TSTP Solution File: ALG071+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG071+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 : n008.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 16:42:09 EDT 2023
% Result : Theorem 0.21s 0.51s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ALG071+1 : TPTP v8.1.2. Released v2.7.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n008.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 03:23:32 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.51 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.51
% 0.21/0.51 % SZS status Theorem
% 0.21/0.51
% 0.21/0.52 % SZS output start Proof
% 0.21/0.52 Take the following subset of the input axioms:
% 0.21/0.52 fof(ax1, axiom, ![U]: (sorti1(U) => ![V]: (sorti1(V) => sorti1(op1(U, V))))).
% 0.21/0.52 fof(ax3, axiom, ![U2]: (sorti1(U2) => ?[V2]: (sorti1(V2) & (op1(V2, op1(V2, U2))!=U2 & op1(U2, op1(V2, U2))=V2)))).
% 0.21/0.52 fof(ax4, axiom, ~![U2]: (sorti2(U2) => ?[V2]: (sorti2(V2) & (op2(V2, op2(V2, U2))!=U2 & op2(U2, op2(V2, U2))=V2)))).
% 0.21/0.52 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))))).
% 0.21/0.52
% 0.21/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.52 fresh(y, y, x1...xn) = u
% 0.21/0.52 C => fresh(s, t, x1...xn) = v
% 0.21/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.52 variables of u and v.
% 0.21/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.52 input problem has no model of domain size 1).
% 0.21/0.52
% 0.21/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.52
% 0.21/0.52 Axiom 1 (ax4): sorti2(u) = true2.
% 0.21/0.52 Axiom 2 (co1_5): fresh(X, X, Y) = Y.
% 0.21/0.52 Axiom 3 (ax3_1): fresh12(X, X, Y) = v(Y).
% 0.21/0.52 Axiom 4 (ax3_2): fresh11(X, X, Y) = true2.
% 0.21/0.52 Axiom 5 (ax4_1): fresh9(X, X, Y) = u.
% 0.21/0.52 Axiom 6 (co1): fresh8(X, X, Y) = true2.
% 0.21/0.52 Axiom 7 (co1_3): fresh5(X, X, Y) = true2.
% 0.21/0.52 Axiom 8 (co1_2): fresh2(X, X, Y) = Y.
% 0.21/0.52 Axiom 9 (ax4_1): fresh10(X, X, Y) = op2(Y, op2(Y, u)).
% 0.21/0.52 Axiom 10 (co1_5): fresh(sorti2(X), true2, X) = h(j(X)).
% 0.21/0.52 Axiom 11 (ax1): fresh16(X, X, Y, Z) = true2.
% 0.21/0.52 Axiom 12 (ax1): fresh15(X, X, Y, Z) = sorti1(op1(Y, Z)).
% 0.21/0.52 Axiom 13 (ax3_2): fresh11(sorti1(X), true2, X) = sorti1(v(X)).
% 0.21/0.52 Axiom 14 (co1): fresh8(sorti1(X), true2, X) = sorti2(h(X)).
% 0.21/0.52 Axiom 15 (co1_1): fresh7(X, X, Y, Z) = op2(h(Y), h(Z)).
% 0.21/0.52 Axiom 16 (co1_1): fresh6(X, X, Y, Z) = h(op1(Y, Z)).
% 0.21/0.52 Axiom 17 (co1_3): fresh5(sorti2(X), true2, X) = sorti1(j(X)).
% 0.21/0.52 Axiom 18 (co1_4): fresh4(X, X, Y, Z) = op1(j(Y), j(Z)).
% 0.21/0.52 Axiom 19 (co1_4): fresh3(X, X, Y, Z) = j(op2(Y, Z)).
% 0.21/0.52 Axiom 20 (co1_2): fresh2(sorti1(X), true2, X) = j(h(X)).
% 0.21/0.52 Axiom 21 (ax3_1): fresh12(sorti1(X), true2, X) = op1(X, op1(v(X), X)).
% 0.21/0.52 Axiom 22 (ax1): fresh15(sorti1(X), true2, Y, X) = fresh16(sorti1(Y), true2, Y, X).
% 0.21/0.52 Axiom 23 (co1_1): fresh7(sorti1(X), true2, Y, X) = fresh6(sorti1(Y), true2, Y, X).
% 0.21/0.52 Axiom 24 (co1_4): fresh4(sorti2(X), true2, Y, X) = fresh3(sorti2(Y), true2, Y, X).
% 0.21/0.52 Axiom 25 (ax4_1): fresh10(sorti2(X), true2, X) = fresh9(op2(u, op2(X, u)), X, X).
% 0.21/0.52
% 0.21/0.52 Lemma 26: sorti1(j(u)) = true2.
% 0.21/0.52 Proof:
% 0.21/0.52 sorti1(j(u))
% 0.21/0.52 = { by axiom 17 (co1_3) R->L }
% 0.21/0.52 fresh5(sorti2(u), true2, u)
% 0.21/0.52 = { by axiom 1 (ax4) }
% 0.21/0.52 fresh5(true2, true2, u)
% 0.21/0.52 = { by axiom 7 (co1_3) }
% 0.21/0.52 true2
% 0.21/0.52
% 0.21/0.52 Lemma 27: h(j(u)) = u.
% 0.21/0.52 Proof:
% 0.21/0.52 h(j(u))
% 0.21/0.52 = { by axiom 10 (co1_5) R->L }
% 0.21/0.52 fresh(sorti2(u), true2, u)
% 0.21/0.52 = { by axiom 1 (ax4) }
% 0.21/0.52 fresh(true2, true2, u)
% 0.21/0.52 = { by axiom 2 (co1_5) }
% 0.21/0.52 u
% 0.21/0.52
% 0.21/0.52 Lemma 28: sorti1(v(j(u))) = true2.
% 0.21/0.52 Proof:
% 0.21/0.52 sorti1(v(j(u)))
% 0.21/0.52 = { by axiom 13 (ax3_2) R->L }
% 0.21/0.52 fresh11(sorti1(j(u)), true2, j(u))
% 0.21/0.52 = { by lemma 26 }
% 0.21/0.52 fresh11(true2, true2, j(u))
% 0.21/0.52 = { by axiom 4 (ax3_2) }
% 0.21/0.52 true2
% 0.21/0.52
% 0.21/0.52 Lemma 29: sorti2(h(v(j(u)))) = true2.
% 0.21/0.52 Proof:
% 0.21/0.52 sorti2(h(v(j(u))))
% 0.21/0.52 = { by axiom 14 (co1) R->L }
% 0.21/0.52 fresh8(sorti1(v(j(u))), true2, v(j(u)))
% 0.21/0.52 = { by lemma 28 }
% 0.21/0.52 fresh8(true2, true2, v(j(u)))
% 0.21/0.52 = { by axiom 6 (co1) }
% 0.21/0.52 true2
% 0.21/0.52
% 0.21/0.52 Lemma 30: sorti1(op1(v(j(u)), j(u))) = true2.
% 0.21/0.53 Proof:
% 0.21/0.53 sorti1(op1(v(j(u)), j(u)))
% 0.21/0.53 = { by axiom 12 (ax1) R->L }
% 0.21/0.53 fresh15(true2, true2, v(j(u)), j(u))
% 0.21/0.53 = { by lemma 26 R->L }
% 0.21/0.53 fresh15(sorti1(j(u)), true2, v(j(u)), j(u))
% 0.21/0.53 = { by axiom 22 (ax1) }
% 0.21/0.53 fresh16(sorti1(v(j(u))), true2, v(j(u)), j(u))
% 0.21/0.53 = { by lemma 28 }
% 0.21/0.53 fresh16(true2, true2, v(j(u)), j(u))
% 0.21/0.53 = { by axiom 11 (ax1) }
% 0.21/0.53 true2
% 0.21/0.53
% 0.21/0.53 Lemma 31: h(op1(v(j(u)), j(u))) = op2(h(v(j(u))), u).
% 0.21/0.53 Proof:
% 0.21/0.53 h(op1(v(j(u)), j(u)))
% 0.21/0.53 = { by axiom 16 (co1_1) R->L }
% 0.21/0.53 fresh6(true2, true2, v(j(u)), j(u))
% 0.21/0.53 = { by lemma 28 R->L }
% 0.21/0.53 fresh6(sorti1(v(j(u))), true2, v(j(u)), j(u))
% 0.21/0.53 = { by axiom 23 (co1_1) R->L }
% 0.21/0.53 fresh7(sorti1(j(u)), true2, v(j(u)), j(u))
% 0.21/0.53 = { by lemma 26 }
% 0.21/0.53 fresh7(true2, true2, v(j(u)), j(u))
% 0.21/0.53 = { by axiom 15 (co1_1) }
% 0.21/0.53 op2(h(v(j(u))), h(j(u)))
% 0.21/0.53 = { by lemma 27 }
% 0.21/0.53 op2(h(v(j(u))), u)
% 0.21/0.53
% 0.21/0.53 Goal 1 (ax3): tuple(op1(v(X), op1(v(X), X)), sorti1(X)) = tuple(X, true2).
% 0.21/0.53 The goal is true when:
% 0.21/0.53 X = j(u)
% 0.21/0.53
% 0.21/0.53 Proof:
% 0.21/0.53 tuple(op1(v(j(u)), op1(v(j(u)), j(u))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 8 (co1_2) R->L }
% 0.21/0.53 tuple(op1(v(j(u)), fresh2(true2, true2, op1(v(j(u)), j(u)))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 30 R->L }
% 0.21/0.53 tuple(op1(v(j(u)), fresh2(sorti1(op1(v(j(u)), j(u))), true2, op1(v(j(u)), j(u)))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 20 (co1_2) }
% 0.21/0.53 tuple(op1(v(j(u)), j(h(op1(v(j(u)), j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 31 }
% 0.21/0.53 tuple(op1(v(j(u)), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 8 (co1_2) R->L }
% 0.21/0.53 tuple(op1(fresh2(true2, true2, v(j(u))), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 28 R->L }
% 0.21/0.53 tuple(op1(fresh2(sorti1(v(j(u))), true2, v(j(u))), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 20 (co1_2) }
% 0.21/0.53 tuple(op1(j(h(v(j(u)))), j(op2(h(v(j(u))), u))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 18 (co1_4) R->L }
% 0.21/0.53 tuple(fresh4(true2, true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by axiom 6 (co1) R->L }
% 0.21/0.53 tuple(fresh4(fresh8(true2, true2, op1(v(j(u)), j(u))), true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by lemma 30 R->L }
% 0.21/0.53 tuple(fresh4(fresh8(sorti1(op1(v(j(u)), j(u))), true2, op1(v(j(u)), j(u))), true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by axiom 14 (co1) }
% 0.21/0.53 tuple(fresh4(sorti2(h(op1(v(j(u)), j(u)))), true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by lemma 31 }
% 0.21/0.53 tuple(fresh4(sorti2(op2(h(v(j(u))), u)), true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by axiom 24 (co1_4) }
% 0.21/0.53 tuple(fresh3(sorti2(h(v(j(u)))), true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by lemma 29 }
% 0.21/0.53 tuple(fresh3(true2, true2, h(v(j(u))), op2(h(v(j(u))), u)), sorti1(j(u)))
% 0.21/0.53 = { by axiom 19 (co1_4) }
% 0.21/0.53 tuple(j(op2(h(v(j(u))), op2(h(v(j(u))), u))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 9 (ax4_1) R->L }
% 0.21/0.53 tuple(j(fresh10(true2, true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 29 R->L }
% 0.21/0.53 tuple(j(fresh10(sorti2(h(v(j(u)))), true2, h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 25 (ax4_1) }
% 0.21/0.53 tuple(j(fresh9(op2(u, op2(h(v(j(u))), u)), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 31 R->L }
% 0.21/0.53 tuple(j(fresh9(op2(u, h(op1(v(j(u)), j(u)))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 27 R->L }
% 0.21/0.53 tuple(j(fresh9(op2(h(j(u)), h(op1(v(j(u)), j(u)))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 15 (co1_1) R->L }
% 0.21/0.53 tuple(j(fresh9(fresh7(true2, true2, j(u), op1(v(j(u)), j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 30 R->L }
% 0.21/0.53 tuple(j(fresh9(fresh7(sorti1(op1(v(j(u)), j(u))), true2, j(u), op1(v(j(u)), j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 23 (co1_1) }
% 0.21/0.53 tuple(j(fresh9(fresh6(sorti1(j(u)), true2, j(u), op1(v(j(u)), j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 26 }
% 0.21/0.53 tuple(j(fresh9(fresh6(true2, true2, j(u), op1(v(j(u)), j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 16 (co1_1) }
% 0.21/0.53 tuple(j(fresh9(h(op1(j(u), op1(v(j(u)), j(u)))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 21 (ax3_1) R->L }
% 0.21/0.53 tuple(j(fresh9(h(fresh12(sorti1(j(u)), true2, j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by lemma 26 }
% 0.21/0.53 tuple(j(fresh9(h(fresh12(true2, true2, j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 3 (ax3_1) }
% 0.21/0.53 tuple(j(fresh9(h(v(j(u))), h(v(j(u))), h(v(j(u))))), sorti1(j(u)))
% 0.21/0.53 = { by axiom 5 (ax4_1) }
% 0.21/0.53 tuple(j(u), sorti1(j(u)))
% 0.21/0.53 = { by lemma 26 }
% 0.21/0.53 tuple(j(u), true2)
% 0.21/0.53 % SZS output end Proof
% 0.21/0.53
% 0.21/0.53 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------