TSTP Solution File: SET573+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET573+3 : TPTP v8.1.2. Released v2.2.0.
% 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 : n020.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 15:32:28 EDT 2023
% Result : Theorem 0.22s 0.40s
% 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 : SET573+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n020.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Sat Aug 26 10:57:14 EDT 2023
% 0.17/0.35 % CPUTime :
% 0.22/0.40 Command-line arguments: --no-flatten-goal
% 0.22/0.40
% 0.22/0.40 % SZS status Theorem
% 0.22/0.40
% 0.22/0.40 % SZS output start Proof
% 0.22/0.40 Take the following subset of the input axioms:
% 0.22/0.40 fof(disjoint_defn, axiom, ![B, C]: (disjoint(B, C) <=> ~intersect(B, C))).
% 0.22/0.40 fof(intersect_defn, axiom, ![B2, C2]: (intersect(B2, C2) <=> ?[D]: (member(D, B2) & member(D, C2)))).
% 0.22/0.40 fof(prove_th12, conjecture, ![B2, C2, D2]: ((member(B2, C2) & disjoint(C2, D2)) => ~member(B2, D2))).
% 0.22/0.40 fof(symmetry_of_intersect, axiom, ![B2, C2]: (intersect(B2, C2) => intersect(C2, B2))).
% 0.22/0.40
% 0.22/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.40 fresh(y, y, x1...xn) = u
% 0.22/0.40 C => fresh(s, t, x1...xn) = v
% 0.22/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.40 variables of u and v.
% 0.22/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.40 input problem has no model of domain size 1).
% 0.22/0.40
% 0.22/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.40
% 0.22/0.40 Axiom 1 (prove_th12_2): disjoint(c, d) = true2.
% 0.22/0.40 Axiom 2 (prove_th12): member(b, c) = true2.
% 0.22/0.40 Axiom 3 (prove_th12_1): member(b, d) = true2.
% 0.22/0.40 Axiom 4 (symmetry_of_intersect): fresh(X, X, Y, Z) = true2.
% 0.22/0.40 Axiom 5 (intersect_defn_2): fresh2(X, X, Y, Z) = true2.
% 0.22/0.40 Axiom 6 (intersect_defn_2): fresh3(X, X, Y, Z, W) = intersect(Y, Z).
% 0.22/0.40 Axiom 7 (symmetry_of_intersect): fresh(intersect(X, Y), true2, X, Y) = intersect(Y, X).
% 0.22/0.40 Axiom 8 (intersect_defn_2): fresh3(member(X, Y), true2, Z, Y, X) = fresh2(member(X, Z), true2, Z, Y).
% 0.22/0.40
% 0.22/0.40 Goal 1 (disjoint_defn_1): tuple(intersect(X, Y), disjoint(X, Y)) = tuple(true2, true2).
% 0.22/0.40 The goal is true when:
% 0.22/0.40 X = c
% 0.22/0.40 Y = d
% 0.22/0.40
% 0.22/0.40 Proof:
% 0.22/0.40 tuple(intersect(c, d), disjoint(c, d))
% 0.22/0.40 = { by axiom 7 (symmetry_of_intersect) R->L }
% 0.22/0.40 tuple(fresh(intersect(d, c), true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 6 (intersect_defn_2) R->L }
% 0.22/0.40 tuple(fresh(fresh3(true2, true2, d, c, b), true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 2 (prove_th12) R->L }
% 0.22/0.40 tuple(fresh(fresh3(member(b, c), true2, d, c, b), true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 8 (intersect_defn_2) }
% 0.22/0.40 tuple(fresh(fresh2(member(b, d), true2, d, c), true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 3 (prove_th12_1) }
% 0.22/0.40 tuple(fresh(fresh2(true2, true2, d, c), true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 5 (intersect_defn_2) }
% 0.22/0.40 tuple(fresh(true2, true2, d, c), disjoint(c, d))
% 0.22/0.40 = { by axiom 4 (symmetry_of_intersect) }
% 0.22/0.40 tuple(true2, disjoint(c, d))
% 0.22/0.40 = { by axiom 1 (prove_th12_2) }
% 0.22/0.40 tuple(true2, true2)
% 0.22/0.40 % SZS output end Proof
% 0.22/0.40
% 0.22/0.40 RESULT: Theorem (the conjecture is true).
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