TSTP Solution File: GRP702+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP702+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 : n014.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 01:19:47 EDT 2023
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP702+1 : 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
% 0.19/0.35 % Computer : n014.cluster.edu
% 0.19/0.35 % Model : x86_64 x86_64
% 0.19/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.35 % Memory : 8042.1875MB
% 0.19/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.19/0.35 % CPULimit : 300
% 0.19/0.35 % WCLimit : 300
% 0.19/0.35 % DateTime : Mon Aug 28 21:14:01 EDT 2023
% 0.19/0.35 % CPUTime :
% 0.21/0.40 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.21/0.40
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 Take the following subset of the input axioms:
% 0.21/0.41 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 0.21/0.42 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 0.21/0.42 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 0.21/0.42 fof(f05, axiom, ![A2]: mult(A2, unit)=A2).
% 0.21/0.42 fof(f06, axiom, ![A2]: mult(unit, A2)=A2).
% 0.21/0.42 fof(f08, axiom, ![B2, A2]: mult(op_c, mult(A2, B2))=mult(mult(op_c, A2), B2)).
% 0.21/0.42 fof(f09, axiom, ![B2, A2]: mult(A2, mult(B2, op_c))=mult(mult(A2, B2), op_c)).
% 0.21/0.42 fof(f11, axiom, ![A2]: op_d=ld(A2, mult(op_c, A2))).
% 0.21/0.42 fof(f12, axiom, ![B2, A2]: op_e=mult(mult(rd(op_c, mult(A2, B2)), B2), A2)).
% 0.21/0.42 fof(f13, axiom, ![B2, A2]: op_f=mult(A2, mult(B2, ld(mult(A2, B2), op_c)))).
% 0.21/0.42 fof(goals, conjecture, ![X0, X1]: (mult(op_d, mult(X0, X1))=mult(mult(op_d, X0), X1) & (mult(X0, mult(X1, op_d))=mult(mult(X0, X1), op_d) & mult(X0, mult(op_d, X1))=mult(mult(X0, op_d), X1)))).
% 0.21/0.42
% 0.21/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.42 fresh(y, y, x1...xn) = u
% 0.21/0.42 C => fresh(s, t, x1...xn) = v
% 0.21/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.42 variables of u and v.
% 0.21/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.42 input problem has no model of domain size 1).
% 0.21/0.42
% 0.21/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.42
% 0.21/0.42 Axiom 1 (f05): mult(X, unit) = X.
% 0.21/0.42 Axiom 2 (f06): mult(unit, X) = X.
% 0.21/0.42 Axiom 3 (f02): ld(X, mult(X, Y)) = Y.
% 0.21/0.42 Axiom 4 (f11): op_d = ld(X, mult(op_c, X)).
% 0.21/0.42 Axiom 5 (f01): mult(X, ld(X, Y)) = Y.
% 0.21/0.42 Axiom 6 (f03): mult(rd(X, Y), Y) = X.
% 0.21/0.42 Axiom 7 (f09): mult(X, mult(Y, op_c)) = mult(mult(X, Y), op_c).
% 0.21/0.42 Axiom 8 (f08): mult(op_c, mult(X, Y)) = mult(mult(op_c, X), Y).
% 0.21/0.42 Axiom 9 (f13): op_f = mult(X, mult(Y, ld(mult(X, Y), op_c))).
% 0.21/0.42 Axiom 10 (f12): op_e = mult(mult(rd(op_c, mult(X, Y)), Y), X).
% 0.21/0.42
% 0.21/0.42 Lemma 11: op_c = op_d.
% 0.21/0.42 Proof:
% 0.21/0.42 op_c
% 0.21/0.42 = { by axiom 3 (f02) R->L }
% 0.21/0.42 ld(op_c, mult(op_c, op_c))
% 0.21/0.42 = { by axiom 4 (f11) R->L }
% 0.21/0.42 op_d
% 0.21/0.42
% 0.21/0.42 Lemma 12: op_d = op_f.
% 0.21/0.42 Proof:
% 0.21/0.42 op_d
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 op_c
% 0.21/0.42 = { by axiom 5 (f01) R->L }
% 0.21/0.42 mult(X, ld(X, op_c))
% 0.21/0.42 = { by axiom 2 (f06) R->L }
% 0.21/0.42 mult(X, mult(unit, ld(X, op_c)))
% 0.21/0.42 = { by axiom 1 (f05) R->L }
% 0.21/0.42 mult(X, mult(unit, ld(mult(X, unit), op_c)))
% 0.21/0.42 = { by axiom 9 (f13) R->L }
% 0.21/0.42 op_f
% 0.21/0.42
% 0.21/0.42 Lemma 13: op_f = op_e.
% 0.21/0.42 Proof:
% 0.21/0.42 op_f
% 0.21/0.42 = { by axiom 6 (f03) R->L }
% 0.21/0.42 mult(rd(op_f, X), X)
% 0.21/0.42 = { by axiom 2 (f06) R->L }
% 0.21/0.42 mult(rd(op_f, mult(unit, X)), X)
% 0.21/0.42 = { by lemma 12 R->L }
% 0.21/0.42 mult(rd(op_d, mult(unit, X)), X)
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 mult(rd(op_c, mult(unit, X)), X)
% 0.21/0.42 = { by axiom 1 (f05) R->L }
% 0.21/0.42 mult(mult(rd(op_c, mult(unit, X)), X), unit)
% 0.21/0.42 = { by axiom 10 (f12) R->L }
% 0.21/0.42 op_e
% 0.21/0.42
% 0.21/0.42 Lemma 14: mult(op_e, X) = mult(X, op_e).
% 0.21/0.42 Proof:
% 0.21/0.42 mult(op_e, X)
% 0.21/0.42 = { by lemma 13 R->L }
% 0.21/0.42 mult(op_f, X)
% 0.21/0.42 = { by lemma 12 R->L }
% 0.21/0.42 mult(op_d, X)
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 mult(op_c, X)
% 0.21/0.42 = { by axiom 5 (f01) R->L }
% 0.21/0.42 mult(X, ld(X, mult(op_c, X)))
% 0.21/0.42 = { by axiom 4 (f11) R->L }
% 0.21/0.42 mult(X, op_d)
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 mult(X, op_f)
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 mult(X, op_e)
% 0.21/0.42
% 0.21/0.42 Lemma 15: mult(op_e, mult(X, Y)) = mult(X, mult(Y, op_e)).
% 0.21/0.42 Proof:
% 0.21/0.42 mult(op_e, mult(X, Y))
% 0.21/0.42 = { by lemma 14 }
% 0.21/0.42 mult(mult(X, Y), op_e)
% 0.21/0.42 = { by lemma 13 R->L }
% 0.21/0.42 mult(mult(X, Y), op_f)
% 0.21/0.42 = { by lemma 12 R->L }
% 0.21/0.42 mult(mult(X, Y), op_d)
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 mult(mult(X, Y), op_c)
% 0.21/0.42 = { by axiom 7 (f09) R->L }
% 0.21/0.42 mult(X, mult(Y, op_c))
% 0.21/0.42 = { by lemma 11 }
% 0.21/0.42 mult(X, mult(Y, op_d))
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 mult(X, mult(Y, op_f))
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 mult(X, mult(Y, op_e))
% 0.21/0.42
% 0.21/0.42 Lemma 16: mult(op_e, mult(X, Y)) = mult(X, mult(op_e, Y)).
% 0.21/0.42 Proof:
% 0.21/0.42 mult(op_e, mult(X, Y))
% 0.21/0.42 = { by lemma 15 }
% 0.21/0.42 mult(X, mult(Y, op_e))
% 0.21/0.42 = { by lemma 14 R->L }
% 0.21/0.42 mult(X, mult(op_e, Y))
% 0.21/0.42
% 0.21/0.42 Lemma 17: mult(mult(op_e, X), Y) = mult(X, mult(Y, op_e)).
% 0.21/0.42 Proof:
% 0.21/0.42 mult(mult(op_e, X), Y)
% 0.21/0.42 = { by lemma 13 R->L }
% 0.21/0.42 mult(mult(op_f, X), Y)
% 0.21/0.42 = { by lemma 12 R->L }
% 0.21/0.42 mult(mult(op_d, X), Y)
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 mult(mult(op_c, X), Y)
% 0.21/0.42 = { by axiom 8 (f08) R->L }
% 0.21/0.42 mult(op_c, mult(X, Y))
% 0.21/0.42 = { by lemma 11 }
% 0.21/0.42 mult(op_d, mult(X, Y))
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 mult(op_f, mult(X, Y))
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 mult(op_e, mult(X, Y))
% 0.21/0.42 = { by lemma 15 }
% 0.21/0.42 mult(X, mult(Y, op_e))
% 0.21/0.42
% 0.21/0.42 Goal 1 (goals): tuple(mult(op_d, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_d)), mult(x0, mult(op_d, x1))) = tuple(mult(mult(op_d, x0_3), x1_3), mult(mult(x0_2, x1_2), op_d), mult(mult(x0, op_d), x1)).
% 0.21/0.42 Proof:
% 0.21/0.42 tuple(mult(op_d, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_d)), mult(x0, mult(op_d, x1)))
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 tuple(mult(op_f, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_d)), mult(x0, mult(op_d, x1)))
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 tuple(mult(op_f, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_f)), mult(x0, mult(op_d, x1)))
% 0.21/0.42 = { by lemma 12 }
% 0.21/0.42 tuple(mult(op_f, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_f)), mult(x0, mult(op_f, x1)))
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 tuple(mult(op_e, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_f)), mult(x0, mult(op_f, x1)))
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 tuple(mult(op_e, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_e)), mult(x0, mult(op_f, x1)))
% 0.21/0.42 = { by lemma 13 }
% 0.21/0.42 tuple(mult(op_e, mult(x0_3, x1_3)), mult(x0_2, mult(x1_2, op_e)), mult(x0, mult(op_e, x1)))
% 0.21/0.43 = { by lemma 14 R->L }
% 0.21/0.43 tuple(mult(op_e, mult(x0_3, x1_3)), mult(x0_2, mult(op_e, x1_2)), mult(x0, mult(op_e, x1)))
% 0.21/0.43 = { by lemma 16 R->L }
% 0.21/0.43 tuple(mult(op_e, mult(x0_3, x1_3)), mult(op_e, mult(x0_2, x1_2)), mult(x0, mult(op_e, x1)))
% 0.21/0.43 = { by lemma 16 R->L }
% 0.21/0.43 tuple(mult(op_e, mult(x0_3, x1_3)), mult(op_e, mult(x0_2, x1_2)), mult(op_e, mult(x0, x1)))
% 0.21/0.43 = { by lemma 15 }
% 0.21/0.43 tuple(mult(op_e, mult(x0_3, x1_3)), mult(op_e, mult(x0_2, x1_2)), mult(x0, mult(x1, op_e)))
% 0.21/0.43 = { by lemma 17 R->L }
% 0.21/0.43 tuple(mult(op_e, mult(x0_3, x1_3)), mult(op_e, mult(x0_2, x1_2)), mult(mult(op_e, x0), x1))
% 0.21/0.43 = { by lemma 15 }
% 0.21/0.43 tuple(mult(x0_3, mult(x1_3, op_e)), mult(op_e, mult(x0_2, x1_2)), mult(mult(op_e, x0), x1))
% 0.21/0.43 = { by lemma 17 R->L }
% 0.21/0.43 tuple(mult(mult(op_e, x0_3), x1_3), mult(op_e, mult(x0_2, x1_2)), mult(mult(op_e, x0), x1))
% 0.21/0.43 = { by lemma 14 }
% 0.21/0.43 tuple(mult(mult(op_e, x0_3), x1_3), mult(op_e, mult(x0_2, x1_2)), mult(mult(x0, op_e), x1))
% 0.21/0.43 = { by lemma 14 }
% 0.21/0.43 tuple(mult(mult(op_e, x0_3), x1_3), mult(mult(x0_2, x1_2), op_e), mult(mult(x0, op_e), x1))
% 0.21/0.43 = { by lemma 13 R->L }
% 0.21/0.43 tuple(mult(mult(op_f, x0_3), x1_3), mult(mult(x0_2, x1_2), op_e), mult(mult(x0, op_e), x1))
% 0.21/0.43 = { by lemma 13 R->L }
% 0.21/0.43 tuple(mult(mult(op_f, x0_3), x1_3), mult(mult(x0_2, x1_2), op_f), mult(mult(x0, op_e), x1))
% 0.21/0.43 = { by lemma 13 R->L }
% 0.21/0.43 tuple(mult(mult(op_f, x0_3), x1_3), mult(mult(x0_2, x1_2), op_f), mult(mult(x0, op_f), x1))
% 0.21/0.43 = { by lemma 12 R->L }
% 0.21/0.43 tuple(mult(mult(op_d, x0_3), x1_3), mult(mult(x0_2, x1_2), op_f), mult(mult(x0, op_f), x1))
% 0.21/0.43 = { by lemma 12 R->L }
% 0.21/0.43 tuple(mult(mult(op_d, x0_3), x1_3), mult(mult(x0_2, x1_2), op_d), mult(mult(x0, op_f), x1))
% 0.21/0.43 = { by lemma 12 R->L }
% 0.21/0.43 tuple(mult(mult(op_d, x0_3), x1_3), mult(mult(x0_2, x1_2), op_d), mult(mult(x0, op_d), x1))
% 0.21/0.43 % SZS output end Proof
% 0.21/0.43
% 0.21/0.43 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------