TSTP Solution File: SET642+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET642+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 : n022.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 15:32:50 EDT 2023
% Result : Theorem 0.22s 0.49s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SET642+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n022.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sat Aug 26 11:34:40 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.22/0.49 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.22/0.49
% 0.22/0.49 % SZS status Theorem
% 0.22/0.49
% 0.22/0.49 % SZS output start Proof
% 0.22/0.49 Take the following subset of the input axioms:
% 0.22/0.49 fof(p1, axiom, ![B]: (ilf_type(B, set_type) => ![C]: (ilf_type(C, set_type) => ![D]: (ilf_type(D, set_type) => ![E]: (ilf_type(E, relation_type(C, D)) => (subset(B, E) => subset(B, cross_product(C, D)))))))).
% 0.22/0.49 fof(p19, axiom, ![B2]: ilf_type(B2, set_type)).
% 0.22/0.49 fof(p2, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, set_type) => (subset(B2, cross_product(C2, D2)) => ilf_type(B2, relation_type(C2, D2))))))).
% 0.22/0.49 fof(prove_relset_1_4, conjecture, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, set_type) => ![E2]: (ilf_type(E2, relation_type(C2, D2)) => (subset(B2, E2) => ilf_type(B2, relation_type(C2, D2)))))))).
% 0.22/0.49
% 0.22/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.49 fresh(y, y, x1...xn) = u
% 0.22/0.49 C => fresh(s, t, x1...xn) = v
% 0.22/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.49 variables of u and v.
% 0.22/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.49 input problem has no model of domain size 1).
% 0.22/0.49
% 0.22/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.49
% 0.22/0.49 Axiom 1 (p19): ilf_type(X, set_type) = true2.
% 0.22/0.49 Axiom 2 (prove_relset_1_4_4): subset(b, e) = true2.
% 0.22/0.49 Axiom 3 (prove_relset_1_4_3): ilf_type(e, relation_type(c, d)) = true2.
% 0.22/0.49 Axiom 4 (p1): fresh67(X, X, Y, Z, W) = true2.
% 0.22/0.49 Axiom 5 (p1): fresh65(X, X, Y, Z, W) = subset(Y, cross_product(Z, W)).
% 0.22/0.49 Axiom 6 (p2): fresh62(X, X, Y, Z, W) = true2.
% 0.22/0.49 Axiom 7 (p2): fresh60(X, X, Y, Z, W) = ilf_type(Y, relation_type(Z, W)).
% 0.22/0.49 Axiom 8 (p1): fresh66(X, X, Y, Z, W) = fresh67(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.22/0.49 Axiom 9 (p1): fresh63(X, X, Y, Z, W, V) = fresh66(ilf_type(W, set_type), true2, Y, Z, W).
% 0.22/0.49 Axiom 10 (p1): fresh64(X, X, Y, Z, W) = fresh65(ilf_type(Z, set_type), true2, Y, Z, W).
% 0.22/0.49 Axiom 11 (p2): fresh61(X, X, Y, Z, W) = fresh62(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.22/0.49 Axiom 12 (p2): fresh59(X, X, Y, Z, W) = fresh60(ilf_type(Z, set_type), true2, Y, Z, W).
% 0.22/0.49 Axiom 13 (p1): fresh63(subset(X, Y), true2, X, Z, W, Y) = fresh64(ilf_type(Y, relation_type(Z, W)), true2, X, Z, W).
% 0.22/0.49 Axiom 14 (p2): fresh59(subset(X, cross_product(Y, Z)), true2, X, Y, Z) = fresh61(ilf_type(Z, set_type), true2, X, Y, Z).
% 0.22/0.49
% 0.22/0.49 Goal 1 (prove_relset_1_4_5): ilf_type(b, relation_type(c, d)) = true2.
% 0.22/0.49 Proof:
% 0.22/0.49 ilf_type(b, relation_type(c, d))
% 0.22/0.49 = { by axiom 7 (p2) R->L }
% 0.22/0.49 fresh60(true2, true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) R->L }
% 0.22/0.49 fresh60(ilf_type(c, set_type), true2, b, c, d)
% 0.22/0.49 = { by axiom 12 (p2) R->L }
% 0.22/0.49 fresh59(true2, true2, b, c, d)
% 0.22/0.49 = { by axiom 4 (p1) R->L }
% 0.22/0.49 fresh59(fresh67(true2, true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) R->L }
% 0.22/0.49 fresh59(fresh67(ilf_type(b, set_type), true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 8 (p1) R->L }
% 0.22/0.49 fresh59(fresh66(true2, true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) R->L }
% 0.22/0.49 fresh59(fresh66(ilf_type(d, set_type), true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 9 (p1) R->L }
% 0.22/0.49 fresh59(fresh63(true2, true2, b, c, d, e), true2, b, c, d)
% 0.22/0.49 = { by axiom 2 (prove_relset_1_4_4) R->L }
% 0.22/0.49 fresh59(fresh63(subset(b, e), true2, b, c, d, e), true2, b, c, d)
% 0.22/0.49 = { by axiom 13 (p1) }
% 0.22/0.49 fresh59(fresh64(ilf_type(e, relation_type(c, d)), true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 3 (prove_relset_1_4_3) }
% 0.22/0.49 fresh59(fresh64(true2, true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 10 (p1) }
% 0.22/0.49 fresh59(fresh65(ilf_type(c, set_type), true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) }
% 0.22/0.49 fresh59(fresh65(true2, true2, b, c, d), true2, b, c, d)
% 0.22/0.49 = { by axiom 5 (p1) }
% 0.22/0.49 fresh59(subset(b, cross_product(c, d)), true2, b, c, d)
% 0.22/0.49 = { by axiom 14 (p2) }
% 0.22/0.49 fresh61(ilf_type(d, set_type), true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) }
% 0.22/0.49 fresh61(true2, true2, b, c, d)
% 0.22/0.49 = { by axiom 11 (p2) }
% 0.22/0.49 fresh62(ilf_type(b, set_type), true2, b, c, d)
% 0.22/0.49 = { by axiom 1 (p19) }
% 0.22/0.49 fresh62(true2, true2, b, c, d)
% 0.22/0.49 = { by axiom 6 (p2) }
% 0.22/0.49 true2
% 0.22/0.49 % SZS output end Proof
% 0.22/0.49
% 0.22/0.49 RESULT: Theorem (the conjecture is true).
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