TSTP Solution File: SWC417+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWC417+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 : n011.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 20:55:28 EDT 2023
% Result : Theorem 58.70s 7.93s
% Output : Proof 59.06s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SWC417+1 : TPTP v8.1.2. Released v2.4.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.38 % Computer : n011.cluster.edu
% 0.14/0.38 % Model : x86_64 x86_64
% 0.14/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.38 % Memory : 8042.1875MB
% 0.14/0.38 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.38 % CPULimit : 300
% 0.14/0.38 % WCLimit : 300
% 0.14/0.38 % DateTime : Mon Aug 28 16:51:41 EDT 2023
% 0.14/0.38 % CPUTime :
% 58.70/7.93 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 58.70/7.93
% 58.70/7.93 % SZS status Theorem
% 58.70/7.93
% 59.06/7.94 % SZS output start Proof
% 59.06/7.94 Take the following subset of the input axioms:
% 59.06/7.95 fof(ax1, axiom, ![U]: (ssItem(U) => ![V]: (ssItem(V) => (neq(U, V) <=> U!=V)))).
% 59.06/7.95 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))))))))).
% 59.06/7.95 fof(ax15, axiom, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => (neq(U2, V2) <=> U2!=V2)))).
% 59.06/7.95 fof(ax16, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => ssList(cons(V2, U2))))).
% 59.06/7.95 fof(ax17, axiom, ssList(nil)).
% 59.06/7.95 fof(ax18, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => cons(V2, U2)!=U2))).
% 59.06/7.95 fof(ax21, axiom, ![U2]: (ssList(U2) => ![V2]: (ssItem(V2) => nil!=cons(V2, U2)))).
% 59.06/7.95 fof(ax28, axiom, ![U2]: (ssList(U2) => app(nil, U2)=U2)).
% 59.06/7.95 fof(ax33, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) => ~lt(V2, U2))))).
% 59.06/7.95 fof(ax38, axiom, ![U2]: (ssItem(U2) => ~memberP(nil, U2))).
% 59.06/7.95 fof(ax8, axiom, ![U2]: (ssList(U2) => (cyclefreeP(U2) <=> ![V2]: (ssItem(V2) => ![W2]: (ssItem(W2) => ![X4]: (ssList(X4) => ![Y2]: (ssList(Y2) => ![Z2]: (ssList(Z2) => (app(app(X4, cons(V2, Y2)), cons(W2, Z2))=U2 => ~(leq(V2, W2) & leq(W2, V2))))))))))).
% 59.06/7.95 fof(ax84, axiom, ![U2]: (ssList(U2) => app(U2, nil)=U2)).
% 59.06/7.95 fof(ax90, axiom, ![U2]: (ssItem(U2) => ~lt(U2, U2))).
% 59.06/7.95 fof(ax93, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (lt(U2, V2) <=> (U2!=V2 & leq(U2, V2)))))).
% 59.06/7.95 fof(ax94, axiom, ![U2]: (ssItem(U2) => ![V2]: (ssItem(V2) => (gt(U2, V2) => ~gt(V2, U2))))).
% 59.06/7.95 fof(co1, conjecture, ![U2]: (ssList(U2) => ![V2]: (ssList(V2) => ![W2]: (ssList(W2) => ![X4]: (ssList(X4) => (V2!=X4 | (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) => (app(cons(X2, nil), X3)!=X4 | app(X3, cons(X2, nil))!=W2))))) & (~neq(V2, nil) | neq(X4, nil)))))))))).
% 59.06/7.95
% 59.06/7.95 Now clausify the problem and encode Horn clauses using encoding 3 of
% 59.06/7.95 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 59.06/7.95 We repeatedly replace C & s=t => u=v by the two clauses:
% 59.06/7.95 fresh(y, y, x1...xn) = u
% 59.06/7.95 C => fresh(s, t, x1...xn) = v
% 59.06/7.95 where fresh is a fresh function symbol and x1..xn are the free
% 59.06/7.95 variables of u and v.
% 59.06/7.95 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 59.06/7.95 input problem has no model of domain size 1).
% 59.06/7.95
% 59.06/7.95 The encoding turns the above axioms into the following unit equations and goals:
% 59.06/7.95
% 59.06/7.95 Axiom 1 (co1_3): v = x.
% 59.06/7.95 Axiom 2 (co1_2): u = w.
% 59.06/7.95 Axiom 3 (ax17): ssList(nil) = true2.
% 59.06/7.95 Axiom 4 (co1_7): ssList(u) = true2.
% 59.06/7.95 Axiom 5 (co1_5): neq(v, nil) = true2.
% 59.06/7.95 Axiom 6 (co1_13): fresh18(X, X) = x.
% 59.06/7.95 Axiom 7 (co1_14): fresh17(X, X) = w.
% 59.06/7.95 Axiom 8 (co1_15): fresh16(X, X) = true2.
% 59.06/7.95 Axiom 9 (co1_17): fresh14(X, X) = true2.
% 59.06/7.95 Axiom 10 (ax84): fresh(X, X, Y) = Y.
% 59.06/7.95 Axiom 11 (ax28): fresh7(X, X, Y) = Y.
% 59.06/7.95 Axiom 12 (ax84): fresh(ssList(X), true2, X) = app(X, nil).
% 59.06/7.95 Axiom 13 (ax16): fresh82(X, X, Y, Z) = ssList(cons(Z, Y)).
% 59.06/7.95 Axiom 14 (ax16): fresh81(X, X, Y, Z) = true2.
% 59.06/7.95 Axiom 15 (co1_13): fresh18(neq(x, nil), true2) = app(cons(x2, nil), x3).
% 59.06/7.95 Axiom 16 (co1_14): fresh17(neq(x, nil), true2) = app(x3, cons(x2, nil)).
% 59.06/7.95 Axiom 17 (co1_15): fresh16(neq(x, nil), true2) = ssItem(x2).
% 59.06/7.95 Axiom 18 (co1_17): fresh14(neq(x, nil), true2) = ssList(x3).
% 59.06/7.95 Axiom 19 (ax28): fresh7(ssList(X), true2, X) = app(nil, X).
% 59.06/7.95 Axiom 20 (ax16): fresh82(ssList(X), true2, X, Y) = fresh81(ssItem(Y), true2, X, Y).
% 59.06/7.95
% 59.06/7.95 Lemma 21: neq(x, nil) = true2.
% 59.06/7.95 Proof:
% 59.06/7.95 neq(x, nil)
% 59.06/7.95 = { by axiom 1 (co1_3) R->L }
% 59.06/7.95 neq(v, nil)
% 59.06/7.95 = { by axiom 5 (co1_5) }
% 59.06/7.95 true2
% 59.06/7.95
% 59.06/7.95 Lemma 22: ssItem(x2) = true2.
% 59.06/7.95 Proof:
% 59.06/7.95 ssItem(x2)
% 59.06/7.95 = { by axiom 17 (co1_15) R->L }
% 59.06/7.95 fresh16(neq(x, nil), true2)
% 59.06/7.95 = { by lemma 21 }
% 59.06/7.95 fresh16(true2, true2)
% 59.06/7.95 = { by axiom 8 (co1_15) }
% 59.06/7.95 true2
% 59.06/7.95
% 59.06/7.95 Goal 1 (co1_11): 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).
% 59.06/7.95 The goal is true when:
% 59.06/7.95 X = nil
% 59.06/7.95 Y = x2
% 59.06/7.95 Z = x3
% 59.06/7.95
% 59.06/7.95 Proof:
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(app(x3, cons(x2, nil)), nil), ssItem(x2), neq(x, nil), ssList(nil), ssList(x3))
% 59.06/7.95 = { by lemma 21 }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(app(x3, cons(x2, nil)), nil), ssItem(x2), true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by axiom 16 (co1_14) R->L }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(fresh17(neq(x, nil), true2), nil), ssItem(x2), true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by lemma 21 }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(fresh17(true2, true2), nil), ssItem(x2), true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by axiom 7 (co1_14) }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(w, nil), ssItem(x2), true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by axiom 2 (co1_2) R->L }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), ssItem(x2), true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by lemma 22 }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), true2, true2, ssList(nil), ssList(x3))
% 59.06/7.95 = { by axiom 18 (co1_17) R->L }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), true2, true2, ssList(nil), fresh14(neq(x, nil), true2))
% 59.06/7.95 = { by lemma 21 }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), true2, true2, ssList(nil), fresh14(true2, true2))
% 59.06/7.95 = { by axiom 9 (co1_17) }
% 59.06/7.95 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), true2, true2, ssList(nil), true2)
% 59.06/7.95 = { by axiom 3 (ax17) }
% 59.06/7.96 tuple6(app(app(nil, cons(x2, nil)), x3), app(u, nil), true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 12 (ax84) R->L }
% 59.06/7.96 tuple6(app(app(nil, cons(x2, nil)), x3), fresh(ssList(u), true2, u), true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 4 (co1_7) }
% 59.06/7.96 tuple6(app(app(nil, cons(x2, nil)), x3), fresh(true2, true2, u), true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 10 (ax84) }
% 59.06/7.96 tuple6(app(app(nil, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 19 (ax28) R->L }
% 59.06/7.96 tuple6(app(fresh7(ssList(cons(x2, nil)), true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 13 (ax16) R->L }
% 59.06/7.96 tuple6(app(fresh7(fresh82(true2, true2, nil, x2), true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 3 (ax17) R->L }
% 59.06/7.96 tuple6(app(fresh7(fresh82(ssList(nil), true2, nil, x2), true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 20 (ax16) }
% 59.06/7.96 tuple6(app(fresh7(fresh81(ssItem(x2), true2, nil, x2), true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by lemma 22 }
% 59.06/7.96 tuple6(app(fresh7(fresh81(true2, true2, nil, x2), true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 14 (ax16) }
% 59.06/7.96 tuple6(app(fresh7(true2, true2, cons(x2, nil)), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 11 (ax28) }
% 59.06/7.96 tuple6(app(cons(x2, nil), x3), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 15 (co1_13) R->L }
% 59.06/7.96 tuple6(fresh18(neq(x, nil), true2), u, true2, true2, true2, true2)
% 59.06/7.96 = { by lemma 21 }
% 59.06/7.96 tuple6(fresh18(true2, true2), u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 6 (co1_13) }
% 59.06/7.96 tuple6(x, u, true2, true2, true2, true2)
% 59.06/7.96 = { by axiom 1 (co1_3) R->L }
% 59.06/7.96 tuple6(v, u, true2, true2, true2, true2)
% 59.06/7.96 % SZS output end Proof
% 59.06/7.96
% 59.06/7.96 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------