TSTP Solution File: SET971+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET971+1 : TPTP v8.1.2. Released v3.2.0.
% 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 : n018.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 15:33:52 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 : SET971+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n018.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Aug 26 11:37:14 EDT 2023
% 0.14/0.35 % 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
% 0.20/0.39 % SZS output start Proof
% 0.20/0.39 Take the following subset of the input axioms:
% 0.20/0.39 fof(commutativity_k3_xboole_0, axiom, ![A, B]: set_intersection2(A, B)=set_intersection2(B, A)).
% 0.20/0.39 fof(t123_zfmisc_1, axiom, ![C, D, B2, A3]: cartesian_product2(set_intersection2(A3, B2), set_intersection2(C, D))=set_intersection2(cartesian_product2(A3, C), cartesian_product2(B2, D))).
% 0.20/0.39 fof(t124_zfmisc_1, conjecture, ![B2, A3, C2, D2]: ((subset(A3, B2) & subset(C2, D2)) => set_intersection2(cartesian_product2(A3, D2), cartesian_product2(B2, C2))=cartesian_product2(A3, C2))).
% 0.20/0.39 fof(t28_xboole_1, axiom, ![A2, B2]: (subset(A2, B2) => set_intersection2(A2, B2)=A2)).
% 0.20/0.39
% 0.20/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.39 fresh(y, y, x1...xn) = u
% 0.20/0.39 C => fresh(s, t, x1...xn) = v
% 0.20/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.39 variables of u and v.
% 0.20/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.39 input problem has no model of domain size 1).
% 0.20/0.39
% 0.20/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.39
% 0.20/0.40 Axiom 1 (t124_zfmisc_1): subset(c, d) = true.
% 0.20/0.40 Axiom 2 (t124_zfmisc_1_1): subset(a, b) = true.
% 0.20/0.40 Axiom 3 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.20/0.40 Axiom 4 (t28_xboole_1): fresh(X, X, Y, Z) = Y.
% 0.20/0.40 Axiom 5 (t28_xboole_1): fresh(subset(X, Y), true, X, Y) = set_intersection2(X, Y).
% 0.20/0.40 Axiom 6 (t123_zfmisc_1): cartesian_product2(set_intersection2(X, Y), set_intersection2(Z, W)) = set_intersection2(cartesian_product2(X, Z), cartesian_product2(Y, W)).
% 0.20/0.40
% 0.20/0.40 Goal 1 (t124_zfmisc_1_2): set_intersection2(cartesian_product2(a, d), cartesian_product2(b, c)) = cartesian_product2(a, c).
% 0.20/0.40 Proof:
% 0.20/0.40 set_intersection2(cartesian_product2(a, d), cartesian_product2(b, c))
% 0.20/0.40 = { by axiom 3 (commutativity_k3_xboole_0) }
% 0.20/0.40 set_intersection2(cartesian_product2(b, c), cartesian_product2(a, d))
% 0.20/0.40 = { by axiom 6 (t123_zfmisc_1) R->L }
% 0.20/0.40 cartesian_product2(set_intersection2(b, a), set_intersection2(c, d))
% 0.20/0.40 = { by axiom 5 (t28_xboole_1) R->L }
% 0.20/0.40 cartesian_product2(set_intersection2(b, a), fresh(subset(c, d), true, c, d))
% 0.20/0.40 = { by axiom 1 (t124_zfmisc_1) }
% 0.20/0.40 cartesian_product2(set_intersection2(b, a), fresh(true, true, c, d))
% 0.20/0.40 = { by axiom 4 (t28_xboole_1) }
% 0.20/0.40 cartesian_product2(set_intersection2(b, a), c)
% 0.20/0.40 = { by axiom 3 (commutativity_k3_xboole_0) R->L }
% 0.20/0.40 cartesian_product2(set_intersection2(a, b), c)
% 0.20/0.40 = { by axiom 5 (t28_xboole_1) R->L }
% 0.20/0.40 cartesian_product2(fresh(subset(a, b), true, a, b), c)
% 0.20/0.40 = { by axiom 2 (t124_zfmisc_1_1) }
% 0.20/0.40 cartesian_product2(fresh(true, true, a, b), c)
% 0.20/0.40 = { by axiom 4 (t28_xboole_1) }
% 0.20/0.40 cartesian_product2(a, c)
% 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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