(a))), true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(true, true, ifeq(product(inverse(a), inverse(a), multiply(inverse(a), inverse(a))), true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 12 (total_function1) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(product(d, inverse(a), multiply(d, inverse(a))), true, ifeq(product(inverse(a), inverse(a), multiply(inverse(a), inverse(a))), true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(product(d, inverse(a), multiply(d, inverse(a))), true, ifeq(product(inverse(a), inverse(a), multiply(inverse(a), inverse(a))), true, ifeq(true, true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 6 (c_times_inverse_a_is_d) R->L }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(ifeq(product(d, inverse(a), multiply(d, inverse(a))), true, ifeq(product(inverse(a), inverse(a), multiply(inverse(a), inverse(a))), true, ifeq(product(c, inverse(a), d), true, product(c, multiply(inverse(a), inverse(a)), multiply(d, inverse(a))), true), true), true), true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 15 (associativity1) }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), ifeq2(true, true, multiply(c, multiply(inverse(a), inverse(a))), multiply(d, inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by axiom 11 (ifeq_axiom) }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), multiply(c, multiply(inverse(a), inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.53 = { by lemma 23 }
% 54.53/7.53 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), multiply(c, inverse(inverse(a)))))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 22 }
% 54.53/7.54 ifeq(product(c, inverse(multiply(d, multiply(inverse(a), multiply(c, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 30 }
% 54.53/7.54 ifeq(product(c, inverse(multiply(d, multiply(b, a))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 24 R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(product(k, c, multiply(d, multiply(b, a))), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(true, true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 15 (associativity1) R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(j, a, c), true, ifeq(product(b, a, multiply(b, a)), true, ifeq(product(h, b, j), true, product(h, multiply(b, a), c), true), true), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 3 (h_times_b_is_j) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(j, a, c), true, ifeq(product(b, a, multiply(b, a)), true, ifeq(true, true, product(h, multiply(b, a), c), true), true), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 18 }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(d, a, c), true, ifeq(product(b, a, multiply(b, a)), true, ifeq(true, true, product(h, multiply(b, a), c), true), true), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(d, a, c), true, ifeq(product(b, a, multiply(b, a)), true, product(h, multiply(b, a), c), true), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 12 (total_function1) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(d, a, c), true, ifeq(true, true, product(h, multiply(b, a), c), true), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(d, a, c), true, product(h, multiply(b, a), c), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 22 R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(product(d, inverse(inverse(a)), c), true, product(h, multiply(b, a), c), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(ifeq(true, true, product(d, inverse(inverse(a)), c), true), true, product(h, multiply(b, a), c), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 6 (c_times_inverse_a_is_d) R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(ifeq(product(c, inverse(a), d), true, product(d, inverse(inverse(a)), c), true), true, product(h, multiply(b, a), c), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 31 }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(ifeq(true, true, product(h, multiply(b, a), c), true), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, product(k, c, multiply(d, multiply(b, a))), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(true, true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 31 R->L }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(ifeq(product(j, inverse(h), k), true, product(k, inverse(inverse(h)), j), true), true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 8 (j_times_inverse_h_is_k) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(ifeq(true, true, product(k, inverse(inverse(h)), j), true), true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(product(k, inverse(inverse(h)), j), true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 22 }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(product(k, h, j), true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 18 }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(ifeq(product(h, multiply(b, a), c), true, ifeq(product(k, h, d), true, product(k, c, multiply(d, multiply(b, a))), true), true), true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by lemma 25 }
% 54.53/7.54 ifeq(product(c, inverse(ifeq2(true, true, multiply(k, c), multiply(d, multiply(b, a)))), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 11 (ifeq_axiom) }
% 54.53/7.54 ifeq(product(c, inverse(multiply(k, c)), inverse(b)), true, product(k, inverse(b), identity), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.54 ifeq(product(c, inverse(multiply(k, c)), inverse(b)), true, ifeq(true, true, product(k, inverse(b), identity), true), true)
% 54.53/7.54 = { by axiom 12 (total_function1) R->L }
% 54.53/7.54 ifeq(product(c, inverse(multiply(k, c)), inverse(b)), true, ifeq(product(k, c, multiply(k, c)), true, product(k, inverse(b), identity), true), true)
% 54.53/7.54 = { by axiom 10 (ifeq_axiom_001) R->L }
% 54.53/7.54 ifeq(true, true, ifeq(product(c, inverse(multiply(k, c)), inverse(b)), true, ifeq(product(k, c, multiply(k, c)), true, product(k, inverse(b), identity), true), true), true)
% 54.53/7.54 = { by axiom 5 (right_inverse) R->L }
% 54.53/7.54 ifeq(product(multiply(k, c), inverse(multiply(k, c)), identity), true, ifeq(product(c, inverse(multiply(k, c)), inverse(b)), true, ifeq(product(k, c, multiply(k, c)), true, product(k, inverse(b), identity), true), true), true)
% 54.53/7.54 = { by axiom 15 (associativity1) }
% 54.53/7.54 true
% 54.53/7.54 % SZS output end Proof
% 54.53/7.54
% 54.53/7.54 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------