TSTP Solution File: SET585+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET585+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 15:32:31 EDT 2023
% Result : Theorem 0.19s 0.40s
% 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
% 0.07/0.12 % Problem : SET585+3 : TPTP v8.1.2. Released v2.2.0.
% 0.07/0.13 % 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 : n020.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 12:31:14 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.40 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.40
% 0.19/0.40 % SZS status Theorem
% 0.19/0.40
% 0.19/0.40 % SZS output start Proof
% 0.19/0.40 Take the following subset of the input axioms:
% 0.19/0.41 fof(intersection_is_subset, axiom, ![B, C]: subset(intersection(B, C), B)).
% 0.19/0.41 fof(prove_intersection_subset_of_union, conjecture, ![D, B2, C2]: subset(intersection(B2, C2), union(B2, D))).
% 0.19/0.41 fof(subset_of_union, axiom, ![B2, C2]: subset(B2, union(B2, C2))).
% 0.19/0.41 fof(transitivity_of_subset, axiom, ![B2, C2, D2]: ((subset(B2, C2) & subset(C2, D2)) => subset(B2, D2))).
% 0.19/0.41
% 0.19/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.41 fresh(y, y, x1...xn) = u
% 0.19/0.41 C => fresh(s, t, x1...xn) = v
% 0.19/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.41 variables of u and v.
% 0.19/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.41 input problem has no model of domain size 1).
% 0.19/0.41
% 0.19/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.41
% 0.19/0.41 Axiom 1 (transitivity_of_subset): fresh5(X, X, Y, Z) = true.
% 0.19/0.41 Axiom 2 (subset_of_union): subset(X, union(X, Y)) = true.
% 0.19/0.41 Axiom 3 (intersection_is_subset): subset(intersection(X, Y), X) = true.
% 0.19/0.41 Axiom 4 (transitivity_of_subset): fresh6(X, X, Y, Z, W) = subset(Y, W).
% 0.19/0.41 Axiom 5 (transitivity_of_subset): fresh6(subset(X, Y), true, Z, X, Y) = fresh5(subset(Z, X), true, Z, Y).
% 0.19/0.41
% 0.19/0.41 Goal 1 (prove_intersection_subset_of_union): subset(intersection(b, c), union(b, d)) = true.
% 0.19/0.41 Proof:
% 0.19/0.41 subset(intersection(b, c), union(b, d))
% 0.19/0.41 = { by axiom 4 (transitivity_of_subset) R->L }
% 0.19/0.41 fresh6(true, true, intersection(b, c), b, union(b, d))
% 0.19/0.41 = { by axiom 2 (subset_of_union) R->L }
% 0.19/0.41 fresh6(subset(b, union(b, d)), true, intersection(b, c), b, union(b, d))
% 0.19/0.41 = { by axiom 5 (transitivity_of_subset) }
% 0.19/0.41 fresh5(subset(intersection(b, c), b), true, intersection(b, c), union(b, d))
% 0.19/0.41 = { by axiom 3 (intersection_is_subset) }
% 0.19/0.41 fresh5(true, true, intersection(b, c), union(b, d))
% 0.19/0.41 = { by axiom 1 (transitivity_of_subset) }
% 0.19/0.41 true
% 0.19/0.41 % SZS output end Proof
% 0.19/0.41
% 0.19/0.41 RESULT: Theorem (the conjecture is true).
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