TSTP Solution File: SET638+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET638+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 : n009.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 15:32:49 EDT 2023
% Result : Theorem 0.19s 0.41s
% 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 : SET638+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
% 0.12/0.34 % Computer : n009.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 14:24:05 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.41 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.41
% 0.19/0.41 % SZS status Theorem
% 0.19/0.41
% 0.19/0.41 % SZS output start Proof
% 0.19/0.41 Take the following subset of the input axioms:
% 0.19/0.41 fof(commutativity_of_intersection, axiom, ![B, C]: intersection(B, C)=intersection(C, B)).
% 0.19/0.41 fof(intersection_distributes_over_union, axiom, ![D, B2, C2]: intersection(B2, union(C2, D))=union(intersection(B2, C2), intersection(B2, D))).
% 0.19/0.41 fof(intersection_is_subset, axiom, ![B2, C2]: subset(intersection(B2, C2), B2)).
% 0.19/0.41 fof(prove_th120, conjecture, ![B2, C2, D2]: ((subset(B2, union(C2, D2)) & intersection(B2, D2)=empty_set) => subset(B2, C2))).
% 0.19/0.41 fof(subset_intersection, axiom, ![B2, C2]: (subset(B2, C2) => intersection(B2, C2)=B2)).
% 0.19/0.41 fof(union_empty_set, axiom, ![B2]: union(B2, empty_set)=B2).
% 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 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.19/0.41 Axiom 2 (prove_th120): intersection(b, d) = empty_set.
% 0.19/0.41 Axiom 3 (union_empty_set): union(X, empty_set) = X.
% 0.19/0.41 Axiom 4 (prove_th120_1): subset(b, union(c, d)) = true2.
% 0.19/0.41 Axiom 5 (intersection_is_subset): subset(intersection(X, Y), X) = true2.
% 0.19/0.41 Axiom 6 (subset_intersection): fresh5(X, X, Y, Z) = Y.
% 0.19/0.41 Axiom 7 (intersection_distributes_over_union): intersection(X, union(Y, Z)) = union(intersection(X, Y), intersection(X, Z)).
% 0.19/0.41 Axiom 8 (subset_intersection): fresh5(subset(X, Y), true2, X, Y) = intersection(X, Y).
% 0.19/0.41
% 0.19/0.41 Goal 1 (prove_th120_2): subset(b, c) = true2.
% 0.19/0.41 Proof:
% 0.19/0.41 subset(b, c)
% 0.19/0.41 = { by axiom 6 (subset_intersection) R->L }
% 0.19/0.41 subset(fresh5(true2, true2, b, union(c, d)), c)
% 0.19/0.41 = { by axiom 4 (prove_th120_1) R->L }
% 0.19/0.41 subset(fresh5(subset(b, union(c, d)), true2, b, union(c, d)), c)
% 0.19/0.41 = { by axiom 8 (subset_intersection) }
% 0.19/0.41 subset(intersection(b, union(c, d)), c)
% 0.19/0.41 = { by axiom 7 (intersection_distributes_over_union) }
% 0.19/0.41 subset(union(intersection(b, c), intersection(b, d)), c)
% 0.19/0.41 = { by axiom 2 (prove_th120) }
% 0.19/0.41 subset(union(intersection(b, c), empty_set), c)
% 0.19/0.41 = { by axiom 3 (union_empty_set) }
% 0.19/0.41 subset(intersection(b, c), c)
% 0.19/0.41 = { by axiom 1 (commutativity_of_intersection) R->L }
% 0.19/0.41 subset(intersection(c, b), c)
% 0.19/0.41 = { by axiom 5 (intersection_is_subset) }
% 0.19/0.41 true2
% 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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