TSTP Solution File: SET803+4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET803+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 : n007.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 15:33:17 EDT 2023
% Result : Theorem 0.20s 0.49s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : SET803+4 : TPTP v8.1.2. Released v3.2.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n007.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Aug 26 15:03:56 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.49 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.49
% 0.20/0.49 % SZS status Theorem
% 0.20/0.49
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Take the following subset of the input axioms:
% 0.20/0.50 fof(greatest, axiom, ![R, E, M]: (greatest(M, R, E) <=> (member(M, E) & ![X]: (member(X, E) => apply(R, X, M))))).
% 0.20/0.50 fof(max, axiom, ![M3, R2, E2]: (max(M3, R2, E2) <=> (member(M3, E2) & ![X2]: ((member(X2, E2) & apply(R2, M3, X2)) => M3=X2)))).
% 0.20/0.50 fof(thIV15, conjecture, ![R2, E2]: (order(R2, E2) => ![M1, M2]: ((max(M1, R2, E2) & (max(M2, R2, E2) & M1!=M2)) => ~?[M3]: greatest(M3, R2, E2)))).
% 0.20/0.50
% 0.20/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.50 fresh(y, y, x1...xn) = u
% 0.20/0.50 C => fresh(s, t, x1...xn) = v
% 0.20/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.50 variables of u and v.
% 0.20/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.50 input problem has no model of domain size 1).
% 0.20/0.50
% 0.20/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.50
% 0.20/0.50 Axiom 1 (thIV15_2): max(m2, r, e) = true2.
% 0.20/0.50 Axiom 2 (thIV15_3): max(m1, r, e) = true2.
% 0.20/0.50 Axiom 3 (thIV15_1): greatest(m, r, e) = true2.
% 0.20/0.50 Axiom 4 (max_3): fresh56(X, X, Y, Z) = Z.
% 0.20/0.50 Axiom 5 (greatest_3): fresh40(X, X, Y, Z) = true2.
% 0.20/0.50 Axiom 6 (max_4): fresh21(X, X, Y, Z) = true2.
% 0.20/0.50 Axiom 7 (greatest_2): fresh41(X, X, Y, Z, W) = true2.
% 0.20/0.50 Axiom 8 (max_3): fresh2(X, X, Y, Z, W) = Z.
% 0.20/0.50 Axiom 9 (max_3): fresh55(X, X, Y, Z, W, V) = fresh56(member(V, Z), true2, W, V).
% 0.20/0.50 Axiom 10 (greatest_2): fresh42(X, X, Y, Z, W, V) = apply(Y, V, W).
% 0.20/0.50 Axiom 11 (greatest_3): fresh40(greatest(X, Y, Z), true2, Z, X) = member(X, Z).
% 0.20/0.50 Axiom 12 (max_4): fresh21(max(X, Y, Z), true2, Z, X) = member(X, Z).
% 0.20/0.50 Axiom 13 (max_3): fresh55(max(X, Y, Z), true2, Y, Z, X, W) = fresh2(apply(Y, X, W), true2, Z, X, W).
% 0.20/0.50 Axiom 14 (greatest_2): fresh42(greatest(X, Y, Z), true2, Y, Z, X, W) = fresh41(member(W, Z), true2, Y, X, W).
% 0.20/0.50
% 0.20/0.50 Lemma 15: member(m2, e) = true2.
% 0.20/0.50 Proof:
% 0.20/0.50 member(m2, e)
% 0.20/0.50 = { by axiom 12 (max_4) R->L }
% 0.20/0.50 fresh21(max(m2, r, e), true2, e, m2)
% 0.20/0.50 = { by axiom 1 (thIV15_2) }
% 0.20/0.50 fresh21(true2, true2, e, m2)
% 0.20/0.50 = { by axiom 6 (max_4) }
% 0.20/0.50 true2
% 0.20/0.50
% 0.20/0.50 Lemma 16: fresh41(member(X, e), true2, r, m, X) = apply(r, X, m).
% 0.20/0.50 Proof:
% 0.20/0.50 fresh41(member(X, e), true2, r, m, X)
% 0.20/0.50 = { by axiom 14 (greatest_2) R->L }
% 0.20/0.50 fresh42(greatest(m, r, e), true2, r, e, m, X)
% 0.20/0.50 = { by axiom 3 (thIV15_1) }
% 0.20/0.50 fresh42(true2, true2, r, e, m, X)
% 0.20/0.50 = { by axiom 10 (greatest_2) }
% 0.20/0.50 apply(r, X, m)
% 0.20/0.50
% 0.20/0.50 Goal 1 (thIV15_4): m1 = m2.
% 0.20/0.50 Proof:
% 0.20/0.50 m1
% 0.20/0.50 = { by axiom 8 (max_3) R->L }
% 0.20/0.50 fresh2(true2, true2, e, m1, m)
% 0.20/0.50 = { by axiom 7 (greatest_2) R->L }
% 0.20/0.50 fresh2(fresh41(true2, true2, r, m, m1), true2, e, m1, m)
% 0.20/0.50 = { by axiom 6 (max_4) R->L }
% 0.20/0.50 fresh2(fresh41(fresh21(true2, true2, e, m1), true2, r, m, m1), true2, e, m1, m)
% 0.20/0.50 = { by axiom 2 (thIV15_3) R->L }
% 0.20/0.50 fresh2(fresh41(fresh21(max(m1, r, e), true2, e, m1), true2, r, m, m1), true2, e, m1, m)
% 0.20/0.50 = { by axiom 12 (max_4) }
% 0.20/0.50 fresh2(fresh41(member(m1, e), true2, r, m, m1), true2, e, m1, m)
% 0.20/0.50 = { by lemma 16 }
% 0.20/0.50 fresh2(apply(r, m1, m), true2, e, m1, m)
% 0.20/0.50 = { by axiom 13 (max_3) R->L }
% 0.20/0.50 fresh55(max(m1, r, e), true2, r, e, m1, m)
% 0.20/0.50 = { by axiom 2 (thIV15_3) }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, m)
% 0.20/0.50 = { by axiom 4 (max_3) R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh56(true2, true2, m2, m))
% 0.20/0.50 = { by axiom 5 (greatest_3) R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh56(fresh40(true2, true2, e, m), true2, m2, m))
% 0.20/0.50 = { by axiom 3 (thIV15_1) R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh56(fresh40(greatest(m, r, e), true2, e, m), true2, m2, m))
% 0.20/0.50 = { by axiom 11 (greatest_3) }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh56(member(m, e), true2, m2, m))
% 0.20/0.50 = { by axiom 9 (max_3) R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh55(true2, true2, r, e, m2, m))
% 0.20/0.50 = { by axiom 1 (thIV15_2) R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh55(max(m2, r, e), true2, r, e, m2, m))
% 0.20/0.50 = { by axiom 13 (max_3) }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh2(apply(r, m2, m), true2, e, m2, m))
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh2(fresh41(member(m2, e), true2, r, m, m2), true2, e, m2, m))
% 0.20/0.50 = { by lemma 15 }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh2(fresh41(true2, true2, r, m, m2), true2, e, m2, m))
% 0.20/0.50 = { by axiom 7 (greatest_2) }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, fresh2(true2, true2, e, m2, m))
% 0.20/0.50 = { by axiom 8 (max_3) }
% 0.20/0.50 fresh55(true2, true2, r, e, m1, m2)
% 0.20/0.50 = { by axiom 9 (max_3) }
% 0.20/0.50 fresh56(member(m2, e), true2, m1, m2)
% 0.20/0.50 = { by lemma 15 }
% 0.20/0.50 fresh56(true2, true2, m1, m2)
% 0.20/0.50 = { by axiom 4 (max_3) }
% 0.20/0.50 m2
% 0.20/0.50 % SZS output end Proof
% 0.20/0.50
% 0.20/0.50 RESULT: Theorem (the conjecture is true).
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