TSTP Solution File: GRP703+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP703+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 01:19:48 EDT 2023
% Result : Theorem 0.17s 0.37s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : GRP703+1 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n021.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Tue Aug 29 00:05:28 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.17/0.37 Command-line arguments: --no-flatten-goal
% 0.17/0.37
% 0.17/0.37 % SZS status Theorem
% 0.17/0.37
% 0.17/0.38 % SZS output start Proof
% 0.17/0.38 Take the following subset of the input axioms:
% 0.17/0.38 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 0.17/0.38 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 0.17/0.38 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 0.17/0.38 fof(f05, axiom, ![A2]: mult(A2, unit)=A2).
% 0.17/0.38 fof(f06, axiom, ![A2]: mult(unit, A2)=A2).
% 0.17/0.38 fof(f08, axiom, ![B2, A2]: mult(op_c, mult(A2, B2))=mult(mult(op_c, A2), B2)).
% 0.17/0.38 fof(f09, axiom, ![B2, A2]: mult(A2, mult(B2, op_c))=mult(mult(A2, B2), op_c)).
% 0.17/0.38 fof(f11, axiom, ![A2]: op_d=ld(A2, mult(op_c, A2))).
% 0.17/0.38 fof(f12, axiom, ![B2, A2]: op_e=mult(mult(rd(op_c, mult(A2, B2)), B2), A2)).
% 0.17/0.38 fof(goals, conjecture, ![X2, X3]: (mult(op_e, mult(X2, X3))=mult(mult(op_e, X2), X3) & (mult(X2, mult(X3, op_e))=mult(mult(X2, X3), op_e) & mult(X2, mult(op_e, X3))=mult(mult(X2, op_e), X3)))).
% 0.17/0.38
% 0.17/0.38 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.38 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.38 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.38 fresh(y, y, x1...xn) = u
% 0.17/0.38 C => fresh(s, t, x1...xn) = v
% 0.17/0.38 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.38 variables of u and v.
% 0.17/0.38 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.38 input problem has no model of domain size 1).
% 0.17/0.38
% 0.17/0.38 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.38
% 0.17/0.38 Axiom 1 (f05): mult(X, unit) = X.
% 0.17/0.38 Axiom 2 (f06): mult(unit, X) = X.
% 0.17/0.38 Axiom 3 (f02): ld(X, mult(X, Y)) = Y.
% 0.17/0.38 Axiom 4 (f11): op_d = ld(X, mult(op_c, X)).
% 0.17/0.38 Axiom 5 (f01): mult(X, ld(X, Y)) = Y.
% 0.17/0.38 Axiom 6 (f03): mult(rd(X, Y), Y) = X.
% 0.17/0.38 Axiom 7 (f09): mult(X, mult(Y, op_c)) = mult(mult(X, Y), op_c).
% 0.17/0.38 Axiom 8 (f08): mult(op_c, mult(X, Y)) = mult(mult(op_c, X), Y).
% 0.17/0.38 Axiom 9 (f12): op_e = mult(mult(rd(op_c, mult(X, Y)), Y), X).
% 0.17/0.38
% 0.17/0.38 Lemma 10: op_c = op_d.
% 0.17/0.38 Proof:
% 0.17/0.38 op_c
% 0.17/0.38 = { by axiom 3 (f02) R->L }
% 0.17/0.38 ld(op_c, mult(op_c, op_c))
% 0.17/0.38 = { by axiom 4 (f11) R->L }
% 0.17/0.38 op_d
% 0.17/0.38
% 0.17/0.38 Lemma 11: op_e = op_d.
% 0.17/0.38 Proof:
% 0.17/0.38 op_e
% 0.17/0.38 = { by axiom 9 (f12) }
% 0.17/0.38 mult(mult(rd(op_c, mult(unit, X)), X), unit)
% 0.17/0.38 = { by axiom 1 (f05) }
% 0.17/0.38 mult(rd(op_c, mult(unit, X)), X)
% 0.17/0.38 = { by lemma 10 }
% 0.17/0.38 mult(rd(op_d, mult(unit, X)), X)
% 0.17/0.38 = { by axiom 2 (f06) }
% 0.17/0.38 mult(rd(op_d, X), X)
% 0.17/0.38 = { by axiom 6 (f03) }
% 0.17/0.38 op_d
% 0.17/0.38
% 0.17/0.38 Lemma 12: mult(op_d, X) = mult(X, op_d).
% 0.17/0.38 Proof:
% 0.17/0.38 mult(op_d, X)
% 0.17/0.38 = { by lemma 10 R->L }
% 0.17/0.38 mult(op_c, X)
% 0.17/0.38 = { by axiom 5 (f01) R->L }
% 0.17/0.38 mult(X, ld(X, mult(op_c, X)))
% 0.17/0.38 = { by axiom 4 (f11) R->L }
% 0.17/0.38 mult(X, op_d)
% 0.17/0.38
% 0.17/0.38 Lemma 13: mult(op_d, mult(X, Y)) = mult(X, mult(Y, op_d)).
% 0.17/0.38 Proof:
% 0.17/0.38 mult(op_d, mult(X, Y))
% 0.17/0.38 = { by lemma 12 }
% 0.17/0.38 mult(mult(X, Y), op_d)
% 0.17/0.38 = { by lemma 10 R->L }
% 0.17/0.38 mult(mult(X, Y), op_c)
% 0.17/0.38 = { by axiom 7 (f09) R->L }
% 0.17/0.38 mult(X, mult(Y, op_c))
% 0.17/0.38 = { by lemma 10 }
% 0.17/0.38 mult(X, mult(Y, op_d))
% 0.17/0.38
% 0.17/0.38 Lemma 14: mult(op_d, mult(X, Y)) = mult(X, mult(op_d, Y)).
% 0.17/0.38 Proof:
% 0.17/0.38 mult(op_d, mult(X, Y))
% 0.17/0.38 = { by lemma 13 }
% 0.17/0.38 mult(X, mult(Y, op_d))
% 0.17/0.38 = { by lemma 12 R->L }
% 0.17/0.38 mult(X, mult(op_d, Y))
% 0.17/0.38
% 0.17/0.38 Lemma 15: mult(mult(op_d, X), Y) = mult(X, mult(Y, op_d)).
% 0.17/0.38 Proof:
% 0.17/0.38 mult(mult(op_d, X), Y)
% 0.17/0.38 = { by lemma 10 R->L }
% 0.17/0.38 mult(mult(op_c, X), Y)
% 0.17/0.38 = { by axiom 8 (f08) R->L }
% 0.17/0.38 mult(op_c, mult(X, Y))
% 0.17/0.38 = { by lemma 10 }
% 0.17/0.38 mult(op_d, mult(X, Y))
% 0.17/0.38 = { by lemma 13 }
% 0.17/0.38 mult(X, mult(Y, op_d))
% 0.17/0.38
% 0.17/0.38 Goal 1 (goals): tuple(mult(op_e, mult(x2_3, x3_3)), mult(x2_2, mult(x3_2, op_e)), mult(x2, mult(op_e, x3))) = tuple(mult(mult(op_e, x2_3), x3_3), mult(mult(x2_2, x3_2), op_e), mult(mult(x2, op_e), x3)).
% 0.17/0.38 Proof:
% 0.17/0.38 tuple(mult(op_e, mult(x2_3, x3_3)), mult(x2_2, mult(x3_2, op_e)), mult(x2, mult(op_e, x3)))
% 0.17/0.38 = { by lemma 11 }
% 0.17/0.38 tuple(mult(op_d, mult(x2_3, x3_3)), mult(x2_2, mult(x3_2, op_e)), mult(x2, mult(op_e, x3)))
% 0.17/0.38 = { by lemma 11 }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(x2_2, mult(x3_2, op_d)), mult(x2, mult(op_e, x3)))
% 0.17/0.39 = { by lemma 11 }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(x2_2, mult(x3_2, op_d)), mult(x2, mult(op_d, x3)))
% 0.17/0.39 = { by lemma 12 R->L }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(x2_2, mult(op_d, x3_2)), mult(x2, mult(op_d, x3)))
% 0.17/0.39 = { by lemma 14 R->L }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(op_d, mult(x2_2, x3_2)), mult(x2, mult(op_d, x3)))
% 0.17/0.39 = { by lemma 14 R->L }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(op_d, mult(x2_2, x3_2)), mult(op_d, mult(x2, x3)))
% 0.17/0.39 = { by lemma 13 }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(op_d, mult(x2_2, x3_2)), mult(x2, mult(x3, op_d)))
% 0.17/0.39 = { by lemma 15 R->L }
% 0.17/0.39 tuple(mult(op_d, mult(x2_3, x3_3)), mult(op_d, mult(x2_2, x3_2)), mult(mult(op_d, x2), x3))
% 0.17/0.39 = { by lemma 13 }
% 0.17/0.39 tuple(mult(x2_3, mult(x3_3, op_d)), mult(op_d, mult(x2_2, x3_2)), mult(mult(op_d, x2), x3))
% 0.17/0.39 = { by lemma 15 R->L }
% 0.17/0.39 tuple(mult(mult(op_d, x2_3), x3_3), mult(op_d, mult(x2_2, x3_2)), mult(mult(op_d, x2), x3))
% 0.17/0.39 = { by lemma 12 }
% 0.17/0.39 tuple(mult(mult(op_d, x2_3), x3_3), mult(op_d, mult(x2_2, x3_2)), mult(mult(x2, op_d), x3))
% 0.17/0.39 = { by lemma 12 }
% 0.17/0.39 tuple(mult(mult(op_d, x2_3), x3_3), mult(mult(x2_2, x3_2), op_d), mult(mult(x2, op_d), x3))
% 0.17/0.39 = { by lemma 11 R->L }
% 0.17/0.39 tuple(mult(mult(op_e, x2_3), x3_3), mult(mult(x2_2, x3_2), op_d), mult(mult(x2, op_d), x3))
% 0.17/0.39 = { by lemma 11 R->L }
% 0.17/0.39 tuple(mult(mult(op_e, x2_3), x3_3), mult(mult(x2_2, x3_2), op_e), mult(mult(x2, op_d), x3))
% 0.17/0.39 = { by lemma 11 R->L }
% 0.17/0.39 tuple(mult(mult(op_e, x2_3), x3_3), mult(mult(x2_2, x3_2), op_e), mult(mult(x2, op_e), x3))
% 0.17/0.39 % SZS output end Proof
% 0.17/0.39
% 0.17/0.39 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------