TSTP Solution File: SWC008+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWC008+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 : 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 : Thu Aug 31 20:53:23 EDT 2023
% Result : Theorem 9.40s 1.56s
% Output : Proof 9.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SWC008+1 : TPTP v8.1.2. Released v2.4.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n006.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Mon Aug 28 16:20:51 EDT 2023
% 0.13/0.34 % CPUTime :
% 9.40/1.56 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 9.40/1.56
% 9.40/1.56 % SZS status Theorem
% 9.40/1.56
% 9.40/1.57 % SZS output start Proof
% 9.40/1.57 Take the following subset of the input axioms:
% 9.40/1.59 fof(ax1, axiom, ![U]: (ssItem(U) => ![V]: (ssItem(V) => (neq(U, V) <=> U!=V)))).
% 9.40/1.59 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))))))))).
% 9.40/1.59 fof(ax15, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (neq(U2, V2) <=> U2!=V2)))).
% 9.40/1.59 fof(ax17, axiom, ssList(nil)).
% 9.40/1.59 fof(ax18, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => cons(V2, U2)!=U2))).
% 9.40/1.60 fof(ax21, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => nil!=cons(V2, U2)))).
% 9.40/1.60 fof(ax33, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) => ~lt(V2, U2))))).
% 9.40/1.60 fof(ax38, axiom, ![U2]: (ssItem(U2) => ~memberP(nil, U2))).
% 9.40/1.60 fof(ax5, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (frontsegP(U2, V2) <=> ?[W2]: (ssList(W2) & app(V2, W2)=U2))))).
% 9.40/1.60 fof(ax8, axiom, ![U2]: (ssList(U2) => (cyclefreeP(U2) <=> ![V2]: (ssItem(V2) => ![W2]: (ssItem(W2) => ![X3]: (ssList(X3) => ![Y2]: (ssList(Y2) => ![Z2]: (ssList(Z2) => (app(app(X3, cons(V2, Y2)), cons(W2, Z2))=U2 => ~(leq(V2, W2) & leq(W2, V2))))))))))).
% 9.40/1.60 fof(ax84, axiom, ![U2]: (ssList(U2) => app(U2, nil)=U2)).
% 9.40/1.60 fof(ax90, axiom, ![U2]: (ssItem(U2) => ~lt(U2, U2))).
% 9.40/1.60 fof(ax93, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) <=> (U2!=V2 & leq(U2, V2)))))).
% 9.40/1.60 fof(ax94, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (gt(U2, V2) => ~gt(V2, U2))))).
% 9.40/1.60 fof(co1, conjecture, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => ![X3]: (ssList(X3) => (V2!=X3 | (U2!=W2 | (~frontsegP(X3, W2) | (~equalelemsP(W2) | (?[Y2]: (ssList(Y2) & ?[Z2]: (ssList(Z2) & ?[X1]: (ssList(X1) & (app(app(Y2, Z2), X1)=V2 & app(Y2, X1)=U2)))) | ?[X2]: (ssList(X2) & (neq(W2, X2) & (frontsegP(X3, X2) & (segmentP(X2, W2) & equalelemsP(X2))))))))))))))).
% 9.40/1.60
% 9.40/1.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 9.40/1.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 9.40/1.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 9.40/1.60 fresh(y, y, x1...xn) = u
% 9.40/1.60 C => fresh(s, t, x1...xn) = v
% 9.40/1.60 where fresh is a fresh function symbol and x1..xn are the free
% 9.40/1.60 variables of u and v.
% 9.40/1.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 9.40/1.60 input problem has no model of domain size 1).
% 9.40/1.60
% 9.40/1.60 The encoding turns the above axioms into the following unit equations and goals:
% 9.40/1.60
% 9.40/1.60 Axiom 1 (co1): u = w.
% 9.40/1.60 Axiom 2 (co1_1): v = x.
% 9.40/1.60 Axiom 3 (ax17): ssList(nil) = true2.
% 9.40/1.60 Axiom 4 (co1_4): ssList(w) = true2.
% 9.40/1.60 Axiom 5 (co1_3): ssList(v) = true2.
% 9.40/1.60 Axiom 6 (co1_6): frontsegP(x, w) = true2.
% 9.40/1.60 Axiom 7 (ax84): fresh(X, X, Y) = Y.
% 9.40/1.60 Axiom 8 (ax84): fresh(ssList(X), true2, X) = app(X, nil).
% 9.40/1.60 Axiom 9 (ax5_2): fresh274(X, X, Y, Z) = true2.
% 9.40/1.60 Axiom 10 (ax5_1): fresh272(X, X, Y, Z) = Y.
% 9.40/1.60 Axiom 11 (ax5_1): fresh39(X, X, Y, Z) = app(Z, w11(Y, Z)).
% 9.40/1.60 Axiom 12 (ax5_2): fresh38(X, X, Y, Z) = ssList(w11(Y, Z)).
% 9.40/1.60 Axiom 13 (ax5_2): fresh273(X, X, Y, Z) = fresh274(ssList(Y), true2, Y, Z).
% 9.40/1.60 Axiom 14 (ax5_1): fresh271(X, X, Y, Z) = fresh272(ssList(Y), true2, Y, Z).
% 9.40/1.60 Axiom 15 (ax5_2): fresh273(frontsegP(X, Y), true2, X, Y) = fresh38(ssList(Y), true2, X, Y).
% 9.40/1.60 Axiom 16 (ax5_1): fresh271(frontsegP(X, Y), true2, X, Y) = fresh39(ssList(Y), true2, X, Y).
% 9.40/1.60
% 9.40/1.60 Lemma 17: frontsegP(v, w) = true2.
% 9.40/1.60 Proof:
% 9.40/1.60 frontsegP(v, w)
% 9.40/1.60 = { by axiom 2 (co1_1) }
% 9.40/1.60 frontsegP(x, w)
% 9.40/1.60 = { by axiom 6 (co1_6) }
% 9.40/1.60 true2
% 9.40/1.60
% 9.40/1.60 Goal 1 (co1_8): tuple7(app(X, Y), app(app(X, Z), Y), ssList(X), ssList(Z), ssList(Y)) = tuple7(u, v, true2, true2, true2).
% 9.40/1.60 The goal is true when:
% 9.40/1.60 X = w
% 9.40/1.60 Y = nil
% 9.40/1.60 Z = w11(v, w)
% 9.40/1.60
% 9.40/1.60 Proof:
% 9.40/1.60 tuple7(app(w, nil), app(app(w, w11(v, w)), nil), ssList(w), ssList(w11(v, w)), ssList(nil))
% 9.40/1.60 = { by axiom 3 (ax17) }
% 9.40/1.60 tuple7(app(w, nil), app(app(w, w11(v, w)), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 11 (ax5_1) R->L }
% 9.40/1.60 tuple7(app(w, nil), app(fresh39(true2, true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 4 (co1_4) R->L }
% 9.40/1.60 tuple7(app(w, nil), app(fresh39(ssList(w), true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 16 (ax5_1) R->L }
% 9.40/1.60 tuple7(app(w, nil), app(fresh271(frontsegP(v, w), true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by lemma 17 }
% 9.40/1.60 tuple7(app(w, nil), app(fresh271(true2, true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 14 (ax5_1) }
% 9.40/1.60 tuple7(app(w, nil), app(fresh272(ssList(v), true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 5 (co1_3) }
% 9.40/1.60 tuple7(app(w, nil), app(fresh272(true2, true2, v, w), nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 10 (ax5_1) }
% 9.40/1.60 tuple7(app(w, nil), app(v, nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 8 (ax84) R->L }
% 9.40/1.60 tuple7(fresh(ssList(w), true2, w), app(v, nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 4 (co1_4) }
% 9.40/1.60 tuple7(fresh(true2, true2, w), app(v, nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 7 (ax84) }
% 9.40/1.60 tuple7(w, app(v, nil), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 8 (ax84) R->L }
% 9.40/1.60 tuple7(w, fresh(ssList(v), true2, v), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 5 (co1_3) }
% 9.40/1.60 tuple7(w, fresh(true2, true2, v), ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 7 (ax84) }
% 9.40/1.60 tuple7(w, v, ssList(w), ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 4 (co1_4) }
% 9.40/1.60 tuple7(w, v, true2, ssList(w11(v, w)), true2)
% 9.40/1.60 = { by axiom 12 (ax5_2) R->L }
% 9.40/1.60 tuple7(w, v, true2, fresh38(true2, true2, v, w), true2)
% 9.40/1.60 = { by axiom 4 (co1_4) R->L }
% 9.40/1.60 tuple7(w, v, true2, fresh38(ssList(w), true2, v, w), true2)
% 9.40/1.60 = { by axiom 15 (ax5_2) R->L }
% 9.40/1.60 tuple7(w, v, true2, fresh273(frontsegP(v, w), true2, v, w), true2)
% 9.40/1.60 = { by lemma 17 }
% 9.40/1.60 tuple7(w, v, true2, fresh273(true2, true2, v, w), true2)
% 9.40/1.60 = { by axiom 13 (ax5_2) }
% 9.40/1.60 tuple7(w, v, true2, fresh274(ssList(v), true2, v, w), true2)
% 9.40/1.60 = { by axiom 5 (co1_3) }
% 9.40/1.60 tuple7(w, v, true2, fresh274(true2, true2, v, w), true2)
% 9.40/1.60 = { by axiom 9 (ax5_2) }
% 9.40/1.60 tuple7(w, v, true2, true2, true2)
% 9.40/1.60 = { by axiom 1 (co1) R->L }
% 9.40/1.61 tuple7(u, v, true2, true2, true2)
% 9.40/1.61 % SZS output end Proof
% 9.40/1.61
% 9.40/1.61 RESULT: Theorem (the conjecture is true).
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