TSTP Solution File: SET631+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET631+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 : n012.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 15:32:47 EDT 2023
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% 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 : SET631+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n012.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Aug 26 09:41:40 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.38 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.38
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38
% 0.19/0.39 % SZS output start Proof
% 0.19/0.39 Take the following subset of the input axioms:
% 0.19/0.39 fof(difference_defn, axiom, ![B, C, D]: (member(D, difference(B, C)) <=> (member(D, B) & ~member(D, C)))).
% 0.19/0.39 fof(intersect_defn, axiom, ![B2, C2]: (intersect(B2, C2) <=> ?[D2]: (member(D2, B2) & member(D2, C2)))).
% 0.19/0.39 fof(prove_th113, conjecture, ![B2, C2, D2]: (intersect(B2, difference(C2, D2)) => intersect(B2, C2))).
% 0.19/0.39 fof(symmetry_of_intersect, axiom, ![B2, C2]: (intersect(B2, C2) => intersect(C2, B2))).
% 0.19/0.39
% 0.19/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.39 fresh(y, y, x1...xn) = u
% 0.19/0.39 C => fresh(s, t, x1...xn) = v
% 0.19/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.39 variables of u and v.
% 0.19/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.39 input problem has no model of domain size 1).
% 0.19/0.39
% 0.19/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.39
% 0.19/0.39 Axiom 1 (prove_th113): intersect(b, difference(c, d)) = true2.
% 0.19/0.39 Axiom 2 (symmetry_of_intersect): fresh(X, X, Y, Z) = true2.
% 0.19/0.39 Axiom 3 (difference_defn_2): fresh5(X, X, Y, Z) = true2.
% 0.19/0.39 Axiom 4 (intersect_defn): fresh4(X, X, Y, Z) = true2.
% 0.19/0.39 Axiom 5 (intersect_defn_1): fresh3(X, X, Y, Z) = true2.
% 0.19/0.39 Axiom 6 (intersect_defn_2): fresh2(X, X, Y, Z) = true2.
% 0.19/0.39 Axiom 7 (intersect_defn): fresh6(X, X, Y, Z, W) = intersect(Y, Z).
% 0.19/0.39 Axiom 8 (symmetry_of_intersect): fresh(intersect(X, Y), true2, X, Y) = intersect(Y, X).
% 0.19/0.39 Axiom 9 (intersect_defn_1): fresh3(intersect(X, Y), true2, X, Y) = member(d2(X, Y), X).
% 0.19/0.39 Axiom 10 (intersect_defn_2): fresh2(intersect(X, Y), true2, X, Y) = member(d2(X, Y), Y).
% 0.19/0.39 Axiom 11 (intersect_defn): fresh6(member(X, Y), true2, Z, Y, X) = fresh4(member(X, Z), true2, Z, Y).
% 0.19/0.39 Axiom 12 (difference_defn_2): fresh5(member(X, difference(Y, Z)), true2, Y, X) = member(X, Y).
% 0.19/0.39
% 0.19/0.39 Goal 1 (prove_th113_1): intersect(b, c) = true2.
% 0.19/0.39 Proof:
% 0.19/0.39 intersect(b, c)
% 0.19/0.39 = { by axiom 8 (symmetry_of_intersect) R->L }
% 0.19/0.39 fresh(intersect(c, b), true2, c, b)
% 0.19/0.39 = { by axiom 7 (intersect_defn) R->L }
% 0.19/0.39 fresh(fresh6(true2, true2, c, b, d2(b, difference(c, d))), true2, c, b)
% 0.19/0.39 = { by axiom 5 (intersect_defn_1) R->L }
% 0.19/0.39 fresh(fresh6(fresh3(true2, true2, b, difference(c, d)), true2, c, b, d2(b, difference(c, d))), true2, c, b)
% 0.19/0.39 = { by axiom 1 (prove_th113) R->L }
% 0.19/0.39 fresh(fresh6(fresh3(intersect(b, difference(c, d)), true2, b, difference(c, d)), true2, c, b, d2(b, difference(c, d))), true2, c, b)
% 0.19/0.39 = { by axiom 9 (intersect_defn_1) }
% 0.19/0.39 fresh(fresh6(member(d2(b, difference(c, d)), b), true2, c, b, d2(b, difference(c, d))), true2, c, b)
% 0.19/0.39 = { by axiom 11 (intersect_defn) }
% 0.19/0.39 fresh(fresh4(member(d2(b, difference(c, d)), c), true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 12 (difference_defn_2) R->L }
% 0.19/0.39 fresh(fresh4(fresh5(member(d2(b, difference(c, d)), difference(c, d)), true2, c, d2(b, difference(c, d))), true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 10 (intersect_defn_2) R->L }
% 0.19/0.39 fresh(fresh4(fresh5(fresh2(intersect(b, difference(c, d)), true2, b, difference(c, d)), true2, c, d2(b, difference(c, d))), true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 1 (prove_th113) }
% 0.19/0.39 fresh(fresh4(fresh5(fresh2(true2, true2, b, difference(c, d)), true2, c, d2(b, difference(c, d))), true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 6 (intersect_defn_2) }
% 0.19/0.39 fresh(fresh4(fresh5(true2, true2, c, d2(b, difference(c, d))), true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 3 (difference_defn_2) }
% 0.19/0.39 fresh(fresh4(true2, true2, c, b), true2, c, b)
% 0.19/0.39 = { by axiom 4 (intersect_defn) }
% 0.19/0.39 fresh(true2, true2, c, b)
% 0.19/0.39 = { by axiom 2 (symmetry_of_intersect) }
% 0.19/0.39 true2
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39
% 0.19/0.39 RESULT: Theorem (the conjecture is true).
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