TSTP Solution File: GRP774+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP774+1 : TPTP v8.1.2. Released v4.1.0.
% 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 : n018.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 01:20:04 EDT 2023
% Result : Theorem 0.19s 0.48s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP774+1 : TPTP v8.1.2. Released v4.1.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n018.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 19:51:47 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.48 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.48
% 0.19/0.48 % SZS status Theorem
% 0.19/0.48
% 0.19/0.50 % SZS output start Proof
% 0.19/0.50 Take the following subset of the input axioms:
% 0.19/0.50 fof(goals, conjecture, ![X2, X3, X4, X5]: ((d(X2, X3) & d(X4, X5)) => d(product(X2, X4), product(X3, X5)))).
% 0.19/0.50 fof(sos01, axiom, ![C, B, A]: product(product(A, B), C)=product(A, product(B, C))).
% 0.19/0.50 fof(sos02, axiom, ![A2]: product(A2, A2)=A2).
% 0.19/0.50 fof(sos03, axiom, ![X0, X1]: (d(X0, X1) <=> (product(X0, product(X1, X0))=X0 & product(X1, product(X0, X1))=X1))).
% 0.19/0.50
% 0.19/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.50 fresh(y, y, x1...xn) = u
% 0.19/0.50 C => fresh(s, t, x1...xn) = v
% 0.19/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.50 variables of u and v.
% 0.19/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.50 input problem has no model of domain size 1).
% 0.19/0.50
% 0.19/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.50
% 0.19/0.50 Axiom 1 (sos02): product(X, X) = X.
% 0.19/0.50 Axiom 2 (goals): d(x4, x5) = true.
% 0.19/0.50 Axiom 3 (goals_1): d(x2, x3) = true.
% 0.19/0.50 Axiom 4 (sos01): product(product(X, Y), Z) = product(X, product(Y, Z)).
% 0.19/0.50 Axiom 5 (sos03_1): fresh(X, X, Y, Z) = Y.
% 0.19/0.50 Axiom 6 (sos03): fresh4(X, X, Y, Z) = true.
% 0.19/0.50 Axiom 7 (sos03): fresh3(X, X, Y, Z) = d(Y, Z).
% 0.19/0.50 Axiom 8 (sos03_2): fresh2(X, X, Y, Z) = Z.
% 0.19/0.50 Axiom 9 (sos03_1): fresh(d(X, Y), true, X, Y) = product(X, product(Y, X)).
% 0.19/0.50 Axiom 10 (sos03_2): fresh2(d(X, Y), true, X, Y) = product(Y, product(X, Y)).
% 0.19/0.50 Axiom 11 (sos03): fresh3(product(X, product(Y, X)), X, Y, X) = fresh4(product(Y, product(X, Y)), Y, Y, X).
% 0.19/0.50
% 0.19/0.50 Lemma 12: product(X, product(Y, product(X, Y))) = product(X, Y).
% 0.19/0.50 Proof:
% 0.19/0.50 product(X, product(Y, product(X, Y)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(X, Y), product(X, Y))
% 0.19/0.50 = { by axiom 1 (sos02) }
% 0.19/0.50 product(X, Y)
% 0.19/0.50
% 0.19/0.50 Lemma 13: product(x4, product(x5, product(x4, X))) = product(x4, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(x4, product(x5, product(x4, X)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x4, product(product(x5, x4), X))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(x4, product(x5, x4)), X)
% 0.19/0.50 = { by axiom 9 (sos03_1) R->L }
% 0.19/0.50 product(fresh(d(x4, x5), true, x4, x5), X)
% 0.19/0.50 = { by axiom 2 (goals) }
% 0.19/0.50 product(fresh(true, true, x4, x5), X)
% 0.19/0.50 = { by axiom 5 (sos03_1) }
% 0.19/0.50 product(x4, X)
% 0.19/0.50
% 0.19/0.50 Lemma 14: product(x5, product(x4, product(x5, X))) = product(x5, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(x5, product(x4, product(x5, X)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x5, product(product(x4, x5), X))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(x5, product(x4, x5)), X)
% 0.19/0.50 = { by axiom 10 (sos03_2) R->L }
% 0.19/0.50 product(fresh2(d(x4, x5), true, x4, x5), X)
% 0.19/0.50 = { by axiom 2 (goals) }
% 0.19/0.50 product(fresh2(true, true, x4, x5), X)
% 0.19/0.50 = { by axiom 8 (sos03_2) }
% 0.19/0.50 product(x5, X)
% 0.19/0.50
% 0.19/0.50 Lemma 15: product(x2, product(x3, product(x2, X))) = product(x2, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(x2, product(x3, product(x2, X)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x2, product(product(x3, x2), X))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(x2, product(x3, x2)), X)
% 0.19/0.50 = { by axiom 9 (sos03_1) R->L }
% 0.19/0.50 product(fresh(d(x2, x3), true, x2, x3), X)
% 0.19/0.50 = { by axiom 3 (goals_1) }
% 0.19/0.50 product(fresh(true, true, x2, x3), X)
% 0.19/0.50 = { by axiom 5 (sos03_1) }
% 0.19/0.50 product(x2, X)
% 0.19/0.50
% 0.19/0.50 Lemma 16: product(x3, product(x2, product(x3, X))) = product(x3, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(x3, product(x2, product(x3, X)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x3, product(product(x2, x3), X))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(x3, product(x2, x3)), X)
% 0.19/0.50 = { by axiom 10 (sos03_2) R->L }
% 0.19/0.50 product(fresh2(d(x2, x3), true, x2, x3), X)
% 0.19/0.50 = { by axiom 3 (goals_1) }
% 0.19/0.50 product(fresh2(true, true, x2, x3), X)
% 0.19/0.50 = { by axiom 8 (sos03_2) }
% 0.19/0.50 product(x3, X)
% 0.19/0.50
% 0.19/0.50 Lemma 17: product(product(x2, X), product(x3, product(x2, X))) = product(x2, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(product(x2, X), product(x3, product(x2, X)))
% 0.19/0.50 = { by axiom 4 (sos01) }
% 0.19/0.50 product(x2, product(X, product(x3, product(x2, X))))
% 0.19/0.50 = { by lemma 15 R->L }
% 0.19/0.50 product(x2, product(x3, product(x2, product(X, product(x3, product(x2, X))))))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x2, product(x3, product(product(x2, X), product(x3, product(x2, X)))))
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 product(x2, product(x3, product(x2, X)))
% 0.19/0.50 = { by lemma 15 }
% 0.19/0.50 product(x2, X)
% 0.19/0.50
% 0.19/0.50 Lemma 18: product(product(x3, X), product(x2, product(x3, X))) = product(x3, X).
% 0.19/0.50 Proof:
% 0.19/0.50 product(product(x3, X), product(x2, product(x3, X)))
% 0.19/0.50 = { by axiom 4 (sos01) }
% 0.19/0.50 product(x3, product(X, product(x2, product(x3, X))))
% 0.19/0.50 = { by lemma 16 R->L }
% 0.19/0.50 product(x3, product(x2, product(x3, product(X, product(x2, product(x3, X))))))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(x3, product(x2, product(product(x3, X), product(x2, product(x3, X)))))
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 product(x3, product(x2, product(x3, X)))
% 0.19/0.50 = { by lemma 16 }
% 0.19/0.50 product(x3, X)
% 0.19/0.50
% 0.19/0.50 Lemma 19: product(X, product(Y, product(Z, product(X, product(Y, Z))))) = product(X, product(Y, Z)).
% 0.19/0.50 Proof:
% 0.19/0.50 product(X, product(Y, product(Z, product(X, product(Y, Z)))))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(X, product(Y, product(Z, product(product(X, Y), Z))))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(X, Y), product(Z, product(product(X, Y), Z)))
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 product(product(X, Y), Z)
% 0.19/0.50 = { by axiom 4 (sos01) }
% 0.19/0.50 product(X, product(Y, Z))
% 0.19/0.50
% 0.19/0.50 Lemma 20: product(product(X, Y), product(Z, product(Y, product(Z, W)))) = product(product(X, Y), product(Z, W)).
% 0.19/0.50 Proof:
% 0.19/0.50 product(product(X, Y), product(Z, product(Y, product(Z, W))))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(X, Y), product(Z, product(product(Y, Z), W)))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(X, Y), product(product(Z, product(Y, Z)), W))
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(product(X, Y), product(Z, product(Y, Z))), W)
% 0.19/0.50 = { by axiom 4 (sos01) }
% 0.19/0.50 product(product(X, product(Y, product(Z, product(Y, Z)))), W)
% 0.19/0.50 = { by lemma 12 }
% 0.19/0.50 product(product(X, product(Y, Z)), W)
% 0.19/0.50 = { by axiom 4 (sos01) R->L }
% 0.19/0.50 product(product(product(X, Y), Z), W)
% 0.19/0.50 = { by axiom 4 (sos01) }
% 0.19/0.50 product(product(X, Y), product(Z, W))
% 0.19/0.50
% 0.19/0.50 Lemma 21: product(product(X, x4), product(product(Y, x5), product(x4, Y))) = product(product(X, x4), Y).
% 0.19/0.50 Proof:
% 0.19/0.50 product(product(X, x4), product(product(Y, x5), product(x4, Y)))
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(X, product(x4, product(product(Y, x5), product(x4, Y))))
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(X, product(x4, product(Y, product(x5, product(x4, Y)))))
% 0.19/0.51 = { by lemma 13 R->L }
% 0.19/0.51 product(X, product(x4, product(x5, product(x4, product(Y, product(x5, product(x4, Y)))))))
% 0.19/0.51 = { by lemma 19 }
% 0.19/0.51 product(X, product(x4, product(x5, product(x4, Y))))
% 0.19/0.51 = { by lemma 13 }
% 0.19/0.51 product(X, product(x4, Y))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(X, x4), Y)
% 0.19/0.51
% 0.19/0.51 Lemma 22: product(product(x2, X), product(x3, product(product(x2, X), Y))) = product(product(x2, X), Y).
% 0.19/0.51 Proof:
% 0.19/0.51 product(product(x2, X), product(x3, product(product(x2, X), Y)))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(x2, X), product(product(x3, product(x2, X)), Y))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(x2, X), product(x3, product(x2, X))), Y)
% 0.19/0.51 = { by lemma 17 }
% 0.19/0.51 product(product(x2, X), Y)
% 0.19/0.51
% 0.19/0.51 Lemma 23: product(product(X, x5), product(product(Y, x4), product(x5, product(Y, Z)))) = product(product(X, x5), product(Y, Z)).
% 0.19/0.51 Proof:
% 0.19/0.51 product(product(X, x5), product(product(Y, x4), product(x5, product(Y, Z))))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(X, x5), product(product(Y, x4), product(product(x5, Y), Z)))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(X, x5), product(product(product(Y, x4), product(x5, Y)), Z))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(X, x5), product(product(Y, x4), product(x5, Y))), Z)
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(product(X, product(x5, product(product(Y, x4), product(x5, Y)))), Z)
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(product(X, product(x5, product(Y, product(x4, product(x5, Y))))), Z)
% 0.19/0.51 = { by lemma 14 R->L }
% 0.19/0.51 product(product(X, product(x5, product(x4, product(x5, product(Y, product(x4, product(x5, Y))))))), Z)
% 0.19/0.51 = { by lemma 19 }
% 0.19/0.51 product(product(X, product(x5, product(x4, product(x5, Y)))), Z)
% 0.19/0.51 = { by lemma 14 }
% 0.19/0.51 product(product(X, product(x5, Y)), Z)
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(X, x5), Y), Z)
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(product(X, x5), product(Y, Z))
% 0.19/0.51
% 0.19/0.51 Lemma 24: product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(Y, Z)))) = product(product(X, x4), product(Y, Z)).
% 0.19/0.51 Proof:
% 0.19/0.51 product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(Y, Z))))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(X, x4), product(product(Y, x5), product(product(product(X, x4), Y), Z)))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(X, x4), product(product(product(Y, x5), product(product(X, x4), Y)), Z))
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(X, x4), product(product(Y, x5), product(product(X, x4), Y))), Z)
% 0.19/0.51 = { by lemma 21 R->L }
% 0.19/0.51 product(product(product(X, x4), product(product(Y, x5), product(product(X, x4), product(product(Y, x5), product(x4, Y))))), Z)
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(X, x4), product(product(Y, x5), product(product(product(X, x4), product(Y, x5)), product(x4, Y)))), Z)
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(product(X, x4), product(Y, x5)), product(product(product(X, x4), product(Y, x5)), product(x4, Y))), Z)
% 0.19/0.51 = { by axiom 4 (sos01) R->L }
% 0.19/0.51 product(product(product(product(product(X, x4), product(Y, x5)), product(product(X, x4), product(Y, x5))), product(x4, Y)), Z)
% 0.19/0.51 = { by axiom 1 (sos02) }
% 0.19/0.51 product(product(product(product(X, x4), product(Y, x5)), product(x4, Y)), Z)
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(product(product(X, x4), product(product(Y, x5), product(x4, Y))), Z)
% 0.19/0.51 = { by lemma 21 }
% 0.19/0.51 product(product(product(X, x4), Y), Z)
% 0.19/0.51 = { by axiom 4 (sos01) }
% 0.19/0.51 product(product(X, x4), product(Y, Z))
% 0.19/0.51
% 0.19/0.51 Goal 1 (goals_2): d(product(x2, x4), product(x3, x5)) = true.
% 0.19/0.51 Proof:
% 0.19/0.51 d(product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by axiom 7 (sos03) R->L }
% 0.19/0.51 fresh3(product(x3, x5), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 18 R->L }
% 0.19/0.51 fresh3(product(product(x3, x5), product(x2, product(x3, x5))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 23 R->L }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(x2, product(x3, x5))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 20 R->L }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 22 R->L }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x3, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 24 R->L }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x3, product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 22 }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x5, product(x2, product(x3, x5))))))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 23 }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(product(x3, x5), product(x2, product(x3, x5))))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 18 }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x5, product(product(x2, x4), product(x3, x5))))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 20 }
% 0.19/0.51 fresh3(product(product(x3, x5), product(product(x2, x4), product(x3, x5))), product(x3, x5), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by axiom 11 (sos03) }
% 0.19/0.51 fresh4(product(product(x2, x4), product(product(x3, x5), product(x2, x4))), product(x2, x4), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 17 R->L }
% 0.19/0.51 fresh4(product(product(x2, x4), product(product(x3, x5), product(product(x2, x4), product(x3, product(x2, x4))))), product(x2, x4), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 24 }
% 0.19/0.51 fresh4(product(product(x2, x4), product(x3, product(x2, x4))), product(x2, x4), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by lemma 17 }
% 0.19/0.51 fresh4(product(x2, x4), product(x2, x4), product(x2, x4), product(x3, x5))
% 0.19/0.51 = { by axiom 6 (sos03) }
% 0.19/0.51 true
% 0.19/0.51 % SZS output end Proof
% 0.19/0.51
% 0.19/0.51 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------