TSTP Solution File: GRP710+1 by Twee---2.4.2
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- Process Solution
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% File : Twee---2.4.2
% Problem : GRP710+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 : n020.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 01:19:51 EDT 2023
% Result : Theorem 0.20s 0.42s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : GRP710+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.14 % 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 : n020.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 22:50:59 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.42 Command-line arguments: --flatten
% 0.20/0.42
% 0.20/0.42 % SZS status Theorem
% 0.20/0.42
% 0.20/0.44 % SZS output start Proof
% 0.20/0.44 Take the following subset of the input axioms:
% 0.20/0.44 fof(f01, axiom, ![A]: mult(A, unit)=A).
% 0.20/0.44 fof(f02, axiom, ![A2]: mult(unit, A2)=A2).
% 0.20/0.44 fof(f03, axiom, ![C, B, A2]: mult(A2, mult(B, mult(B, C)))=mult(mult(mult(A2, B), B), C)).
% 0.20/0.44 fof(f04, axiom, ![A2]: mult(A2, i(A2))=unit).
% 0.20/0.44 fof(f05, axiom, ![A2]: mult(i(A2), A2)=unit).
% 0.20/0.44 fof(goals, conjecture, ![X0, X1]: ?[X2]: mult(X0, X2)=X1 & ![X3, X4]: ?[X5]: mult(X5, X4)=X3).
% 0.20/0.44
% 0.20/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.44 fresh(y, y, x1...xn) = u
% 0.20/0.44 C => fresh(s, t, x1...xn) = v
% 0.20/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.44 variables of u and v.
% 0.20/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.44 input problem has no model of domain size 1).
% 0.20/0.44
% 0.20/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.44
% 0.20/0.44 Axiom 1 (f01): mult(X, unit) = X.
% 0.20/0.44 Axiom 2 (f02): mult(unit, X) = X.
% 0.20/0.44 Axiom 3 (f04): mult(X, i(X)) = unit.
% 0.20/0.44 Axiom 4 (f05): mult(i(X), X) = unit.
% 0.20/0.44 Axiom 5 (f03): mult(X, mult(Y, mult(Y, Z))) = mult(mult(mult(X, Y), Y), Z).
% 0.20/0.44
% 0.20/0.44 Lemma 6: mult(mult(X, Y), Y) = mult(X, mult(Y, Y)).
% 0.20/0.44 Proof:
% 0.20/0.44 mult(mult(X, Y), Y)
% 0.20/0.44 = { by axiom 1 (f01) R->L }
% 0.20/0.44 mult(mult(mult(X, Y), Y), unit)
% 0.20/0.44 = { by axiom 5 (f03) R->L }
% 0.20/0.44 mult(X, mult(Y, mult(Y, unit)))
% 0.20/0.44 = { by axiom 1 (f01) }
% 0.20/0.44 mult(X, mult(Y, Y))
% 0.20/0.44
% 0.20/0.44 Lemma 7: mult(i(mult(X, X)), mult(X, mult(X, Y))) = Y.
% 0.20/0.44 Proof:
% 0.20/0.44 mult(i(mult(X, X)), mult(X, mult(X, Y)))
% 0.20/0.44 = { by axiom 5 (f03) }
% 0.20/0.44 mult(mult(mult(i(mult(X, X)), X), X), Y)
% 0.20/0.44 = { by lemma 6 }
% 0.20/0.44 mult(mult(i(mult(X, X)), mult(X, X)), Y)
% 0.20/0.44 = { by axiom 4 (f05) }
% 0.20/0.44 mult(unit, Y)
% 0.20/0.44 = { by axiom 2 (f02) }
% 0.20/0.44 Y
% 0.20/0.44
% 0.20/0.44 Lemma 8: mult(X, mult(i(X), mult(i(X), Y))) = mult(i(X), Y).
% 0.20/0.44 Proof:
% 0.20/0.44 mult(X, mult(i(X), mult(i(X), Y)))
% 0.20/0.44 = { by axiom 5 (f03) }
% 0.20/0.44 mult(mult(mult(X, i(X)), i(X)), Y)
% 0.20/0.44 = { by axiom 3 (f04) }
% 0.20/0.44 mult(mult(unit, i(X)), Y)
% 0.20/0.44 = { by axiom 2 (f02) }
% 0.20/0.44 mult(i(X), Y)
% 0.20/0.44
% 0.20/0.44 Lemma 9: mult(X, mult(i(X), Y)) = Y.
% 0.20/0.44 Proof:
% 0.20/0.44 mult(X, mult(i(X), Y))
% 0.20/0.44 = { by lemma 7 R->L }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(i(X), mult(X, mult(i(X), Y)))))
% 0.20/0.44 = { by lemma 8 R->L }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(i(X), mult(X, mult(X, mult(i(X), mult(i(X), Y)))))))
% 0.20/0.44 = { by axiom 5 (f03) }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(mult(mult(i(X), X), X), mult(i(X), mult(i(X), Y)))))
% 0.20/0.44 = { by axiom 4 (f05) }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(mult(unit, X), mult(i(X), mult(i(X), Y)))))
% 0.20/0.44 = { by axiom 2 (f02) }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(X, mult(i(X), mult(i(X), Y)))))
% 0.20/0.44 = { by lemma 8 }
% 0.20/0.44 mult(i(mult(i(X), i(X))), mult(i(X), mult(i(X), Y)))
% 0.20/0.44 = { by lemma 7 }
% 0.20/0.44 Y
% 0.20/0.44
% 0.20/0.44 Goal 1 (goals): tuple(mult(X, x4), mult(x0, Y)) = tuple(x3, x1).
% 0.20/0.44 The goal is true when:
% 0.20/0.44 X = mult(x3, i(x4))
% 0.20/0.44 Y = mult(i(x0), mult(i(x0), mult(x0, x1)))
% 0.20/0.44
% 0.20/0.44 Proof:
% 0.20/0.44 tuple(mult(mult(x3, i(x4)), x4), mult(x0, mult(i(x0), mult(i(x0), mult(x0, x1)))))
% 0.20/0.44 = { by lemma 8 }
% 0.20/0.44 tuple(mult(mult(x3, i(x4)), x4), mult(i(x0), mult(x0, x1)))
% 0.20/0.44 = { by lemma 7 R->L }
% 0.20/0.44 tuple(mult(mult(x3, i(x4)), x4), mult(i(mult(x0, x0)), mult(x0, mult(x0, mult(i(x0), mult(x0, x1))))))
% 0.20/0.44 = { by lemma 9 }
% 0.20/0.44 tuple(mult(mult(x3, i(x4)), x4), mult(i(mult(x0, x0)), mult(x0, mult(x0, x1))))
% 0.20/0.44 = { by lemma 7 }
% 0.20/0.44 tuple(mult(mult(x3, i(x4)), x4), x1)
% 0.20/0.44 = { by axiom 1 (f01) R->L }
% 0.20/0.44 tuple(mult(mult(x3, mult(i(x4), unit)), x4), x1)
% 0.20/0.44 = { by axiom 4 (f05) R->L }
% 0.20/0.44 tuple(mult(mult(x3, mult(i(x4), mult(i(x4), x4))), x4), x1)
% 0.20/0.44 = { by axiom 5 (f03) }
% 0.20/0.44 tuple(mult(mult(mult(mult(x3, i(x4)), i(x4)), x4), x4), x1)
% 0.20/0.44 = { by lemma 6 }
% 0.20/0.44 tuple(mult(mult(mult(x3, i(x4)), i(x4)), mult(x4, x4)), x1)
% 0.20/0.44 = { by axiom 5 (f03) R->L }
% 0.20/0.44 tuple(mult(x3, mult(i(x4), mult(i(x4), mult(x4, x4)))), x1)
% 0.20/0.44 = { by axiom 2 (f02) R->L }
% 0.20/0.44 tuple(mult(x3, mult(unit, mult(i(x4), mult(i(x4), mult(x4, x4))))), x1)
% 0.20/0.44 = { by axiom 5 (f03) }
% 0.20/0.44 tuple(mult(x3, mult(mult(mult(unit, i(x4)), i(x4)), mult(x4, x4))), x1)
% 0.20/0.44 = { by axiom 2 (f02) }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(x4), i(x4)), mult(x4, x4))), x1)
% 0.20/0.44 = { by lemma 7 R->L }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(mult(x4, x4)), mult(x4, mult(x4, mult(i(x4), i(x4))))), mult(x4, x4))), x1)
% 0.20/0.44 = { by lemma 6 R->L }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(mult(x4, x4)), mult(x4, mult(mult(x4, i(x4)), i(x4)))), mult(x4, x4))), x1)
% 0.20/0.44 = { by axiom 3 (f04) }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(mult(x4, x4)), mult(x4, mult(unit, i(x4)))), mult(x4, x4))), x1)
% 0.20/0.44 = { by axiom 2 (f02) }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(mult(x4, x4)), mult(x4, i(x4))), mult(x4, x4))), x1)
% 0.20/0.44 = { by axiom 3 (f04) }
% 0.20/0.44 tuple(mult(x3, mult(mult(i(mult(x4, x4)), unit), mult(x4, x4))), x1)
% 0.20/0.44 = { by axiom 1 (f01) }
% 0.20/0.44 tuple(mult(x3, mult(i(mult(x4, x4)), mult(x4, x4))), x1)
% 0.20/0.44 = { by lemma 9 R->L }
% 0.20/0.45 tuple(mult(x3, mult(i(mult(x4, x4)), mult(x4, mult(x4, mult(i(x4), x4))))), x1)
% 0.20/0.45 = { by lemma 7 }
% 0.20/0.45 tuple(mult(x3, mult(i(x4), x4)), x1)
% 0.20/0.45 = { by axiom 4 (f05) }
% 0.20/0.45 tuple(mult(x3, unit), x1)
% 0.20/0.45 = { by axiom 1 (f01) }
% 0.20/0.45 tuple(x3, x1)
% 0.20/0.45 % SZS output end Proof
% 0.20/0.45
% 0.20/0.45 RESULT: Theorem (the conjecture is true).
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