TSTP Solution File: GRP685+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP685+1 : 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 : n007.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 01:19:42 EDT 2023
% Result : Theorem 0.20s 0.54s
% Output : Proof 1.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP685+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % 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 : n007.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 00:50:28 EDT 2023
% 0.19/0.34 % CPUTime :
% 0.20/0.54 Command-line arguments: --no-flatten-goal
% 0.20/0.54
% 0.20/0.54 % SZS status Theorem
% 0.20/0.54
% 1.86/0.59 % SZS output start Proof
% 1.86/0.59 Take the following subset of the input axioms:
% 1.86/0.59 fof(f01, axiom, ![A]: ld(A, mult(A, A))=A).
% 1.86/0.59 fof(f02, axiom, ![A2]: rd(mult(A2, A2), A2)=A2).
% 1.86/0.59 fof(f03, axiom, ![B, A2]: mult(A2, ld(A2, B))=ld(A2, mult(A2, B))).
% 1.86/0.59 fof(f04, axiom, ![A2, B2]: mult(rd(A2, B2), B2)=rd(mult(A2, B2), B2)).
% 1.86/0.59 fof(f05, axiom, ![D, C, A2, B2]: ld(ld(A2, B2), mult(ld(A2, B2), mult(C, D)))=mult(ld(A2, mult(A2, C)), D)).
% 1.86/0.59 fof(f06, axiom, ![A2, B2, D2, C2]: rd(mult(mult(A2, B2), rd(C2, D2)), rd(C2, D2))=mult(A2, rd(mult(B2, D2), D2))).
% 1.86/0.59 fof(f07, axiom, ![A2, B2]: ld(A2, mult(A2, ld(B2, B2)))=rd(mult(rd(A2, A2), B2), B2)).
% 1.86/0.59 fof(goals, conjecture, ![X6, X7, X8]: (rd(mult(X6, ld(X7, X8)), ld(X7, X8))=rd(mult(X6, X8), X8) & rd(mult(X6, rd(X7, X8)), rd(X7, X8))=rd(mult(X6, X8), X8))).
% 1.86/0.59
% 1.86/0.59 Now clausify the problem and encode Horn clauses using encoding 3 of
% 1.86/0.59 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 1.86/0.59 We repeatedly replace C & s=t => u=v by the two clauses:
% 1.86/0.59 fresh(y, y, x1...xn) = u
% 1.86/0.59 C => fresh(s, t, x1...xn) = v
% 1.86/0.59 where fresh is a fresh function symbol and x1..xn are the free
% 1.86/0.59 variables of u and v.
% 1.86/0.59 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 1.86/0.59 input problem has no model of domain size 1).
% 1.86/0.59
% 1.86/0.59 The encoding turns the above axioms into the following unit equations and goals:
% 1.86/0.59
% 1.86/0.59 Axiom 1 (f01): ld(X, mult(X, X)) = X.
% 1.86/0.59 Axiom 2 (f02): rd(mult(X, X), X) = X.
% 1.86/0.59 Axiom 3 (f03): mult(X, ld(X, Y)) = ld(X, mult(X, Y)).
% 1.86/0.59 Axiom 4 (f04): mult(rd(X, Y), Y) = rd(mult(X, Y), Y).
% 1.86/0.59 Axiom 5 (f07): ld(X, mult(X, ld(Y, Y))) = rd(mult(rd(X, X), Y), Y).
% 1.86/0.59 Axiom 6 (f05): ld(ld(X, Y), mult(ld(X, Y), mult(Z, W))) = mult(ld(X, mult(X, Z)), W).
% 1.86/0.59 Axiom 7 (f06): rd(mult(mult(X, Y), rd(Z, W)), rd(Z, W)) = mult(X, rd(mult(Y, W), W)).
% 1.86/0.59
% 1.86/0.59 Lemma 8: rd(X, X) = ld(X, X).
% 1.86/0.59 Proof:
% 1.86/0.59 rd(X, X)
% 1.86/0.59 = { by axiom 2 (f02) R->L }
% 1.86/0.59 rd(rd(mult(X, X), X), X)
% 1.86/0.59 = { by axiom 4 (f04) R->L }
% 1.86/0.59 rd(mult(rd(X, X), X), X)
% 1.86/0.59 = { by axiom 5 (f07) R->L }
% 1.86/0.59 ld(X, mult(X, ld(X, X)))
% 1.86/0.59 = { by axiom 3 (f03) }
% 1.86/0.59 ld(X, ld(X, mult(X, X)))
% 1.86/0.59 = { by axiom 1 (f01) }
% 1.86/0.59 ld(X, X)
% 1.86/0.59
% 1.86/0.59 Lemma 9: mult(ld(X, mult(X, Y)), Z) = ld(X, mult(X, mult(Y, Z))).
% 1.86/0.59 Proof:
% 1.86/0.59 mult(ld(X, mult(X, Y)), Z)
% 1.86/0.59 = { by axiom 6 (f05) R->L }
% 1.86/0.59 ld(ld(X, mult(X, X)), mult(ld(X, mult(X, X)), mult(Y, Z)))
% 1.86/0.59 = { by axiom 1 (f01) }
% 1.86/0.59 ld(X, mult(ld(X, mult(X, X)), mult(Y, Z)))
% 1.86/0.59 = { by axiom 1 (f01) }
% 1.86/0.59 ld(X, mult(X, mult(Y, Z)))
% 1.86/0.59
% 1.86/0.59 Lemma 10: ld(X, mult(X, mult(X, Y))) = mult(X, Y).
% 1.86/0.59 Proof:
% 1.86/0.59 ld(X, mult(X, mult(X, Y)))
% 1.86/0.59 = { by lemma 9 R->L }
% 1.86/0.59 mult(ld(X, mult(X, X)), Y)
% 1.86/0.59 = { by axiom 1 (f01) }
% 1.86/0.59 mult(X, Y)
% 1.86/0.59
% 1.86/0.59 Lemma 11: mult(X, rd(mult(Y, Z), Z)) = rd(mult(mult(X, Y), Z), Z).
% 1.86/0.59 Proof:
% 1.86/0.59 mult(X, rd(mult(Y, Z), Z))
% 1.86/0.59 = { by axiom 7 (f06) R->L }
% 1.86/0.59 rd(mult(mult(X, Y), rd(mult(Z, Z), Z)), rd(mult(Z, Z), Z))
% 1.86/0.59 = { by axiom 2 (f02) }
% 1.86/0.59 rd(mult(mult(X, Y), Z), rd(mult(Z, Z), Z))
% 1.86/0.59 = { by axiom 2 (f02) }
% 1.86/0.59 rd(mult(mult(X, Y), Z), Z)
% 1.86/0.59
% 1.86/0.59 Lemma 12: mult(X, ld(Y, Y)) = rd(mult(X, Y), Y).
% 1.86/0.59 Proof:
% 1.86/0.59 mult(X, ld(Y, Y))
% 1.86/0.59 = { by lemma 10 R->L }
% 1.86/0.59 ld(X, mult(X, mult(X, ld(Y, Y))))
% 1.86/0.59 = { by axiom 3 (f03) R->L }
% 1.86/0.59 mult(X, ld(X, mult(X, ld(Y, Y))))
% 1.86/0.59 = { by axiom 5 (f07) }
% 1.86/0.59 mult(X, rd(mult(rd(X, X), Y), Y))
% 1.86/0.59 = { by lemma 11 }
% 1.86/0.59 rd(mult(mult(X, rd(X, X)), Y), Y)
% 1.86/0.59 = { by lemma 8 }
% 1.86/0.59 rd(mult(mult(X, ld(X, X)), Y), Y)
% 1.86/0.59 = { by axiom 3 (f03) }
% 1.86/0.59 rd(mult(ld(X, mult(X, X)), Y), Y)
% 1.86/0.59 = { by lemma 9 }
% 1.86/0.59 rd(ld(X, mult(X, mult(X, Y))), Y)
% 1.86/0.59 = { by lemma 10 }
% 1.86/0.59 rd(mult(X, Y), Y)
% 1.86/0.59
% 1.86/0.59 Lemma 13: mult(ld(X, X), X) = X.
% 1.86/0.59 Proof:
% 1.86/0.59 mult(ld(X, X), X)
% 1.86/0.59 = { by lemma 8 R->L }
% 1.86/0.59 mult(rd(X, X), X)
% 1.86/0.59 = { by axiom 4 (f04) }
% 1.86/0.59 rd(mult(X, X), X)
% 1.86/0.59 = { by axiom 2 (f02) }
% 1.86/0.59 X
% 1.86/0.59
% 1.86/0.59 Lemma 14: rd(mult(mult(X, Y), Y), Y) = mult(X, Y).
% 1.86/0.59 Proof:
% 1.86/0.59 rd(mult(mult(X, Y), Y), Y)
% 1.86/0.59 = { by lemma 11 R->L }
% 1.86/0.59 mult(X, rd(mult(Y, Y), Y))
% 1.86/0.59 = { by axiom 2 (f02) }
% 1.86/0.59 mult(X, Y)
% 1.86/0.59
% 1.86/0.59 Lemma 15: rd(mult(ld(X, X), Y), Y) = ld(X, mult(X, ld(Y, Y))).
% 1.86/0.59 Proof:
% 1.86/0.59 rd(mult(ld(X, X), Y), Y)
% 1.86/0.59 = { by lemma 8 R->L }
% 1.86/0.59 rd(mult(rd(X, X), Y), Y)
% 1.86/0.59 = { by axiom 5 (f07) R->L }
% 1.86/0.59 ld(X, mult(X, ld(Y, Y)))
% 1.86/0.60
% 1.86/0.60 Lemma 16: mult(ld(X, X), Y) = ld(X, mult(X, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 mult(ld(X, X), Y)
% 1.86/0.60 = { by lemma 14 R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), Y), Y), Y)
% 1.86/0.60 = { by axiom 4 (f04) R->L }
% 1.86/0.60 mult(rd(mult(ld(X, X), Y), Y), Y)
% 1.86/0.60 = { by lemma 15 }
% 1.86/0.60 mult(ld(X, mult(X, ld(Y, Y))), Y)
% 1.86/0.60 = { by lemma 9 }
% 1.86/0.60 ld(X, mult(X, mult(ld(Y, Y), Y)))
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 ld(X, mult(X, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 17: ld(ld(X, Y), mult(ld(X, Y), Z)) = ld(X, mult(X, Z)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(ld(X, Y), mult(ld(X, Y), Z))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 ld(ld(X, Y), mult(ld(X, Y), mult(ld(Z, Z), Z)))
% 1.86/0.60 = { by axiom 6 (f05) }
% 1.86/0.60 mult(ld(X, mult(X, ld(Z, Z))), Z)
% 1.86/0.60 = { by lemma 9 }
% 1.86/0.60 ld(X, mult(X, mult(ld(Z, Z), Z)))
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 ld(X, mult(X, Z))
% 1.86/0.60
% 1.86/0.60 Lemma 18: ld(X, ld(X, mult(X, Y))) = ld(X, Y).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(X, ld(X, mult(X, Y)))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 ld(X, mult(X, ld(X, Y)))
% 1.86/0.60 = { by lemma 17 R->L }
% 1.86/0.60 ld(ld(X, Y), mult(ld(X, Y), ld(X, Y)))
% 1.86/0.60 = { by axiom 1 (f01) }
% 1.86/0.60 ld(X, Y)
% 1.86/0.60
% 1.86/0.60 Lemma 19: ld(ld(X, X), ld(X, X)) = ld(X, X).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(ld(X, X), ld(X, X))
% 1.86/0.60 = { by axiom 1 (f01) R->L }
% 1.86/0.60 ld(ld(X, X), ld(X, ld(X, mult(X, X))))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 ld(ld(X, X), ld(X, mult(X, ld(X, X))))
% 1.86/0.60 = { by lemma 16 R->L }
% 1.86/0.60 ld(ld(X, X), mult(ld(X, X), ld(X, X)))
% 1.86/0.60 = { by axiom 1 (f01) }
% 1.86/0.60 ld(X, X)
% 1.86/0.60
% 1.86/0.60 Lemma 20: ld(ld(X, X), ld(X, mult(X, Y))) = ld(X, mult(X, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(ld(X, X), ld(X, mult(X, Y)))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 ld(ld(X, X), mult(X, ld(X, Y)))
% 1.86/0.60 = { by lemma 10 R->L }
% 1.86/0.60 ld(ld(X, X), ld(X, mult(X, mult(X, ld(X, Y)))))
% 1.86/0.60 = { by lemma 16 R->L }
% 1.86/0.60 ld(ld(X, X), mult(ld(X, X), mult(X, ld(X, Y))))
% 1.86/0.60 = { by lemma 9 R->L }
% 1.86/0.60 mult(ld(ld(X, X), mult(ld(X, X), X)), ld(X, Y))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 mult(ld(ld(X, X), mult(ld(X, X), mult(ld(X, X), X))), ld(X, Y))
% 1.86/0.60 = { by lemma 10 }
% 1.86/0.60 mult(mult(ld(X, X), X), ld(X, Y))
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 mult(X, ld(X, Y))
% 1.86/0.60 = { by axiom 3 (f03) }
% 1.86/0.60 ld(X, mult(X, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 21: ld(ld(X, X), Y) = ld(X, mult(X, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(ld(X, X), Y)
% 1.86/0.60 = { by lemma 18 R->L }
% 1.86/0.60 ld(ld(X, X), ld(ld(X, X), mult(ld(X, X), Y)))
% 1.86/0.60 = { by lemma 19 R->L }
% 1.86/0.60 ld(ld(ld(X, X), ld(X, X)), ld(ld(X, X), mult(ld(X, X), Y)))
% 1.86/0.60 = { by lemma 20 }
% 1.86/0.60 ld(ld(X, X), mult(ld(X, X), Y))
% 1.86/0.60 = { by lemma 17 }
% 1.86/0.60 ld(X, mult(X, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 22: rd(mult(X, ld(Y, Y)), ld(Y, Y)) = rd(mult(X, Y), Y).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(mult(X, ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 8 R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), ld(Y, Y)), rd(Y, Y))
% 1.86/0.60 = { by lemma 8 R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), rd(Y, Y)), rd(Y, Y))
% 1.86/0.60 = { by axiom 7 (f06) }
% 1.86/0.60 mult(ld(X, X), rd(mult(X, Y), Y))
% 1.86/0.60 = { by lemma 11 }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), Y), Y)
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 rd(mult(X, Y), Y)
% 1.86/0.60
% 1.86/0.60 Lemma 23: rd(mult(rd(X, ld(Y, Y)), Y), Y) = rd(X, ld(Y, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(mult(rd(X, ld(Y, Y)), Y), Y)
% 1.86/0.60 = { by lemma 22 R->L }
% 1.86/0.60 rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 rd(mult(mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), rd(X, ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 11 R->L }
% 1.86/0.60 mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)))
% 1.86/0.60 = { by axiom 7 (f06) R->L }
% 1.86/0.60 rd(mult(mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), rd(X, ld(Y, Y))), rd(X, ld(Y, Y))), rd(X, ld(Y, Y)))
% 1.86/0.60 = { by lemma 14 }
% 1.86/0.60 mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), rd(X, ld(Y, Y)))
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 rd(X, ld(Y, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 24: rd(rd(X, ld(Y, Y)), ld(Y, Y)) = rd(X, ld(Y, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(rd(X, ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 23 R->L }
% 1.86/0.60 rd(rd(mult(rd(X, ld(Y, Y)), Y), Y), ld(Y, Y))
% 1.86/0.60 = { by lemma 19 R->L }
% 1.86/0.60 rd(rd(mult(rd(X, ld(ld(Y, Y), ld(Y, Y))), Y), Y), ld(Y, Y))
% 1.86/0.60 = { by lemma 12 R->L }
% 1.86/0.60 rd(mult(rd(X, ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 23 }
% 1.86/0.60 rd(X, ld(ld(Y, Y), ld(Y, Y)))
% 1.86/0.60 = { by lemma 19 }
% 1.86/0.60 rd(X, ld(Y, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 25: rd(mult(X, Y), Y) = rd(X, ld(Y, Y)).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(mult(X, Y), Y)
% 1.86/0.60 = { by lemma 22 R->L }
% 1.86/0.60 rd(mult(X, ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by axiom 4 (f04) R->L }
% 1.86/0.60 mult(rd(X, ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 24 R->L }
% 1.86/0.60 mult(rd(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by axiom 4 (f04) }
% 1.86/0.60 rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
% 1.86/0.60 = { by lemma 22 }
% 1.86/0.60 rd(mult(rd(X, ld(Y, Y)), Y), Y)
% 1.86/0.60 = { by lemma 23 }
% 1.86/0.60 rd(X, ld(Y, Y))
% 1.86/0.60
% 1.86/0.60 Lemma 26: ld(mult(X, Y), mult(mult(X, Y), Z)) = ld(X, mult(X, Z)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(mult(X, Y), mult(mult(X, Y), Z))
% 1.86/0.60 = { by lemma 10 R->L }
% 1.86/0.60 ld(mult(X, Y), mult(ld(X, mult(X, mult(X, Y))), Z))
% 1.86/0.60 = { by lemma 10 R->L }
% 1.86/0.60 ld(ld(X, mult(X, mult(X, Y))), mult(ld(X, mult(X, mult(X, Y))), Z))
% 1.86/0.60 = { by lemma 17 }
% 1.86/0.60 ld(X, mult(X, Z))
% 1.86/0.60
% 1.86/0.60 Lemma 27: ld(X, mult(X, ld(Y, mult(Y, Z)))) = ld(X, mult(X, Z)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(X, mult(X, ld(Y, mult(Y, Z))))
% 1.86/0.60 = { by lemma 26 R->L }
% 1.86/0.60 ld(mult(X, ld(Y, Y)), mult(mult(X, ld(Y, Y)), ld(Y, mult(Y, Z))))
% 1.86/0.60 = { by lemma 26 R->L }
% 1.86/0.60 ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, mult(Y, Z))))
% 1.86/0.60 = { by lemma 16 R->L }
% 1.86/0.60 ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), Z)))
% 1.86/0.60 = { by lemma 9 R->L }
% 1.86/0.60 mult(ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 12 }
% 1.86/0.60 mult(ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), rd(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), Y), Y)), Z)
% 1.86/0.60 = { by lemma 25 }
% 1.86/0.60 mult(ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), rd(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 14 }
% 1.86/0.60 mult(ld(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(X, ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 12 }
% 1.86/0.60 mult(ld(rd(mult(mult(X, ld(Y, Y)), Y), Y), mult(X, ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 11 R->L }
% 1.86/0.60 mult(ld(mult(X, rd(mult(ld(Y, Y), Y), Y)), mult(X, ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 mult(ld(mult(X, rd(Y, Y)), mult(X, ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 8 }
% 1.86/0.60 mult(ld(mult(X, ld(Y, Y)), mult(X, ld(Y, Y))), Z)
% 1.86/0.60 = { by lemma 16 }
% 1.86/0.60 ld(mult(X, ld(Y, Y)), mult(mult(X, ld(Y, Y)), Z))
% 1.86/0.60 = { by lemma 26 }
% 1.86/0.60 ld(X, mult(X, Z))
% 1.86/0.60
% 1.86/0.60 Lemma 28: ld(X, ld(Y, mult(Y, Z))) = ld(X, Z).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(X, ld(Y, mult(Y, Z)))
% 1.86/0.60 = { by lemma 18 R->L }
% 1.86/0.60 ld(X, ld(X, mult(X, ld(Y, mult(Y, Z)))))
% 1.86/0.60 = { by lemma 27 }
% 1.86/0.60 ld(X, ld(X, mult(X, Z)))
% 1.86/0.60 = { by lemma 18 }
% 1.86/0.60 ld(X, Z)
% 1.86/0.60
% 1.86/0.60 Lemma 29: ld(rd(X, Y), mult(rd(X, Y), Z)) = ld(X, mult(X, Z)).
% 1.86/0.60 Proof:
% 1.86/0.60 ld(rd(X, Y), mult(rd(X, Y), Z))
% 1.86/0.60 = { by lemma 26 R->L }
% 1.86/0.60 ld(mult(rd(X, Y), Y), mult(mult(rd(X, Y), Y), Z))
% 1.86/0.60 = { by axiom 4 (f04) }
% 1.86/0.60 ld(rd(mult(X, Y), Y), mult(mult(rd(X, Y), Y), Z))
% 1.86/0.60 = { by lemma 25 }
% 1.86/0.60 ld(rd(X, ld(Y, Y)), mult(mult(rd(X, Y), Y), Z))
% 1.86/0.60 = { by axiom 4 (f04) }
% 1.86/0.60 ld(rd(X, ld(Y, Y)), mult(rd(mult(X, Y), Y), Z))
% 1.86/0.60 = { by lemma 12 R->L }
% 1.86/0.60 ld(rd(X, ld(Y, Y)), mult(mult(X, ld(Y, Y)), Z))
% 1.86/0.60 = { by lemma 25 R->L }
% 1.86/0.60 ld(rd(mult(X, Y), Y), mult(mult(X, ld(Y, Y)), Z))
% 1.86/0.60 = { by lemma 12 R->L }
% 1.86/0.60 ld(mult(X, ld(Y, Y)), mult(mult(X, ld(Y, Y)), Z))
% 1.86/0.60 = { by lemma 26 }
% 1.86/0.60 ld(X, mult(X, Z))
% 1.86/0.60
% 1.86/0.60 Lemma 30: rd(mult(X, mult(Y, Z)), mult(Y, Z)) = rd(mult(X, Z), Z).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(mult(X, mult(Y, Z)), mult(Y, Z))
% 1.86/0.60 = { by lemma 14 R->L }
% 1.86/0.60 rd(mult(X, mult(Y, Z)), rd(mult(mult(Y, Z), Z), Z))
% 1.86/0.60 = { by lemma 14 R->L }
% 1.86/0.60 rd(mult(X, rd(mult(mult(Y, Z), Z), Z)), rd(mult(mult(Y, Z), Z), Z))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), rd(mult(mult(Y, Z), Z), Z)), rd(mult(mult(Y, Z), Z), Z))
% 1.86/0.60 = { by axiom 7 (f06) }
% 1.86/0.60 mult(ld(X, X), rd(mult(X, Z), Z))
% 1.86/0.60 = { by axiom 7 (f06) R->L }
% 1.86/0.60 rd(mult(mult(ld(X, X), X), rd(mult(Z, Z), Z)), rd(mult(Z, Z), Z))
% 1.86/0.60 = { by lemma 13 }
% 1.86/0.60 rd(mult(X, rd(mult(Z, Z), Z)), rd(mult(Z, Z), Z))
% 1.86/0.60 = { by axiom 2 (f02) }
% 1.86/0.60 rd(mult(X, Z), rd(mult(Z, Z), Z))
% 1.86/0.60 = { by axiom 2 (f02) }
% 1.86/0.60 rd(mult(X, Z), Z)
% 1.86/0.60
% 1.86/0.60 Lemma 31: rd(mult(X, Y), ld(Z, mult(Z, Y))) = rd(X, ld(ld(Z, Y), ld(Z, Y))).
% 1.86/0.60 Proof:
% 1.86/0.60 rd(mult(X, Y), ld(Z, mult(Z, Y)))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 rd(mult(X, Y), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by lemma 10 R->L }
% 1.86/0.60 rd(ld(X, mult(X, mult(X, Y))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 rd(mult(X, ld(X, mult(X, Y))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by lemma 27 R->L }
% 1.86/0.60 rd(mult(X, ld(X, mult(X, ld(Z, mult(Z, Y))))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by axiom 3 (f03) }
% 1.86/0.60 rd(ld(X, mult(X, mult(X, ld(Z, mult(Z, Y))))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by lemma 10 }
% 1.86/0.60 rd(mult(X, ld(Z, mult(Z, Y))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by axiom 3 (f03) R->L }
% 1.86/0.60 rd(mult(X, mult(Z, ld(Z, Y))), mult(Z, ld(Z, Y)))
% 1.86/0.60 = { by lemma 30 }
% 1.86/0.60 rd(mult(X, ld(Z, Y)), ld(Z, Y))
% 1.86/0.60 = { by lemma 25 }
% 1.86/0.60 rd(X, ld(ld(Z, Y), ld(Z, Y)))
% 1.86/0.60
% 1.86/0.60 Goal 1 (goals): tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), rd(mult(x6, rd(x7, x8)), rd(x7, x8))) = tuple(rd(mult(x6_2, x8_2), x8_2), rd(mult(x6, x8), x8)).
% 1.86/0.60 Proof:
% 1.86/0.60 tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), rd(mult(x6, rd(x7, x8)), rd(x7, x8)))
% 1.86/0.60 = { by lemma 13 R->L }
% 1.86/0.60 tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), rd(mult(mult(ld(x6, x6), x6), rd(x7, x8)), rd(x7, x8)))
% 1.86/0.60 = { by axiom 7 (f06) }
% 1.86/0.60 tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), mult(ld(x6, x6), rd(mult(x6, x8), x8)))
% 1.86/0.60 = { by lemma 16 }
% 1.86/0.60 tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), ld(x6, mult(x6, rd(mult(x6, x8), x8))))
% 1.86/0.60 = { by lemma 11 }
% 1.86/0.60 tuple(rd(mult(x6_2, ld(x7_2, x8_2)), ld(x7_2, x8_2)), ld(x6, rd(mult(mult(x6, x6), x8), x8)))
% 1.86/0.60 = { by lemma 25 }
% 1.86/0.60 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), ld(x6, rd(mult(mult(x6, x6), x8), x8)))
% 1.86/0.60 = { by lemma 11 R->L }
% 1.86/0.60 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), ld(x6, mult(x6, rd(mult(x6, x8), x8))))
% 1.86/0.61 = { by lemma 12 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), ld(x6, mult(x6, mult(x6, ld(x8, x8)))))
% 1.86/0.61 = { by lemma 10 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), mult(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 12 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), rd(mult(x6, x8), x8))
% 1.86/0.61 = { by lemma 25 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, x8_2), ld(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 31 R->L }
% 1.86/0.61 tuple(rd(mult(x6_2, x8_2), ld(x7_2, mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 20 R->L }
% 1.86/0.61 tuple(rd(mult(x6_2, x8_2), ld(ld(x7_2, x7_2), ld(x7_2, mult(x7_2, x8_2)))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 16 R->L }
% 1.86/0.61 tuple(rd(mult(x6_2, x8_2), ld(ld(x7_2, x7_2), mult(ld(x7_2, x7_2), x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 31 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(ld(x7_2, x7_2), x8_2), ld(ld(x7_2, x7_2), x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 21 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), ld(ld(x7_2, x7_2), x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 21 }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), ld(x7_2, mult(x7_2, x8_2)))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 28 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), ld(x8_2, mult(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by axiom 3 (f03) R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), mult(x8_2, ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 28 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), ld(rd(x7_2, X), mult(rd(x7_2, X), mult(x8_2, ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 9 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(x7_2, mult(x7_2, x8_2)), mult(ld(rd(x7_2, X), mult(rd(x7_2, X), x8_2)), ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 29 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(ld(rd(x7_2, X), mult(rd(x7_2, X), x8_2)), mult(ld(rd(x7_2, X), mult(rd(x7_2, X), x8_2)), ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 17 }
% 1.86/0.61 tuple(rd(x6_2, ld(rd(x7_2, X), mult(rd(x7_2, X), ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 29 }
% 1.86/0.61 tuple(rd(x6_2, ld(x7_2, mult(x7_2, ld(x8_2, ld(x7_2, mult(x7_2, x8_2)))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 28 }
% 1.86/0.61 tuple(rd(x6_2, ld(x7_2, mult(x7_2, ld(x8_2, x8_2)))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 26 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(mult(x7_2, x8_2), mult(mult(x7_2, x8_2), ld(x8_2, x8_2)))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 15 R->L }
% 1.86/0.61 tuple(rd(x6_2, rd(mult(ld(mult(x7_2, x8_2), mult(x7_2, x8_2)), x8_2), x8_2)), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 30 R->L }
% 1.86/0.61 tuple(rd(x6_2, rd(mult(ld(mult(x7_2, x8_2), mult(x7_2, x8_2)), mult(x7_2, x8_2)), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 13 }
% 1.86/0.61 tuple(rd(x6_2, rd(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 8 }
% 1.86/0.61 tuple(rd(x6_2, ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 24 R->L }
% 1.86/0.61 tuple(rd(rd(x6_2, ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 25 R->L }
% 1.86/0.61 tuple(rd(rd(mult(x6_2, mult(x7_2, x8_2)), mult(x7_2, x8_2)), ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 12 R->L }
% 1.86/0.61 tuple(rd(mult(x6_2, ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 25 R->L }
% 1.86/0.61 tuple(rd(mult(mult(x6_2, ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), mult(x7_2, x8_2)), mult(x7_2, x8_2)), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 11 R->L }
% 1.86/0.61 tuple(mult(x6_2, rd(mult(ld(mult(x7_2, x8_2), mult(x7_2, x8_2)), mult(x7_2, x8_2)), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 15 }
% 1.86/0.61 tuple(mult(x6_2, ld(mult(x7_2, x8_2), mult(mult(x7_2, x8_2), ld(mult(x7_2, x8_2), mult(x7_2, x8_2))))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 12 }
% 1.86/0.61 tuple(mult(x6_2, ld(mult(x7_2, x8_2), rd(mult(mult(x7_2, x8_2), mult(x7_2, x8_2)), mult(x7_2, x8_2)))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by axiom 2 (f02) }
% 1.86/0.61 tuple(mult(x6_2, ld(mult(x7_2, x8_2), mult(x7_2, x8_2))), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 12 }
% 1.86/0.61 tuple(rd(mult(x6_2, mult(x7_2, x8_2)), mult(x7_2, x8_2)), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 30 }
% 1.86/0.61 tuple(rd(mult(x6_2, x8_2), x8_2), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 25 }
% 1.86/0.61 tuple(rd(x6_2, ld(x8_2, x8_2)), rd(x6, ld(x8, x8)))
% 1.86/0.61 = { by lemma 25 R->L }
% 1.86/0.61 tuple(rd(x6_2, ld(x8_2, x8_2)), rd(mult(x6, x8), x8))
% 1.86/0.61 = { by lemma 25 R->L }
% 1.86/0.61 tuple(rd(mult(x6_2, x8_2), x8_2), rd(mult(x6, x8), x8))
% 1.86/0.61 % SZS output end Proof
% 1.86/0.61
% 1.86/0.61 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------