TSTP Solution File: SWC363+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWC363+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 : n031.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 20:55:11 EDT 2023
% Result : Theorem 96.59s 12.83s
% Output : Proof 96.59s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : SWC363+1 : TPTP v8.1.2. Released v2.4.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n031.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Mon Aug 28 18:25:55 EDT 2023
% 0.13/0.35 % CPUTime :
% 96.59/12.83 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 96.59/12.83
% 96.59/12.83 % SZS status Theorem
% 96.59/12.83
% 96.59/12.85 % SZS output start Proof
% 96.59/12.85 Take the following subset of the input axioms:
% 96.59/12.85 fof(ax3, axiom, ![U]: (ssList(U) => ![V]: (ssItem(V) => (memberP(U, V) <=> ?[W]: (ssList(W) & ?[X]: (ssList(X) & app(W, cons(V, X))=U)))))).
% 96.59/12.85 fof(ax7, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (segmentP(U2, V2) <=> ?[W2]: (ssList(W2) & ?[X2]: (ssList(X2) & app(app(W2, V2), X2)=U2)))))).
% 96.59/12.85 fof(ax81, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => cons(V2, U2)=app(cons(V2, nil), U2)))).
% 96.59/12.85 fof(ax82, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => app(app(U2, V2), W2)=app(U2, app(V2, W2)))))).
% 96.59/12.85 fof(co1, conjecture, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => ![X2]: (ssList(X2) => (V2!=X2 | (U2!=W2 | (~neq(V2, nil) | (segmentP(V2, U2) | ((nil!=W2 & nil=X2) | (![Y]: (ssItem(Y) => (cons(Y, nil)!=W2 | ~memberP(X2, Y))) & neq(X2, nil)))))))))))).
% 96.59/12.85
% 96.59/12.85 Now clausify the problem and encode Horn clauses using encoding 3 of
% 96.59/12.85 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 96.59/12.85 We repeatedly replace C & s=t => u=v by the two clauses:
% 96.59/12.85 fresh(y, y, x1...xn) = u
% 96.59/12.85 C => fresh(s, t, x1...xn) = v
% 96.59/12.85 where fresh is a fresh function symbol and x1..xn are the free
% 96.59/12.85 variables of u and v.
% 96.59/12.85 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 96.59/12.85 input problem has no model of domain size 1).
% 96.59/12.85
% 96.59/12.85 The encoding turns the above axioms into the following unit equations and goals:
% 96.59/12.85
% 96.59/12.85 Axiom 1 (co1_1): v = x.
% 96.59/12.85 Axiom 2 (co1): u = w.
% 96.59/12.85 Axiom 3 (co1_6): ssList(x) = true2.
% 96.59/12.85 Axiom 4 (co1_3): ssList(u) = true2.
% 96.59/12.85 Axiom 5 (co1_2): neq(v, nil) = true2.
% 96.59/12.85 Axiom 6 (co1_10): fresh17(X, X) = true2.
% 96.59/12.85 Axiom 7 (co1_8): fresh15(X, X) = w.
% 96.59/12.85 Axiom 8 (co1_9): fresh14(X, X) = true2.
% 96.59/12.85 Axiom 9 (ax3_3): fresh291(X, X, Y, Z) = true2.
% 96.59/12.85 Axiom 10 (ax3_2): fresh289(X, X, Y, Z) = true2.
% 96.59/12.85 Axiom 11 (ax3_1): fresh287(X, X, Y, Z) = Y.
% 96.59/12.85 Axiom 12 (ax3_2): fresh65(X, X, Y, Z) = ssList(w12(Y, Z)).
% 96.59/12.85 Axiom 13 (ax3_3): fresh64(X, X, Y, Z) = ssList(x10(Y, Z)).
% 96.59/12.85 Axiom 14 (ax7): fresh34(X, X, Y, Z) = true2.
% 96.59/12.85 Axiom 15 (ax81): fresh28(X, X, Y, Z) = app(cons(Z, nil), Y).
% 96.59/12.85 Axiom 16 (ax81): fresh27(X, X, Y, Z) = cons(Z, Y).
% 96.59/12.85 Axiom 17 (co1_10): fresh17(neq(x, nil), true2) = memberP(x, y).
% 96.59/12.85 Axiom 18 (co1_8): fresh15(neq(x, nil), true2) = cons(y, nil).
% 96.59/12.85 Axiom 19 (co1_9): fresh14(neq(x, nil), true2) = ssItem(y).
% 96.59/12.85 Axiom 20 (ax3_3): fresh290(X, X, Y, Z) = fresh291(ssItem(Z), true2, Y, Z).
% 96.59/12.85 Axiom 21 (ax3_2): fresh288(X, X, Y, Z) = fresh289(ssItem(Z), true2, Y, Z).
% 96.59/12.85 Axiom 22 (ax3_1): fresh286(X, X, Y, Z) = fresh287(ssItem(Z), true2, Y, Z).
% 96.59/12.85 Axiom 23 (ax3_1): fresh285(X, X, Y, Z) = fresh286(ssList(Y), true2, Y, Z).
% 96.59/12.85 Axiom 24 (ax82): fresh123(X, X, Y, Z, W) = app(Y, app(Z, W)).
% 96.59/12.85 Axiom 25 (ax81): fresh28(ssList(X), true2, X, Y) = fresh27(ssItem(Y), true2, X, Y).
% 96.59/12.85 Axiom 26 (ax82): fresh26(X, X, Y, Z, W) = app(app(Y, Z), W).
% 96.59/12.85 Axiom 27 (ax3_3): fresh290(memberP(X, Y), true2, X, Y) = fresh64(ssList(X), true2, X, Y).
% 96.59/12.85 Axiom 28 (ax3_2): fresh288(memberP(X, Y), true2, X, Y) = fresh65(ssList(X), true2, X, Y).
% 96.59/12.85 Axiom 29 (ax7): fresh255(X, X, Y, Z, W, V) = segmentP(Y, Z).
% 96.59/12.85 Axiom 30 (ax82): fresh122(X, X, Y, Z, W) = fresh123(ssList(Y), true2, Y, Z, W).
% 96.59/12.85 Axiom 31 (ax82): fresh122(ssList(X), true2, Y, Z, X) = fresh26(ssList(Z), true2, Y, Z, X).
% 96.59/12.85 Axiom 32 (ax7): fresh254(X, X, Y, Z, W, V) = fresh255(ssList(Y), true2, Y, Z, W, V).
% 96.59/12.85 Axiom 33 (ax7): fresh253(X, X, Y, Z, W, V) = fresh254(ssList(Z), true2, Y, Z, W, V).
% 96.59/12.85 Axiom 34 (ax7): fresh252(X, X, Y, Z, W, V) = fresh253(ssList(W), true2, Y, Z, W, V).
% 96.59/12.85 Axiom 35 (ax3_1): fresh285(memberP(X, Y), true2, X, Y) = app(w12(X, Y), cons(Y, x10(X, Y))).
% 96.59/12.85 Axiom 36 (ax7): fresh252(ssList(X), true2, Y, Z, W, X) = fresh34(app(app(W, Z), X), Y, Y, Z).
% 96.59/12.85
% 96.59/12.85 Lemma 37: neq(x, nil) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 neq(x, nil)
% 96.59/12.85 = { by axiom 1 (co1_1) R->L }
% 96.59/12.85 neq(v, nil)
% 96.59/12.85 = { by axiom 5 (co1_2) }
% 96.59/12.85 true2
% 96.59/12.85
% 96.59/12.85 Lemma 38: ssItem(y) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 ssItem(y)
% 96.59/12.85 = { by axiom 19 (co1_9) R->L }
% 96.59/12.85 fresh14(neq(x, nil), true2)
% 96.59/12.85 = { by lemma 37 }
% 96.59/12.85 fresh14(true2, true2)
% 96.59/12.85 = { by axiom 8 (co1_9) }
% 96.59/12.85 true2
% 96.59/12.85
% 96.59/12.85 Lemma 39: memberP(x, y) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 memberP(x, y)
% 96.59/12.85 = { by axiom 17 (co1_10) R->L }
% 96.59/12.85 fresh17(neq(x, nil), true2)
% 96.59/12.85 = { by lemma 37 }
% 96.59/12.85 fresh17(true2, true2)
% 96.59/12.85 = { by axiom 6 (co1_10) }
% 96.59/12.85 true2
% 96.59/12.85
% 96.59/12.85 Lemma 40: ssList(w12(x, y)) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 ssList(w12(x, y))
% 96.59/12.85 = { by axiom 12 (ax3_2) R->L }
% 96.59/12.85 fresh65(true2, true2, x, y)
% 96.59/12.85 = { by axiom 3 (co1_6) R->L }
% 96.59/12.85 fresh65(ssList(x), true2, x, y)
% 96.59/12.85 = { by axiom 28 (ax3_2) R->L }
% 96.59/12.85 fresh288(memberP(x, y), true2, x, y)
% 96.59/12.85 = { by lemma 39 }
% 96.59/12.85 fresh288(true2, true2, x, y)
% 96.59/12.85 = { by axiom 21 (ax3_2) }
% 96.59/12.85 fresh289(ssItem(y), true2, x, y)
% 96.59/12.85 = { by lemma 38 }
% 96.59/12.85 fresh289(true2, true2, x, y)
% 96.59/12.85 = { by axiom 10 (ax3_2) }
% 96.59/12.85 true2
% 96.59/12.85
% 96.59/12.85 Lemma 41: ssList(x10(x, y)) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 ssList(x10(x, y))
% 96.59/12.85 = { by axiom 13 (ax3_3) R->L }
% 96.59/12.85 fresh64(true2, true2, x, y)
% 96.59/12.85 = { by axiom 3 (co1_6) R->L }
% 96.59/12.85 fresh64(ssList(x), true2, x, y)
% 96.59/12.85 = { by axiom 27 (ax3_3) R->L }
% 96.59/12.85 fresh290(memberP(x, y), true2, x, y)
% 96.59/12.85 = { by lemma 39 }
% 96.59/12.85 fresh290(true2, true2, x, y)
% 96.59/12.85 = { by axiom 20 (ax3_3) }
% 96.59/12.85 fresh291(ssItem(y), true2, x, y)
% 96.59/12.85 = { by lemma 38 }
% 96.59/12.85 fresh291(true2, true2, x, y)
% 96.59/12.85 = { by axiom 9 (ax3_3) }
% 96.59/12.85 true2
% 96.59/12.85
% 96.59/12.85 Goal 1 (co1_11): segmentP(v, u) = true2.
% 96.59/12.85 Proof:
% 96.59/12.85 segmentP(v, u)
% 96.59/12.85 = { by axiom 1 (co1_1) }
% 96.59/12.85 segmentP(x, u)
% 96.59/12.85 = { by axiom 29 (ax7) R->L }
% 96.59/12.85 fresh255(true2, true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 3 (co1_6) R->L }
% 96.59/12.85 fresh255(ssList(x), true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 32 (ax7) R->L }
% 96.59/12.85 fresh254(true2, true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 4 (co1_3) R->L }
% 96.59/12.85 fresh254(ssList(u), true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 33 (ax7) R->L }
% 96.59/12.85 fresh253(true2, true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by lemma 40 R->L }
% 96.59/12.85 fresh253(ssList(w12(x, y)), true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 34 (ax7) R->L }
% 96.59/12.85 fresh252(true2, true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by lemma 41 R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, x, u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 11 (ax3_1) R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh287(true2, true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by lemma 38 R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh287(ssItem(y), true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 22 (ax3_1) R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh286(true2, true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 3 (co1_6) R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh286(ssList(x), true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 23 (ax3_1) R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh285(true2, true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by lemma 39 R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, fresh285(memberP(x, y), true2, x, y), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 35 (ax3_1) }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), cons(y, x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by axiom 16 (ax81) R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), fresh27(true2, true2, x10(x, y), y)), u, w12(x, y), x10(x, y))
% 96.59/12.85 = { by lemma 38 R->L }
% 96.59/12.85 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), fresh27(ssItem(y), true2, x10(x, y), y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 25 (ax81) R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), fresh28(ssList(x10(x, y)), true2, x10(x, y), y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by lemma 41 }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), fresh28(true2, true2, x10(x, y), y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 15 (ax81) }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), app(cons(y, nil), x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 18 (co1_8) R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), app(fresh15(neq(x, nil), true2), x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by lemma 37 }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), app(fresh15(true2, true2), x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 7 (co1_8) }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), app(w, x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 2 (co1) R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(w12(x, y), app(u, x10(x, y))), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 24 (ax82) R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh123(true2, true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by lemma 40 R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh123(ssList(w12(x, y)), true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 30 (ax82) R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh122(true2, true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by lemma 41 R->L }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh122(ssList(x10(x, y)), true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 31 (ax82) }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh26(ssList(u), true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 4 (co1_3) }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, fresh26(true2, true2, w12(x, y), u, x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 26 (ax82) }
% 96.59/12.86 fresh252(ssList(x10(x, y)), true2, app(app(w12(x, y), u), x10(x, y)), u, w12(x, y), x10(x, y))
% 96.59/12.86 = { by axiom 36 (ax7) }
% 96.59/12.86 fresh34(app(app(w12(x, y), u), x10(x, y)), app(app(w12(x, y), u), x10(x, y)), app(app(w12(x, y), u), x10(x, y)), u)
% 96.59/12.86 = { by axiom 14 (ax7) }
% 96.59/12.86 true2
% 96.59/12.86 % SZS output end Proof
% 96.59/12.86
% 96.59/12.86 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------