TSTP Solution File: GRP660+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP660+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 : n021.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 01:19:32 EDT 2023
% Result : Theorem 0.20s 0.48s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP660+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n021.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Mon Aug 28 22:37:58 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.48 Command-line arguments: --no-flatten-goal
% 0.20/0.48
% 0.20/0.48 % SZS status Theorem
% 0.20/0.48
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Take the following subset of the input axioms:
% 0.20/0.50 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 0.20/0.50 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 0.20/0.50 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 0.20/0.50 fof(f04, axiom, ![B2, A2]: rd(mult(A2, B2), B2)=A2).
% 0.20/0.51 fof(f05, axiom, ![C, B2, A2]: mult(mult(mult(A2, B2), C), A2)=mult(A2, mult(B2, mult(C, A2)))).
% 0.20/0.51 fof(goals, conjecture, ?[X0]: ![X1]: (mult(X1, X0)=X1 & mult(X0, X1)=X1)).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (f02): ld(X, mult(X, Y)) = Y.
% 0.20/0.51 Axiom 2 (f04): rd(mult(X, Y), Y) = X.
% 0.20/0.51 Axiom 3 (f01): mult(X, ld(X, Y)) = Y.
% 0.20/0.51 Axiom 4 (f03): mult(rd(X, Y), Y) = X.
% 0.20/0.51 Axiom 5 (f05): mult(mult(mult(X, Y), Z), X) = mult(X, mult(Y, mult(Z, X))).
% 0.20/0.51
% 0.20/0.51 Lemma 6: ld(rd(X, Y), X) = Y.
% 0.20/0.51 Proof:
% 0.20/0.51 ld(rd(X, Y), X)
% 0.20/0.51 = { by axiom 4 (f03) R->L }
% 0.20/0.51 ld(rd(X, Y), mult(rd(X, Y), Y))
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 Y
% 0.20/0.51
% 0.20/0.51 Lemma 7: rd(mult(X, mult(Y, mult(Z, X))), X) = mult(mult(X, Y), Z).
% 0.20/0.51 Proof:
% 0.20/0.51 rd(mult(X, mult(Y, mult(Z, X))), X)
% 0.20/0.51 = { by axiom 5 (f05) R->L }
% 0.20/0.51 rd(mult(mult(mult(X, Y), Z), X), X)
% 0.20/0.51 = { by axiom 2 (f04) }
% 0.20/0.51 mult(mult(X, Y), Z)
% 0.20/0.51
% 0.20/0.51 Lemma 8: mult(mult(X, rd(Y, mult(Z, X))), Z) = rd(mult(X, Y), X).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(mult(X, rd(Y, mult(Z, X))), Z)
% 0.20/0.51 = { by lemma 7 R->L }
% 0.20/0.51 rd(mult(X, mult(rd(Y, mult(Z, X)), mult(Z, X))), X)
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 rd(mult(X, Y), X)
% 0.20/0.51
% 0.20/0.51 Lemma 9: rd(X, mult(Y, X)) = ld(X, rd(X, Y)).
% 0.20/0.51 Proof:
% 0.20/0.51 rd(X, mult(Y, X))
% 0.20/0.51 = { by axiom 1 (f02) R->L }
% 0.20/0.51 ld(X, mult(X, rd(X, mult(Y, X))))
% 0.20/0.51 = { by axiom 2 (f04) R->L }
% 0.20/0.51 ld(X, rd(mult(mult(X, rd(X, mult(Y, X))), Y), Y))
% 0.20/0.51 = { by lemma 8 }
% 0.20/0.51 ld(X, rd(rd(mult(X, X), X), Y))
% 0.20/0.51 = { by axiom 2 (f04) }
% 0.20/0.51 ld(X, rd(X, Y))
% 0.20/0.51
% 0.20/0.51 Lemma 10: ld(X, rd(X, rd(Y, X))) = rd(X, Y).
% 0.20/0.51 Proof:
% 0.20/0.51 ld(X, rd(X, rd(Y, X)))
% 0.20/0.51 = { by lemma 9 R->L }
% 0.20/0.51 rd(X, mult(rd(Y, X), X))
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 rd(X, Y)
% 0.20/0.51
% 0.20/0.51 Lemma 11: mult(mult(X, Y), rd(Z, X)) = rd(mult(X, mult(Y, Z)), X).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(mult(X, Y), rd(Z, X))
% 0.20/0.51 = { by lemma 7 R->L }
% 0.20/0.51 rd(mult(X, mult(Y, mult(rd(Z, X), X))), X)
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 rd(mult(X, mult(Y, Z)), X)
% 0.20/0.51
% 0.20/0.51 Lemma 12: mult(X, mult(ld(X, Y), mult(Z, X))) = mult(mult(Y, Z), X).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(X, mult(ld(X, Y), mult(Z, X)))
% 0.20/0.51 = { by axiom 5 (f05) R->L }
% 0.20/0.51 mult(mult(mult(X, ld(X, Y)), Z), X)
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 mult(mult(Y, Z), X)
% 0.20/0.51
% 0.20/0.51 Lemma 13: rd(ld(X, mult(Y, X)), Z) = ld(X, rd(Y, rd(Z, X))).
% 0.20/0.51 Proof:
% 0.20/0.51 rd(ld(X, mult(Y, X)), Z)
% 0.20/0.51 = { by lemma 6 R->L }
% 0.20/0.51 rd(ld(X, mult(Y, ld(rd(Z, X), Z))), Z)
% 0.20/0.51 = { by lemma 6 R->L }
% 0.20/0.51 rd(ld(ld(rd(Z, X), Z), mult(Y, ld(rd(Z, X), Z))), Z)
% 0.20/0.51 = { by axiom 4 (f03) R->L }
% 0.20/0.51 rd(ld(ld(rd(Z, X), Z), mult(mult(rd(Y, rd(Z, X)), rd(Z, X)), ld(rd(Z, X), Z))), Z)
% 0.20/0.51 = { by lemma 12 R->L }
% 0.20/0.51 rd(ld(ld(rd(Z, X), Z), mult(ld(rd(Z, X), Z), mult(ld(ld(rd(Z, X), Z), rd(Y, rd(Z, X))), mult(rd(Z, X), ld(rd(Z, X), Z))))), Z)
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 rd(mult(ld(ld(rd(Z, X), Z), rd(Y, rd(Z, X))), mult(rd(Z, X), ld(rd(Z, X), Z))), Z)
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 rd(mult(ld(ld(rd(Z, X), Z), rd(Y, rd(Z, X))), Z), Z)
% 0.20/0.51 = { by lemma 6 }
% 0.20/0.51 rd(mult(ld(X, rd(Y, rd(Z, X))), Z), Z)
% 0.20/0.51 = { by axiom 2 (f04) }
% 0.20/0.51 ld(X, rd(Y, rd(Z, X)))
% 0.20/0.51
% 0.20/0.51 Lemma 14: rd(X, X) = ld(X, X).
% 0.20/0.51 Proof:
% 0.20/0.51 rd(X, X)
% 0.20/0.51 = { by lemma 6 R->L }
% 0.20/0.51 ld(rd(X, rd(X, X)), X)
% 0.20/0.51 = { by lemma 6 R->L }
% 0.20/0.51 ld(ld(rd(rd(X, X), rd(X, rd(X, X))), rd(X, X)), X)
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 ld(ld(mult(rd(X, X), ld(rd(X, X), rd(rd(X, X), rd(X, rd(X, X))))), rd(X, X)), X)
% 0.20/0.51 = { by lemma 10 }
% 0.20/0.51 ld(ld(mult(rd(X, X), rd(rd(X, X), X)), rd(X, X)), X)
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 ld(ld(mult(mult(X, ld(X, rd(X, X))), rd(rd(X, X), X)), rd(X, X)), X)
% 0.20/0.51 = { by lemma 11 }
% 0.20/0.51 ld(ld(rd(mult(X, mult(ld(X, rd(X, X)), rd(X, X))), X), rd(X, X)), X)
% 0.20/0.51 = { by axiom 1 (f02) R->L }
% 0.20/0.51 ld(ld(rd(ld(rd(X, X), mult(rd(X, X), mult(X, mult(ld(X, rd(X, X)), rd(X, X))))), X), rd(X, X)), X)
% 0.20/0.51 = { by axiom 5 (f05) R->L }
% 0.20/0.51 ld(ld(rd(ld(rd(X, X), mult(mult(mult(rd(X, X), X), ld(X, rd(X, X))), rd(X, X))), X), rd(X, X)), X)
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 ld(ld(rd(ld(rd(X, X), mult(mult(X, ld(X, rd(X, X))), rd(X, X))), X), rd(X, X)), X)
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 ld(ld(rd(ld(rd(X, X), mult(rd(X, X), rd(X, X))), X), rd(X, X)), X)
% 0.20/0.51 = { by lemma 13 }
% 0.20/0.51 ld(ld(ld(rd(X, X), rd(rd(X, X), rd(X, rd(X, X)))), rd(X, X)), X)
% 0.20/0.51 = { by lemma 9 R->L }
% 0.20/0.51 ld(ld(rd(rd(X, X), mult(rd(X, rd(X, X)), rd(X, X))), rd(X, X)), X)
% 0.20/0.51 = { by lemma 6 }
% 0.20/0.51 ld(mult(rd(X, rd(X, X)), rd(X, X)), X)
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 ld(X, X)
% 0.20/0.51
% 0.20/0.51 Lemma 15: mult(X, mult(Y, mult(ld(mult(X, Y), Z), X))) = mult(Z, X).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(X, mult(Y, mult(ld(mult(X, Y), Z), X)))
% 0.20/0.51 = { by axiom 5 (f05) R->L }
% 0.20/0.51 mult(mult(mult(X, Y), ld(mult(X, Y), Z)), X)
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 mult(Z, X)
% 0.20/0.51
% 0.20/0.51 Lemma 16: mult(X, mult(ld(mult(Y, X), Z), Y)) = ld(Y, mult(Z, Y)).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(X, mult(ld(mult(Y, X), Z), Y))
% 0.20/0.51 = { by axiom 1 (f02) R->L }
% 0.20/0.51 ld(Y, mult(Y, mult(X, mult(ld(mult(Y, X), Z), Y))))
% 0.20/0.51 = { by lemma 15 }
% 0.20/0.51 ld(Y, mult(Z, Y))
% 0.20/0.51
% 0.20/0.51 Lemma 17: mult(ld(mult(X, Y), Z), X) = ld(Y, ld(X, mult(Z, X))).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(ld(mult(X, Y), Z), X)
% 0.20/0.51 = { by axiom 1 (f02) R->L }
% 0.20/0.51 ld(Y, mult(Y, mult(ld(mult(X, Y), Z), X)))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 ld(Y, ld(X, mult(Z, X)))
% 0.20/0.51
% 0.20/0.51 Lemma 18: mult(rd(X, X), rd(X, X)) = ld(X, X).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(rd(X, X), rd(X, X))
% 0.20/0.51 = { by lemma 10 R->L }
% 0.20/0.51 mult(ld(X, rd(X, rd(X, X))), rd(X, X))
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(mult(rd(X, X), ld(rd(X, X), X)), rd(X, rd(X, X))), rd(X, X))
% 0.20/0.51 = { by lemma 17 }
% 0.20/0.51 ld(ld(rd(X, X), X), ld(rd(X, X), mult(rd(X, rd(X, X)), rd(X, X))))
% 0.20/0.51 = { by axiom 4 (f03) }
% 0.20/0.51 ld(ld(rd(X, X), X), ld(rd(X, X), X))
% 0.20/0.51 = { by lemma 6 }
% 0.20/0.51 ld(X, ld(rd(X, X), X))
% 0.20/0.51 = { by lemma 6 }
% 0.20/0.51 ld(X, X)
% 0.20/0.51
% 0.20/0.51 Lemma 19: ld(ld(X, X), ld(X, X)) = ld(X, X).
% 0.20/0.51 Proof:
% 0.20/0.51 ld(ld(X, X), ld(X, X))
% 0.20/0.51 = { by lemma 14 R->L }
% 0.20/0.51 ld(rd(X, X), ld(X, X))
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 ld(rd(X, X), mult(rd(X, X), rd(X, X)))
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 rd(X, X)
% 0.20/0.51 = { by lemma 14 }
% 0.20/0.51 ld(X, X)
% 0.20/0.51
% 0.20/0.51 Lemma 20: mult(ld(X, X), Y) = rd(Y, ld(X, X)).
% 0.20/0.51 Proof:
% 0.20/0.51 mult(ld(X, X), Y)
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(X, X), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 19 R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(X, X)), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(X, ld(X, X)), X)), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(X, ld(X, X)), mult(X, ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(mult(X, ld(X, X)), ld(X, X)), mult(X, ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 15 R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(mult(X, ld(X, X)), ld(X, X)), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 15 R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))), ld(X, X)), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 2 (f04) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))), ld(X, X)), rd(mult(mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))))), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 11 R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))), ld(X, X)), mult(mult(mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))), ld(X, X)), rd(mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), rd(mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))), mult(ld(X, X), mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X)))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 9 }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))), rd(mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))), ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(ld(X, X), mult(X, ld(X, X))), rd(mult(Z, mult(ld(mult(ld(X, X), Z), X), ld(X, X))), ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(ld(X, X), mult(X, ld(X, X))), rd(ld(ld(X, X), mult(X, ld(X, X))), ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 9 R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), rd(ld(ld(X, X), mult(X, ld(X, X))), mult(ld(X, X), ld(ld(X, X), mult(X, ld(X, X)))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), rd(ld(ld(X, X), mult(X, ld(X, X))), mult(X, ld(X, X)))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 13 }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(X, X), rd(X, rd(mult(X, ld(X, X)), ld(X, X))))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 2 (f04) }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(X, X), rd(X, X))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 14 }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(X, X), ld(X, X))), ld(ld(X, X), Y)))
% 0.20/0.51 = { by axiom 4 (f03) R->L }
% 0.20/0.51 mult(ld(X, X), mult(ld(ld(X, X), ld(ld(X, X), ld(X, X))), mult(rd(ld(ld(X, X), Y), ld(X, X)), ld(X, X))))
% 0.20/0.51 = { by lemma 12 }
% 0.20/0.51 mult(mult(ld(ld(X, X), ld(X, X)), rd(ld(ld(X, X), Y), ld(X, X))), ld(X, X))
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(mult(ld(ld(X, X), ld(X, X)), rd(ld(ld(X, X), Y), mult(ld(X, X), ld(ld(X, X), ld(X, X))))), ld(X, X))
% 0.20/0.51 = { by lemma 8 }
% 0.20/0.51 rd(mult(ld(ld(X, X), ld(X, X)), ld(ld(X, X), Y)), ld(ld(X, X), ld(X, X)))
% 0.20/0.51 = { by lemma 19 }
% 0.20/0.51 rd(mult(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(X, X)))
% 0.20/0.51 = { by lemma 19 }
% 0.20/0.51 rd(mult(ld(X, X), ld(ld(X, X), Y)), ld(X, X))
% 0.20/0.51 = { by axiom 3 (f01) }
% 0.20/0.51 rd(Y, ld(X, X))
% 0.20/0.51
% 0.20/0.51 Lemma 21: ld(ld(X, X), Y) = Y.
% 0.20/0.51 Proof:
% 0.20/0.51 ld(ld(X, X), Y)
% 0.20/0.51 = { by axiom 3 (f01) R->L }
% 0.20/0.51 mult(ld(X, X), ld(ld(X, X), ld(ld(X, X), Y)))
% 0.20/0.51 = { by lemma 20 }
% 0.20/0.51 rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(X, X))
% 0.20/0.51 = { by axiom 4 (f03) R->L }
% 0.20/0.51 rd(ld(ld(X, X), ld(ld(X, X), mult(rd(Y, ld(X, X)), ld(X, X)))), ld(X, X))
% 0.20/0.51 = { by lemma 17 R->L }
% 0.20/0.51 rd(mult(ld(mult(ld(X, X), ld(X, X)), rd(Y, ld(X, X))), ld(X, X)), ld(X, X))
% 0.20/0.51 = { by axiom 2 (f04) }
% 0.20/0.51 ld(mult(ld(X, X), ld(X, X)), rd(Y, ld(X, X)))
% 0.20/0.51 = { by lemma 14 R->L }
% 0.20/0.51 ld(mult(ld(X, X), rd(X, X)), rd(Y, ld(X, X)))
% 0.20/0.51 = { by lemma 14 R->L }
% 0.20/0.51 ld(mult(rd(X, X), rd(X, X)), rd(Y, ld(X, X)))
% 0.20/0.51 = { by lemma 18 }
% 0.20/0.51 ld(ld(X, X), rd(Y, ld(X, X)))
% 0.20/0.51 = { by lemma 20 R->L }
% 0.20/0.51 ld(ld(X, X), mult(ld(X, X), Y))
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 Y
% 0.20/0.51
% 0.20/0.51 Goal 1 (goals): tuple(mult(X, x1(X)), mult(x1_2(X), X)) = tuple(x1(X), x1_2(X)).
% 0.20/0.51 The goal is true when:
% 0.20/0.51 X = ld(X, X)
% 0.20/0.51
% 0.20/0.51 Proof:
% 0.20/0.51 tuple(mult(ld(X, X), x1(ld(X, X))), mult(x1_2(ld(X, X)), ld(X, X)))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 tuple(ld(ld(X, X), mult(ld(X, X), x1(ld(X, X)))), mult(x1_2(ld(X, X)), ld(X, X)))
% 0.20/0.51 = { by axiom 1 (f02) }
% 0.20/0.51 tuple(x1(ld(X, X)), mult(x1_2(ld(X, X)), ld(X, X)))
% 0.20/0.51 = { by axiom 2 (f04) R->L }
% 0.20/0.51 tuple(x1(ld(X, X)), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), ld(X, X)), ld(X, X)))
% 0.20/0.52 = { by lemma 19 R->L }
% 0.20/0.52 tuple(x1(ld(X, X)), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), ld(ld(X, X), ld(X, X))), ld(X, X)))
% 0.20/0.52 = { by lemma 20 R->L }
% 0.20/0.52 tuple(x1(ld(X, X)), mult(ld(X, X), mult(mult(x1_2(ld(X, X)), ld(X, X)), ld(ld(X, X), ld(X, X)))))
% 0.20/0.52 = { by lemma 12 R->L }
% 0.20/0.52 tuple(x1(ld(X, X)), mult(ld(X, X), mult(ld(ld(X, X), ld(X, X)), mult(ld(ld(ld(X, X), ld(X, X)), x1_2(ld(X, X))), mult(ld(X, X), ld(ld(X, X), ld(X, X)))))))
% 0.20/0.52 = { by axiom 3 (f01) }
% 0.20/0.52 tuple(x1(ld(X, X)), mult(ld(X, X), mult(ld(ld(X, X), ld(X, X)), mult(ld(ld(ld(X, X), ld(X, X)), x1_2(ld(X, X))), ld(X, X)))))
% 0.20/0.52 = { by lemma 12 }
% 0.20/0.52 tuple(x1(ld(X, X)), mult(mult(ld(X, X), ld(ld(ld(X, X), ld(X, X)), x1_2(ld(X, X)))), ld(X, X)))
% 0.20/0.52 = { by lemma 20 }
% 0.20/0.52 tuple(x1(ld(X, X)), mult(rd(ld(ld(ld(X, X), ld(X, X)), x1_2(ld(X, X))), ld(X, X)), ld(X, X)))
% 0.20/0.52 = { by axiom 4 (f03) }
% 0.20/0.52 tuple(x1(ld(X, X)), ld(ld(ld(X, X), ld(X, X)), x1_2(ld(X, X))))
% 0.20/0.52 = { by lemma 19 }
% 0.20/0.52 tuple(x1(ld(X, X)), ld(ld(X, X), x1_2(ld(X, X))))
% 0.20/0.52 = { by lemma 21 }
% 0.20/0.52 tuple(x1(ld(X, X)), x1_2(ld(X, X)))
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52
% 0.20/0.52 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------