TSTP Solution File: GRP617-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP617-1 : TPTP v8.1.2. Released v3.1.0.
% 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 : n026.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 01:19:08 EDT 2023
% Result : Unsatisfiable 4.55s 0.94s
% Output : Proof 4.70s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP617-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n026.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Mon Aug 28 20:14:35 EDT 2023
% 0.13/0.34 % CPUTime :
% 4.55/0.94 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.55/0.94
% 4.55/0.94 % SZS status Unsatisfiable
% 4.55/0.94
% 4.70/0.97 % SZS output start Proof
% 4.70/0.97 Take the following subset of the input axioms:
% 4.70/0.97 fof(associativity1, axiom, ![X, Y, Z, W, U, V]: (~product(X, Y, U) | (~product(Y, Z, V) | (~product(U, Z, W) | product(X, V, W))))).
% 4.70/0.97 fof(associativity2, axiom, ![X2, Y2, Z2, W2, U2, V2]: (~product(X2, Y2, U2) | (~product(Y2, Z2, V2) | (~product(X2, V2, W2) | product(U2, Z2, W2))))).
% 4.70/0.97 fof(closure_of_inverse1, axiom, ![X2]: (~subgroup1_member(X2) | subgroup1_member(inverse(X2)))).
% 4.70/0.97 fof(closure_of_inverse2, axiom, ![X2]: (~subgroup2_member(X2) | subgroup2_member(inverse(X2)))).
% 4.70/0.97 fof(closure_of_product1, axiom, ![B, C, A2]: (~subgroup1_member(A2) | (~subgroup1_member(B) | (~product(A2, B, C) | subgroup1_member(C))))).
% 4.70/0.97 fof(closure_of_product2, axiom, ![A2_2, B2, C2]: (~subgroup2_member(A2_2) | (~subgroup2_member(B2) | (~product(A2_2, B2, C2) | subgroup2_member(C2))))).
% 4.70/0.97 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 4.70/0.97 fof(left_inverse, axiom, ![X2]: product(inverse(X2), X2, identity)).
% 4.70/0.97 fof(normality1, hypothesis, ![X2, A2_2]: (~subgroup1_member(X2) | subgroup1_member(multiply(A2_2, multiply(X2, inverse(A2_2)))))).
% 4.70/0.97 fof(normality2, hypothesis, ![X2, A2_2]: (~subgroup2_member(X2) | subgroup2_member(multiply(A2_2, multiply(X2, inverse(A2_2)))))).
% 4.70/0.97 fof(prove_vu_equals_uv, negated_conjecture, multiply(v, u)!=multiply(u, v)).
% 4.70/0.97 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 4.70/0.97 fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 4.70/0.97 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 4.70/0.97 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 4.70/0.97 fof(trivial_intersection, hypothesis, ![X2]: (~subgroup1_member(X2) | (~subgroup2_member(X2) | X2=identity))).
% 4.70/0.97 fof(u_in_G2, hypothesis, subgroup2_member(u)).
% 4.70/0.97 fof(v_in_G1, hypothesis, subgroup1_member(v)).
% 4.70/0.97
% 4.70/0.97 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.70/0.97 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.70/0.97 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.70/0.97 fresh(y, y, x1...xn) = u
% 4.70/0.97 C => fresh(s, t, x1...xn) = v
% 4.70/0.97 where fresh is a fresh function symbol and x1..xn are the free
% 4.70/0.97 variables of u and v.
% 4.70/0.97 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.70/0.97 input problem has no model of domain size 1).
% 4.70/0.97
% 4.70/0.97 The encoding turns the above axioms into the following unit equations and goals:
% 4.70/0.97
% 4.70/0.97 Axiom 1 (v_in_G1): subgroup1_member(v) = true.
% 4.70/0.97 Axiom 2 (u_in_G2): subgroup2_member(u) = true.
% 4.70/0.97 Axiom 3 (right_identity): product(X, identity, X) = true.
% 4.70/0.97 Axiom 4 (left_identity): product(identity, X, X) = true.
% 4.70/0.97 Axiom 5 (trivial_intersection): fresh(X, X, Y) = Y.
% 4.70/0.97 Axiom 6 (closure_of_product1): fresh16(X, X, Y) = true.
% 4.70/0.97 Axiom 7 (closure_of_product2): fresh14(X, X, Y) = true.
% 4.70/0.97 Axiom 8 (closure_of_inverse1): fresh10(X, X, Y) = true.
% 4.70/0.97 Axiom 9 (closure_of_inverse2): fresh9(X, X, Y) = true.
% 4.70/0.97 Axiom 10 (trivial_intersection): fresh4(X, X, Y) = identity.
% 4.70/0.97 Axiom 11 (right_inverse): product(X, inverse(X), identity) = true.
% 4.70/0.97 Axiom 12 (left_inverse): product(inverse(X), X, identity) = true.
% 4.70/0.97 Axiom 13 (closure_of_inverse1): fresh10(subgroup1_member(X), true, X) = subgroup1_member(inverse(X)).
% 4.70/0.97 Axiom 14 (closure_of_inverse2): fresh9(subgroup2_member(X), true, X) = subgroup2_member(inverse(X)).
% 4.70/0.97 Axiom 15 (normality1): fresh6(X, X, Y, Z) = true.
% 4.70/0.97 Axiom 16 (normality2): fresh5(X, X, Y, Z) = true.
% 4.70/0.97 Axiom 17 (trivial_intersection): fresh(subgroup2_member(X), true, X) = fresh4(subgroup1_member(X), true, X).
% 4.70/0.97 Axiom 18 (total_function2): fresh2(X, X, Y, Z) = Z.
% 4.70/0.97 Axiom 19 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 4.70/0.97 Axiom 20 (associativity1): fresh20(X, X, Y, Z, W) = true.
% 4.70/0.97 Axiom 21 (associativity2): fresh18(X, X, Y, Z, W) = true.
% 4.70/0.97 Axiom 22 (closure_of_product1): fresh8(X, X, Y, Z, W) = subgroup1_member(W).
% 4.70/0.97 Axiom 23 (closure_of_product2): fresh7(X, X, Y, Z, W) = subgroup2_member(W).
% 4.70/0.97 Axiom 24 (normality1): fresh6(subgroup1_member(X), true, X, Y) = subgroup1_member(multiply(Y, multiply(X, inverse(Y)))).
% 4.70/0.97 Axiom 25 (normality2): fresh5(subgroup2_member(X), true, X, Y) = subgroup2_member(multiply(Y, multiply(X, inverse(Y)))).
% 4.70/0.97 Axiom 26 (closure_of_product1): fresh15(X, X, Y, Z, W) = fresh16(product(Y, Z, W), true, W).
% 4.70/0.97 Axiom 27 (closure_of_product2): fresh13(X, X, Y, Z, W) = fresh14(product(Y, Z, W), true, W).
% 4.70/0.97 Axiom 28 (closure_of_product1): fresh15(subgroup1_member(X), true, Y, X, Z) = fresh8(subgroup1_member(Y), true, Y, X, Z).
% 4.70/0.97 Axiom 29 (closure_of_product2): fresh13(subgroup2_member(X), true, Y, X, Z) = fresh7(subgroup2_member(Y), true, Y, X, Z).
% 4.70/0.97 Axiom 30 (total_function2): fresh3(X, X, Y, Z, W, V) = W.
% 4.70/0.97 Axiom 31 (associativity1): fresh12(X, X, Y, Z, W, V, U) = product(Y, V, U).
% 4.70/0.97 Axiom 32 (associativity2): fresh11(X, X, Y, Z, W, V, U) = product(W, V, U).
% 4.70/0.97 Axiom 33 (associativity1): fresh19(X, X, Y, Z, W, V, U, T) = fresh20(product(Y, Z, W), true, Y, U, T).
% 4.70/0.97 Axiom 34 (associativity2): fresh17(X, X, Y, Z, W, V, U, T) = fresh18(product(Y, Z, W), true, W, V, T).
% 4.70/0.97 Axiom 35 (total_function2): fresh3(product(X, Y, Z), true, X, Y, W, Z) = fresh2(product(X, Y, W), true, W, Z).
% 4.70/0.97 Axiom 36 (associativity1): fresh19(product(X, Y, Z), true, W, V, X, Y, U, Z) = fresh12(product(V, Y, U), true, W, V, X, U, Z).
% 4.70/0.97 Axiom 37 (associativity2): fresh17(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh11(product(W, Z, U), true, W, X, V, Y, U).
% 4.70/0.97
% 4.70/0.97 Lemma 38: multiply(identity, X) = X.
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(identity, X)
% 4.70/0.97 = { by axiom 30 (total_function2) R->L }
% 4.70/0.97 fresh3(true, true, identity, X, multiply(identity, X), X)
% 4.70/0.97 = { by axiom 4 (left_identity) R->L }
% 4.70/0.97 fresh3(product(identity, X, X), true, identity, X, multiply(identity, X), X)
% 4.70/0.97 = { by axiom 35 (total_function2) }
% 4.70/0.97 fresh2(product(identity, X, multiply(identity, X)), true, multiply(identity, X), X)
% 4.70/0.97 = { by axiom 19 (total_function1) }
% 4.70/0.97 fresh2(true, true, multiply(identity, X), X)
% 4.70/0.97 = { by axiom 18 (total_function2) }
% 4.70/0.97 X
% 4.70/0.97
% 4.70/0.97 Lemma 39: fresh17(X, X, Y, Z, multiply(Y, Z), W, V, U) = true.
% 4.70/0.97 Proof:
% 4.70/0.97 fresh17(X, X, Y, Z, multiply(Y, Z), W, V, U)
% 4.70/0.97 = { by axiom 34 (associativity2) }
% 4.70/0.97 fresh18(product(Y, Z, multiply(Y, Z)), true, multiply(Y, Z), W, U)
% 4.70/0.97 = { by axiom 19 (total_function1) }
% 4.70/0.97 fresh18(true, true, multiply(Y, Z), W, U)
% 4.70/0.97 = { by axiom 21 (associativity2) }
% 4.70/0.97 true
% 4.70/0.97
% 4.70/0.97 Lemma 40: fresh2(product(X, Y, Z), true, Z, multiply(X, Y)) = Z.
% 4.70/0.97 Proof:
% 4.70/0.97 fresh2(product(X, Y, Z), true, Z, multiply(X, Y))
% 4.70/0.97 = { by axiom 35 (total_function2) R->L }
% 4.70/0.97 fresh3(product(X, Y, multiply(X, Y)), true, X, Y, Z, multiply(X, Y))
% 4.70/0.97 = { by axiom 19 (total_function1) }
% 4.70/0.97 fresh3(true, true, X, Y, Z, multiply(X, Y))
% 4.70/0.97 = { by axiom 30 (total_function2) }
% 4.70/0.97 Z
% 4.70/0.97
% 4.70/0.97 Lemma 41: multiply(multiply(inverse(multiply(X, Y)), X), Y) = identity.
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(multiply(inverse(multiply(X, Y)), X), Y)
% 4.70/0.97 = { by axiom 18 (total_function2) R->L }
% 4.70/0.97 fresh2(true, true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by lemma 39 R->L }
% 4.70/0.97 fresh2(fresh17(true, true, inverse(multiply(X, Y)), X, multiply(inverse(multiply(X, Y)), X), Y, multiply(X, Y), identity), true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by axiom 19 (total_function1) R->L }
% 4.70/0.97 fresh2(fresh17(product(X, Y, multiply(X, Y)), true, inverse(multiply(X, Y)), X, multiply(inverse(multiply(X, Y)), X), Y, multiply(X, Y), identity), true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by axiom 37 (associativity2) }
% 4.70/0.97 fresh2(fresh11(product(inverse(multiply(X, Y)), multiply(X, Y), identity), true, inverse(multiply(X, Y)), X, multiply(inverse(multiply(X, Y)), X), Y, identity), true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by axiom 12 (left_inverse) }
% 4.70/0.97 fresh2(fresh11(true, true, inverse(multiply(X, Y)), X, multiply(inverse(multiply(X, Y)), X), Y, identity), true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by axiom 32 (associativity2) }
% 4.70/0.97 fresh2(product(multiply(inverse(multiply(X, Y)), X), Y, identity), true, identity, multiply(multiply(inverse(multiply(X, Y)), X), Y))
% 4.70/0.97 = { by lemma 40 }
% 4.70/0.97 identity
% 4.70/0.97
% 4.70/0.97 Lemma 42: fresh17(product(X, Y, identity), true, Z, X, W, Y, identity, Z) = product(W, Y, Z).
% 4.70/0.97 Proof:
% 4.70/0.97 fresh17(product(X, Y, identity), true, Z, X, W, Y, identity, Z)
% 4.70/0.97 = { by axiom 37 (associativity2) }
% 4.70/0.97 fresh11(product(Z, identity, Z), true, Z, X, W, Y, Z)
% 4.70/0.97 = { by axiom 3 (right_identity) }
% 4.70/0.97 fresh11(true, true, Z, X, W, Y, Z)
% 4.70/0.97 = { by axiom 32 (associativity2) }
% 4.70/0.97 product(W, Y, Z)
% 4.70/0.97
% 4.70/0.97 Lemma 43: multiply(multiply(X, Y), inverse(Y)) = X.
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(multiply(X, Y), inverse(Y))
% 4.70/0.97 = { by axiom 18 (total_function2) R->L }
% 4.70/0.97 fresh2(true, true, X, multiply(multiply(X, Y), inverse(Y)))
% 4.70/0.97 = { by lemma 39 R->L }
% 4.70/0.97 fresh2(fresh17(true, true, X, Y, multiply(X, Y), inverse(Y), identity, X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 4.70/0.97 = { by axiom 11 (right_inverse) R->L }
% 4.70/0.97 fresh2(fresh17(product(Y, inverse(Y), identity), true, X, Y, multiply(X, Y), inverse(Y), identity, X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 4.70/0.97 = { by lemma 42 }
% 4.70/0.97 fresh2(product(multiply(X, Y), inverse(Y), X), true, X, multiply(multiply(X, Y), inverse(Y)))
% 4.70/0.97 = { by lemma 40 }
% 4.70/0.97 X
% 4.70/0.97
% 4.70/0.97 Lemma 44: multiply(inverse(X), inverse(Y)) = inverse(multiply(Y, X)).
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(inverse(X), inverse(Y))
% 4.70/0.97 = { by lemma 38 R->L }
% 4.70/0.97 multiply(multiply(identity, inverse(X)), inverse(Y))
% 4.70/0.97 = { by lemma 41 R->L }
% 4.70/0.97 multiply(multiply(multiply(multiply(inverse(multiply(Y, X)), Y), X), inverse(X)), inverse(Y))
% 4.70/0.97 = { by lemma 43 }
% 4.70/0.97 multiply(multiply(inverse(multiply(Y, X)), Y), inverse(Y))
% 4.70/0.97 = { by lemma 43 }
% 4.70/0.97 inverse(multiply(Y, X))
% 4.70/0.97
% 4.70/0.97 Lemma 45: multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(multiply(X, Y), Z)
% 4.70/0.97 = { by lemma 40 R->L }
% 4.70/0.97 fresh2(product(X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 31 (associativity1) R->L }
% 4.70/0.97 fresh2(fresh12(true, true, X, Y, multiply(X, Y), multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 19 (total_function1) R->L }
% 4.70/0.97 fresh2(fresh12(product(Y, Z, multiply(Y, Z)), true, X, Y, multiply(X, Y), multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 36 (associativity1) R->L }
% 4.70/0.97 fresh2(fresh19(product(multiply(X, Y), Z, multiply(multiply(X, Y), Z)), true, X, Y, multiply(X, Y), Z, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 19 (total_function1) }
% 4.70/0.97 fresh2(fresh19(true, true, X, Y, multiply(X, Y), Z, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 33 (associativity1) }
% 4.70/0.97 fresh2(fresh20(product(X, Y, multiply(X, Y)), true, X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 19 (total_function1) }
% 4.70/0.97 fresh2(fresh20(true, true, X, multiply(Y, Z), multiply(multiply(X, Y), Z)), true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 20 (associativity1) }
% 4.70/0.97 fresh2(true, true, multiply(multiply(X, Y), Z), multiply(X, multiply(Y, Z)))
% 4.70/0.97 = { by axiom 18 (total_function2) }
% 4.70/0.97 multiply(X, multiply(Y, Z))
% 4.70/0.97
% 4.70/0.97 Lemma 46: multiply(multiply(X, inverse(Y)), Y) = X.
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(multiply(X, inverse(Y)), Y)
% 4.70/0.97 = { by axiom 18 (total_function2) R->L }
% 4.70/0.97 fresh2(true, true, X, multiply(multiply(X, inverse(Y)), Y))
% 4.70/0.97 = { by lemma 39 R->L }
% 4.70/0.97 fresh2(fresh17(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 4.70/0.97 = { by axiom 12 (left_inverse) R->L }
% 4.70/0.97 fresh2(fresh17(product(inverse(Y), Y, identity), true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 4.70/0.97 = { by lemma 42 }
% 4.70/0.97 fresh2(product(multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 4.70/0.97 = { by lemma 40 }
% 4.70/0.97 X
% 4.70/0.97
% 4.70/0.97 Goal 1 (prove_vu_equals_uv): multiply(v, u) = multiply(u, v).
% 4.70/0.97 Proof:
% 4.70/0.97 multiply(v, u)
% 4.70/0.97 = { by lemma 38 R->L }
% 4.70/0.97 multiply(identity, multiply(v, u))
% 4.70/0.97 = { by axiom 10 (trivial_intersection) R->L }
% 4.70/0.97 multiply(fresh4(true, true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 8 (closure_of_inverse1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(true, true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 6 (closure_of_product1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh16(true, true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 19 (total_function1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh16(product(v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 26 (closure_of_product1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(true, true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 15 (normality1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(fresh6(true, true, inverse(v), u), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 8 (closure_of_inverse1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(fresh6(fresh10(true, true, v), true, inverse(v), u), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 1 (v_in_G1) R->L }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(fresh6(fresh10(subgroup1_member(v), true, v), true, inverse(v), u), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 13 (closure_of_inverse1) }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(fresh6(subgroup1_member(inverse(v)), true, inverse(v), u), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 24 (normality1) }
% 4.70/0.97 multiply(fresh4(fresh10(fresh15(subgroup1_member(multiply(u, multiply(inverse(v), inverse(u)))), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 28 (closure_of_product1) }
% 4.70/0.97 multiply(fresh4(fresh10(fresh8(subgroup1_member(v), true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 1 (v_in_G1) }
% 4.70/0.97 multiply(fresh4(fresh10(fresh8(true, true, v, multiply(u, multiply(inverse(v), inverse(u))), multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.97 = { by axiom 22 (closure_of_product1) }
% 4.70/0.97 multiply(fresh4(fresh10(subgroup1_member(multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(v, multiply(u, multiply(inverse(v), inverse(u))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 13 (closure_of_inverse1) }
% 4.70/0.98 multiply(fresh4(subgroup1_member(inverse(multiply(v, multiply(u, multiply(inverse(v), inverse(u)))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 44 }
% 4.70/0.98 multiply(fresh4(subgroup1_member(inverse(multiply(v, multiply(u, inverse(multiply(u, v)))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 45 R->L }
% 4.70/0.98 multiply(fresh4(subgroup1_member(inverse(multiply(multiply(v, u), inverse(multiply(u, v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 43 R->L }
% 4.70/0.98 multiply(fresh4(subgroup1_member(multiply(multiply(inverse(multiply(multiply(v, u), inverse(multiply(u, v)))), multiply(v, u)), inverse(multiply(v, u)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 46 R->L }
% 4.70/0.98 multiply(fresh4(subgroup1_member(multiply(multiply(multiply(multiply(inverse(multiply(multiply(v, u), inverse(multiply(u, v)))), multiply(v, u)), inverse(multiply(u, v))), multiply(u, v)), inverse(multiply(v, u)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 41 }
% 4.70/0.98 multiply(fresh4(subgroup1_member(multiply(multiply(identity, multiply(u, v)), inverse(multiply(v, u)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 38 }
% 4.70/0.98 multiply(fresh4(subgroup1_member(multiply(multiply(u, v), inverse(multiply(v, u)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 17 (trivial_intersection) R->L }
% 4.70/0.98 multiply(fresh(subgroup2_member(multiply(multiply(u, v), inverse(multiply(v, u)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 44 R->L }
% 4.70/0.98 multiply(fresh(subgroup2_member(multiply(multiply(u, v), multiply(inverse(u), inverse(v)))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by lemma 45 }
% 4.70/0.98 multiply(fresh(subgroup2_member(multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 23 (closure_of_product2) R->L }
% 4.70/0.98 multiply(fresh(fresh7(true, true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 2 (u_in_G2) R->L }
% 4.70/0.98 multiply(fresh(fresh7(subgroup2_member(u), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 29 (closure_of_product2) R->L }
% 4.70/0.98 multiply(fresh(fresh13(subgroup2_member(multiply(v, multiply(inverse(u), inverse(v)))), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 25 (normality2) R->L }
% 4.70/0.98 multiply(fresh(fresh13(fresh5(subgroup2_member(inverse(u)), true, inverse(u), v), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 14 (closure_of_inverse2) R->L }
% 4.70/0.98 multiply(fresh(fresh13(fresh5(fresh9(subgroup2_member(u), true, u), true, inverse(u), v), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 2 (u_in_G2) }
% 4.70/0.98 multiply(fresh(fresh13(fresh5(fresh9(true, true, u), true, inverse(u), v), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 9 (closure_of_inverse2) }
% 4.70/0.98 multiply(fresh(fresh13(fresh5(true, true, inverse(u), v), true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 16 (normality2) }
% 4.70/0.98 multiply(fresh(fresh13(true, true, u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 27 (closure_of_product2) }
% 4.70/0.98 multiply(fresh(fresh14(product(u, multiply(v, multiply(inverse(u), inverse(v))), multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 19 (total_function1) }
% 4.70/0.98 multiply(fresh(fresh14(true, true, multiply(u, multiply(v, multiply(inverse(u), inverse(v))))), true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 7 (closure_of_product2) }
% 4.70/0.98 multiply(fresh(true, true, multiply(multiply(u, v), inverse(multiply(v, u)))), multiply(v, u))
% 4.70/0.98 = { by axiom 5 (trivial_intersection) }
% 4.70/0.98 multiply(multiply(multiply(u, v), inverse(multiply(v, u))), multiply(v, u))
% 4.70/0.98 = { by lemma 46 }
% 4.70/0.98 multiply(u, v)
% 4.70/0.98 % SZS output end Proof
% 4.70/0.98
% 4.70/0.98 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------