TSTP Solution File: GRP654+3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP654+3 : TPTP v8.1.2. Released v4.0.0.
% 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 : n015.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 01:19:28 EDT 2023
% Result : Theorem 0.20s 0.61s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP654+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n015.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 00:51:40 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.61 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.61
% 0.20/0.61 % SZS status Theorem
% 0.20/0.61
% 0.20/0.68 % SZS output start Proof
% 0.20/0.68 Take the following subset of the input axioms:
% 0.20/0.68 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 0.20/0.68 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 0.20/0.68 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 0.20/0.68 fof(f04, axiom, ![B2, A2]: rd(mult(A2, B2), B2)=A2).
% 0.20/0.68 fof(f05, axiom, ![C, B2, A2]: mult(A2, mult(B2, mult(A2, C)))=mult(mult(mult(A2, B2), A2), C)).
% 0.20/0.68 fof(goals, conjecture, ![X0, X1]: (mult(X0, ld(X1, X1))=X0 & mult(ld(X1, X1), X0)=X0)).
% 0.20/0.68
% 0.20/0.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.68 fresh(y, y, x1...xn) = u
% 0.20/0.68 C => fresh(s, t, x1...xn) = v
% 0.20/0.68 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.68 variables of u and v.
% 0.20/0.68 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.68 input problem has no model of domain size 1).
% 0.20/0.68
% 0.20/0.68 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.68
% 0.20/0.68 Axiom 1 (f04): rd(mult(X, Y), Y) = X.
% 0.20/0.68 Axiom 2 (f02): ld(X, mult(X, Y)) = Y.
% 0.20/0.68 Axiom 3 (f01): mult(X, ld(X, Y)) = Y.
% 0.20/0.68 Axiom 4 (f03): mult(rd(X, Y), Y) = X.
% 0.20/0.68 Axiom 5 (f05): mult(X, mult(Y, mult(X, Z))) = mult(mult(mult(X, Y), X), Z).
% 0.20/0.68
% 0.20/0.68 Lemma 6: ld(mult(X, Y), mult(Y, mult(ld(Y, X), Z))) = ld(Y, Z).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(mult(X, Y), mult(Y, mult(ld(Y, X), Z)))
% 0.20/0.68 = { by axiom 3 (f01) R->L }
% 0.20/0.68 ld(mult(mult(Y, ld(Y, X)), Y), mult(Y, mult(ld(Y, X), Z)))
% 0.20/0.68 = { by axiom 3 (f01) R->L }
% 0.20/0.68 ld(mult(mult(Y, ld(Y, X)), Y), mult(Y, mult(ld(Y, X), mult(Y, ld(Y, Z)))))
% 0.20/0.68 = { by axiom 5 (f05) }
% 0.20/0.68 ld(mult(mult(Y, ld(Y, X)), Y), mult(mult(mult(Y, ld(Y, X)), Y), ld(Y, Z)))
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 ld(Y, Z)
% 0.20/0.68
% 0.20/0.68 Lemma 7: rd(ld(X, mult(Y, ld(X, Z))), Z) = ld(X, rd(Y, X)).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(ld(X, mult(Y, ld(X, Z))), Z)
% 0.20/0.68 = { by axiom 4 (f03) R->L }
% 0.20/0.68 rd(ld(X, mult(mult(rd(Y, X), X), ld(X, Z))), Z)
% 0.20/0.68 = { by lemma 6 R->L }
% 0.20/0.68 rd(ld(X, mult(mult(rd(Y, X), X), ld(mult(rd(Y, X), X), mult(X, mult(ld(X, rd(Y, X)), Z))))), Z)
% 0.20/0.68 = { by axiom 3 (f01) }
% 0.20/0.68 rd(ld(X, mult(X, mult(ld(X, rd(Y, X)), Z))), Z)
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 rd(mult(ld(X, rd(Y, X)), Z), Z)
% 0.20/0.68 = { by axiom 1 (f04) }
% 0.20/0.68 ld(X, rd(Y, X))
% 0.20/0.68
% 0.20/0.68 Lemma 8: ld(X, mult(Y, mult(ld(Y, rd(X, Y)), Z))) = ld(Y, Z).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(X, mult(Y, mult(ld(Y, rd(X, Y)), Z)))
% 0.20/0.68 = { by axiom 4 (f03) R->L }
% 0.20/0.68 ld(mult(rd(X, Y), Y), mult(Y, mult(ld(Y, rd(X, Y)), Z)))
% 0.20/0.68 = { by lemma 6 }
% 0.20/0.68 ld(Y, Z)
% 0.20/0.68
% 0.20/0.68 Lemma 9: mult(ld(X, rd(X, X)), Y) = ld(X, Y).
% 0.20/0.68 Proof:
% 0.20/0.68 mult(ld(X, rd(X, X)), Y)
% 0.20/0.68 = { by axiom 2 (f02) R->L }
% 0.20/0.68 ld(X, mult(X, mult(ld(X, rd(X, X)), Y)))
% 0.20/0.68 = { by lemma 8 }
% 0.20/0.68 ld(X, Y)
% 0.20/0.68
% 0.20/0.68 Lemma 10: rd(ld(X, Z), Z) = rd(ld(X, Y), Y).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(ld(X, Z), Z)
% 0.20/0.68 = { by lemma 9 R->L }
% 0.20/0.68 rd(mult(ld(X, rd(X, X)), Z), Z)
% 0.20/0.68 = { by axiom 1 (f04) }
% 0.20/0.68 ld(X, rd(X, X))
% 0.20/0.68 = { by axiom 1 (f04) R->L }
% 0.20/0.68 rd(mult(ld(X, rd(X, X)), Y), Y)
% 0.20/0.68 = { by lemma 9 }
% 0.20/0.68 rd(ld(X, Y), Y)
% 0.20/0.68
% 0.20/0.68 Lemma 11: rd(ld(X, Y), Y) = rd(Z, mult(X, Z)).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(ld(X, Y), Y)
% 0.20/0.68 = { by lemma 10 }
% 0.20/0.68 rd(ld(X, mult(X, Z)), mult(X, Z))
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 rd(Z, mult(X, Z))
% 0.20/0.68
% 0.20/0.68 Lemma 12: ld(X, rd(X, X)) = rd(ld(X, Y), Y).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(X, rd(X, X))
% 0.20/0.68 = { by axiom 1 (f04) R->L }
% 0.20/0.68 rd(mult(ld(X, rd(X, X)), Y), Y)
% 0.20/0.68 = { by lemma 9 }
% 0.20/0.68 rd(ld(X, Y), Y)
% 0.20/0.68
% 0.20/0.68 Lemma 13: mult(rd(X, mult(Y, X)), Z) = ld(Y, Z).
% 0.20/0.68 Proof:
% 0.20/0.68 mult(rd(X, mult(Y, X)), Z)
% 0.20/0.68 = { by lemma 11 R->L }
% 0.20/0.68 mult(rd(ld(Y, W), W), Z)
% 0.20/0.68 = { by axiom 2 (f02) R->L }
% 0.20/0.68 ld(Y, mult(Y, mult(rd(ld(Y, W), W), Z)))
% 0.20/0.68 = { by lemma 12 R->L }
% 0.20/0.68 ld(Y, mult(Y, mult(ld(Y, rd(Y, Y)), Z)))
% 0.20/0.68 = { by lemma 8 }
% 0.20/0.68 ld(Y, Z)
% 0.20/0.68
% 0.20/0.68 Lemma 14: ld(rd(X, Y), X) = Y.
% 0.20/0.68 Proof:
% 0.20/0.68 ld(rd(X, Y), X)
% 0.20/0.68 = { by axiom 4 (f03) R->L }
% 0.20/0.68 ld(rd(X, Y), mult(rd(X, Y), Y))
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 Y
% 0.20/0.68
% 0.20/0.68 Lemma 15: ld(rd(X, mult(Y, X)), Z) = mult(Y, Z).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(rd(X, mult(Y, X)), Z)
% 0.20/0.68 = { by axiom 2 (f02) R->L }
% 0.20/0.68 ld(rd(ld(Y, mult(Y, X)), mult(Y, X)), Z)
% 0.20/0.68 = { by lemma 10 R->L }
% 0.20/0.68 ld(rd(ld(Y, mult(Y, Z)), mult(Y, Z)), Z)
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 ld(rd(Z, mult(Y, Z)), Z)
% 0.20/0.68 = { by lemma 14 }
% 0.20/0.68 mult(Y, Z)
% 0.20/0.68
% 0.20/0.68 Lemma 16: rd(ld(X, mult(Y, Z)), mult(X, Z)) = ld(X, rd(Y, X)).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(ld(X, mult(Y, Z)), mult(X, Z))
% 0.20/0.68 = { by axiom 2 (f02) R->L }
% 0.20/0.68 rd(ld(X, mult(Y, ld(X, mult(X, Z)))), mult(X, Z))
% 0.20/0.68 = { by lemma 7 }
% 0.20/0.68 ld(X, rd(Y, X))
% 0.20/0.68
% 0.20/0.68 Lemma 17: ld(X, rd(rd(Y, Z), X)) = rd(ld(X, Y), mult(X, Z)).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(X, rd(rd(Y, Z), X))
% 0.20/0.68 = { by lemma 16 R->L }
% 0.20/0.68 rd(ld(X, mult(rd(Y, Z), Z)), mult(X, Z))
% 0.20/0.68 = { by axiom 4 (f03) }
% 0.20/0.68 rd(ld(X, Y), mult(X, Z))
% 0.20/0.68
% 0.20/0.68 Lemma 18: rd(X, ld(Y, X)) = Y.
% 0.20/0.68 Proof:
% 0.20/0.68 rd(X, ld(Y, X))
% 0.20/0.68 = { by axiom 3 (f01) R->L }
% 0.20/0.68 rd(mult(Y, ld(Y, X)), ld(Y, X))
% 0.20/0.68 = { by axiom 1 (f04) }
% 0.20/0.68 Y
% 0.20/0.68
% 0.20/0.68 Lemma 19: rd(mult(X, X), ld(X, X)) = mult(X, X).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(mult(X, X), ld(X, X))
% 0.20/0.68 = { by lemma 13 R->L }
% 0.20/0.68 rd(mult(X, X), mult(rd(Y, mult(X, Y)), X))
% 0.20/0.68 = { by lemma 15 R->L }
% 0.20/0.68 rd(ld(rd(Y, mult(X, Y)), X), mult(rd(Y, mult(X, Y)), X))
% 0.20/0.68 = { by lemma 17 R->L }
% 0.20/0.68 ld(rd(Y, mult(X, Y)), rd(rd(X, X), rd(Y, mult(X, Y))))
% 0.20/0.68 = { by lemma 11 R->L }
% 0.20/0.68 ld(rd(Y, mult(X, Y)), rd(rd(X, X), rd(ld(X, Z), Z)))
% 0.20/0.68 = { by lemma 12 R->L }
% 0.20/0.68 ld(rd(Y, mult(X, Y)), rd(rd(X, X), ld(X, rd(X, X))))
% 0.20/0.68 = { by lemma 18 }
% 0.20/0.68 ld(rd(Y, mult(X, Y)), X)
% 0.20/0.68 = { by lemma 15 }
% 0.20/0.68 mult(X, X)
% 0.20/0.68
% 0.20/0.68 Lemma 20: mult(mult(X, X), ld(X, X)) = mult(X, X).
% 0.20/0.68 Proof:
% 0.20/0.68 mult(mult(X, X), ld(X, X))
% 0.20/0.68 = { by lemma 19 R->L }
% 0.20/0.68 mult(rd(mult(X, X), ld(X, X)), ld(X, X))
% 0.20/0.68 = { by axiom 4 (f03) }
% 0.20/0.68 mult(X, X)
% 0.20/0.68
% 0.20/0.68 Lemma 21: ld(X, X) = rd(X, X).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(X, X)
% 0.20/0.68 = { by axiom 1 (f04) R->L }
% 0.20/0.68 ld(X, rd(mult(X, X), X))
% 0.20/0.68 = { by lemma 7 R->L }
% 0.20/0.68 rd(ld(X, mult(mult(X, X), ld(X, X))), X)
% 0.20/0.68 = { by lemma 20 }
% 0.20/0.68 rd(ld(X, mult(X, X)), X)
% 0.20/0.68 = { by axiom 2 (f02) }
% 0.20/0.68 rd(X, X)
% 0.20/0.68
% 0.20/0.68 Lemma 22: rd(X, mult(rd(Y, Z), X)) = rd(Z, Y).
% 0.20/0.68 Proof:
% 0.20/0.68 rd(X, mult(rd(Y, Z), X))
% 0.20/0.68 = { by lemma 11 R->L }
% 0.20/0.68 rd(ld(rd(Y, Z), Y), Y)
% 0.20/0.68 = { by lemma 14 }
% 0.20/0.68 rd(Z, Y)
% 0.20/0.68
% 0.20/0.68 Lemma 23: mult(rd(X, Y), Z) = ld(rd(Y, X), Z).
% 0.20/0.68 Proof:
% 0.20/0.68 mult(rd(X, Y), Z)
% 0.20/0.68 = { by lemma 22 R->L }
% 0.20/0.68 mult(rd(W, mult(rd(Y, X), W)), Z)
% 0.20/0.68 = { by lemma 13 }
% 0.20/0.68 ld(rd(Y, X), Z)
% 0.20/0.68
% 0.20/0.68 Lemma 24: ld(mult(X, Y), mult(Y, Z)) = ld(Y, ld(ld(Y, X), Z)).
% 0.20/0.68 Proof:
% 0.20/0.68 ld(mult(X, Y), mult(Y, Z))
% 0.20/0.68 = { by axiom 3 (f01) R->L }
% 0.20/0.68 ld(mult(X, Y), mult(Y, mult(ld(Y, X), ld(ld(Y, X), Z))))
% 0.20/0.68 = { by lemma 6 }
% 0.20/0.68 ld(Y, ld(ld(Y, X), Z))
% 0.20/0.68
% 0.20/0.68 Lemma 25: mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z))) = ld(Y, Z).
% 0.20/0.68 Proof:
% 0.20/0.68 mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z)))
% 0.20/0.68 = { by axiom 2 (f02) R->L }
% 0.20/0.68 ld(Y, mult(Y, mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z)))))
% 0.20/0.68 = { by axiom 5 (f05) }
% 0.20/0.68 ld(Y, mult(mult(mult(Y, X), Y), ld(mult(mult(Y, X), Y), Z)))
% 0.20/0.68 = { by axiom 3 (f01) }
% 0.20/0.68 ld(Y, Z)
% 0.20/0.68
% 0.20/0.69 Lemma 26: mult(ld(X, Y), mult(X, ld(mult(Y, X), Z))) = ld(X, Z).
% 0.20/0.69 Proof:
% 0.20/0.69 mult(ld(X, Y), mult(X, ld(mult(Y, X), Z)))
% 0.20/0.69 = { by axiom 3 (f01) R->L }
% 0.20/0.69 mult(ld(X, Y), mult(X, ld(mult(mult(X, ld(X, Y)), X), Z)))
% 0.20/0.69 = { by lemma 25 }
% 0.20/0.69 ld(X, Z)
% 0.20/0.69
% 0.20/0.69 Lemma 27: mult(X, ld(mult(Y, X), Z)) = ld(ld(X, Y), ld(X, Z)).
% 0.20/0.69 Proof:
% 0.20/0.69 mult(X, ld(mult(Y, X), Z))
% 0.20/0.69 = { by axiom 2 (f02) R->L }
% 0.20/0.69 ld(ld(X, Y), mult(ld(X, Y), mult(X, ld(mult(Y, X), Z))))
% 0.20/0.69 = { by lemma 26 }
% 0.20/0.69 ld(ld(X, Y), ld(X, Z))
% 0.20/0.69
% 0.20/0.69 Lemma 28: ld(X, ld(ld(X, Y), ld(X, Z))) = ld(mult(Y, X), Z).
% 0.20/0.69 Proof:
% 0.20/0.69 ld(X, ld(ld(X, Y), ld(X, Z)))
% 0.20/0.69 = { by lemma 27 R->L }
% 0.20/0.69 ld(X, mult(X, ld(mult(Y, X), Z)))
% 0.20/0.69 = { by axiom 2 (f02) }
% 0.20/0.69 ld(mult(Y, X), Z)
% 0.20/0.69
% 0.20/0.69 Lemma 29: rd(rd(X, X), rd(X, X)) = rd(X, X).
% 0.20/0.69 Proof:
% 0.20/0.69 rd(rd(X, X), rd(X, X))
% 0.20/0.69 = { by lemma 21 R->L }
% 0.20/0.69 ld(rd(X, X), rd(X, X))
% 0.20/0.69 = { by lemma 22 R->L }
% 0.20/0.69 ld(rd(X, X), rd(Y, mult(rd(X, X), Y)))
% 0.20/0.69 = { by lemma 21 R->L }
% 0.20/0.69 ld(ld(X, X), rd(Y, mult(rd(X, X), Y)))
% 0.20/0.69 = { by lemma 21 R->L }
% 0.20/0.69 ld(ld(X, X), rd(Y, mult(ld(X, X), Y)))
% 0.20/0.69 = { by lemma 6 R->L }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), mult(ld(ld(X, X), mult(X, X)), rd(Y, mult(ld(X, X), Y)))))
% 0.20/0.69 = { by lemma 11 R->L }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), mult(ld(ld(X, X), mult(X, X)), rd(ld(ld(X, X), mult(X, X)), mult(X, X)))))
% 0.20/0.69 = { by lemma 18 R->L }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), ld(ld(X, X), mult(X, X))), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z))), ld(mult(ld(ld(X, X), mult(X, X)), rd(ld(ld(X, X), mult(X, X)), mult(X, X))), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), ld(ld(X, X), mult(X, X))), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z)))))))
% 0.20/0.69 = { by lemma 6 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), ld(ld(X, X), mult(X, X))), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z))), ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z)))))
% 0.20/0.69 = { by lemma 14 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(mult(X, X), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z))), ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z)))))
% 0.20/0.69 = { by lemma 23 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(ld(rd(mult(X, X), ld(ld(X, X), mult(X, X))), mult(mult(X, X), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z))), ld(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z)))))
% 0.20/0.69 = { by axiom 2 (f02) }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(ld(rd(mult(X, X), ld(ld(X, X), mult(X, X))), mult(mult(X, X), mult(rd(ld(ld(X, X), mult(X, X)), mult(X, X)), Z))), Z)))
% 0.20/0.69 = { by lemma 23 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), rd(ld(rd(mult(X, X), ld(ld(X, X), mult(X, X))), mult(mult(X, X), ld(rd(mult(X, X), ld(ld(X, X), mult(X, X))), Z))), Z)))
% 0.20/0.69 = { by lemma 7 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), ld(rd(mult(X, X), ld(ld(X, X), mult(X, X))), rd(mult(X, X), rd(mult(X, X), ld(ld(X, X), mult(X, X)))))))
% 0.20/0.69 = { by lemma 18 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), ld(ld(X, X), rd(mult(X, X), rd(mult(X, X), ld(ld(X, X), mult(X, X)))))))
% 0.20/0.69 = { by lemma 18 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(ld(X, X), ld(ld(X, X), rd(mult(X, X), ld(X, X)))))
% 0.20/0.69 = { by lemma 24 }
% 0.20/0.69 ld(ld(X, X), ld(ld(ld(X, X), mult(X, X)), ld(ld(X, X), rd(mult(X, X), ld(X, X)))))
% 0.20/0.69 = { by lemma 28 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), rd(mult(X, X), ld(X, X)))
% 0.20/0.69 = { by lemma 19 }
% 0.20/0.69 ld(mult(mult(X, X), ld(X, X)), mult(X, X))
% 0.20/0.69 = { by lemma 20 }
% 0.20/0.69 ld(mult(X, X), mult(X, X))
% 0.20/0.69 = { by lemma 24 }
% 0.20/0.69 ld(X, ld(ld(X, X), X))
% 0.20/0.69 = { by axiom 2 (f02) R->L }
% 0.20/0.69 ld(X, ld(ld(X, X), ld(X, mult(X, X))))
% 0.20/0.69 = { by lemma 20 R->L }
% 0.20/0.69 ld(X, ld(ld(X, X), ld(X, mult(mult(X, X), ld(X, X)))))
% 0.20/0.69 = { by lemma 27 R->L }
% 0.20/0.69 ld(X, mult(X, ld(mult(X, X), mult(mult(X, X), ld(X, X)))))
% 0.20/0.69 = { by axiom 2 (f02) }
% 0.20/0.69 ld(X, mult(X, ld(X, X)))
% 0.20/0.69 = { by axiom 3 (f01) }
% 0.20/0.69 ld(X, X)
% 0.20/0.69 = { by lemma 21 }
% 0.20/0.69 rd(X, X)
% 0.20/0.69
% 0.20/0.69 Lemma 30: ld(X, rd(ld(X, Y), ld(X, Y))) = ld(mult(Y, X), Y).
% 0.20/0.69 Proof:
% 0.20/0.69 ld(X, rd(ld(X, Y), ld(X, Y)))
% 0.20/0.69 = { by lemma 21 R->L }
% 0.20/0.69 ld(X, ld(ld(X, Y), ld(X, Y)))
% 0.20/0.69 = { by lemma 28 }
% 0.20/0.69 ld(mult(Y, X), Y)
% 0.20/0.69
% 0.20/0.69 Lemma 31: mult(X, rd(ld(X, Y), Y)) = rd(X, X).
% 0.20/0.69 Proof:
% 0.20/0.69 mult(X, rd(ld(X, Y), Y))
% 0.20/0.69 = { by lemma 12 R->L }
% 0.20/0.69 mult(X, ld(X, rd(X, X)))
% 0.20/0.69 = { by axiom 3 (f01) }
% 0.20/0.69 rd(X, X)
% 0.20/0.69
% 0.20/0.69 Lemma 32: rd(rd(X, mult(Y, X)), rd(X, mult(Y, X))) = ld(Y, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 rd(rd(X, mult(Y, X)), rd(X, mult(Y, X)))
% 0.20/0.69 = { by lemma 31 R->L }
% 0.20/0.69 mult(rd(X, mult(Y, X)), rd(ld(rd(X, mult(Y, X)), Z), Z))
% 0.20/0.69 = { by lemma 13 }
% 0.20/0.69 ld(Y, rd(ld(rd(X, mult(Y, X)), Z), Z))
% 0.20/0.69 = { by lemma 11 }
% 0.20/0.69 ld(Y, rd(W, mult(rd(X, mult(Y, X)), W)))
% 0.20/0.69 = { by lemma 22 }
% 0.20/0.69 ld(Y, rd(mult(Y, X), X))
% 0.20/0.69 = { by axiom 1 (f04) }
% 0.20/0.69 ld(Y, Y)
% 0.20/0.69
% 0.20/0.69 Lemma 33: mult(ld(X, X), Y) = ld(ld(X, X), Y).
% 0.20/0.69 Proof:
% 0.20/0.69 mult(ld(X, X), Y)
% 0.20/0.69 = { by lemma 32 R->L }
% 0.20/0.69 mult(rd(rd(Z, mult(X, Z)), rd(Z, mult(X, Z))), Y)
% 0.20/0.69 = { by lemma 23 }
% 0.20/0.69 ld(rd(rd(Z, mult(X, Z)), rd(Z, mult(X, Z))), Y)
% 0.20/0.69 = { by lemma 32 }
% 0.20/0.69 ld(ld(X, X), Y)
% 0.20/0.69
% 0.20/0.69 Lemma 34: ld(mult(X, rd(Y, Y)), X) = rd(Y, Y).
% 0.20/0.69 Proof:
% 0.20/0.69 ld(mult(X, rd(Y, Y)), X)
% 0.20/0.69 = { by axiom 1 (f04) R->L }
% 0.20/0.69 rd(mult(ld(mult(X, rd(Y, Y)), X), Z), Z)
% 0.20/0.69 = { by lemma 22 R->L }
% 0.20/0.69 rd(W, mult(rd(Z, mult(ld(mult(X, rd(Y, Y)), X), Z)), W))
% 0.20/0.69 = { by lemma 11 R->L }
% 0.20/0.69 rd(W, mult(rd(ld(ld(mult(X, rd(Y, Y)), X), V), V), W))
% 0.20/0.69 = { by lemma 12 R->L }
% 0.20/0.69 rd(W, mult(ld(ld(mult(X, rd(Y, Y)), X), rd(ld(mult(X, rd(Y, Y)), X), ld(mult(X, rd(Y, Y)), X))), W))
% 0.20/0.69 = { by lemma 30 R->L }
% 0.20/0.69 rd(W, mult(ld(ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(ld(mult(X, rd(Y, Y)), X), ld(mult(X, rd(Y, Y)), X))), W))
% 0.20/0.69 = { by lemma 30 R->L }
% 0.20/0.69 rd(W, mult(ld(ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(ld(mult(X, rd(Y, Y)), X), ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.69 = { by lemma 29 R->L }
% 0.20/0.69 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(mult(X, rd(Y, Y)), X), ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 30 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 23 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), ld(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 13 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(rd(Y, Y), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(rd(U, mult(rd(Y, Y), U)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 15 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(T, mult(rd(Y, Y), T)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(rd(U, mult(rd(Y, Y), U)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 11 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(T, mult(rd(Y, Y), T)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(rd(ld(rd(Y, Y), S), S), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 11 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(ld(rd(Y, Y), X2), X2), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(rd(ld(rd(Y, Y), S), S), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 12 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(ld(rd(Y, Y), X2), X2), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 12 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), W))
% 0.20/0.70 = { by lemma 17 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y)))))), W))
% 0.20/0.70 = { by lemma 16 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), mult(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(ld(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), Y2), Y2))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(ld(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), Y2), Y2)))), W))
% 0.20/0.70 = { by lemma 31 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(ld(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), Y2), Y2)))), W))
% 0.20/0.70 = { by lemma 11 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(Z2, mult(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), Z2))))), W))
% 0.20/0.70 = { by lemma 29 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(Z2, mult(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), Z2))))), W))
% 0.20/0.70 = { by lemma 22 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), mult(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 9 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(ld(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 12 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(ld(rd(Y, Y), W2), W2), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 11 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(ld(rd(V2, mult(rd(Y, Y), V2)), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 15 }
% 0.20/0.70 rd(W, mult(ld(ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 21 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.70 = { by lemma 21 R->L }
% 0.20/0.70 rd(W, mult(ld(ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(rd(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by lemma 21 R->L }
% 0.20/0.71 rd(W, mult(ld(ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by lemma 33 R->L }
% 0.20/0.71 rd(W, mult(ld(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by lemma 29 R->L }
% 0.20/0.71 rd(W, mult(ld(mult(ld(Y, Y), rd(rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by axiom 2 (f02) R->L }
% 0.20/0.71 rd(W, mult(ld(mult(ld(Y, Y), rd(ld(ld(Y, Y), mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by axiom 3 (f01) R->L }
% 0.20/0.71 rd(W, mult(ld(mult(ld(Y, Y), rd(ld(ld(Y, Y), mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), mult(ld(Y, Y), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by lemma 17 R->L }
% 0.20/0.71 rd(W, mult(ld(mult(ld(Y, Y), ld(ld(Y, Y), rd(rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), ld(Y, Y)))), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by axiom 3 (f01) }
% 0.20/0.71 rd(W, mult(ld(rd(rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X))))), ld(Y, Y)), rd(mult(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))), ld(ld(Y, Y), rd(rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)), rd(ld(rd(Y, Y), X), ld(rd(Y, Y), X)))))), W))
% 0.20/0.71 = { by lemma 14 }
% 0.20/0.71 rd(W, mult(ld(Y, Y), W))
% 0.20/0.71 = { by lemma 21 }
% 0.20/0.71 rd(W, mult(rd(Y, Y), W))
% 0.20/0.71 = { by lemma 22 }
% 0.20/0.71 rd(Y, Y)
% 0.20/0.71
% 0.20/0.71 Lemma 35: mult(X, rd(Y, Y)) = X.
% 0.20/0.71 Proof:
% 0.20/0.71 mult(X, rd(Y, Y))
% 0.20/0.71 = { by lemma 29 R->L }
% 0.20/0.71 mult(X, rd(rd(Y, Y), rd(Y, Y)))
% 0.20/0.71 = { by lemma 21 R->L }
% 0.20/0.71 mult(X, ld(rd(Y, Y), rd(Y, Y)))
% 0.20/0.71 = { by lemma 23 R->L }
% 0.20/0.71 mult(X, mult(rd(Y, Y), rd(Y, Y)))
% 0.20/0.71 = { by lemma 34 R->L }
% 0.20/0.71 mult(X, mult(rd(Y, Y), ld(mult(mult(rd(Y, Y), X), rd(Y, Y)), mult(rd(Y, Y), X))))
% 0.20/0.71 = { by lemma 25 }
% 0.20/0.71 ld(rd(Y, Y), mult(rd(Y, Y), X))
% 0.20/0.71 = { by axiom 2 (f02) }
% 0.20/0.71 X
% 0.20/0.71
% 0.20/0.71 Goal 1 (goals): tuple(mult(ld(x1, x1), x0), mult(x0_2, ld(x1_2, x1_2))) = tuple(x0, x0_2).
% 0.20/0.71 Proof:
% 0.20/0.71 tuple(mult(ld(x1, x1), x0), mult(x0_2, ld(x1_2, x1_2)))
% 0.20/0.71 = { by lemma 33 }
% 0.20/0.71 tuple(ld(ld(x1, x1), x0), mult(x0_2, ld(x1_2, x1_2)))
% 0.20/0.71 = { by lemma 21 }
% 0.20/0.71 tuple(ld(rd(x1, x1), x0), mult(x0_2, ld(x1_2, x1_2)))
% 0.20/0.71 = { by lemma 21 }
% 0.20/0.71 tuple(ld(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2)))
% 0.20/0.71 = { by lemma 35 }
% 0.20/0.71 tuple(ld(rd(x1, x1), x0), x0_2)
% 0.20/0.71 = { by lemma 35 R->L }
% 0.20/0.71 tuple(ld(rd(x1, x1), mult(x0, rd(x1, x1))), x0_2)
% 0.20/0.71 = { by lemma 23 R->L }
% 0.20/0.71 tuple(mult(rd(x1, x1), mult(x0, rd(x1, x1))), x0_2)
% 0.20/0.71 = { by lemma 34 R->L }
% 0.20/0.71 tuple(mult(ld(mult(x0, rd(x1, x1)), x0), mult(x0, rd(x1, x1))), x0_2)
% 0.20/0.71 = { by lemma 35 R->L }
% 0.20/0.71 tuple(mult(ld(mult(x0, rd(x1, x1)), x0), mult(mult(x0, rd(x1, x1)), rd(X, X))), x0_2)
% 0.20/0.71 = { by axiom 2 (f02) R->L }
% 0.20/0.71 tuple(mult(ld(mult(x0, rd(x1, x1)), x0), mult(mult(x0, rd(x1, x1)), ld(mult(x0, mult(x0, rd(x1, x1))), mult(mult(x0, mult(x0, rd(x1, x1))), rd(X, X))))), x0_2)
% 0.20/0.71 = { by lemma 26 }
% 0.20/0.71 tuple(ld(mult(x0, rd(x1, x1)), mult(mult(x0, mult(x0, rd(x1, x1))), rd(X, X))), x0_2)
% 0.20/0.71 = { by lemma 35 }
% 0.20/0.71 tuple(ld(mult(x0, rd(x1, x1)), mult(x0, mult(x0, rd(x1, x1)))), x0_2)
% 0.20/0.71 = { by lemma 35 }
% 0.20/0.71 tuple(ld(x0, mult(x0, mult(x0, rd(x1, x1)))), x0_2)
% 0.20/0.71 = { by lemma 35 }
% 0.20/0.71 tuple(ld(x0, mult(x0, x0)), x0_2)
% 0.20/0.71 = { by axiom 2 (f02) }
% 0.20/0.71 tuple(x0, x0_2)
% 0.20/0.71 % SZS output end Proof
% 0.20/0.71
% 0.20/0.71 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------