TSTP Solution File: SET625+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET625+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 : n003.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 15:32:45 EDT 2023
% Result : Theorem 0.20s 0.39s
% 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
% 0.00/0.12 % Problem : SET625+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
% 0.12/0.34 % Computer : n003.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 12:05:09 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.39 Command-line arguments: --no-flatten-goal
% 0.20/0.39
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Take the following subset of the input axioms:
% 0.20/0.40 fof(intersect_defn, axiom, ![B, C]: (intersect(B, C) <=> ?[D]: (member(D, B) & member(D, C)))).
% 0.20/0.40 fof(prove_th101, conjecture, ![B2, C2, D2]: ((intersect(B2, C2) & subset(C2, D2)) => intersect(B2, D2))).
% 0.20/0.40 fof(subset_defn, axiom, ![B2, C2]: (subset(B2, C2) <=> ![D2]: (member(D2, B2) => member(D2, C2)))).
% 0.20/0.40 fof(symmetry_of_intersect, axiom, ![B2, C2]: (intersect(B2, C2) => intersect(C2, B2))).
% 0.20/0.40
% 0.20/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40 fresh(y, y, x1...xn) = u
% 0.20/0.40 C => fresh(s, t, x1...xn) = v
% 0.20/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40 variables of u and v.
% 0.20/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40 input problem has no model of domain size 1).
% 0.20/0.40
% 0.20/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40
% 0.20/0.40 Axiom 1 (prove_th101_1): subset(c, d) = true.
% 0.20/0.40 Axiom 2 (prove_th101): intersect(b, c) = true.
% 0.20/0.40 Axiom 3 (symmetry_of_intersect): fresh(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 4 (intersect_defn_1): fresh8(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 5 (intersect_defn): fresh7(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 6 (intersect_defn_2): fresh5(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 7 (subset_defn_1): fresh3(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 8 (intersect_defn_2): fresh6(X, X, Y, Z, W) = intersect(Y, Z).
% 0.20/0.40 Axiom 9 (subset_defn_1): fresh4(X, X, Y, Z, W) = member(W, Z).
% 0.20/0.40 Axiom 10 (symmetry_of_intersect): fresh(intersect(X, Y), true, X, Y) = intersect(Y, X).
% 0.20/0.40 Axiom 11 (intersect_defn_1): fresh8(intersect(X, Y), true, X, Y) = member(d3(X, Y), Y).
% 0.20/0.40 Axiom 12 (intersect_defn): fresh7(intersect(X, Y), true, X, Y) = member(d3(X, Y), X).
% 0.20/0.40 Axiom 13 (intersect_defn_2): fresh6(member(X, Y), true, Z, Y, X) = fresh5(member(X, Z), true, Z, Y).
% 0.20/0.40 Axiom 14 (subset_defn_1): fresh4(subset(X, Y), true, X, Y, Z) = fresh3(member(Z, X), true, Y, Z).
% 0.20/0.40
% 0.20/0.40 Goal 1 (prove_th101_2): intersect(b, d) = true.
% 0.20/0.40 Proof:
% 0.20/0.40 intersect(b, d)
% 0.20/0.40 = { by axiom 10 (symmetry_of_intersect) R->L }
% 0.20/0.40 fresh(intersect(d, b), true, d, b)
% 0.20/0.40 = { by axiom 8 (intersect_defn_2) R->L }
% 0.20/0.40 fresh(fresh6(true, true, d, b, d3(b, c)), true, d, b)
% 0.20/0.40 = { by axiom 5 (intersect_defn) R->L }
% 0.20/0.40 fresh(fresh6(fresh7(true, true, b, c), true, d, b, d3(b, c)), true, d, b)
% 0.20/0.40 = { by axiom 2 (prove_th101) R->L }
% 0.20/0.40 fresh(fresh6(fresh7(intersect(b, c), true, b, c), true, d, b, d3(b, c)), true, d, b)
% 0.20/0.40 = { by axiom 12 (intersect_defn) }
% 0.20/0.40 fresh(fresh6(member(d3(b, c), b), true, d, b, d3(b, c)), true, d, b)
% 0.20/0.40 = { by axiom 13 (intersect_defn_2) }
% 0.20/0.40 fresh(fresh5(member(d3(b, c), d), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 9 (subset_defn_1) R->L }
% 0.20/0.40 fresh(fresh5(fresh4(true, true, c, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 1 (prove_th101_1) R->L }
% 0.20/0.40 fresh(fresh5(fresh4(subset(c, d), true, c, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 14 (subset_defn_1) }
% 0.20/0.40 fresh(fresh5(fresh3(member(d3(b, c), c), true, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 11 (intersect_defn_1) R->L }
% 0.20/0.40 fresh(fresh5(fresh3(fresh8(intersect(b, c), true, b, c), true, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 2 (prove_th101) }
% 0.20/0.40 fresh(fresh5(fresh3(fresh8(true, true, b, c), true, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 4 (intersect_defn_1) }
% 0.20/0.40 fresh(fresh5(fresh3(true, true, d, d3(b, c)), true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 7 (subset_defn_1) }
% 0.20/0.40 fresh(fresh5(true, true, d, b), true, d, b)
% 0.20/0.40 = { by axiom 6 (intersect_defn_2) }
% 0.20/0.40 fresh(true, true, d, b)
% 0.20/0.40 = { by axiom 3 (symmetry_of_intersect) }
% 0.20/0.40 true
% 0.20/0.40 % SZS output end Proof
% 0.20/0.40
% 0.20/0.40 RESULT: Theorem (the conjecture is true).
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