TSTP Solution File: SEU167+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SEU167+3 : 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 : n027.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 17:51:21 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 : SEU167+3 : 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
% 0.13/0.34 % Computer : n027.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Aug 23 15:35:16 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.39 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.39
% 0.20/0.39 % SZS status Theorem
% 0.20/0.39
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Take the following subset of the input axioms:
% 0.20/0.40 fof(t118_zfmisc_1, axiom, ![B, C, A2]: (subset(A2, B) => (subset(cartesian_product2(A2, C), cartesian_product2(B, C)) & subset(cartesian_product2(C, A2), cartesian_product2(C, B))))).
% 0.20/0.40 fof(t119_zfmisc_1, conjecture, ![A, D, B2, C2]: ((subset(A, B2) & subset(C2, D)) => subset(cartesian_product2(A, C2), cartesian_product2(B2, D)))).
% 0.20/0.40 fof(t1_xboole_1, axiom, ![B2, C2, A2_2]: ((subset(A2_2, B2) & subset(B2, C2)) => subset(A2_2, C2))).
% 0.20/0.40
% 0.20/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40 fresh(y, y, x1...xn) = u
% 0.20/0.40 C => fresh(s, t, x1...xn) = v
% 0.20/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40 variables of u and v.
% 0.20/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40 input problem has no model of domain size 1).
% 0.20/0.40
% 0.20/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40
% 0.20/0.40 Axiom 1 (t119_zfmisc_1): subset(c, d) = true.
% 0.20/0.40 Axiom 2 (t119_zfmisc_1_1): subset(a, b) = true.
% 0.20/0.40 Axiom 3 (t1_xboole_1): fresh(X, X, Y, Z) = true.
% 0.20/0.40 Axiom 4 (t118_zfmisc_1_1): fresh4(X, X, Y, Z, W) = true.
% 0.20/0.40 Axiom 5 (t118_zfmisc_1): fresh3(X, X, Y, Z, W) = true.
% 0.20/0.40 Axiom 6 (t1_xboole_1): fresh2(X, X, Y, Z, W) = subset(Y, W).
% 0.20/0.40 Axiom 7 (t118_zfmisc_1_1): fresh4(subset(X, Y), true, X, Y, Z) = subset(cartesian_product2(Z, X), cartesian_product2(Z, Y)).
% 0.20/0.40 Axiom 8 (t118_zfmisc_1): fresh3(subset(X, Y), true, X, Y, Z) = subset(cartesian_product2(X, Z), cartesian_product2(Y, Z)).
% 0.20/0.40 Axiom 9 (t1_xboole_1): fresh2(subset(X, Y), true, Z, X, Y) = fresh(subset(Z, X), true, Z, Y).
% 0.20/0.40
% 0.20/0.40 Goal 1 (t119_zfmisc_1_2): subset(cartesian_product2(a, c), cartesian_product2(b, d)) = true.
% 0.20/0.40 Proof:
% 0.20/0.40 subset(cartesian_product2(a, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 6 (t1_xboole_1) R->L }
% 0.20/0.40 fresh2(true, true, cartesian_product2(a, c), cartesian_product2(b, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 4 (t118_zfmisc_1_1) R->L }
% 0.20/0.40 fresh2(fresh4(true, true, c, d, b), true, cartesian_product2(a, c), cartesian_product2(b, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 1 (t119_zfmisc_1) R->L }
% 0.20/0.40 fresh2(fresh4(subset(c, d), true, c, d, b), true, cartesian_product2(a, c), cartesian_product2(b, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 7 (t118_zfmisc_1_1) }
% 0.20/0.40 fresh2(subset(cartesian_product2(b, c), cartesian_product2(b, d)), true, cartesian_product2(a, c), cartesian_product2(b, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 9 (t1_xboole_1) }
% 0.20/0.40 fresh(subset(cartesian_product2(a, c), cartesian_product2(b, c)), true, cartesian_product2(a, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 8 (t118_zfmisc_1) R->L }
% 0.20/0.40 fresh(fresh3(subset(a, b), true, a, b, c), true, cartesian_product2(a, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 2 (t119_zfmisc_1_1) }
% 0.20/0.40 fresh(fresh3(true, true, a, b, c), true, cartesian_product2(a, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 5 (t118_zfmisc_1) }
% 0.20/0.40 fresh(true, true, cartesian_product2(a, c), cartesian_product2(b, d))
% 0.20/0.40 = { by axiom 3 (t1_xboole_1) }
% 0.20/0.40 true
% 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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