TSTP Solution File: GRP104-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP104-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 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 : n014.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 01:17:00 EDT 2023
% Result : Unsatisfiable 0.19s 0.44s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP104-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 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 : n014.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 : Tue Aug 29 00:44:15 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.44 Command-line arguments: --no-flatten-goal
% 0.19/0.44
% 0.19/0.44 % SZS status Unsatisfiable
% 0.19/0.44
% 0.19/0.48 % SZS output start Proof
% 0.19/0.48 Take the following subset of the input axioms:
% 0.19/0.49 fof(multiply, axiom, ![X, Y]: multiply(X, Y)=inverse(double_divide(Y, X))).
% 0.19/0.49 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=multiply(inverse(b1), b1) | (multiply(multiply(inverse(b2), b2), a2)!=a2 | (multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)) | multiply(a4, b4)!=multiply(b4, a4)))).
% 0.19/0.49 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(X2, inverse(double_divide(inverse(double_divide(double_divide(X2, Y2), inverse(Z))), Y2)))=Z).
% 0.19/0.49
% 0.19/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.49 fresh(y, y, x1...xn) = u
% 0.19/0.49 C => fresh(s, t, x1...xn) = v
% 0.19/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.49 variables of u and v.
% 0.19/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.49 input problem has no model of domain size 1).
% 0.19/0.49
% 0.19/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.49
% 0.19/0.49 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 0.19/0.49 Axiom 2 (single_axiom): double_divide(X, inverse(double_divide(inverse(double_divide(double_divide(X, Y), inverse(Z))), Y))) = Z.
% 0.19/0.49
% 0.19/0.49 Lemma 3: double_divide(X, multiply(Y, multiply(inverse(Z), double_divide(X, Y)))) = Z.
% 0.19/0.49 Proof:
% 0.19/0.49 double_divide(X, multiply(Y, multiply(inverse(Z), double_divide(X, Y))))
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 double_divide(X, multiply(Y, inverse(double_divide(double_divide(X, Y), inverse(Z)))))
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 double_divide(X, inverse(double_divide(inverse(double_divide(double_divide(X, Y), inverse(Z))), Y)))
% 0.19/0.49 = { by axiom 2 (single_axiom) }
% 0.19/0.49 Z
% 0.19/0.49
% 0.19/0.49 Lemma 4: multiply(multiply(X, multiply(inverse(Y), double_divide(Z, X))), Z) = inverse(Y).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(X, multiply(inverse(Y), double_divide(Z, X))), Z)
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 inverse(double_divide(Z, multiply(X, multiply(inverse(Y), double_divide(Z, X)))))
% 0.19/0.49 = { by lemma 3 }
% 0.19/0.49 inverse(Y)
% 0.19/0.49
% 0.19/0.49 Lemma 5: multiply(multiply(X, multiply(multiply(Y, Z), double_divide(W, X))), W) = multiply(Y, Z).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(X, multiply(multiply(Y, Z), double_divide(W, X))), W)
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 multiply(multiply(X, multiply(inverse(double_divide(Z, Y)), double_divide(W, X))), W)
% 0.19/0.49 = { by lemma 4 }
% 0.19/0.49 inverse(double_divide(Z, Y))
% 0.19/0.49 = { by axiom 1 (multiply) R->L }
% 0.19/0.49 multiply(Y, Z)
% 0.19/0.49
% 0.19/0.49 Lemma 6: multiply(X, multiply(multiply(Y, Z), double_divide(double_divide(W, V), X))) = multiply(multiply(V, multiply(Y, Z)), W).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(X, multiply(multiply(Y, Z), double_divide(double_divide(W, V), X)))
% 0.19/0.49 = { by lemma 5 R->L }
% 0.19/0.49 multiply(multiply(V, multiply(multiply(X, multiply(multiply(Y, Z), double_divide(double_divide(W, V), X))), double_divide(W, V))), W)
% 0.19/0.49 = { by lemma 5 }
% 0.19/0.49 multiply(multiply(V, multiply(Y, Z)), W)
% 0.19/0.49
% 0.19/0.49 Lemma 7: multiply(multiply(multiply(X, multiply(Y, Z)), W), double_divide(W, X)) = multiply(Y, Z).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(multiply(X, multiply(Y, Z)), W), double_divide(W, X))
% 0.19/0.49 = { by lemma 6 R->L }
% 0.19/0.49 multiply(multiply(V, multiply(multiply(Y, Z), double_divide(double_divide(W, X), V))), double_divide(W, X))
% 0.19/0.49 = { by lemma 5 }
% 0.19/0.49 multiply(Y, Z)
% 0.19/0.49
% 0.19/0.49 Lemma 8: double_divide(X, multiply(Y, multiply(multiply(Z, W), double_divide(X, Y)))) = double_divide(W, Z).
% 0.19/0.49 Proof:
% 0.19/0.49 double_divide(X, multiply(Y, multiply(multiply(Z, W), double_divide(X, Y))))
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 double_divide(X, multiply(Y, multiply(inverse(double_divide(W, Z)), double_divide(X, Y))))
% 0.19/0.49 = { by lemma 3 }
% 0.19/0.49 double_divide(W, Z)
% 0.19/0.49
% 0.19/0.49 Lemma 9: multiply(X, multiply(inverse(Y), double_divide(double_divide(Z, W), X))) = multiply(multiply(W, inverse(Y)), Z).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(X, multiply(inverse(Y), double_divide(double_divide(Z, W), X)))
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 inverse(double_divide(multiply(inverse(Y), double_divide(double_divide(Z, W), X)), X))
% 0.19/0.49 = { by lemma 8 R->L }
% 0.19/0.49 inverse(double_divide(Z, multiply(W, multiply(multiply(X, multiply(inverse(Y), double_divide(double_divide(Z, W), X))), double_divide(Z, W)))))
% 0.19/0.49 = { by lemma 4 }
% 0.19/0.49 inverse(double_divide(Z, multiply(W, inverse(Y))))
% 0.19/0.49 = { by axiom 1 (multiply) R->L }
% 0.19/0.49 multiply(multiply(W, inverse(Y)), Z)
% 0.19/0.49
% 0.19/0.49 Lemma 10: multiply(multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)), X) = inverse(Y).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)), X)
% 0.19/0.49 = { by lemma 3 R->L }
% 0.19/0.49 multiply(multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)), double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)))))
% 0.19/0.49 = { by lemma 7 R->L }
% 0.19/0.49 multiply(multiply(multiply(multiply(Z, multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z))), inverse(Y)), double_divide(inverse(Y), Z)), double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)))))
% 0.19/0.49 = { by lemma 9 R->L }
% 0.19/0.49 multiply(multiply(W, multiply(inverse(Y), double_divide(double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)))), W))), double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(X), double_divide(double_divide(inverse(Y), Z), Z)))))
% 0.19/0.49 = { by lemma 4 }
% 0.19/0.49 inverse(Y)
% 0.19/0.49
% 0.19/0.49 Lemma 11: double_divide(double_divide(X, Y), multiply(multiply(Y, multiply(Z, W)), X)) = double_divide(W, Z).
% 0.19/0.49 Proof:
% 0.19/0.49 double_divide(double_divide(X, Y), multiply(multiply(Y, multiply(Z, W)), X))
% 0.19/0.49 = { by lemma 6 R->L }
% 0.19/0.49 double_divide(double_divide(X, Y), multiply(V, multiply(multiply(Z, W), double_divide(double_divide(X, Y), V))))
% 0.19/0.49 = { by lemma 8 }
% 0.19/0.49 double_divide(W, Z)
% 0.19/0.49
% 0.19/0.49 Lemma 12: multiply(multiply(X, Y), double_divide(double_divide(inverse(Z), W), W)) = multiply(multiply(X, inverse(Z)), Y).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(X, Y), double_divide(double_divide(inverse(Z), W), W))
% 0.19/0.49 = { by axiom 1 (multiply) }
% 0.19/0.49 multiply(inverse(double_divide(Y, X)), double_divide(double_divide(inverse(Z), W), W))
% 0.19/0.49 = { by lemma 5 R->L }
% 0.19/0.49 multiply(multiply(X, multiply(multiply(inverse(double_divide(Y, X)), double_divide(double_divide(inverse(Z), W), W)), double_divide(Y, X))), Y)
% 0.19/0.49 = { by lemma 10 }
% 0.19/0.49 multiply(multiply(X, inverse(Z)), Y)
% 0.19/0.49
% 0.19/0.49 Lemma 13: multiply(multiply(X, multiply(Y, Z)), inverse(W)) = multiply(X, multiply(multiply(Y, inverse(W)), Z)).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(multiply(X, multiply(Y, Z)), inverse(W))
% 0.19/0.49 = { by lemma 6 R->L }
% 0.19/0.49 multiply(X, multiply(multiply(Y, Z), double_divide(double_divide(inverse(W), X), X)))
% 0.19/0.49 = { by lemma 12 }
% 0.19/0.49 multiply(X, multiply(multiply(Y, inverse(W)), Z))
% 0.19/0.49
% 0.19/0.49 Lemma 14: multiply(X, multiply(multiply(Y, inverse(W)), Z)) = multiply(X, multiply(multiply(Y, Z), inverse(W))).
% 0.19/0.49 Proof:
% 0.19/0.49 multiply(X, multiply(multiply(Y, inverse(W)), Z))
% 0.19/0.49 = { by lemma 13 R->L }
% 0.19/0.49 multiply(multiply(X, multiply(Y, Z)), inverse(W))
% 0.19/0.49 = { by lemma 5 R->L }
% 0.19/0.49 multiply(multiply(X, multiply(multiply(V, multiply(multiply(Y, Z), double_divide(U, V))), U)), inverse(W))
% 0.19/0.49 = { by lemma 13 }
% 0.19/0.49 multiply(X, multiply(multiply(multiply(V, multiply(multiply(Y, Z), double_divide(U, V))), inverse(W)), U))
% 0.19/0.49 = { by lemma 13 }
% 0.19/0.49 multiply(X, multiply(multiply(V, multiply(multiply(multiply(Y, Z), inverse(W)), double_divide(U, V))), U))
% 0.19/0.49 = { by lemma 5 }
% 0.19/0.49 multiply(X, multiply(multiply(Y, Z), inverse(W)))
% 0.19/0.49
% 0.19/0.49 Lemma 15: double_divide(inverse(X), multiply(Y, Z)) = double_divide(Z, multiply(Y, inverse(X))).
% 0.19/0.49 Proof:
% 0.19/0.49 double_divide(inverse(X), multiply(Y, Z))
% 0.19/0.49 = { by lemma 11 R->L }
% 0.19/0.49 double_divide(double_divide(W, V), multiply(multiply(V, multiply(multiply(Y, Z), inverse(X))), W))
% 0.19/0.49 = { by lemma 14 R->L }
% 0.19/0.49 double_divide(double_divide(W, V), multiply(multiply(V, multiply(multiply(Y, inverse(X)), Z)), W))
% 0.19/0.49 = { by lemma 11 }
% 0.19/0.49 double_divide(Z, multiply(Y, inverse(X)))
% 0.19/0.49
% 0.19/0.49 Lemma 16: double_divide(inverse(X), multiply(inverse(inverse(X)), inverse(Y))) = Y.
% 0.19/0.49 Proof:
% 0.19/0.49 double_divide(inverse(X), multiply(inverse(inverse(X)), inverse(Y)))
% 0.19/0.49 = { by lemma 10 R->L }
% 0.19/0.50 double_divide(inverse(X), multiply(inverse(inverse(X)), multiply(multiply(inverse(double_divide(inverse(X), inverse(inverse(X)))), double_divide(double_divide(inverse(Y), Z), Z)), double_divide(inverse(X), inverse(inverse(X))))))
% 0.19/0.50 = { by lemma 8 }
% 0.19/0.50 double_divide(double_divide(double_divide(inverse(Y), Z), Z), inverse(double_divide(inverse(X), inverse(inverse(X)))))
% 0.19/0.50 = { by axiom 1 (multiply) R->L }
% 0.19/0.50 double_divide(double_divide(double_divide(inverse(Y), Z), Z), multiply(inverse(inverse(X)), inverse(X)))
% 0.19/0.50 = { by lemma 15 R->L }
% 0.19/0.50 double_divide(inverse(X), multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))
% 0.19/0.50 = { by lemma 3 R->L }
% 0.19/0.50 double_divide(double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))), multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))
% 0.19/0.50 = { by lemma 7 R->L }
% 0.19/0.50 double_divide(double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))), multiply(multiply(multiply(Z, multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z))), inverse(Y)), double_divide(inverse(Y), Z)))
% 0.19/0.50 = { by lemma 9 R->L }
% 0.19/0.50 double_divide(double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))), multiply(W, multiply(inverse(Y), double_divide(double_divide(double_divide(inverse(Y), Z), multiply(Z, multiply(inverse(inverse(X)), double_divide(double_divide(inverse(Y), Z), Z)))), W))))
% 0.19/0.50 = { by lemma 3 }
% 0.19/0.50 Y
% 0.19/0.50
% 0.19/0.50 Lemma 17: multiply(multiply(X, inverse(Y)), inverse(Z)) = multiply(X, multiply(inverse(Y), inverse(Z))).
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(multiply(X, inverse(Y)), inverse(Z))
% 0.19/0.50 = { by lemma 4 R->L }
% 0.19/0.50 multiply(multiply(X, multiply(multiply(W, multiply(inverse(Y), double_divide(V, W))), V)), inverse(Z))
% 0.19/0.50 = { by lemma 13 }
% 0.19/0.50 multiply(X, multiply(multiply(multiply(W, multiply(inverse(Y), double_divide(V, W))), inverse(Z)), V))
% 0.19/0.50 = { by lemma 13 }
% 0.19/0.50 multiply(X, multiply(multiply(W, multiply(multiply(inverse(Y), inverse(Z)), double_divide(V, W))), V))
% 0.19/0.50 = { by lemma 5 }
% 0.19/0.50 multiply(X, multiply(inverse(Y), inverse(Z)))
% 0.19/0.50
% 0.19/0.50 Lemma 18: multiply(multiply(X, inverse(Z)), Y) = multiply(multiply(X, Y), inverse(Z)).
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(multiply(X, inverse(Z)), Y)
% 0.19/0.50 = { by lemma 7 R->L }
% 0.19/0.50 multiply(multiply(multiply(W, multiply(multiply(X, inverse(Z)), Y)), V), double_divide(V, W))
% 0.19/0.50 = { by lemma 14 }
% 0.19/0.50 multiply(multiply(multiply(W, multiply(multiply(X, Y), inverse(Z))), V), double_divide(V, W))
% 0.19/0.50 = { by lemma 7 }
% 0.19/0.50 multiply(multiply(X, Y), inverse(Z))
% 0.19/0.50
% 0.19/0.50 Lemma 19: multiply(X, multiply(inverse(Z), inverse(Y))) = multiply(X, multiply(inverse(Y), inverse(Z))).
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(X, multiply(inverse(Z), inverse(Y)))
% 0.19/0.50 = { by lemma 17 R->L }
% 0.19/0.50 multiply(multiply(X, inverse(Z)), inverse(Y))
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 multiply(multiply(X, inverse(Y)), inverse(Z))
% 0.19/0.50 = { by lemma 17 }
% 0.19/0.50 multiply(X, multiply(inverse(Y), inverse(Z)))
% 0.19/0.50
% 0.19/0.50 Lemma 20: inverse(inverse(X)) = X.
% 0.19/0.50 Proof:
% 0.19/0.50 inverse(inverse(X))
% 0.19/0.50 = { by lemma 16 R->L }
% 0.19/0.50 double_divide(inverse(X), multiply(inverse(inverse(X)), inverse(inverse(inverse(X)))))
% 0.19/0.50 = { by lemma 7 R->L }
% 0.19/0.50 double_divide(inverse(X), multiply(multiply(multiply(Y, multiply(inverse(inverse(X)), inverse(inverse(inverse(X))))), Z), double_divide(Z, Y)))
% 0.19/0.50 = { by lemma 19 }
% 0.19/0.50 double_divide(inverse(X), multiply(multiply(multiply(Y, multiply(inverse(inverse(inverse(X))), inverse(inverse(X)))), Z), double_divide(Z, Y)))
% 0.19/0.50 = { by lemma 7 }
% 0.19/0.50 double_divide(inverse(X), multiply(inverse(inverse(inverse(X))), inverse(inverse(X))))
% 0.19/0.50 = { by lemma 15 R->L }
% 0.19/0.50 double_divide(inverse(inverse(X)), multiply(inverse(inverse(inverse(X))), inverse(X)))
% 0.19/0.50 = { by lemma 16 }
% 0.19/0.50 X
% 0.19/0.50
% 0.19/0.50 Lemma 21: double_divide(Y, X) = double_divide(X, Y).
% 0.19/0.50 Proof:
% 0.19/0.50 double_divide(Y, X)
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 double_divide(inverse(inverse(Y)), X)
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 double_divide(inverse(inverse(Y)), inverse(inverse(X)))
% 0.19/0.50 = { by lemma 11 R->L }
% 0.19/0.50 double_divide(double_divide(Z, W), multiply(multiply(W, multiply(inverse(inverse(X)), inverse(inverse(Y)))), Z))
% 0.19/0.50 = { by lemma 19 }
% 0.19/0.50 double_divide(double_divide(Z, W), multiply(multiply(W, multiply(inverse(inverse(Y)), inverse(inverse(X)))), Z))
% 0.19/0.50 = { by lemma 11 }
% 0.19/0.50 double_divide(inverse(inverse(X)), inverse(inverse(Y)))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 double_divide(X, inverse(inverse(Y)))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 double_divide(X, Y)
% 0.19/0.50
% 0.19/0.50 Lemma 22: multiply(Y, X) = multiply(X, Y).
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(Y, X)
% 0.19/0.50 = { by axiom 1 (multiply) }
% 0.19/0.50 inverse(double_divide(X, Y))
% 0.19/0.50 = { by lemma 21 }
% 0.19/0.50 inverse(double_divide(Y, X))
% 0.19/0.50 = { by axiom 1 (multiply) R->L }
% 0.19/0.50 multiply(X, Y)
% 0.19/0.50
% 0.19/0.50 Lemma 23: double_divide(Y, multiply(Z, X)) = double_divide(X, multiply(Y, Z)).
% 0.19/0.50 Proof:
% 0.19/0.50 double_divide(Y, multiply(Z, X))
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 double_divide(inverse(inverse(Y)), multiply(Z, X))
% 0.19/0.50 = { by lemma 15 }
% 0.19/0.50 double_divide(X, multiply(Z, inverse(inverse(Y))))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 double_divide(X, multiply(Z, Y))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 double_divide(X, multiply(Y, Z))
% 0.19/0.50
% 0.19/0.50 Lemma 24: multiply(Y, multiply(X, Z)) = multiply(X, multiply(Y, Z)).
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(Y, multiply(X, Z))
% 0.19/0.50 = { by lemma 22 }
% 0.19/0.50 multiply(Y, multiply(Z, X))
% 0.19/0.50 = { by lemma 22 }
% 0.19/0.50 multiply(multiply(Z, X), Y)
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 multiply(multiply(Z, inverse(inverse(X))), Y)
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 multiply(multiply(Z, Y), inverse(inverse(X)))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 multiply(multiply(Z, Y), X)
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 multiply(X, multiply(Z, Y))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 multiply(X, multiply(Y, Z))
% 0.19/0.50
% 0.19/0.50 Lemma 25: multiply(X, multiply(Y, inverse(X))) = Y.
% 0.19/0.50 Proof:
% 0.19/0.50 multiply(X, multiply(Y, inverse(X)))
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 multiply(inverse(inverse(X)), multiply(Y, inverse(X)))
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 multiply(inverse(inverse(X)), multiply(inverse(inverse(Y)), inverse(X)))
% 0.19/0.50 = { by lemma 17 R->L }
% 0.19/0.50 multiply(multiply(inverse(inverse(X)), inverse(inverse(Y))), inverse(X))
% 0.19/0.50 = { by lemma 12 R->L }
% 0.19/0.50 multiply(multiply(inverse(inverse(X)), inverse(X)), double_divide(double_divide(inverse(inverse(Y)), Z), Z))
% 0.19/0.50 = { by lemma 18 }
% 0.19/0.50 multiply(multiply(inverse(inverse(X)), double_divide(double_divide(inverse(inverse(Y)), Z), Z)), inverse(X))
% 0.19/0.50 = { by lemma 10 }
% 0.19/0.50 inverse(inverse(Y))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 Y
% 0.19/0.50
% 0.19/0.50 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4)) = tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4)).
% 0.19/0.50 Proof:
% 0.19/0.50 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(a2, multiply(inverse(b2), b2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(a2, multiply(b2, inverse(b2))), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(a2, multiply(b2, inverse(b2))), multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 24 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(b2, multiply(a2, inverse(b2))), multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 24 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(b2, multiply(a2, inverse(b2))), multiply(a3, multiply(c3, b3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), multiply(b2, multiply(a2, inverse(b2))), multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 25 }
% 0.19/0.50 tuple(multiply(a1, inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 }
% 0.19/0.50 tuple(multiply(inverse(a1), a1), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by axiom 1 (multiply) }
% 0.19/0.50 tuple(inverse(double_divide(a1, inverse(a1))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 25 R->L }
% 0.19/0.50 tuple(inverse(double_divide(a1, multiply(b1, multiply(inverse(a1), inverse(b1))))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 24 R->L }
% 0.19/0.50 tuple(inverse(double_divide(a1, multiply(inverse(a1), multiply(b1, inverse(b1))))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 23 R->L }
% 0.19/0.50 tuple(inverse(double_divide(inverse(a1), multiply(multiply(b1, inverse(b1)), a1))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 15 }
% 0.19/0.50 tuple(inverse(double_divide(a1, multiply(multiply(b1, inverse(b1)), inverse(a1)))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 23 }
% 0.19/0.50 tuple(inverse(double_divide(inverse(a1), multiply(a1, multiply(b1, inverse(b1))))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 tuple(inverse(double_divide(inverse(a1), multiply(inverse(inverse(a1)), multiply(b1, inverse(b1))))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 20 R->L }
% 0.19/0.50 tuple(inverse(double_divide(inverse(a1), multiply(inverse(inverse(a1)), inverse(inverse(multiply(b1, inverse(b1))))))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 16 }
% 0.19/0.50 tuple(inverse(inverse(multiply(b1, inverse(b1)))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by axiom 1 (multiply) }
% 0.19/0.50 tuple(inverse(inverse(inverse(double_divide(inverse(b1), b1)))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 20 }
% 0.19/0.50 tuple(inverse(double_divide(inverse(b1), b1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 21 R->L }
% 0.19/0.50 tuple(inverse(double_divide(b1, inverse(b1))), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by axiom 1 (multiply) R->L }
% 0.19/0.50 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 R->L }
% 0.19/0.50 tuple(multiply(b1, inverse(b1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.50 = { by lemma 22 }
% 0.19/0.50 tuple(multiply(b1, inverse(b1)), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.19/0.50 = { by lemma 22 }
% 0.19/0.50 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.19/0.50 % SZS output end Proof
% 0.19/0.51
% 0.19/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------