TSTP Solution File: SET789+4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET789+4 : TPTP v8.1.2. Released v3.2.0.
% 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 : n027.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 15:33:15 EDT 2023
% Result : Theorem 0.22s 0.49s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SET789+4 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n027.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sat Aug 26 15:49:35 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.22/0.49 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.22/0.49
% 0.22/0.49 % SZS status Theorem
% 0.22/0.49
% 0.22/0.49 % SZS output start Proof
% 0.22/0.49 Take the following subset of the input axioms:
% 0.22/0.50 fof(greatest, axiom, ![R, E, M]: (greatest(M, R, E) <=> (member(M, E) & ![X]: (member(X, E) => apply(R, X, M))))).
% 0.22/0.50 fof(order, axiom, ![R2, E2]: (order(R2, E2) <=> (![X2]: (member(X2, E2) => apply(R2, X2, X2)) & (![Y, X2]: ((member(X2, E2) & member(Y, E2)) => ((apply(R2, X2, Y) & apply(R2, Y, X2)) => X2=Y)) & ![Z, X2, Y2]: ((member(X2, E2) & (member(Y2, E2) & member(Z, E2))) => ((apply(R2, X2, Y2) & apply(R2, Y2, Z)) => apply(R2, X2, Z))))))).
% 0.22/0.50 fof(thIV1, conjecture, ![R2, E2, M2]: ((order(R2, E2) & greatest(M2, R2, E2)) => ![X2]: (greatest(X2, R2, E2) => M2=X2))).
% 0.22/0.50
% 0.22/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.50 fresh(y, y, x1...xn) = u
% 0.22/0.50 C => fresh(s, t, x1...xn) = v
% 0.22/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.50 variables of u and v.
% 0.22/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.50 input problem has no model of domain size 1).
% 0.22/0.50
% 0.22/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.50
% 0.22/0.50 Axiom 1 (thIV1): order(r, e) = true2.
% 0.22/0.50 Axiom 2 (thIV1_1): greatest(m, r, e) = true2.
% 0.22/0.50 Axiom 3 (thIV1_2): greatest(x, r, e) = true2.
% 0.22/0.50 Axiom 4 (order_7): fresh61(X, X, Y, Z) = Z.
% 0.22/0.50 Axiom 5 (greatest_3): fresh40(X, X, Y, Z) = true2.
% 0.22/0.50 Axiom 6 (greatest_2): fresh41(X, X, Y, Z, W) = true2.
% 0.22/0.50 Axiom 7 (order_7): fresh60(X, X, Y, Z, W, V) = fresh61(order(Y, Z), true2, W, V).
% 0.22/0.50 Axiom 8 (order_7): fresh59(X, X, Y, Z, W, V) = W.
% 0.22/0.50 Axiom 9 (greatest_2): fresh42(X, X, Y, Z, W, V) = apply(Y, V, W).
% 0.22/0.50 Axiom 10 (greatest_3): fresh40(greatest(X, Y, Z), true2, Z, X) = member(X, Z).
% 0.22/0.50 Axiom 11 (order_7): fresh57(X, X, Y, Z, W, V) = fresh60(member(V, Z), true2, Y, Z, W, V).
% 0.22/0.50 Axiom 12 (order_7): fresh58(X, X, Y, Z, W, V) = fresh59(member(W, Z), true2, Y, Z, W, V).
% 0.22/0.50 Axiom 13 (order_7): fresh57(apply(X, Y, Z), true2, X, W, Z, Y) = fresh58(apply(X, Z, Y), true2, X, W, Z, Y).
% 0.22/0.50 Axiom 14 (greatest_2): fresh42(greatest(X, Y, Z), true2, Y, Z, X, W) = fresh41(member(W, Z), true2, Y, X, W).
% 0.22/0.50
% 0.22/0.50 Lemma 15: member(m, e) = true2.
% 0.22/0.50 Proof:
% 0.22/0.50 member(m, e)
% 0.22/0.50 = { by axiom 10 (greatest_3) R->L }
% 0.22/0.50 fresh40(greatest(m, r, e), true2, e, m)
% 0.22/0.50 = { by axiom 2 (thIV1_1) }
% 0.22/0.50 fresh40(true2, true2, e, m)
% 0.22/0.50 = { by axiom 5 (greatest_3) }
% 0.22/0.50 true2
% 0.22/0.50
% 0.22/0.50 Lemma 16: member(x, e) = true2.
% 0.22/0.50 Proof:
% 0.22/0.50 member(x, e)
% 0.22/0.50 = { by axiom 10 (greatest_3) R->L }
% 0.22/0.50 fresh40(greatest(x, r, e), true2, e, x)
% 0.22/0.50 = { by axiom 3 (thIV1_2) }
% 0.22/0.50 fresh40(true2, true2, e, x)
% 0.22/0.50 = { by axiom 5 (greatest_3) }
% 0.22/0.50 true2
% 0.22/0.50
% 0.22/0.50 Goal 1 (thIV1_3): m = x.
% 0.22/0.50 Proof:
% 0.22/0.50 m
% 0.22/0.50 = { by axiom 4 (order_7) R->L }
% 0.22/0.50 fresh61(true2, true2, x, m)
% 0.22/0.50 = { by axiom 1 (thIV1) R->L }
% 0.22/0.50 fresh61(order(r, e), true2, x, m)
% 0.22/0.50 = { by axiom 7 (order_7) R->L }
% 0.22/0.50 fresh60(true2, true2, r, e, x, m)
% 0.22/0.50 = { by lemma 15 R->L }
% 0.22/0.50 fresh60(member(m, e), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 11 (order_7) R->L }
% 0.22/0.50 fresh57(true2, true2, r, e, x, m)
% 0.22/0.50 = { by axiom 6 (greatest_2) R->L }
% 0.22/0.50 fresh57(fresh41(true2, true2, r, x, m), true2, r, e, x, m)
% 0.22/0.50 = { by lemma 15 R->L }
% 0.22/0.50 fresh57(fresh41(member(m, e), true2, r, x, m), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 14 (greatest_2) R->L }
% 0.22/0.50 fresh57(fresh42(greatest(x, r, e), true2, r, e, x, m), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 3 (thIV1_2) }
% 0.22/0.50 fresh57(fresh42(true2, true2, r, e, x, m), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 9 (greatest_2) }
% 0.22/0.50 fresh57(apply(r, m, x), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 13 (order_7) }
% 0.22/0.50 fresh58(apply(r, x, m), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 9 (greatest_2) R->L }
% 0.22/0.50 fresh58(fresh42(true2, true2, r, e, m, x), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 2 (thIV1_1) R->L }
% 0.22/0.50 fresh58(fresh42(greatest(m, r, e), true2, r, e, m, x), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 14 (greatest_2) }
% 0.22/0.50 fresh58(fresh41(member(x, e), true2, r, m, x), true2, r, e, x, m)
% 0.22/0.50 = { by lemma 16 }
% 0.22/0.50 fresh58(fresh41(true2, true2, r, m, x), true2, r, e, x, m)
% 0.22/0.50 = { by axiom 6 (greatest_2) }
% 0.22/0.50 fresh58(true2, true2, r, e, x, m)
% 0.22/0.50 = { by axiom 12 (order_7) }
% 0.22/0.50 fresh59(member(x, e), true2, r, e, x, m)
% 0.22/0.50 = { by lemma 16 }
% 0.22/0.50 fresh59(true2, true2, r, e, x, m)
% 0.22/0.50 = { by axiom 8 (order_7) }
% 0.22/0.50 x
% 0.22/0.50 % SZS output end Proof
% 0.22/0.50
% 0.22/0.50 RESULT: Theorem (the conjecture is true).
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