TSTP Solution File: SET633+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET633+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 : n026.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 15:32:48 EDT 2023
% Result : Theorem 0.20s 0.40s
% 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
% 0.00/0.13 % Problem : SET633+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 : n026.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 09:41:18 EDT 2023
% 0.17/0.35 % CPUTime :
% 0.20/0.40 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
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% 0.20/0.40 % SZS status Theorem
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% 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(prove_th115, conjecture, ![B, C, D]: ((subset(difference(B, C), D) & subset(difference(C, B), D)) => subset(symmetric_difference(B, C), D))).
% 0.20/0.40 fof(symmetric_difference_defn, axiom, ![B2, C2]: symmetric_difference(B2, C2)=union(difference(B2, C2), difference(C2, B2))).
% 0.20/0.40 fof(union_subset, axiom, ![B2, C2, D2]: ((subset(B2, C2) & subset(D2, C2)) => subset(union(B2, D2), C2))).
% 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_th115): subset(difference(b, c), d) = true2.
% 0.20/0.40 Axiom 2 (prove_th115_1): subset(difference(c, b), d) = true2.
% 0.20/0.40 Axiom 3 (union_subset): fresh4(X, X, Y, Z, W) = subset(union(Y, W), Z).
% 0.20/0.40 Axiom 4 (union_subset): fresh3(X, X, Y, Z, W) = true2.
% 0.20/0.40 Axiom 5 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.20/0.40 Axiom 6 (union_subset): fresh4(subset(X, Y), true2, Z, Y, X) = fresh3(subset(Z, Y), true2, Z, Y, X).
% 0.20/0.41
% 0.20/0.41 Goal 1 (prove_th115_2): subset(symmetric_difference(b, c), d) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 subset(symmetric_difference(b, c), d)
% 0.20/0.41 = { by axiom 5 (symmetric_difference_defn) }
% 0.20/0.41 subset(union(difference(b, c), difference(c, b)), d)
% 0.20/0.41 = { by axiom 3 (union_subset) R->L }
% 0.20/0.41 fresh4(true2, true2, difference(b, c), d, difference(c, b))
% 0.20/0.41 = { by axiom 2 (prove_th115_1) R->L }
% 0.20/0.41 fresh4(subset(difference(c, b), d), true2, difference(b, c), d, difference(c, b))
% 0.20/0.41 = { by axiom 6 (union_subset) }
% 0.20/0.41 fresh3(subset(difference(b, c), d), true2, difference(b, c), d, difference(c, b))
% 0.20/0.41 = { by axiom 1 (prove_th115) }
% 0.20/0.41 fresh3(true2, true2, difference(b, c), d, difference(c, b))
% 0.20/0.41 = { by axiom 4 (union_subset) }
% 0.20/0.41 true2
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Theorem (the conjecture is true).
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