TSTP Solution File: ALG073+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG073+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 : n028.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 16:42:10 EDT 2023
% Result : Theorem 0.22s 0.44s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : ALG073+1 : TPTP v8.1.2. Released v2.7.0.
% 0.00/0.14 % 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 : n028.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 03:42:24 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.22/0.44 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.44
% 0.22/0.44 % SZS status Theorem
% 0.22/0.44
% 0.22/0.45 % SZS output start Proof
% 0.22/0.45 Take the following subset of the input axioms:
% 0.22/0.45 fof(ax1, axiom, ![U]: (sorti1(U) => ![V]: (sorti1(V) => sorti1(op1(U, V))))).
% 0.22/0.45 fof(ax3, axiom, ?[U2]: (sorti1(U2) & ?[V2]: (sorti1(V2) & (op1(U2, U2)=V2 & (op1(V2, V2)=U2 & op1(U2, V2)!=U2))))).
% 0.22/0.45 fof(ax4, axiom, ~?[U2]: (sorti2(U2) & ?[V2]: (sorti2(V2) & (op2(U2, U2)=V2 & (op2(V2, V2)=U2 & op2(U2, V2)!=U2))))).
% 0.22/0.45 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))))).
% 0.22/0.45
% 0.22/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.45 fresh(y, y, x1...xn) = u
% 0.22/0.45 C => fresh(s, t, x1...xn) = v
% 0.22/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.45 variables of u and v.
% 0.22/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.45 input problem has no model of domain size 1).
% 0.22/0.45
% 0.22/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.45
% 0.22/0.45 Axiom 1 (ax3_2): sorti1(u) = true.
% 0.22/0.45 Axiom 2 (ax3_3): sorti1(v) = true.
% 0.22/0.45 Axiom 3 (ax3): op1(u, u) = v.
% 0.22/0.45 Axiom 4 (ax3_1): op1(v, v) = u.
% 0.22/0.45 Axiom 5 (co1): fresh8(X, X, Y) = true.
% 0.22/0.45 Axiom 6 (co1_2): fresh2(X, X, Y) = Y.
% 0.22/0.45 Axiom 7 (ax4): fresh16(X, X, Y, Z) = Y.
% 0.22/0.45 Axiom 8 (ax1): fresh13(X, X, Y, Z) = true.
% 0.22/0.45 Axiom 9 (ax1): fresh12(X, X, Y, Z) = sorti1(op1(Y, Z)).
% 0.22/0.45 Axiom 10 (ax4): fresh9(X, X, Y, Z) = op2(Y, Z).
% 0.22/0.45 Axiom 11 (co1): fresh8(sorti1(X), true, X) = sorti2(h(X)).
% 0.22/0.45 Axiom 12 (co1_1): fresh7(X, X, Y, Z) = op2(h(Y), h(Z)).
% 0.22/0.45 Axiom 13 (co1_1): fresh6(X, X, Y, Z) = h(op1(Y, Z)).
% 0.22/0.45 Axiom 14 (co1_2): fresh2(sorti1(X), true, X) = j(h(X)).
% 0.22/0.45 Axiom 15 (ax4): fresh14(X, X, Y, Z) = fresh15(sorti2(Y), true, Y, Z).
% 0.22/0.45 Axiom 16 (ax1): fresh12(sorti1(X), true, Y, X) = fresh13(sorti1(Y), true, Y, X).
% 0.22/0.45 Axiom 17 (co1_1): fresh7(sorti1(X), true, Y, X) = fresh6(sorti1(Y), true, Y, X).
% 0.22/0.45 Axiom 18 (ax4): fresh15(X, X, Y, Z) = fresh16(op2(Z, Z), Y, Y, Z).
% 0.22/0.45 Axiom 19 (ax4): fresh14(sorti2(X), true, Y, X) = fresh9(op2(Y, Y), X, Y, X).
% 0.22/0.45
% 0.22/0.45 Lemma 20: op2(h(v), h(v)) = h(u).
% 0.22/0.45 Proof:
% 0.22/0.45 op2(h(v), h(v))
% 0.22/0.45 = { by axiom 12 (co1_1) R->L }
% 0.22/0.45 fresh7(true, true, v, v)
% 0.22/0.45 = { by axiom 2 (ax3_3) R->L }
% 0.22/0.45 fresh7(sorti1(v), true, v, v)
% 0.22/0.45 = { by axiom 17 (co1_1) }
% 0.22/0.45 fresh6(sorti1(v), true, v, v)
% 0.22/0.45 = { by axiom 2 (ax3_3) }
% 0.22/0.45 fresh6(true, true, v, v)
% 0.22/0.45 = { by axiom 13 (co1_1) }
% 0.22/0.45 h(op1(v, v))
% 0.22/0.45 = { by axiom 4 (ax3_1) }
% 0.22/0.45 h(u)
% 0.22/0.45
% 0.22/0.45 Lemma 21: fresh7(sorti1(X), true, u, X) = h(op1(u, X)).
% 0.22/0.45 Proof:
% 0.22/0.45 fresh7(sorti1(X), true, u, X)
% 0.22/0.45 = { by axiom 17 (co1_1) }
% 0.22/0.45 fresh6(sorti1(u), true, u, X)
% 0.22/0.45 = { by axiom 1 (ax3_2) }
% 0.22/0.45 fresh6(true, true, u, X)
% 0.22/0.45 = { by axiom 13 (co1_1) }
% 0.22/0.46 h(op1(u, X))
% 0.22/0.46
% 0.22/0.46 Lemma 22: op2(h(u), h(u)) = h(v).
% 0.22/0.46 Proof:
% 0.22/0.46 op2(h(u), h(u))
% 0.22/0.46 = { by axiom 12 (co1_1) R->L }
% 0.22/0.46 fresh7(true, true, u, u)
% 0.22/0.46 = { by axiom 1 (ax3_2) R->L }
% 0.22/0.46 fresh7(sorti1(u), true, u, u)
% 0.22/0.46 = { by lemma 21 }
% 0.22/0.46 h(op1(u, u))
% 0.22/0.46 = { by axiom 3 (ax3) }
% 0.22/0.46 h(v)
% 0.22/0.46
% 0.22/0.46 Goal 1 (ax3_4): op1(u, v) = u.
% 0.22/0.46 Proof:
% 0.22/0.46 op1(u, v)
% 0.22/0.46 = { by axiom 6 (co1_2) R->L }
% 0.22/0.46 fresh2(true, true, op1(u, v))
% 0.22/0.46 = { by axiom 8 (ax1) R->L }
% 0.22/0.46 fresh2(fresh13(true, true, u, v), true, op1(u, v))
% 0.22/0.46 = { by axiom 1 (ax3_2) R->L }
% 0.22/0.46 fresh2(fresh13(sorti1(u), true, u, v), true, op1(u, v))
% 0.22/0.46 = { by axiom 16 (ax1) R->L }
% 0.22/0.46 fresh2(fresh12(sorti1(v), true, u, v), true, op1(u, v))
% 0.22/0.46 = { by axiom 2 (ax3_3) }
% 0.22/0.46 fresh2(fresh12(true, true, u, v), true, op1(u, v))
% 0.22/0.46 = { by axiom 9 (ax1) }
% 0.22/0.46 fresh2(sorti1(op1(u, v)), true, op1(u, v))
% 0.22/0.46 = { by axiom 14 (co1_2) }
% 0.22/0.46 j(h(op1(u, v)))
% 0.22/0.46 = { by lemma 21 R->L }
% 0.22/0.46 j(fresh7(sorti1(v), true, u, v))
% 0.22/0.46 = { by axiom 2 (ax3_3) }
% 0.22/0.46 j(fresh7(true, true, u, v))
% 0.22/0.46 = { by axiom 12 (co1_1) }
% 0.22/0.46 j(op2(h(u), h(v)))
% 0.22/0.46 = { by lemma 22 R->L }
% 0.22/0.46 j(op2(h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by axiom 10 (ax4) R->L }
% 0.22/0.46 j(fresh9(op2(h(u), h(u)), op2(h(u), h(u)), h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by axiom 19 (ax4) R->L }
% 0.22/0.46 j(fresh14(sorti2(op2(h(u), h(u))), true, h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by lemma 22 }
% 0.22/0.46 j(fresh14(sorti2(h(v)), true, h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by axiom 11 (co1) R->L }
% 0.22/0.46 j(fresh14(fresh8(sorti1(v), true, v), true, h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by axiom 2 (ax3_3) }
% 0.22/0.46 j(fresh14(fresh8(true, true, v), true, h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by axiom 5 (co1) }
% 0.22/0.46 j(fresh14(true, true, h(u), op2(h(u), h(u))))
% 0.22/0.46 = { by lemma 22 }
% 0.22/0.46 j(fresh14(true, true, h(u), h(v)))
% 0.22/0.46 = { by axiom 15 (ax4) }
% 0.22/0.46 j(fresh15(sorti2(h(u)), true, h(u), h(v)))
% 0.22/0.46 = { by axiom 11 (co1) R->L }
% 0.22/0.46 j(fresh15(fresh8(sorti1(u), true, u), true, h(u), h(v)))
% 0.22/0.46 = { by axiom 1 (ax3_2) }
% 0.22/0.46 j(fresh15(fresh8(true, true, u), true, h(u), h(v)))
% 0.22/0.46 = { by axiom 5 (co1) }
% 0.22/0.46 j(fresh15(true, true, h(u), h(v)))
% 0.22/0.46 = { by lemma 20 R->L }
% 0.22/0.46 j(fresh15(true, true, op2(h(v), h(v)), h(v)))
% 0.22/0.46 = { by axiom 18 (ax4) }
% 0.22/0.46 j(fresh16(op2(h(v), h(v)), op2(h(v), h(v)), op2(h(v), h(v)), h(v)))
% 0.22/0.46 = { by axiom 7 (ax4) }
% 0.22/0.46 j(op2(h(v), h(v)))
% 0.22/0.46 = { by lemma 20 }
% 0.22/0.46 j(h(u))
% 0.22/0.46 = { by axiom 14 (co1_2) R->L }
% 0.22/0.46 fresh2(sorti1(u), true, u)
% 0.22/0.46 = { by axiom 1 (ax3_2) }
% 0.22/0.46 fresh2(true, true, u)
% 0.22/0.46 = { by axiom 6 (co1_2) }
% 0.22/0.46 u
% 0.22/0.46 % SZS output end Proof
% 0.22/0.46
% 0.22/0.46 RESULT: Theorem (the conjecture is true).
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