TSTP Solution File: GRP106-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP106-1 : 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 : n016.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 0.20s 0.52s
% Output : Proof 1.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : GRP106-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.03/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n016.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 01:45:30 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.52 Command-line arguments: --no-flatten-goal
% 0.20/0.52
% 0.20/0.52 % SZS status Unsatisfiable
% 0.20/0.52
% 1.66/0.57 % SZS output start Proof
% 1.66/0.57 Take the following subset of the input axioms:
% 1.66/0.57 fof(multiply, axiom, ![X, Y]: multiply(X, Y)=inverse(double_divide(Y, X))).
% 1.66/0.57 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)))).
% 1.66/0.57 fof(single_axiom, axiom, ![Z, X2, Y2]: inverse(double_divide(double_divide(X2, Y2), inverse(double_divide(X2, inverse(double_divide(Z, Y2))))))=Z).
% 1.66/0.57
% 1.66/0.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 1.66/0.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 1.66/0.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 1.66/0.57 fresh(y, y, x1...xn) = u
% 1.66/0.57 C => fresh(s, t, x1...xn) = v
% 1.66/0.57 where fresh is a fresh function symbol and x1..xn are the free
% 1.66/0.57 variables of u and v.
% 1.66/0.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 1.66/0.57 input problem has no model of domain size 1).
% 1.66/0.57
% 1.66/0.57 The encoding turns the above axioms into the following unit equations and goals:
% 1.66/0.57
% 1.66/0.57 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 1.66/0.57 Axiom 2 (single_axiom): inverse(double_divide(double_divide(X, Y), inverse(double_divide(X, inverse(double_divide(Z, Y)))))) = Z.
% 1.66/0.57
% 1.66/0.57 Lemma 3: multiply(multiply(multiply(X, Y), Z), double_divide(Z, X)) = Y.
% 1.66/0.57 Proof:
% 1.66/0.57 multiply(multiply(multiply(X, Y), Z), double_divide(Z, X))
% 1.66/0.57 = { by axiom 1 (multiply) }
% 1.66/0.57 multiply(multiply(inverse(double_divide(Y, X)), Z), double_divide(Z, X))
% 1.66/0.57 = { by axiom 1 (multiply) }
% 1.66/0.57 multiply(inverse(double_divide(Z, inverse(double_divide(Y, X)))), double_divide(Z, X))
% 1.66/0.57 = { by axiom 1 (multiply) }
% 1.66/0.57 inverse(double_divide(double_divide(Z, X), inverse(double_divide(Z, inverse(double_divide(Y, X))))))
% 1.66/0.57 = { by axiom 2 (single_axiom) }
% 1.66/0.57 Y
% 1.66/0.57
% 1.66/0.57 Lemma 4: multiply(X, double_divide(double_divide(Y, Z), multiply(Z, X))) = Y.
% 1.66/0.57 Proof:
% 1.66/0.57 multiply(X, double_divide(double_divide(Y, Z), multiply(Z, X)))
% 1.66/0.57 = { by lemma 3 R->L }
% 1.66/0.57 multiply(multiply(multiply(multiply(Z, X), Y), double_divide(Y, Z)), double_divide(double_divide(Y, Z), multiply(Z, X)))
% 1.66/0.57 = { by lemma 3 }
% 1.66/0.57 Y
% 1.66/0.57
% 1.66/0.57 Lemma 5: multiply(multiply(X, Y), double_divide(Y, Z)) = double_divide(double_divide(X, W), multiply(W, Z)).
% 1.66/0.57 Proof:
% 1.66/0.57 multiply(multiply(X, Y), double_divide(Y, Z))
% 1.66/0.57 = { by lemma 4 R->L }
% 1.66/0.57 multiply(multiply(multiply(Z, double_divide(double_divide(X, W), multiply(W, Z))), Y), double_divide(Y, Z))
% 1.66/0.57 = { by lemma 3 }
% 1.66/0.57 double_divide(double_divide(X, W), multiply(W, Z))
% 1.66/0.57
% 1.66/0.57 Lemma 6: double_divide(double_divide(X, W), multiply(W, Z)) = double_divide(double_divide(X, Y), multiply(Y, Z)).
% 1.66/0.57 Proof:
% 1.66/0.57 double_divide(double_divide(X, W), multiply(W, Z))
% 1.66/0.57 = { by lemma 3 R->L }
% 1.66/0.57 multiply(multiply(multiply(Z, double_divide(double_divide(X, W), multiply(W, Z))), V), double_divide(V, Z))
% 1.66/0.57 = { by lemma 4 }
% 1.66/0.57 multiply(multiply(X, V), double_divide(V, Z))
% 1.66/0.57 = { by lemma 4 R->L }
% 1.66/0.57 multiply(multiply(multiply(Z, double_divide(double_divide(X, Y), multiply(Y, Z))), V), double_divide(V, Z))
% 1.66/0.57 = { by lemma 3 }
% 1.66/0.57 double_divide(double_divide(X, Y), multiply(Y, Z))
% 1.66/0.57
% 1.66/0.57 Lemma 7: multiply(multiply(W, Y), double_divide(Z, W)) = multiply(multiply(X, Y), double_divide(Z, X)).
% 1.66/0.57 Proof:
% 1.66/0.57 multiply(multiply(W, Y), double_divide(Z, W))
% 1.66/0.57 = { by axiom 1 (multiply) }
% 1.66/0.57 inverse(double_divide(double_divide(Z, W), multiply(W, Y)))
% 1.66/0.57 = { by lemma 6 }
% 1.66/0.57 inverse(double_divide(double_divide(Z, X), multiply(X, Y)))
% 1.66/0.57 = { by axiom 1 (multiply) R->L }
% 1.66/0.57 multiply(multiply(X, Y), double_divide(Z, X))
% 1.66/0.57
% 1.66/0.57 Lemma 8: double_divide(double_divide(Z, W), multiply(W, Z)) = double_divide(double_divide(X, Y), multiply(Y, X)).
% 1.66/0.57 Proof:
% 1.66/0.57 double_divide(double_divide(Z, W), multiply(W, Z))
% 1.66/0.57 = { by lemma 5 R->L }
% 1.66/0.57 multiply(multiply(Z, V), double_divide(V, Z))
% 1.66/0.57 = { by lemma 7 }
% 1.66/0.57 multiply(multiply(X, V), double_divide(V, X))
% 1.66/0.57 = { by lemma 5 }
% 1.66/0.57 double_divide(double_divide(X, Y), multiply(Y, X))
% 1.66/0.58
% 1.66/0.58 Lemma 9: multiply(X, double_divide(double_divide(Y, Z), multiply(Z, Y))) = X.
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, Z), multiply(Z, Y)))
% 1.66/0.58 = { by lemma 8 }
% 1.66/0.58 multiply(X, double_divide(double_divide(X, W), multiply(W, X)))
% 1.66/0.58 = { by lemma 4 }
% 1.66/0.58 X
% 1.66/0.58
% 1.66/0.58 Lemma 10: double_divide(double_divide(X, Y), multiply(Y, multiply(multiply(Z, X), W))) = double_divide(W, Z).
% 1.66/0.58 Proof:
% 1.66/0.58 double_divide(double_divide(X, Y), multiply(Y, multiply(multiply(Z, X), W)))
% 1.66/0.58 = { by lemma 5 R->L }
% 1.66/0.58 multiply(multiply(X, V), double_divide(V, multiply(multiply(Z, X), W)))
% 1.66/0.58 = { by lemma 3 R->L }
% 1.66/0.58 multiply(multiply(multiply(multiply(multiply(Z, X), W), double_divide(W, Z)), V), double_divide(V, multiply(multiply(Z, X), W)))
% 1.66/0.58 = { by lemma 3 }
% 1.66/0.58 double_divide(W, Z)
% 1.66/0.58
% 1.66/0.58 Lemma 11: double_divide(double_divide(X, Z), Z) = double_divide(double_divide(X, Y), Y).
% 1.66/0.58 Proof:
% 1.66/0.58 double_divide(double_divide(X, Z), Z)
% 1.66/0.58 = { by lemma 10 R->L }
% 1.66/0.58 double_divide(double_divide(W, V), multiply(V, multiply(multiply(Z, W), double_divide(X, Z))))
% 1.66/0.58 = { by lemma 7 }
% 1.66/0.58 double_divide(double_divide(W, V), multiply(V, multiply(multiply(Y, W), double_divide(X, Y))))
% 1.66/0.58 = { by lemma 10 }
% 1.66/0.58 double_divide(double_divide(X, Y), Y)
% 1.66/0.58
% 1.66/0.58 Lemma 12: multiply(W, double_divide(double_divide(Y, X), W)) = multiply(X, double_divide(double_divide(Y, Z), Z)).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(W, double_divide(double_divide(Y, X), W))
% 1.66/0.58 = { by axiom 1 (multiply) }
% 1.66/0.58 inverse(double_divide(double_divide(double_divide(Y, X), W), W))
% 1.66/0.58 = { by lemma 11 }
% 1.66/0.58 inverse(double_divide(double_divide(double_divide(Y, X), X), X))
% 1.66/0.58 = { by axiom 1 (multiply) R->L }
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, X), X))
% 1.66/0.58 = { by lemma 11 R->L }
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, Z), Z))
% 1.66/0.58
% 1.66/0.58 Lemma 13: multiply(multiply(X, multiply(multiply(Y, Z), W)), double_divide(Z, X)) = multiply(Y, W).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(multiply(X, multiply(multiply(Y, Z), W)), double_divide(Z, X))
% 1.66/0.58 = { by axiom 1 (multiply) }
% 1.66/0.58 inverse(double_divide(double_divide(Z, X), multiply(X, multiply(multiply(Y, Z), W))))
% 1.66/0.58 = { by lemma 10 }
% 1.66/0.58 inverse(double_divide(W, Y))
% 1.66/0.58 = { by axiom 1 (multiply) R->L }
% 1.66/0.58 multiply(Y, W)
% 1.66/0.58
% 1.66/0.58 Lemma 14: multiply(X, double_divide(double_divide(Y, multiply(Z, W)), X)) = multiply(multiply(Z, Y), W).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, multiply(Z, W)), X))
% 1.66/0.58 = { by lemma 12 }
% 1.66/0.58 multiply(multiply(Z, W), double_divide(double_divide(Y, V), V))
% 1.66/0.58 = { by lemma 13 R->L }
% 1.66/0.58 multiply(multiply(multiply(V, multiply(multiply(Z, Y), W)), double_divide(Y, V)), double_divide(double_divide(Y, V), V))
% 1.66/0.58 = { by lemma 3 }
% 1.66/0.58 multiply(multiply(Z, Y), W)
% 1.66/0.58
% 1.66/0.58 Lemma 15: multiply(X, double_divide(double_divide(Y, Z), X)) = multiply(Z, Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, Z), X))
% 1.66/0.58 = { by lemma 9 R->L }
% 1.66/0.58 multiply(X, double_divide(double_divide(Y, multiply(Z, double_divide(double_divide(W, V), multiply(V, W)))), X))
% 1.66/0.58 = { by lemma 14 }
% 1.66/0.58 multiply(multiply(Z, Y), double_divide(double_divide(W, V), multiply(V, W)))
% 1.66/0.58 = { by lemma 9 }
% 1.66/0.58 multiply(Z, Y)
% 1.66/0.58
% 1.66/0.58 Lemma 16: multiply(multiply(X, Y), Z) = multiply(multiply(X, Z), Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(multiply(X, Y), Z)
% 1.66/0.58 = { by lemma 15 R->L }
% 1.66/0.58 multiply(W, double_divide(double_divide(Z, multiply(X, Y)), W))
% 1.66/0.58 = { by lemma 14 }
% 1.66/0.58 multiply(multiply(X, Z), Y)
% 1.66/0.58
% 1.66/0.58 Lemma 17: double_divide(double_divide(multiply(X, Y), Z), multiply(Z, X)) = Y.
% 1.66/0.58 Proof:
% 1.66/0.58 double_divide(double_divide(multiply(X, Y), Z), multiply(Z, X))
% 1.66/0.58 = { by lemma 5 R->L }
% 1.66/0.58 multiply(multiply(multiply(X, Y), W), double_divide(W, X))
% 1.66/0.58 = { by lemma 3 }
% 1.66/0.58 Y
% 1.66/0.58
% 1.66/0.58 Lemma 18: multiply(multiply(X, Y), double_divide(multiply(Y, Z), X)) = inverse(Z).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(multiply(X, Y), double_divide(multiply(Y, Z), X))
% 1.66/0.58 = { by axiom 1 (multiply) }
% 1.66/0.58 inverse(double_divide(double_divide(multiply(Y, Z), X), multiply(X, Y)))
% 1.66/0.58 = { by lemma 17 }
% 1.66/0.58 inverse(Z)
% 1.66/0.58
% 1.66/0.58 Lemma 19: multiply(X, double_divide(Y, X)) = inverse(Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, double_divide(Y, X))
% 1.66/0.58 = { by lemma 17 R->L }
% 1.66/0.58 multiply(X, double_divide(double_divide(double_divide(multiply(Z, Y), W), multiply(W, Z)), X))
% 1.66/0.58 = { by lemma 15 }
% 1.66/0.58 multiply(multiply(W, Z), double_divide(multiply(Z, Y), W))
% 1.66/0.58 = { by lemma 18 }
% 1.66/0.58 inverse(Y)
% 1.66/0.58
% 1.66/0.58 Lemma 20: multiply(inverse(multiply(X, Y)), X) = inverse(Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(inverse(multiply(X, Y)), X)
% 1.66/0.58 = { by lemma 19 R->L }
% 1.66/0.58 multiply(multiply(Z, double_divide(multiply(X, Y), Z)), X)
% 1.66/0.58 = { by lemma 16 R->L }
% 1.66/0.58 multiply(multiply(Z, X), double_divide(multiply(X, Y), Z))
% 1.66/0.58 = { by lemma 18 }
% 1.66/0.58 inverse(Y)
% 1.66/0.58
% 1.66/0.58 Lemma 21: multiply(inverse(inverse(X)), Y) = multiply(Y, X).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(inverse(inverse(X)), Y)
% 1.66/0.58 = { by lemma 19 R->L }
% 1.66/0.58 multiply(inverse(multiply(Y, double_divide(X, Y))), Y)
% 1.66/0.58 = { by lemma 20 }
% 1.66/0.58 inverse(double_divide(X, Y))
% 1.66/0.58 = { by axiom 1 (multiply) R->L }
% 1.66/0.58 multiply(Y, X)
% 1.66/0.58
% 1.66/0.58 Lemma 22: multiply(multiply(X, Y), Z) = multiply(multiply(Y, X), Z).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(multiply(X, Y), Z)
% 1.66/0.58 = { by lemma 16 R->L }
% 1.66/0.58 multiply(multiply(X, Z), Y)
% 1.66/0.58 = { by lemma 21 R->L }
% 1.66/0.58 multiply(multiply(inverse(inverse(Z)), X), Y)
% 1.66/0.58 = { by lemma 14 R->L }
% 1.66/0.58 multiply(W, double_divide(double_divide(X, multiply(inverse(inverse(Z)), Y)), W))
% 1.66/0.58 = { by lemma 21 }
% 1.66/0.58 multiply(W, double_divide(double_divide(X, multiply(Y, Z)), W))
% 1.66/0.58 = { by lemma 14 }
% 1.66/0.58 multiply(multiply(Y, X), Z)
% 1.66/0.58
% 1.66/0.58 Lemma 23: multiply(Y, X) = multiply(X, Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(Y, X)
% 1.66/0.58 = { by lemma 9 R->L }
% 1.66/0.58 multiply(multiply(Y, X), double_divide(double_divide(Z, W), multiply(W, Z)))
% 1.66/0.58 = { by lemma 22 R->L }
% 1.66/0.58 multiply(multiply(X, Y), double_divide(double_divide(Z, W), multiply(W, Z)))
% 1.66/0.58 = { by lemma 9 }
% 1.66/0.58 multiply(X, Y)
% 1.66/0.58
% 1.66/0.58 Lemma 24: multiply(multiply(X, double_divide(Y, X)), Y) = multiply(inverse(Z), Z).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(multiply(X, double_divide(Y, X)), Y)
% 1.66/0.58 = { by lemma 14 R->L }
% 1.66/0.58 multiply(W, double_divide(double_divide(double_divide(Y, X), multiply(X, Y)), W))
% 1.66/0.58 = { by lemma 8 R->L }
% 1.66/0.58 multiply(W, double_divide(double_divide(double_divide(Z, V), multiply(V, Z)), W))
% 1.66/0.58 = { by lemma 14 }
% 1.66/0.58 multiply(multiply(V, double_divide(Z, V)), Z)
% 1.66/0.58 = { by lemma 19 }
% 1.66/0.58 multiply(inverse(Z), Z)
% 1.66/0.58
% 1.66/0.58 Lemma 25: multiply(inverse(X), multiply(X, Y)) = multiply(Z, multiply(Y, inverse(Z))).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(inverse(X), multiply(X, Y))
% 1.66/0.58 = { by lemma 23 }
% 1.66/0.58 multiply(multiply(X, Y), inverse(X))
% 1.66/0.58 = { by lemma 16 R->L }
% 1.66/0.58 multiply(multiply(X, inverse(X)), Y)
% 1.66/0.58 = { by lemma 22 R->L }
% 1.66/0.58 multiply(multiply(inverse(X), X), Y)
% 1.66/0.58 = { by lemma 24 R->L }
% 1.66/0.58 multiply(multiply(multiply(W, double_divide(Z, W)), Z), Y)
% 1.66/0.58 = { by lemma 16 }
% 1.66/0.58 multiply(multiply(multiply(W, double_divide(Z, W)), Y), Z)
% 1.66/0.58 = { by lemma 19 }
% 1.66/0.58 multiply(multiply(inverse(Z), Y), Z)
% 1.66/0.58 = { by lemma 23 R->L }
% 1.66/0.58 multiply(Z, multiply(inverse(Z), Y))
% 1.66/0.58 = { by lemma 23 R->L }
% 1.66/0.58 multiply(Z, multiply(Y, inverse(Z)))
% 1.66/0.58
% 1.66/0.58 Lemma 26: multiply(double_divide(X, Y), multiply(Z, inverse(Z))) = double_divide(X, Y).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(double_divide(X, Y), multiply(Z, inverse(Z)))
% 1.66/0.58 = { by lemma 23 }
% 1.66/0.58 multiply(multiply(Z, inverse(Z)), double_divide(X, Y))
% 1.66/0.58 = { by lemma 22 R->L }
% 1.66/0.58 multiply(multiply(inverse(Z), Z), double_divide(X, Y))
% 1.66/0.58 = { by lemma 24 R->L }
% 1.66/0.58 multiply(multiply(multiply(Y, double_divide(X, Y)), X), double_divide(X, Y))
% 1.66/0.58 = { by lemma 3 }
% 1.66/0.58 double_divide(X, Y)
% 1.66/0.58
% 1.66/0.58 Lemma 27: multiply(X, multiply(Y, inverse(X))) = Y.
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, multiply(Y, inverse(X)))
% 1.66/0.58 = { by lemma 25 R->L }
% 1.66/0.58 multiply(inverse(Z), multiply(Z, Y))
% 1.66/0.58 = { by lemma 23 }
% 1.66/0.58 multiply(multiply(Z, Y), inverse(Z))
% 1.66/0.58 = { by lemma 14 R->L }
% 1.66/0.58 multiply(multiply(Z, inverse(Z)), double_divide(double_divide(Y, multiply(Z, inverse(Z))), multiply(Z, inverse(Z))))
% 1.66/0.58 = { by lemma 11 }
% 1.66/0.58 multiply(multiply(Z, inverse(Z)), double_divide(double_divide(Y, double_divide(W, V)), double_divide(W, V)))
% 1.66/0.58 = { by lemma 26 R->L }
% 1.66/0.58 multiply(multiply(Z, inverse(Z)), double_divide(double_divide(Y, double_divide(W, V)), multiply(double_divide(W, V), multiply(Z, inverse(Z)))))
% 1.66/0.58 = { by lemma 4 }
% 1.66/0.58 Y
% 1.66/0.58
% 1.66/0.58 Lemma 28: inverse(inverse(X)) = X.
% 1.66/0.58 Proof:
% 1.66/0.58 inverse(inverse(X))
% 1.66/0.58 = { by lemma 19 R->L }
% 1.66/0.58 multiply(Y, double_divide(inverse(X), Y))
% 1.66/0.58 = { by lemma 27 R->L }
% 1.66/0.58 multiply(X, multiply(multiply(Y, double_divide(inverse(X), Y)), inverse(X)))
% 1.66/0.58 = { by lemma 24 }
% 1.66/0.58 multiply(X, multiply(inverse(X), X))
% 1.66/0.58 = { by lemma 23 R->L }
% 1.66/0.58 multiply(X, multiply(X, inverse(X)))
% 1.66/0.58 = { by lemma 27 }
% 1.66/0.58 X
% 1.66/0.58
% 1.66/0.58 Lemma 29: inverse(multiply(X, Y)) = double_divide(X, Y).
% 1.66/0.58 Proof:
% 1.66/0.58 inverse(multiply(X, Y))
% 1.66/0.58 = { by lemma 23 }
% 1.66/0.58 inverse(multiply(Y, X))
% 1.66/0.58 = { by axiom 1 (multiply) }
% 1.66/0.58 inverse(inverse(double_divide(X, Y)))
% 1.66/0.58 = { by lemma 28 }
% 1.66/0.58 double_divide(X, Y)
% 1.66/0.58
% 1.66/0.58 Lemma 30: double_divide(Y, X) = double_divide(X, Y).
% 1.66/0.58 Proof:
% 1.66/0.58 double_divide(Y, X)
% 1.66/0.58 = { by lemma 29 R->L }
% 1.66/0.58 inverse(multiply(Y, X))
% 1.66/0.58 = { by lemma 21 R->L }
% 1.66/0.58 inverse(multiply(inverse(inverse(X)), Y))
% 1.66/0.58 = { by lemma 29 }
% 1.66/0.58 double_divide(inverse(inverse(X)), Y)
% 1.66/0.58 = { by lemma 28 }
% 1.66/0.58 double_divide(X, Y)
% 1.66/0.58
% 1.66/0.58 Lemma 31: multiply(X, double_divide(Y, multiply(multiply(Z, W), X))) = double_divide(multiply(W, Y), Z).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, double_divide(Y, multiply(multiply(Z, W), X)))
% 1.66/0.58 = { by lemma 17 R->L }
% 1.66/0.58 multiply(X, double_divide(double_divide(double_divide(multiply(W, Y), Z), multiply(Z, W)), multiply(multiply(Z, W), X)))
% 1.66/0.58 = { by lemma 4 }
% 1.66/0.58 double_divide(multiply(W, Y), Z)
% 1.66/0.58
% 1.66/0.58 Lemma 32: double_divide(multiply(X, double_divide(Y, multiply(Z, X))), Z) = Y.
% 1.66/0.58 Proof:
% 1.66/0.58 double_divide(multiply(X, double_divide(Y, multiply(Z, X))), Z)
% 1.66/0.58 = { by lemma 31 R->L }
% 1.66/0.58 multiply(W, double_divide(double_divide(Y, multiply(Z, X)), multiply(multiply(Z, X), W)))
% 1.66/0.58 = { by lemma 4 }
% 1.66/0.58 Y
% 1.66/0.58
% 1.66/0.58 Lemma 33: multiply(X, inverse(Y)) = double_divide(Y, inverse(X)).
% 1.66/0.58 Proof:
% 1.66/0.58 multiply(X, inverse(Y))
% 1.66/0.58 = { by lemma 23 }
% 1.66/0.58 multiply(inverse(Y), X)
% 1.66/0.58 = { by lemma 28 R->L }
% 1.66/0.58 multiply(inverse(Y), inverse(inverse(X)))
% 1.66/0.58 = { by lemma 18 R->L }
% 1.66/0.58 multiply(inverse(Y), multiply(multiply(Z, multiply(Y, inverse(Y))), double_divide(multiply(multiply(Y, inverse(Y)), inverse(X)), Z)))
% 1.66/0.58 = { by lemma 32 R->L }
% 1.66/0.58 multiply(inverse(Y), multiply(multiply(double_divide(multiply(W, double_divide(Z, multiply(V, W))), V), multiply(Y, inverse(Y))), double_divide(multiply(multiply(Y, inverse(Y)), inverse(X)), Z)))
% 1.66/0.59 = { by lemma 26 }
% 1.66/0.59 multiply(inverse(Y), multiply(double_divide(multiply(W, double_divide(Z, multiply(V, W))), V), double_divide(multiply(multiply(Y, inverse(Y)), inverse(X)), Z)))
% 1.66/0.59 = { by lemma 32 }
% 1.66/0.59 multiply(inverse(Y), multiply(Z, double_divide(multiply(multiply(Y, inverse(Y)), inverse(X)), Z)))
% 1.66/0.59 = { by lemma 19 }
% 1.66/0.59 multiply(inverse(Y), inverse(multiply(multiply(Y, inverse(Y)), inverse(X))))
% 1.66/0.59 = { by lemma 16 }
% 1.66/0.59 multiply(inverse(Y), inverse(multiply(multiply(Y, inverse(X)), inverse(Y))))
% 1.66/0.59 = { by lemma 20 R->L }
% 1.66/0.59 multiply(inverse(Y), multiply(inverse(multiply(Y, multiply(multiply(Y, inverse(X)), inverse(Y)))), Y))
% 1.66/0.59 = { by lemma 27 }
% 1.66/0.59 multiply(inverse(Y), multiply(inverse(multiply(Y, inverse(X))), Y))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(inverse(Y), multiply(Y, inverse(multiply(Y, inverse(X)))))
% 1.66/0.59 = { by lemma 25 }
% 1.66/0.59 multiply(U, multiply(inverse(multiply(Y, inverse(X))), inverse(U)))
% 1.66/0.59 = { by lemma 27 }
% 1.66/0.59 inverse(multiply(Y, inverse(X)))
% 1.66/0.59 = { by lemma 29 }
% 1.66/0.59 double_divide(Y, inverse(X))
% 1.66/0.59
% 1.66/0.59 Lemma 34: multiply(inverse(X), Y) = double_divide(X, inverse(Y)).
% 1.66/0.59 Proof:
% 1.66/0.59 multiply(inverse(X), Y)
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 multiply(Y, inverse(X))
% 1.66/0.59 = { by lemma 33 }
% 1.66/0.59 double_divide(X, inverse(Y))
% 1.66/0.59
% 1.66/0.59 Lemma 35: double_divide(X, double_divide(X, Y)) = Y.
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, double_divide(X, Y))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(double_divide(X, Y), X)
% 1.66/0.59 = { by lemma 29 R->L }
% 1.66/0.59 double_divide(inverse(multiply(X, Y)), X)
% 1.66/0.59 = { by lemma 29 R->L }
% 1.66/0.59 inverse(multiply(inverse(multiply(X, Y)), X))
% 1.66/0.59 = { by lemma 20 }
% 1.66/0.59 inverse(inverse(Y))
% 1.66/0.59 = { by lemma 28 }
% 1.66/0.59 Y
% 1.66/0.59
% 1.66/0.59 Lemma 36: double_divide(X, double_divide(Y, X)) = Y.
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, double_divide(Y, X))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(X, double_divide(X, Y))
% 1.66/0.59 = { by lemma 35 }
% 1.66/0.59 Y
% 1.66/0.59
% 1.66/0.59 Lemma 37: multiply(X, double_divide(X, Y)) = inverse(Y).
% 1.66/0.59 Proof:
% 1.66/0.59 multiply(X, double_divide(X, Y))
% 1.66/0.59 = { by lemma 28 R->L }
% 1.66/0.59 multiply(X, double_divide(X, inverse(inverse(Y))))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 multiply(double_divide(X, inverse(inverse(Y))), X)
% 1.66/0.59 = { by lemma 34 R->L }
% 1.66/0.59 multiply(multiply(inverse(X), inverse(Y)), X)
% 1.66/0.59 = { by lemma 4 R->L }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(multiply(Z, W), double_divide(X, inverse(X)))))
% 1.66/0.59 = { by lemma 34 R->L }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(multiply(Z, W), multiply(inverse(X), X))))
% 1.66/0.59 = { by lemma 14 R->L }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(V, double_divide(double_divide(W, multiply(Z, multiply(inverse(X), X))), V))))
% 1.66/0.59 = { by lemma 12 }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(multiply(Z, multiply(inverse(X), X)), double_divide(double_divide(W, Z), Z))))
% 1.66/0.59 = { by lemma 24 R->L }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(multiply(Z, multiply(multiply(Z, double_divide(W, Z)), W)), double_divide(double_divide(W, Z), Z))))
% 1.66/0.59 = { by lemma 13 }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), multiply(Z, W)), multiply(Z, W)))
% 1.66/0.59 = { by lemma 11 R->L }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), double_divide(double_divide(multiply(multiply(inverse(X), inverse(Y)), X), double_divide(X, inverse(X))), double_divide(X, inverse(X))))
% 1.66/0.59 = { by lemma 15 }
% 1.66/0.59 multiply(double_divide(X, inverse(X)), multiply(multiply(inverse(X), inverse(Y)), X))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(multiply(multiply(inverse(X), inverse(Y)), X), double_divide(X, inverse(X)))
% 1.66/0.59 = { by lemma 3 }
% 1.66/0.59 inverse(Y)
% 1.66/0.59
% 1.66/0.59 Lemma 38: double_divide(X, multiply(Y, double_divide(X, Z))) = double_divide(Y, inverse(Z)).
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, multiply(Y, double_divide(X, Z)))
% 1.66/0.59 = { by axiom 1 (multiply) }
% 1.66/0.59 double_divide(X, inverse(double_divide(double_divide(X, Z), Y)))
% 1.66/0.59 = { by lemma 34 R->L }
% 1.66/0.59 multiply(inverse(X), double_divide(double_divide(X, Z), Y))
% 1.66/0.59 = { by lemma 37 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(double_divide(X, Z), Y))
% 1.66/0.59 = { by lemma 29 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(inverse(multiply(X, Z)), Y))
% 1.66/0.59 = { by lemma 19 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(W, double_divide(multiply(X, Z), W)), Y))
% 1.66/0.59 = { by lemma 13 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(multiply(Y, multiply(multiply(W, X), double_divide(multiply(X, Z), W))), double_divide(X, Y)), Y))
% 1.66/0.59 = { by lemma 18 }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(multiply(Y, inverse(Z)), double_divide(X, Y)), Y))
% 1.66/0.59 = { by lemma 33 }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(double_divide(Z, inverse(Y)), double_divide(X, Y)), Y))
% 1.66/0.59 = { by lemma 30 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(double_divide(Z, inverse(Y)), double_divide(Y, X)), Y))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(Y, X)), double_divide(multiply(double_divide(Y, X), double_divide(Z, inverse(Y))), Y))
% 1.66/0.59 = { by lemma 18 }
% 1.66/0.59 inverse(double_divide(Z, inverse(Y)))
% 1.66/0.59 = { by axiom 1 (multiply) R->L }
% 1.66/0.59 multiply(inverse(Y), Z)
% 1.66/0.59 = { by lemma 34 }
% 1.66/0.59 double_divide(Y, inverse(Z))
% 1.66/0.59
% 1.66/0.59 Lemma 39: double_divide(X, double_divide(Y, multiply(X, Z))) = multiply(Y, Z).
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, double_divide(Y, multiply(X, Z)))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 double_divide(X, double_divide(Y, multiply(Z, X)))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(X, double_divide(multiply(Z, X), Y))
% 1.66/0.59 = { by lemma 31 R->L }
% 1.66/0.59 double_divide(X, multiply(W, double_divide(X, multiply(multiply(Y, Z), W))))
% 1.66/0.59 = { by lemma 38 }
% 1.66/0.59 double_divide(W, inverse(multiply(multiply(Y, Z), W)))
% 1.66/0.59 = { by lemma 29 }
% 1.66/0.59 double_divide(W, double_divide(multiply(Y, Z), W))
% 1.66/0.59 = { by lemma 36 }
% 1.66/0.59 multiply(Y, Z)
% 1.66/0.59
% 1.66/0.59 Lemma 40: double_divide(Y, multiply(X, Z)) = double_divide(X, multiply(Y, Z)).
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(Y, multiply(X, Z))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(multiply(X, Z), Y)
% 1.66/0.59 = { by lemma 29 R->L }
% 1.66/0.59 inverse(multiply(multiply(X, Z), Y))
% 1.66/0.59 = { by lemma 37 R->L }
% 1.66/0.59 multiply(V, double_divide(V, multiply(multiply(X, Z), Y)))
% 1.66/0.59 = { by lemma 14 R->L }
% 1.66/0.59 multiply(V, double_divide(V, multiply(V, double_divide(double_divide(Z, multiply(X, Y)), V))))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 multiply(double_divide(V, multiply(V, double_divide(double_divide(Z, multiply(X, Y)), V))), V)
% 1.66/0.59 = { by axiom 1 (multiply) }
% 1.66/0.59 inverse(double_divide(V, double_divide(V, multiply(V, double_divide(double_divide(Z, multiply(X, Y)), V)))))
% 1.66/0.59 = { by lemma 39 }
% 1.66/0.59 inverse(multiply(V, double_divide(double_divide(Z, multiply(X, Y)), V)))
% 1.66/0.59 = { by lemma 29 }
% 1.66/0.59 double_divide(V, double_divide(double_divide(Z, multiply(X, Y)), V))
% 1.66/0.59 = { by lemma 36 }
% 1.66/0.59 double_divide(Z, multiply(X, Y))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 double_divide(Z, multiply(Y, X))
% 1.66/0.59 = { by axiom 1 (multiply) }
% 1.66/0.59 double_divide(Z, inverse(double_divide(X, Y)))
% 1.66/0.59 = { by lemma 34 R->L }
% 1.66/0.59 multiply(inverse(Z), double_divide(X, Y))
% 1.66/0.59 = { by lemma 15 R->L }
% 1.66/0.59 multiply(W, double_divide(double_divide(double_divide(X, Y), inverse(Z)), W))
% 1.66/0.59 = { by lemma 19 R->L }
% 1.66/0.59 multiply(W, double_divide(double_divide(double_divide(X, Y), multiply(Y, double_divide(Z, Y))), W))
% 1.66/0.59 = { by lemma 14 }
% 1.66/0.59 multiply(multiply(Y, double_divide(X, Y)), double_divide(Z, Y))
% 1.66/0.59 = { by lemma 30 R->L }
% 1.66/0.59 multiply(multiply(Y, double_divide(X, Y)), double_divide(Y, Z))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(double_divide(Y, Z), multiply(Y, double_divide(X, Y)))
% 1.66/0.59 = { by lemma 19 }
% 1.66/0.59 multiply(double_divide(Y, Z), inverse(X))
% 1.66/0.59 = { by lemma 33 }
% 1.66/0.59 double_divide(X, inverse(double_divide(Y, Z)))
% 1.66/0.59 = { by lemma 37 R->L }
% 1.66/0.59 double_divide(X, multiply(Y, double_divide(Y, double_divide(Y, Z))))
% 1.66/0.59 = { by lemma 35 }
% 1.66/0.59 double_divide(X, multiply(Y, Z))
% 1.66/0.59
% 1.66/0.59 Lemma 41: double_divide(X, double_divide(Y, multiply(X, Z))) = multiply(Z, Y).
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, double_divide(Y, multiply(X, Z)))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 double_divide(X, double_divide(Y, multiply(Z, X)))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(X, double_divide(multiply(Z, X), Y))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(double_divide(multiply(Z, X), Y), X)
% 1.66/0.59 = { by lemma 29 R->L }
% 1.66/0.59 double_divide(inverse(multiply(multiply(Z, X), Y)), X)
% 1.66/0.59 = { by lemma 3 R->L }
% 1.66/0.59 double_divide(inverse(multiply(multiply(Z, X), Y)), multiply(multiply(multiply(Z, X), Y), double_divide(Y, Z)))
% 1.66/0.59 = { by axiom 1 (multiply) }
% 1.66/0.59 double_divide(inverse(multiply(multiply(Z, X), Y)), inverse(double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y))))
% 1.66/0.59 = { by lemma 34 R->L }
% 1.66/0.59 multiply(inverse(inverse(multiply(multiply(Z, X), Y))), double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y)))
% 1.66/0.59 = { by lemma 19 R->L }
% 1.66/0.59 multiply(multiply(inverse(multiply(multiply(Z, X), Y)), double_divide(inverse(multiply(multiply(Z, X), Y)), inverse(multiply(multiply(Z, X), Y)))), double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y)))
% 1.66/0.59 = { by lemma 16 R->L }
% 1.66/0.59 multiply(multiply(inverse(multiply(multiply(Z, X), Y)), double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y))), double_divide(inverse(multiply(multiply(Z, X), Y)), inverse(multiply(multiply(Z, X), Y))))
% 1.66/0.59 = { by lemma 38 R->L }
% 1.66/0.59 multiply(multiply(inverse(multiply(multiply(Z, X), Y)), double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y))), double_divide(double_divide(Y, Z), multiply(inverse(multiply(multiply(Z, X), Y)), double_divide(double_divide(Y, Z), multiply(multiply(Z, X), Y)))))
% 1.66/0.59 = { by lemma 19 }
% 1.66/0.59 inverse(double_divide(Y, Z))
% 1.66/0.59 = { by axiom 1 (multiply) R->L }
% 1.66/0.59 multiply(Z, Y)
% 1.66/0.59
% 1.66/0.59 Lemma 42: double_divide(X, multiply(double_divide(X, Y), double_divide(Z, W))) = multiply(W, multiply(Y, Z)).
% 1.66/0.59 Proof:
% 1.66/0.59 double_divide(X, multiply(double_divide(X, Y), double_divide(Z, W)))
% 1.66/0.59 = { by lemma 30 }
% 1.66/0.59 double_divide(X, multiply(double_divide(Y, X), double_divide(Z, W)))
% 1.66/0.59 = { by lemma 40 }
% 1.66/0.59 double_divide(double_divide(Y, X), multiply(X, double_divide(Z, W)))
% 1.66/0.59 = { by lemma 6 }
% 1.66/0.59 double_divide(double_divide(Y, multiply(multiply(W, Z), Z)), multiply(multiply(multiply(W, Z), Z), double_divide(Z, W)))
% 1.66/0.59 = { by lemma 3 }
% 1.66/0.59 double_divide(double_divide(Y, multiply(multiply(W, Z), Z)), Z)
% 1.66/0.59 = { by lemma 30 R->L }
% 1.66/0.59 double_divide(Z, double_divide(Y, multiply(multiply(W, Z), Z)))
% 1.66/0.59 = { by lemma 22 }
% 1.66/0.59 double_divide(Z, double_divide(Y, multiply(multiply(Z, W), Z)))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 double_divide(Z, double_divide(Y, multiply(Z, multiply(Z, W))))
% 1.66/0.59 = { by lemma 41 }
% 1.66/0.59 multiply(multiply(Z, W), Y)
% 1.66/0.59 = { by lemma 16 }
% 1.66/0.59 multiply(multiply(Z, Y), W)
% 1.66/0.59 = { by lemma 22 }
% 1.66/0.59 multiply(multiply(Y, Z), W)
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(W, multiply(Y, Z))
% 1.66/0.59
% 1.66/0.59 Lemma 43: multiply(X, multiply(Y, Z)) = multiply(Z, multiply(Y, X)).
% 1.66/0.59 Proof:
% 1.66/0.59 multiply(X, multiply(Y, Z))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 multiply(multiply(Y, Z), X)
% 1.66/0.59 = { by lemma 22 R->L }
% 1.66/0.59 multiply(multiply(Z, Y), X)
% 1.66/0.59 = { by lemma 16 R->L }
% 1.66/0.59 multiply(multiply(Z, X), Y)
% 1.66/0.59 = { by lemma 22 R->L }
% 1.66/0.59 multiply(multiply(X, Z), Y)
% 1.66/0.59 = { by lemma 41 R->L }
% 1.66/0.59 double_divide(double_divide(W, X), double_divide(Y, multiply(double_divide(W, X), multiply(X, Z))))
% 1.66/0.59 = { by lemma 42 R->L }
% 1.66/0.59 double_divide(double_divide(W, X), double_divide(Y, double_divide(W, multiply(double_divide(W, X), double_divide(Z, double_divide(W, X))))))
% 1.66/0.59 = { by lemma 19 }
% 1.66/0.59 double_divide(double_divide(W, X), double_divide(Y, double_divide(W, inverse(Z))))
% 1.66/0.59 = { by lemma 19 R->L }
% 1.66/0.59 double_divide(double_divide(W, X), double_divide(Y, double_divide(W, multiply(Y, double_divide(Z, Y)))))
% 1.66/0.59 = { by lemma 39 }
% 1.66/0.59 double_divide(double_divide(W, X), multiply(W, double_divide(Z, Y)))
% 1.66/0.59 = { by lemma 30 R->L }
% 1.66/0.59 double_divide(double_divide(W, X), multiply(W, double_divide(Y, Z)))
% 1.66/0.59 = { by lemma 40 R->L }
% 1.66/0.59 double_divide(W, multiply(double_divide(W, X), double_divide(Y, Z)))
% 1.66/0.59 = { by lemma 42 }
% 1.66/0.59 multiply(Z, multiply(X, Y))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 multiply(Z, multiply(Y, X))
% 1.66/0.59
% 1.66/0.59 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)).
% 1.66/0.59 Proof:
% 1.66/0.59 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.66/0.59 = { by lemma 22 }
% 1.66/0.59 tuple(multiply(inverse(a1), a1), multiply(multiply(b2, inverse(b2)), a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.66/0.59 = { by lemma 16 }
% 1.66/0.59 tuple(multiply(inverse(a1), a1), multiply(multiply(b2, a2), inverse(b2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), multiply(multiply(b2, a2), inverse(b2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), multiply(inverse(b2), multiply(b2, a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.66/0.59 = { by lemma 23 R->L }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), multiply(inverse(b2), multiply(b2, a2)), multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 25 }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), multiply(X, multiply(a2, inverse(X))), multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 27 }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 33 }
% 1.66/0.59 tuple(double_divide(a1, inverse(a1)), a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 43 }
% 1.66/0.59 tuple(double_divide(a1, inverse(a1)), a2, multiply(b3, multiply(a3, c3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 21 R->L }
% 1.66/0.59 tuple(double_divide(a1, inverse(a1)), a2, multiply(b3, multiply(inverse(inverse(c3)), a3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 43 R->L }
% 1.66/0.59 tuple(double_divide(a1, inverse(a1)), a2, multiply(a3, multiply(inverse(inverse(c3)), b3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 21 }
% 1.66/0.59 tuple(double_divide(a1, inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 33 R->L }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 tuple(multiply(a1, inverse(a1)), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 1.66/0.59 = { by lemma 23 }
% 1.66/0.59 tuple(multiply(inverse(a1), a1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 1.66/0.59 = { by lemma 24 R->L }
% 1.66/0.59 tuple(multiply(multiply(Y, double_divide(b1, Y)), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 1.66/0.59 = { by lemma 19 }
% 1.66/0.59 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 1.66/0.59 % SZS output end Proof
% 1.66/0.59
% 1.66/0.59 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------