TSTP Solution File: SET804+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET804+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 : n013.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 0.21s 0.51s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET804+4 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n013.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 12:15:32 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.51 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.21/0.51
% 0.21/0.51 % SZS status Theorem
% 0.21/0.51
% 0.21/0.52 % SZS output start Proof
% 0.21/0.52 Take the following subset of the input axioms:
% 0.21/0.52 fof(least, axiom, ![R, E, M]: (least(M, R, E) <=> (member(M, E) & ![X]: (member(X, E) => apply(R, M, X))))).
% 0.21/0.52 fof(min, axiom, ![M3, R2, E2]: (min(M3, R2, E2) <=> (member(M3, E2) & ![X2]: ((member(X2, E2) & apply(R2, X2, M3)) => M3=X2)))).
% 0.21/0.52 fof(thIV16, conjecture, ![R2, E2]: (order(R2, E2) => ![M1, M2]: ((min(M1, R2, E2) & (min(M2, R2, E2) & M1!=M2)) => ~?[M3]: least(M3, R2, E2)))).
% 0.21/0.52
% 0.21/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.52 fresh(y, y, x1...xn) = u
% 0.21/0.52 C => fresh(s, t, x1...xn) = v
% 0.21/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.52 variables of u and v.
% 0.21/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.52 input problem has no model of domain size 1).
% 0.21/0.52
% 0.21/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.52
% 0.21/0.52 Axiom 1 (thIV16_1): least(m, r, e) = true2.
% 0.21/0.52 Axiom 2 (thIV16_2): min(m2, r, e) = true2.
% 0.21/0.52 Axiom 3 (thIV16_3): min(m1, r, e) = true2.
% 0.21/0.52 Axiom 4 (min_3): fresh54(X, X, Y, Z) = Z.
% 0.21/0.52 Axiom 5 (least_3): fresh31(X, X, Y, Z) = true2.
% 0.21/0.52 Axiom 6 (min_4): fresh18(X, X, Y, Z) = true2.
% 0.21/0.52 Axiom 7 (min_3): fresh(X, X, Y, Z, W) = Z.
% 0.21/0.52 Axiom 8 (least_2): fresh32(X, X, Y, Z, W) = true2.
% 0.21/0.52 Axiom 9 (min_3): fresh53(X, X, Y, Z, W, V) = fresh54(member(V, Z), true2, W, V).
% 0.21/0.52 Axiom 10 (least_2): fresh33(X, X, Y, Z, W, V) = apply(Y, W, V).
% 0.21/0.52 Axiom 11 (least_3): fresh31(least(X, Y, Z), true2, Z, X) = member(X, Z).
% 0.21/0.52 Axiom 12 (min_4): fresh18(min(X, Y, Z), true2, Z, X) = member(X, Z).
% 0.21/0.52 Axiom 13 (min_3): fresh53(min(X, Y, Z), true2, Y, Z, X, W) = fresh(apply(Y, W, X), true2, Z, X, W).
% 0.21/0.52 Axiom 14 (least_2): fresh33(least(X, Y, Z), true2, Y, Z, X, W) = fresh32(member(W, Z), true2, Y, X, W).
% 0.21/0.52
% 0.21/0.52 Lemma 15: member(m, e) = true2.
% 0.21/0.52 Proof:
% 0.21/0.52 member(m, e)
% 0.21/0.52 = { by axiom 11 (least_3) R->L }
% 0.21/0.52 fresh31(least(m, r, e), true2, e, m)
% 0.21/0.52 = { by axiom 1 (thIV16_1) }
% 0.21/0.52 fresh31(true2, true2, e, m)
% 0.21/0.52 = { by axiom 5 (least_3) }
% 0.21/0.52 true2
% 0.21/0.52
% 0.21/0.52 Lemma 16: fresh32(member(X, e), true2, r, m, X) = apply(r, m, X).
% 0.21/0.52 Proof:
% 0.21/0.52 fresh32(member(X, e), true2, r, m, X)
% 0.21/0.52 = { by axiom 14 (least_2) R->L }
% 0.21/0.52 fresh33(least(m, r, e), true2, r, e, m, X)
% 0.21/0.52 = { by axiom 1 (thIV16_1) }
% 0.21/0.52 fresh33(true2, true2, r, e, m, X)
% 0.21/0.52 = { by axiom 10 (least_2) }
% 0.21/0.52 apply(r, m, X)
% 0.21/0.52
% 0.21/0.52 Goal 1 (thIV16_4): m1 = m2.
% 0.21/0.52 Proof:
% 0.21/0.52 m1
% 0.21/0.52 = { by axiom 7 (min_3) R->L }
% 0.21/0.52 fresh(true2, true2, e, m1, m)
% 0.21/0.52 = { by axiom 8 (least_2) R->L }
% 0.21/0.52 fresh(fresh32(true2, true2, r, m, m1), true2, e, m1, m)
% 0.21/0.52 = { by axiom 6 (min_4) R->L }
% 0.21/0.52 fresh(fresh32(fresh18(true2, true2, e, m1), true2, r, m, m1), true2, e, m1, m)
% 0.21/0.52 = { by axiom 3 (thIV16_3) R->L }
% 0.21/0.52 fresh(fresh32(fresh18(min(m1, r, e), true2, e, m1), true2, r, m, m1), true2, e, m1, m)
% 0.21/0.52 = { by axiom 12 (min_4) }
% 0.21/0.52 fresh(fresh32(member(m1, e), true2, r, m, m1), true2, e, m1, m)
% 0.21/0.52 = { by lemma 16 }
% 0.21/0.52 fresh(apply(r, m, m1), true2, e, m1, m)
% 0.21/0.52 = { by axiom 13 (min_3) R->L }
% 0.21/0.52 fresh53(min(m1, r, e), true2, r, e, m1, m)
% 0.21/0.52 = { by axiom 3 (thIV16_3) }
% 0.21/0.53 fresh53(true2, true2, r, e, m1, m)
% 0.21/0.53 = { by axiom 9 (min_3) }
% 0.21/0.53 fresh54(member(m, e), true2, m1, m)
% 0.21/0.53 = { by lemma 15 }
% 0.21/0.53 fresh54(true2, true2, m1, m)
% 0.21/0.53 = { by axiom 4 (min_3) }
% 0.21/0.53 m
% 0.21/0.53 = { by axiom 4 (min_3) R->L }
% 0.21/0.53 fresh54(true2, true2, m2, m)
% 0.21/0.53 = { by lemma 15 R->L }
% 0.21/0.53 fresh54(member(m, e), true2, m2, m)
% 0.21/0.53 = { by axiom 9 (min_3) R->L }
% 0.21/0.53 fresh53(true2, true2, r, e, m2, m)
% 0.21/0.53 = { by axiom 2 (thIV16_2) R->L }
% 0.21/0.53 fresh53(min(m2, r, e), true2, r, e, m2, m)
% 0.21/0.53 = { by axiom 13 (min_3) }
% 0.21/0.53 fresh(apply(r, m, m2), true2, e, m2, m)
% 0.21/0.53 = { by lemma 16 R->L }
% 0.21/0.53 fresh(fresh32(member(m2, e), true2, r, m, m2), true2, e, m2, m)
% 0.21/0.53 = { by axiom 12 (min_4) R->L }
% 0.21/0.53 fresh(fresh32(fresh18(min(m2, r, e), true2, e, m2), true2, r, m, m2), true2, e, m2, m)
% 0.21/0.53 = { by axiom 2 (thIV16_2) }
% 0.21/0.53 fresh(fresh32(fresh18(true2, true2, e, m2), true2, r, m, m2), true2, e, m2, m)
% 0.21/0.53 = { by axiom 6 (min_4) }
% 0.21/0.53 fresh(fresh32(true2, true2, r, m, m2), true2, e, m2, m)
% 0.21/0.53 = { by axiom 8 (least_2) }
% 0.21/0.53 fresh(true2, true2, e, m2, m)
% 0.21/0.53 = { by axiom 7 (min_3) }
% 0.21/0.53 m2
% 0.21/0.53 % SZS output end Proof
% 0.21/0.53
% 0.21/0.53 RESULT: Theorem (the conjecture is true).
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