TSTP Solution File: SWC050+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWC050+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 : n025.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 20:53:38 EDT 2023
% Result : Theorem 0.20s 0.79s
% Output : Proof 3.46s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWC050+1 : TPTP v8.1.2. Released v2.4.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.33 % Computer : n025.cluster.edu
% 0.17/0.33 % Model : x86_64 x86_64
% 0.17/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.33 % Memory : 8042.1875MB
% 0.17/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Mon Aug 28 17:55:24 EDT 2023
% 0.17/0.34 % CPUTime :
% 0.20/0.79 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.79
% 0.20/0.79 % SZS status Theorem
% 0.20/0.79
% 0.20/0.79 % SZS output start Proof
% 0.20/0.79 Take the following subset of the input axioms:
% 3.46/0.80 fof(ax1, axiom, ![U]: (ssItem(U) => ![V]: (ssItem(V) => (neq(U, V) <=> U!=V)))).
% 3.46/0.80 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))))))))).
% 3.46/0.80 fof(ax15, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (neq(U2, V2) <=> U2!=V2)))).
% 3.46/0.80 fof(ax18, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => cons(V2, U2)!=U2))).
% 3.46/0.80 fof(ax21, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => nil!=cons(V2, U2)))).
% 3.46/0.80 fof(ax33, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) => ~lt(V2, U2))))).
% 3.46/0.80 fof(ax38, axiom, ![U2]: (ssItem(U2) => ~memberP(nil, U2))).
% 3.46/0.80 fof(ax8, axiom, ![U2]: (ssList(U2) => (cyclefreeP(U2) <=> ![V2]: (ssItem(V2) => ![W2]: (ssItem(W2) => ![X2]: (ssList(X2) => ![Y2]: (ssList(Y2) => ![Z2]: (ssList(Z2) => (app(app(X2, cons(V2, Y2)), cons(W2, Z2))=U2 => ~(leq(V2, W2) & leq(W2, V2))))))))))).
% 3.46/0.80 fof(ax90, axiom, ![U2]: (ssItem(U2) => ~lt(U2, U2))).
% 3.46/0.80 fof(ax93, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) <=> (U2!=V2 & leq(U2, V2)))))).
% 3.46/0.80 fof(ax94, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (gt(U2, V2) => ~gt(V2, U2))))).
% 3.46/0.80 fof(co1, conjecture, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => ![X2]: (ssList(X2) => (V2!=X2 | (U2!=W2 | ((~neq(V2, nil) | (?[Y2]: (ssList(Y2) & (neq(Y2, nil) & (rearsegP(V2, Y2) & rearsegP(U2, Y2)))) | ![Z2]: (ssList(Z2) => (~neq(Z2, nil) | (~rearsegP(X2, Z2) | ~rearsegP(W2, Z2)))))) & (~neq(V2, nil) | neq(X2, nil)))))))))).
% 3.46/0.80
% 3.46/0.80 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.46/0.80 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.46/0.80 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.46/0.80 fresh(y, y, x1...xn) = u
% 3.46/0.80 C => fresh(s, t, x1...xn) = v
% 3.46/0.80 where fresh is a fresh function symbol and x1..xn are the free
% 3.46/0.80 variables of u and v.
% 3.46/0.80 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.46/0.80 input problem has no model of domain size 1).
% 3.46/0.80
% 3.46/0.80 The encoding turns the above axioms into the following unit equations and goals:
% 3.46/0.80
% 3.46/0.80 Axiom 1 (co1_1): v = x.
% 3.46/0.80 Axiom 2 (co1): u = w.
% 3.46/0.80 Axiom 3 (co1_2): neq(v, nil) = true2.
% 3.46/0.80 Axiom 4 (co1_14): fresh17(X, X) = true2.
% 3.46/0.80 Axiom 5 (co1_15): fresh16(X, X) = true2.
% 3.46/0.80 Axiom 6 (co1_16): fresh15(X, X) = true2.
% 3.46/0.80 Axiom 7 (co1_17): fresh14(X, X) = true2.
% 3.46/0.80 Axiom 8 (co1_14): fresh17(neq(x, nil), true2) = neq(z, nil).
% 3.46/0.80 Axiom 9 (co1_15): fresh16(neq(x, nil), true2) = ssList(z).
% 3.46/0.80 Axiom 10 (co1_16): fresh15(neq(x, nil), true2) = rearsegP(w, z).
% 3.46/0.80 Axiom 11 (co1_17): fresh14(neq(x, nil), true2) = rearsegP(x, z).
% 3.46/0.80
% 3.46/0.80 Lemma 12: neq(x, nil) = true2.
% 3.46/0.80 Proof:
% 3.46/0.80 neq(x, nil)
% 3.46/0.80 = { by axiom 1 (co1_1) R->L }
% 3.46/0.80 neq(v, nil)
% 3.46/0.80 = { by axiom 3 (co1_2) }
% 3.46/0.80 true2
% 3.46/0.80
% 3.46/0.80 Goal 1 (co1_11): tuple(neq(X, nil), neq(x, nil), ssList(X), rearsegP(u, X), rearsegP(v, X)) = tuple(true2, true2, true2, true2, true2).
% 3.46/0.80 The goal is true when:
% 3.46/0.80 X = z
% 3.46/0.80
% 3.46/0.80 Proof:
% 3.46/0.80 tuple(neq(z, nil), neq(x, nil), ssList(z), rearsegP(u, z), rearsegP(v, z))
% 3.46/0.80 = { by lemma 12 }
% 3.46/0.80 tuple(neq(z, nil), true2, ssList(z), rearsegP(u, z), rearsegP(v, z))
% 3.46/0.80 = { by axiom 1 (co1_1) }
% 3.46/0.80 tuple(neq(z, nil), true2, ssList(z), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 8 (co1_14) R->L }
% 3.46/0.80 tuple(fresh17(neq(x, nil), true2), true2, ssList(z), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by lemma 12 }
% 3.46/0.80 tuple(fresh17(true2, true2), true2, ssList(z), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 4 (co1_14) }
% 3.46/0.80 tuple(true2, true2, ssList(z), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 9 (co1_15) R->L }
% 3.46/0.80 tuple(true2, true2, fresh16(neq(x, nil), true2), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by lemma 12 }
% 3.46/0.80 tuple(true2, true2, fresh16(true2, true2), rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 5 (co1_15) }
% 3.46/0.80 tuple(true2, true2, true2, rearsegP(u, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 2 (co1) }
% 3.46/0.80 tuple(true2, true2, true2, rearsegP(w, z), rearsegP(x, z))
% 3.46/0.80 = { by axiom 10 (co1_16) R->L }
% 3.46/0.80 tuple(true2, true2, true2, fresh15(neq(x, nil), true2), rearsegP(x, z))
% 3.46/0.80 = { by lemma 12 }
% 3.46/0.80 tuple(true2, true2, true2, fresh15(true2, true2), rearsegP(x, z))
% 3.46/0.80 = { by axiom 6 (co1_16) }
% 3.46/0.80 tuple(true2, true2, true2, true2, rearsegP(x, z))
% 3.46/0.80 = { by axiom 11 (co1_17) R->L }
% 3.46/0.80 tuple(true2, true2, true2, true2, fresh14(neq(x, nil), true2))
% 3.46/0.80 = { by lemma 12 }
% 3.46/0.80 tuple(true2, true2, true2, true2, fresh14(true2, true2))
% 3.46/0.80 = { by axiom 7 (co1_17) }
% 3.46/0.80 tuple(true2, true2, true2, true2, true2)
% 3.46/0.80 % SZS output end Proof
% 3.46/0.80
% 3.46/0.80 RESULT: Theorem (the conjecture is true).
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