TSTP Solution File: SET983+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET983+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 : n014.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 15:33:53 EDT 2023
% Result : Theorem 0.20s 0.41s
% 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 : SET983+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 : n014.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 12:43:32 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.41 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.41
% 0.20/0.41 % SZS status Theorem
% 0.20/0.41
% 0.20/0.41 % SZS output start Proof
% 0.20/0.41 Take the following subset of the input axioms:
% 0.20/0.41 fof(commutativity_k3_xboole_0, axiom, ![A, B]: set_intersection2(A, B)=set_intersection2(B, A)).
% 0.20/0.41 fof(d3_xboole_0, axiom, ![C, A2, B2]: (C=set_intersection2(A2, B2) <=> ![D]: (in(D, C) <=> (in(D, A2) & in(D, B2))))).
% 0.20/0.41 fof(t123_zfmisc_1, axiom, ![A3, B2, C2, D2]: cartesian_product2(set_intersection2(A3, B2), set_intersection2(C2, D2))=set_intersection2(cartesian_product2(A3, C2), cartesian_product2(B2, D2))).
% 0.20/0.41 fof(t137_zfmisc_1, conjecture, ![E, A3, B2, C2, D2]: ((in(A3, cartesian_product2(B2, C2)) & in(A3, cartesian_product2(D2, E))) => in(A3, cartesian_product2(set_intersection2(B2, D2), set_intersection2(C2, E))))).
% 0.20/0.41
% 0.20/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.41 fresh(y, y, x1...xn) = u
% 0.20/0.41 C => fresh(s, t, x1...xn) = v
% 0.20/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.41 variables of u and v.
% 0.20/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.41 input problem has no model of domain size 1).
% 0.20/0.41
% 0.20/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.41
% 0.20/0.41 Axiom 1 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.20/0.41 Axiom 2 (d3_xboole_0_2): fresh7(X, X, Y, Z) = true2.
% 0.20/0.41 Axiom 3 (t137_zfmisc_1): in(a, cartesian_product2(d, e)) = true2.
% 0.20/0.41 Axiom 4 (t137_zfmisc_1_1): in(a, cartesian_product2(b, c)) = true2.
% 0.20/0.41 Axiom 5 (d3_xboole_0_3): fresh6(X, X, Y, Z, W) = equiv(Y, Z, W).
% 0.20/0.41 Axiom 6 (d3_xboole_0_3): fresh5(X, X, Y, Z, W) = true2.
% 0.20/0.41 Axiom 7 (d3_xboole_0_2): fresh8(X, X, Y, Z, W, V) = in(V, W).
% 0.20/0.41 Axiom 8 (t123_zfmisc_1): cartesian_product2(set_intersection2(X, Y), set_intersection2(Z, W)) = set_intersection2(cartesian_product2(X, Z), cartesian_product2(Y, W)).
% 0.20/0.41 Axiom 9 (d3_xboole_0_3): fresh6(in(X, Y), true2, Z, Y, X) = fresh5(in(X, Z), true2, Z, Y, X).
% 0.20/0.41 Axiom 10 (d3_xboole_0_2): fresh8(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh7(W, set_intersection2(X, Y), W, Z).
% 0.20/0.41
% 0.20/0.41 Goal 1 (t137_zfmisc_1_2): in(a, cartesian_product2(set_intersection2(b, d), set_intersection2(c, e))) = true2.
% 0.20/0.41 Proof:
% 0.20/0.41 in(a, cartesian_product2(set_intersection2(b, d), set_intersection2(c, e)))
% 0.20/0.41 = { by axiom 1 (commutativity_k3_xboole_0) }
% 0.20/0.41 in(a, cartesian_product2(set_intersection2(d, b), set_intersection2(c, e)))
% 0.20/0.41 = { by axiom 1 (commutativity_k3_xboole_0) }
% 0.20/0.41 in(a, cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)))
% 0.20/0.41 = { by axiom 7 (d3_xboole_0_2) R->L }
% 0.20/0.41 fresh8(true2, true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 6 (d3_xboole_0_3) R->L }
% 0.20/0.41 fresh8(fresh5(true2, true2, cartesian_product2(d, e), cartesian_product2(b, c), a), true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 3 (t137_zfmisc_1) R->L }
% 0.20/0.41 fresh8(fresh5(in(a, cartesian_product2(d, e)), true2, cartesian_product2(d, e), cartesian_product2(b, c), a), true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 9 (d3_xboole_0_3) R->L }
% 0.20/0.41 fresh8(fresh6(in(a, cartesian_product2(b, c)), true2, cartesian_product2(d, e), cartesian_product2(b, c), a), true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 4 (t137_zfmisc_1_1) }
% 0.20/0.41 fresh8(fresh6(true2, true2, cartesian_product2(d, e), cartesian_product2(b, c), a), true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 5 (d3_xboole_0_3) }
% 0.20/0.41 fresh8(equiv(cartesian_product2(d, e), cartesian_product2(b, c), a), true2, cartesian_product2(d, e), cartesian_product2(b, c), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 10 (d3_xboole_0_2) }
% 0.20/0.41 fresh7(cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), set_intersection2(cartesian_product2(d, e), cartesian_product2(b, c)), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 8 (t123_zfmisc_1) R->L }
% 0.20/0.41 fresh7(cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), cartesian_product2(set_intersection2(d, b), set_intersection2(e, c)), a)
% 0.20/0.41 = { by axiom 2 (d3_xboole_0_2) }
% 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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