TSTP Solution File: SWC422+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWC422+1 : TPTP v8.1.2. Released v2.4.0.
% 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 : n010.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 20:55:30 EDT 2023
% Result : Theorem 4.01s 0.88s
% Output : Proof 4.17s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SWC422+1 : TPTP v8.1.2. Released v2.4.0.
% 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 : n010.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 17:38:19 EDT 2023
% 0.14/0.35 % CPUTime :
% 4.01/0.88 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.01/0.88
% 4.01/0.88 % SZS status Theorem
% 4.01/0.88
% 4.01/0.88 % SZS output start Proof
% 4.01/0.88 Take the following subset of the input axioms:
% 4.01/0.90 fof(ax1, axiom, ![U]: (ssItem(U) => ![V]: (ssItem(V) => (neq(U, V) <=> U!=V)))).
% 4.17/0.90 fof(ax13, axiom, ![U2]: (ssList(U2) => (duplicatefreeP(U2) <=> ![V2]: (ssItem(V2) => ![W]: (ssItem(W) => ![X]: (ssList(X) => ![Y]: (ssList(Y) => ![Z]: (ssList(Z) => (app(app(X, cons(V2, Y)), cons(W, Z))=U2 => V2!=W))))))))).
% 4.17/0.90 fof(ax15, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (neq(U2, V2) <=> U2!=V2)))).
% 4.17/0.90 fof(ax18, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => cons(V2, U2)!=U2))).
% 4.17/0.90 fof(ax21, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => nil!=cons(V2, U2)))).
% 4.17/0.90 fof(ax33, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) => ~lt(V2, U2))))).
% 4.17/0.90 fof(ax38, axiom, ![U2]: (ssItem(U2) => ~memberP(nil, U2))).
% 4.17/0.90 fof(ax8, axiom, ![U2]: (ssList(U2) => (cyclefreeP(U2) <=> ![V2]: (ssItem(V2) => ![W2]: (ssItem(W2) => ![X5]: (ssList(X5) => ![Y2]: (ssList(Y2) => ![Z2]: (ssList(Z2) => (app(app(X5, cons(V2, Y2)), cons(W2, Z2))=U2 => ~(leq(V2, W2) & leq(W2, V2))))))))))).
% 4.17/0.90 fof(ax90, axiom, ![U2]: (ssItem(U2) => ~lt(U2, U2))).
% 4.17/0.90 fof(ax93, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) <=> (U2!=V2 & leq(U2, V2)))))).
% 4.17/0.90 fof(ax94, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (gt(U2, V2) => ~gt(V2, U2))))).
% 4.17/0.90 fof(co1, conjecture, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => ![X5]: (ssList(X5) => (V2!=X5 | (U2!=W2 | ((~neq(V2, nil) | (?[Y2]: (ssItem(Y2) & ?[Z2]: (ssList(Z2) & ?[X1]: (ssList(X1) & (app(app(Z2, cons(Y2, nil)), X1)=V2 & app(app(X1, cons(Y2, nil)), Z2)=U2)))) | ![X2]: (ssItem(X2) => ![X3]: (ssList(X3) => ![X4]: (ssList(X4) => (app(app(X3, cons(X2, nil)), X4)!=X5 | app(app(X4, cons(X2, nil)), X3)!=W2)))))) & (~neq(V2, nil) | neq(X5, nil)))))))))).
% 4.17/0.90
% 4.17/0.90 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.17/0.90 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.17/0.90 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.17/0.90 fresh(y, y, x1...xn) = u
% 4.17/0.90 C => fresh(s, t, x1...xn) = v
% 4.17/0.90 where fresh is a fresh function symbol and x1..xn are the free
% 4.17/0.90 variables of u and v.
% 4.17/0.90 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.17/0.90 input problem has no model of domain size 1).
% 4.17/0.90
% 4.17/0.90 The encoding turns the above axioms into the following unit equations and goals:
% 4.17/0.90
% 4.17/0.90 Axiom 1 (co1_3): v = x.
% 4.17/0.90 Axiom 2 (co1_2): u = w.
% 4.17/0.90 Axiom 3 (co1_5): neq(v, nil) = true2.
% 4.17/0.90 Axiom 4 (co1_14): fresh19(X, X) = x.
% 4.17/0.90 Axiom 5 (co1_15): fresh18(X, X) = w.
% 4.17/0.90 Axiom 6 (co1_16): fresh17(X, X) = true2.
% 4.17/0.90 Axiom 7 (co1_18): fresh15(X, X) = true2.
% 4.17/0.90 Axiom 8 (co1_19): fresh14(X, X) = true2.
% 4.17/0.90 Axiom 9 (co1_16): fresh17(neq(x, nil), true2) = ssItem(x2).
% 4.17/0.90 Axiom 10 (co1_18): fresh15(neq(x, nil), true2) = ssList(x3).
% 4.17/0.90 Axiom 11 (co1_19): fresh14(neq(x, nil), true2) = ssList(x4).
% 4.17/0.90 Axiom 12 (co1_14): fresh19(neq(x, nil), true2) = app(app(x3, cons(x2, nil)), x4).
% 4.17/0.90 Axiom 13 (co1_15): fresh18(neq(x, nil), true2) = app(app(x4, cons(x2, nil)), x3).
% 4.17/0.90
% 4.17/0.90 Lemma 14: neq(x, nil) = true2.
% 4.17/0.90 Proof:
% 4.17/0.90 neq(x, nil)
% 4.17/0.90 = { by axiom 1 (co1_3) R->L }
% 4.17/0.90 neq(v, nil)
% 4.17/0.90 = { by axiom 3 (co1_5) }
% 4.17/0.90 true2
% 4.17/0.90
% 4.17/0.90 Goal 1 (co1_12): tuple6(app(app(X, cons(Y, nil)), Z), app(app(Z, cons(Y, nil)), X), ssItem(Y), neq(x, nil), ssList(X), ssList(Z)) = tuple6(v, u, true2, true2, true2, true2).
% 4.17/0.90 The goal is true when:
% 4.17/0.90 X = x3
% 4.17/0.90 Y = x2
% 4.17/0.90 Z = x4
% 4.17/0.90
% 4.17/0.90 Proof:
% 4.17/0.90 tuple6(app(app(x3, cons(x2, nil)), x4), app(app(x4, cons(x2, nil)), x3), ssItem(x2), neq(x, nil), ssList(x3), ssList(x4))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(app(app(x3, cons(x2, nil)), x4), app(app(x4, cons(x2, nil)), x3), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 12 (co1_14) R->L }
% 4.17/0.90 tuple6(fresh19(neq(x, nil), true2), app(app(x4, cons(x2, nil)), x3), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(fresh19(true2, true2), app(app(x4, cons(x2, nil)), x3), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 4 (co1_14) }
% 4.17/0.90 tuple6(x, app(app(x4, cons(x2, nil)), x3), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 13 (co1_15) R->L }
% 4.17/0.90 tuple6(x, fresh18(neq(x, nil), true2), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(x, fresh18(true2, true2), ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 5 (co1_15) }
% 4.17/0.90 tuple6(x, w, ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 2 (co1_2) R->L }
% 4.17/0.90 tuple6(x, u, ssItem(x2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 9 (co1_16) R->L }
% 4.17/0.90 tuple6(x, u, fresh17(neq(x, nil), true2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(x, u, fresh17(true2, true2), true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 6 (co1_16) }
% 4.17/0.90 tuple6(x, u, true2, true2, ssList(x3), ssList(x4))
% 4.17/0.90 = { by axiom 10 (co1_18) R->L }
% 4.17/0.90 tuple6(x, u, true2, true2, fresh15(neq(x, nil), true2), ssList(x4))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(x, u, true2, true2, fresh15(true2, true2), ssList(x4))
% 4.17/0.90 = { by axiom 7 (co1_18) }
% 4.17/0.90 tuple6(x, u, true2, true2, true2, ssList(x4))
% 4.17/0.90 = { by axiom 11 (co1_19) R->L }
% 4.17/0.90 tuple6(x, u, true2, true2, true2, fresh14(neq(x, nil), true2))
% 4.17/0.90 = { by lemma 14 }
% 4.17/0.90 tuple6(x, u, true2, true2, true2, fresh14(true2, true2))
% 4.17/0.90 = { by axiom 8 (co1_19) }
% 4.17/0.90 tuple6(x, u, true2, true2, true2, true2)
% 4.17/0.90 = { by axiom 1 (co1_3) R->L }
% 4.17/0.90 tuple6(v, u, true2, true2, true2, true2)
% 4.17/0.90 % SZS output end Proof
% 4.17/0.90
% 4.17/0.90 RESULT: Theorem (the conjecture is true).
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