TSTP Solution File: ALG041+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG041+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 : n006.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 16:42:05 EDT 2023
% Result : Theorem 0.21s 0.42s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14 % Problem : ALG041+1 : TPTP v8.1.2. Released v2.7.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n006.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Mon Aug 28 04:50:36 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.21/0.42 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.42
% 0.21/0.42 % SZS status Theorem
% 0.21/0.42
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Take the following subset of the input axioms:
% 0.21/0.43 fof(ax1, axiom, ![U]: (sorti1(U) => ![V]: (sorti1(V) => sorti1(op1(U, V))))).
% 0.21/0.44 fof(ax3, axiom, ~?[U2]: (sorti1(U2) & ![V2]: (sorti1(V2) => op1(V2, V2)=U2))).
% 0.21/0.44 fof(ax4, axiom, ~~?[U2]: (sorti2(U2) & ![V2]: (sorti2(V2) => op2(V2, V2)=U2))).
% 0.21/0.44 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))))).
% 0.21/0.44
% 0.21/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.44 fresh(y, y, x1...xn) = u
% 0.21/0.44 C => fresh(s, t, x1...xn) = v
% 0.21/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.44 variables of u and v.
% 0.21/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.44 input problem has no model of domain size 1).
% 0.21/0.44
% 0.21/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.44
% 0.21/0.44 Axiom 1 (ax4): sorti2(u) = true2.
% 0.21/0.44 Axiom 2 (ax3_1): fresh10(X, X, Y) = true2.
% 0.21/0.44 Axiom 3 (ax4_1): fresh9(X, X, Y) = u.
% 0.21/0.44 Axiom 4 (co1): fresh8(X, X, Y) = true2.
% 0.21/0.44 Axiom 5 (co1_3): fresh5(X, X, Y) = true2.
% 0.21/0.44 Axiom 6 (co1_2): fresh2(X, X, Y) = Y.
% 0.21/0.44 Axiom 7 (ax1): fresh14(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 8 (ax1): fresh13(X, X, Y, Z) = sorti1(op1(Y, Z)).
% 0.21/0.44 Axiom 9 (ax3_1): fresh10(sorti1(X), true2, X) = sorti1(v(X)).
% 0.21/0.44 Axiom 10 (ax4_1): fresh9(sorti2(X), true2, X) = op2(X, X).
% 0.21/0.44 Axiom 11 (co1): fresh8(sorti1(X), true2, X) = sorti2(h(X)).
% 0.21/0.44 Axiom 12 (co1_1): fresh7(X, X, Y, Z) = op2(h(Y), h(Z)).
% 0.21/0.44 Axiom 13 (co1_1): fresh6(X, X, Y, Z) = h(op1(Y, Z)).
% 0.21/0.44 Axiom 14 (co1_3): fresh5(sorti2(X), true2, X) = sorti1(j(X)).
% 0.21/0.44 Axiom 15 (co1_2): fresh2(sorti1(X), true2, X) = j(h(X)).
% 0.21/0.44 Axiom 16 (ax1): fresh13(sorti1(X), true2, Y, X) = fresh14(sorti1(Y), true2, Y, X).
% 0.21/0.44 Axiom 17 (co1_1): fresh7(sorti1(X), true2, Y, X) = fresh6(sorti1(Y), true2, Y, X).
% 0.21/0.44
% 0.21/0.44 Lemma 18: sorti1(j(u)) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 sorti1(j(u))
% 0.21/0.44 = { by axiom 14 (co1_3) R->L }
% 0.21/0.44 fresh5(sorti2(u), true2, u)
% 0.21/0.44 = { by axiom 1 (ax4) }
% 0.21/0.44 fresh5(true2, true2, u)
% 0.21/0.44 = { by axiom 5 (co1_3) }
% 0.21/0.44 true2
% 0.21/0.44
% 0.21/0.44 Lemma 19: sorti1(v(j(u))) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 sorti1(v(j(u)))
% 0.21/0.44 = { by axiom 9 (ax3_1) R->L }
% 0.21/0.44 fresh10(sorti1(j(u)), true2, j(u))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 fresh10(true2, true2, j(u))
% 0.21/0.44 = { by axiom 2 (ax3_1) }
% 0.21/0.44 true2
% 0.21/0.44
% 0.21/0.44 Goal 1 (ax3): tuple(op1(v(X), v(X)), sorti1(X)) = tuple(X, true2).
% 0.21/0.44 The goal is true when:
% 0.21/0.44 X = j(u)
% 0.21/0.44
% 0.21/0.44 Proof:
% 0.21/0.44 tuple(op1(v(j(u)), v(j(u))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 6 (co1_2) R->L }
% 0.21/0.44 tuple(fresh2(true2, true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 7 (ax1) R->L }
% 0.21/0.44 tuple(fresh2(fresh14(true2, true2, v(j(u)), v(j(u))), true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by lemma 19 R->L }
% 0.21/0.44 tuple(fresh2(fresh14(sorti1(v(j(u))), true2, v(j(u)), v(j(u))), true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 16 (ax1) R->L }
% 0.21/0.44 tuple(fresh2(fresh13(sorti1(v(j(u))), true2, v(j(u)), v(j(u))), true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by lemma 19 }
% 0.21/0.44 tuple(fresh2(fresh13(true2, true2, v(j(u)), v(j(u))), true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 8 (ax1) }
% 0.21/0.44 tuple(fresh2(sorti1(op1(v(j(u)), v(j(u)))), true2, op1(v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 15 (co1_2) }
% 0.21/0.44 tuple(j(h(op1(v(j(u)), v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 13 (co1_1) R->L }
% 0.21/0.44 tuple(j(fresh6(true2, true2, v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by lemma 19 R->L }
% 0.21/0.44 tuple(j(fresh6(sorti1(v(j(u))), true2, v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 17 (co1_1) R->L }
% 0.21/0.44 tuple(j(fresh7(sorti1(v(j(u))), true2, v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by lemma 19 }
% 0.21/0.44 tuple(j(fresh7(true2, true2, v(j(u)), v(j(u)))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 12 (co1_1) }
% 0.21/0.44 tuple(j(op2(h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 10 (ax4_1) R->L }
% 0.21/0.44 tuple(j(fresh9(sorti2(h(v(j(u)))), true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 11 (co1) R->L }
% 0.21/0.44 tuple(j(fresh9(fresh8(sorti1(v(j(u))), true2, v(j(u))), true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by lemma 19 }
% 0.21/0.44 tuple(j(fresh9(fresh8(true2, true2, v(j(u))), true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 4 (co1) }
% 0.21/0.44 tuple(j(fresh9(true2, true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.44 = { by axiom 3 (ax4_1) }
% 0.21/0.44 tuple(j(u), sorti1(j(u)))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 tuple(j(u), true2)
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44
% 0.21/0.44 RESULT: Theorem (the conjecture is true).
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