TSTP Solution File: SET968+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET968+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 : n025.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 15:33:51 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
% 0.00/0.12 % Problem : SET968+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
% 0.12/0.34 % Computer : n025.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 15:16:08 EDT 2023
% 0.19/0.34 % 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
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% 0.19/0.40 % SZS output start Proof
% 0.19/0.40 Take the following subset of the input axioms:
% 0.19/0.40 fof(commutativity_k2_xboole_0, axiom, ![A, B]: set_union2(A, B)=set_union2(B, A)).
% 0.19/0.40 fof(t120_zfmisc_1, axiom, ![C, A2, B2]: (cartesian_product2(set_union2(A2, B2), C)=set_union2(cartesian_product2(A2, C), cartesian_product2(B2, C)) & cartesian_product2(C, set_union2(A2, B2))=set_union2(cartesian_product2(C, A2), cartesian_product2(C, B2)))).
% 0.19/0.40 fof(t121_zfmisc_1, conjecture, ![D, A2, B2, C2]: cartesian_product2(set_union2(A2, B2), set_union2(C2, D))=set_union2(set_union2(set_union2(cartesian_product2(A2, C2), cartesian_product2(A2, D)), cartesian_product2(B2, C2)), cartesian_product2(B2, D))).
% 0.19/0.40 fof(t4_xboole_1, axiom, ![A2, B2, C2]: set_union2(set_union2(A2, B2), C2)=set_union2(A2, set_union2(B2, C2))).
% 0.19/0.40
% 0.19/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.40 fresh(y, y, x1...xn) = u
% 0.19/0.40 C => fresh(s, t, x1...xn) = v
% 0.19/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.40 variables of u and v.
% 0.19/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.40 input problem has no model of domain size 1).
% 0.19/0.40
% 0.19/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.40
% 0.19/0.40 Axiom 1 (commutativity_k2_xboole_0): set_union2(X, Y) = set_union2(Y, X).
% 0.19/0.40 Axiom 2 (t4_xboole_1): set_union2(set_union2(X, Y), Z) = set_union2(X, set_union2(Y, Z)).
% 0.19/0.40 Axiom 3 (t120_zfmisc_1_1): cartesian_product2(X, set_union2(Y, Z)) = set_union2(cartesian_product2(X, Y), cartesian_product2(X, Z)).
% 0.19/0.40 Axiom 4 (t120_zfmisc_1): cartesian_product2(set_union2(X, Y), Z) = set_union2(cartesian_product2(X, Z), cartesian_product2(Y, Z)).
% 0.19/0.40
% 0.19/0.40 Goal 1 (t121_zfmisc_1): cartesian_product2(set_union2(a, b), set_union2(c, d)) = set_union2(set_union2(set_union2(cartesian_product2(a, c), cartesian_product2(a, d)), cartesian_product2(b, c)), cartesian_product2(b, d)).
% 0.19/0.40 Proof:
% 0.19/0.40 cartesian_product2(set_union2(a, b), set_union2(c, d))
% 0.19/0.40 = { by axiom 4 (t120_zfmisc_1) }
% 0.19/0.40 set_union2(cartesian_product2(a, set_union2(c, d)), cartesian_product2(b, set_union2(c, d)))
% 0.19/0.40 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.19/0.40 set_union2(cartesian_product2(b, set_union2(c, d)), cartesian_product2(a, set_union2(c, d)))
% 0.19/0.40 = { by axiom 3 (t120_zfmisc_1_1) }
% 0.19/0.40 set_union2(set_union2(cartesian_product2(b, c), cartesian_product2(b, d)), cartesian_product2(a, set_union2(c, d)))
% 0.19/0.40 = { by axiom 2 (t4_xboole_1) }
% 0.19/0.40 set_union2(cartesian_product2(b, c), set_union2(cartesian_product2(b, d), cartesian_product2(a, set_union2(c, d))))
% 0.19/0.40 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.19/0.40 set_union2(cartesian_product2(b, c), set_union2(cartesian_product2(a, set_union2(c, d)), cartesian_product2(b, d)))
% 0.19/0.40 = { by axiom 2 (t4_xboole_1) R->L }
% 0.19/0.40 set_union2(set_union2(cartesian_product2(b, c), cartesian_product2(a, set_union2(c, d))), cartesian_product2(b, d))
% 0.19/0.40 = { by axiom 1 (commutativity_k2_xboole_0) }
% 0.19/0.40 set_union2(cartesian_product2(b, d), set_union2(cartesian_product2(b, c), cartesian_product2(a, set_union2(c, d))))
% 0.19/0.40 = { by axiom 3 (t120_zfmisc_1_1) }
% 0.19/0.40 set_union2(cartesian_product2(b, d), set_union2(cartesian_product2(b, c), set_union2(cartesian_product2(a, c), cartesian_product2(a, d))))
% 0.19/0.40 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.19/0.40 set_union2(cartesian_product2(b, d), set_union2(set_union2(cartesian_product2(a, c), cartesian_product2(a, d)), cartesian_product2(b, c)))
% 0.19/0.40 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.19/0.40 set_union2(set_union2(set_union2(cartesian_product2(a, c), cartesian_product2(a, d)), cartesian_product2(b, c)), cartesian_product2(b, d))
% 0.19/0.40 % SZS output end Proof
% 0.19/0.40
% 0.19/0.40 RESULT: Theorem (the conjecture is true).
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