TSTP Solution File: ALG178+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG178+1 : TPTP v8.1.2. Released v2.7.0.
% 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 : n023.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 : Wed Aug 30 16:42:24 EDT 2023
% Result : Theorem 0.18s 0.40s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : ALG178+1 : TPTP v8.1.2. Released v2.7.0.
% 0.04/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n023.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Mon Aug 28 04:37:10 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.18/0.40 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.40
% 0.18/0.40 % SZS status Theorem
% 0.18/0.40
% 0.18/0.41 % SZS output start Proof
% 0.18/0.41 Take the following subset of the input axioms:
% 0.18/0.41 fof(ax2, axiom, ![U]: (sorti2(U) => ![V]: (sorti2(V) => sorti2(op2(U, V))))).
% 0.18/0.41 fof(ax3, axiom, ![U2]: (sorti1(U2) => ![V2]: (sorti1(V2) => op1(U2, op1(U2, V2))=V2))).
% 0.18/0.41 fof(ax4, axiom, ~![U2]: (sorti2(U2) => ![V2]: (sorti2(V2) => op2(U2, op2(U2, V2))=V2))).
% 0.18/0.41 fof(co1, conjecture, (![U2]: (sorti1(U2) => sorti2(h(U2))) & ![V2]: (sorti2(V2) => sorti1(j(V2)))) => ~(![W]: (sorti1(W) => ![X]: (sorti1(X) => h(op1(W, X))=op2(h(W), h(X)))) & (![Y]: (sorti2(Y) => ![Z]: (sorti2(Z) => j(op2(Y, Z))=op1(j(Y), j(Z)))) & (![X1]: (sorti2(X1) => h(j(X1))=X1) & ![X2]: (sorti1(X2) => j(h(X2))=X2))))).
% 0.18/0.41
% 0.18/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.41 fresh(y, y, x1...xn) = u
% 0.18/0.41 C => fresh(s, t, x1...xn) = v
% 0.18/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.41 variables of u and v.
% 0.18/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.41 input problem has no model of domain size 1).
% 0.18/0.41
% 0.18/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.41
% 0.18/0.41 Axiom 1 (ax4): sorti2(u) = true.
% 0.18/0.41 Axiom 2 (ax4_1): sorti2(v) = true.
% 0.18/0.41 Axiom 3 (co1_5): fresh(X, X, Y) = Y.
% 0.18/0.41 Axiom 4 (co1_3): fresh6(X, X, Y) = true.
% 0.18/0.42 Axiom 5 (co1_5): fresh(sorti2(X), true, X) = h(j(X)).
% 0.18/0.42 Axiom 6 (ax2): fresh12(X, X, Y, Z) = sorti2(op2(Y, Z)).
% 0.18/0.42 Axiom 7 (ax2): fresh11(X, X, Y, Z) = true.
% 0.18/0.42 Axiom 8 (ax3): fresh10(X, X, Y, Z) = op1(Y, op1(Y, Z)).
% 0.18/0.42 Axiom 9 (co1_3): fresh6(sorti2(X), true, X) = sorti1(j(X)).
% 0.18/0.42 Axiom 10 (co1_4): fresh5(X, X, Y, Z) = op1(j(Y), j(Z)).
% 0.18/0.42 Axiom 11 (co1_4): fresh4(X, X, Y, Z) = j(op2(Y, Z)).
% 0.18/0.42 Axiom 12 (ax3): fresh3(X, X, Y, Z) = Z.
% 0.18/0.42 Axiom 13 (ax2): fresh12(sorti2(X), true, Y, X) = fresh11(sorti2(Y), true, Y, X).
% 0.18/0.42 Axiom 14 (co1_4): fresh5(sorti2(X), true, Y, X) = fresh4(sorti2(Y), true, Y, X).
% 0.18/0.42 Axiom 15 (ax3): fresh10(sorti1(X), true, Y, X) = fresh3(sorti1(Y), true, Y, X).
% 0.18/0.42
% 0.18/0.42 Lemma 16: fresh12(sorti2(X), true, u, X) = true.
% 0.18/0.42 Proof:
% 0.18/0.42 fresh12(sorti2(X), true, u, X)
% 0.18/0.42 = { by axiom 13 (ax2) }
% 0.18/0.42 fresh11(sorti2(u), true, u, X)
% 0.18/0.42 = { by axiom 1 (ax4) }
% 0.18/0.42 fresh11(true, true, u, X)
% 0.18/0.42 = { by axiom 7 (ax2) }
% 0.18/0.42 true
% 0.18/0.42
% 0.18/0.42 Lemma 17: sorti2(op2(u, v)) = true.
% 0.18/0.42 Proof:
% 0.18/0.42 sorti2(op2(u, v))
% 0.18/0.42 = { by axiom 6 (ax2) R->L }
% 0.18/0.42 fresh12(true, true, u, v)
% 0.18/0.42 = { by axiom 2 (ax4_1) R->L }
% 0.18/0.42 fresh12(sorti2(v), true, u, v)
% 0.18/0.42 = { by lemma 16 }
% 0.18/0.42 true
% 0.18/0.42
% 0.18/0.42 Lemma 18: fresh5(sorti2(X), true, u, X) = j(op2(u, X)).
% 0.18/0.42 Proof:
% 0.18/0.42 fresh5(sorti2(X), true, u, X)
% 0.18/0.42 = { by axiom 14 (co1_4) }
% 0.18/0.42 fresh4(sorti2(u), true, u, X)
% 0.18/0.42 = { by axiom 1 (ax4) }
% 0.18/0.42 fresh4(true, true, u, X)
% 0.18/0.42 = { by axiom 11 (co1_4) }
% 0.18/0.42 j(op2(u, X))
% 0.18/0.42
% 0.18/0.42 Goal 1 (ax4_2): op2(u, op2(u, v)) = v.
% 0.18/0.42 Proof:
% 0.18/0.42 op2(u, op2(u, v))
% 0.18/0.42 = { by axiom 3 (co1_5) R->L }
% 0.18/0.42 fresh(true, true, op2(u, op2(u, v)))
% 0.18/0.42 = { by lemma 16 R->L }
% 0.18/0.42 fresh(fresh12(sorti2(op2(u, v)), true, u, op2(u, v)), true, op2(u, op2(u, v)))
% 0.18/0.42 = { by lemma 17 }
% 0.18/0.42 fresh(fresh12(true, true, u, op2(u, v)), true, op2(u, op2(u, v)))
% 0.18/0.42 = { by axiom 6 (ax2) }
% 0.18/0.42 fresh(sorti2(op2(u, op2(u, v))), true, op2(u, op2(u, v)))
% 0.18/0.42 = { by axiom 5 (co1_5) }
% 0.18/0.42 h(j(op2(u, op2(u, v))))
% 0.18/0.42 = { by lemma 18 R->L }
% 0.18/0.42 h(fresh5(sorti2(op2(u, v)), true, u, op2(u, v)))
% 0.18/0.42 = { by lemma 17 }
% 0.18/0.42 h(fresh5(true, true, u, op2(u, v)))
% 0.18/0.42 = { by axiom 10 (co1_4) }
% 0.18/0.42 h(op1(j(u), j(op2(u, v))))
% 0.18/0.42 = { by lemma 18 R->L }
% 0.18/0.42 h(op1(j(u), fresh5(sorti2(v), true, u, v)))
% 0.18/0.42 = { by axiom 2 (ax4_1) }
% 0.18/0.42 h(op1(j(u), fresh5(true, true, u, v)))
% 0.18/0.42 = { by axiom 10 (co1_4) }
% 0.18/0.42 h(op1(j(u), op1(j(u), j(v))))
% 0.18/0.42 = { by axiom 8 (ax3) R->L }
% 0.18/0.42 h(fresh10(true, true, j(u), j(v)))
% 0.18/0.42 = { by axiom 4 (co1_3) R->L }
% 0.18/0.42 h(fresh10(fresh6(true, true, v), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 2 (ax4_1) R->L }
% 0.18/0.42 h(fresh10(fresh6(sorti2(v), true, v), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 9 (co1_3) }
% 0.18/0.42 h(fresh10(sorti1(j(v)), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 15 (ax3) }
% 0.18/0.42 h(fresh3(sorti1(j(u)), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 9 (co1_3) R->L }
% 0.18/0.42 h(fresh3(fresh6(sorti2(u), true, u), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 1 (ax4) }
% 0.18/0.42 h(fresh3(fresh6(true, true, u), true, j(u), j(v)))
% 0.18/0.42 = { by axiom 4 (co1_3) }
% 0.18/0.42 h(fresh3(true, true, j(u), j(v)))
% 0.18/0.42 = { by axiom 12 (ax3) }
% 0.18/0.42 h(j(v))
% 0.18/0.42 = { by axiom 5 (co1_5) R->L }
% 0.18/0.42 fresh(sorti2(v), true, v)
% 0.18/0.42 = { by axiom 2 (ax4_1) }
% 0.18/0.42 fresh(true, true, v)
% 0.18/0.42 = { by axiom 3 (co1_5) }
% 0.18/0.42 v
% 0.18/0.42 % SZS output end Proof
% 0.18/0.42
% 0.18/0.42 RESULT: Theorem (the conjecture is true).
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