TSTP Solution File: SET592+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET592+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 : n011.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 15:32:33 EDT 2023
% Result : Theorem 0.23s 0.43s
% Output : Proof 0.23s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.14 % Problem : SET592+3 : TPTP v8.1.2. Released v2.2.0.
% 0.12/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n011.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Sat Aug 26 11:47:24 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.23/0.43 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.23/0.43
% 0.23/0.43 % SZS status Theorem
% 0.23/0.43
% 0.23/0.43 % SZS output start Proof
% 0.23/0.43 Take the following subset of the input axioms:
% 0.23/0.43 fof(intersection_of_subsets, axiom, ![B, C, D]: ((subset(B, C) & subset(B, D)) => subset(B, intersection(C, D)))).
% 0.23/0.43 fof(prove_th51, conjecture, ![B2, C2, D2]: ((subset(B2, C2) & (subset(B2, D2) & intersection(C2, D2)=empty_set)) => B2=empty_set)).
% 0.23/0.43 fof(subset_of_empty_set_is_empty_set, axiom, ![B2]: (subset(B2, empty_set) => B2=empty_set)).
% 0.23/0.43
% 0.23/0.43 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.23/0.43 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.23/0.43 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.23/0.43 fresh(y, y, x1...xn) = u
% 0.23/0.43 C => fresh(s, t, x1...xn) = v
% 0.23/0.43 where fresh is a fresh function symbol and x1..xn are the free
% 0.23/0.43 variables of u and v.
% 0.23/0.43 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.23/0.43 input problem has no model of domain size 1).
% 0.23/0.43
% 0.23/0.43 The encoding turns the above axioms into the following unit equations and goals:
% 0.23/0.43
% 0.23/0.43 Axiom 1 (prove_th51_1): subset(b, c) = true2.
% 0.23/0.43 Axiom 2 (prove_th51_2): subset(b, d) = true2.
% 0.23/0.43 Axiom 3 (prove_th51): intersection(c, d) = empty_set.
% 0.23/0.43 Axiom 4 (subset_of_empty_set_is_empty_set): fresh5(X, X, Y) = empty_set.
% 0.23/0.43 Axiom 5 (intersection_of_subsets): fresh10(X, X, Y, Z, W) = subset(Y, intersection(Z, W)).
% 0.23/0.43 Axiom 6 (intersection_of_subsets): fresh9(X, X, Y, Z, W) = true2.
% 0.23/0.43 Axiom 7 (subset_of_empty_set_is_empty_set): fresh5(subset(X, empty_set), true2, X) = X.
% 0.23/0.43 Axiom 8 (intersection_of_subsets): fresh10(subset(X, Y), true2, X, Z, Y) = fresh9(subset(X, Z), true2, X, Z, Y).
% 0.23/0.43
% 0.23/0.43 Goal 1 (prove_th51_3): b = empty_set.
% 0.23/0.43 Proof:
% 0.23/0.43 b
% 0.23/0.43 = { by axiom 7 (subset_of_empty_set_is_empty_set) R->L }
% 0.23/0.43 fresh5(subset(b, empty_set), true2, b)
% 0.23/0.43 = { by axiom 3 (prove_th51) R->L }
% 0.23/0.43 fresh5(subset(b, intersection(c, d)), true2, b)
% 0.23/0.43 = { by axiom 5 (intersection_of_subsets) R->L }
% 0.23/0.43 fresh5(fresh10(true2, true2, b, c, d), true2, b)
% 0.23/0.43 = { by axiom 2 (prove_th51_2) R->L }
% 0.23/0.43 fresh5(fresh10(subset(b, d), true2, b, c, d), true2, b)
% 0.23/0.43 = { by axiom 8 (intersection_of_subsets) }
% 0.23/0.43 fresh5(fresh9(subset(b, c), true2, b, c, d), true2, b)
% 0.23/0.43 = { by axiom 1 (prove_th51_1) }
% 0.23/0.43 fresh5(fresh9(true2, true2, b, c, d), true2, b)
% 0.23/0.43 = { by axiom 6 (intersection_of_subsets) }
% 0.23/0.43 fresh5(true2, true2, b)
% 0.23/0.43 = { by axiom 4 (subset_of_empty_set_is_empty_set) }
% 0.23/0.43 empty_set
% 0.23/0.43 % SZS output end Proof
% 0.23/0.43
% 0.23/0.43 RESULT: Theorem (the conjecture is true).
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