TSTP Solution File: SET643+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET643+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 : n031.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 0.20s 0.49s
% 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 : SET643+3 : TPTP v8.1.2. Released v2.2.0.
% 0.13/0.13 % 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 : n031.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 : Sat Aug 26 11:02:23 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.49 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.49
% 0.20/0.49 % SZS status Theorem
% 0.20/0.49
% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(p1, axiom, ![B]: (ilf_type(B, set_type) => ![C]: (ilf_type(C, set_type) => ![D]: (ilf_type(D, set_type) => (subset(B, cross_product(C, D)) => ilf_type(B, relation_type(C, D))))))).
% 0.20/0.49 fof(p10, axiom, ![B2]: (ilf_type(B2, set_type) => subset(B2, B2))).
% 0.20/0.49 fof(p19, axiom, ![B2]: ilf_type(B2, set_type)).
% 0.20/0.49 fof(prove_relset_1_5, conjecture, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ilf_type(cross_product(B2, C2), relation_type(B2, C2))))).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (p19): ilf_type(X, set_type) = true2.
% 0.20/0.49 Axiom 2 (p10): fresh18(X, X, Y) = true2.
% 0.20/0.49 Axiom 3 (p1): fresh90(X, X, Y, Z, W) = true2.
% 0.20/0.49 Axiom 4 (p1): fresh88(X, X, Y, Z, W) = ilf_type(Y, relation_type(Z, W)).
% 0.20/0.49 Axiom 5 (p10): fresh18(ilf_type(X, set_type), true2, X) = subset(X, X).
% 0.20/0.49 Axiom 6 (p1): fresh89(X, X, Y, Z, W) = fresh90(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.20/0.49 Axiom 7 (p1): fresh87(X, X, Y, Z, W) = fresh88(ilf_type(Z, set_type), true2, Y, Z, W).
% 0.20/0.49 Axiom 8 (p1): fresh87(subset(X, cross_product(Y, Z)), true2, X, Y, Z) = fresh89(ilf_type(Z, set_type), true2, X, Y, Z).
% 0.20/0.50
% 0.20/0.50 Goal 1 (prove_relset_1_5_2): ilf_type(cross_product(b, c), relation_type(b, c)) = true2.
% 0.20/0.50 Proof:
% 0.20/0.50 ilf_type(cross_product(b, c), relation_type(b, c))
% 0.20/0.50 = { by axiom 4 (p1) R->L }
% 0.20/0.50 fresh88(true2, true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 1 (p19) R->L }
% 0.20/0.50 fresh88(ilf_type(b, set_type), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 7 (p1) R->L }
% 0.20/0.50 fresh87(true2, true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 2 (p10) R->L }
% 0.20/0.50 fresh87(fresh18(true2, true2, cross_product(b, c)), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 1 (p19) R->L }
% 0.20/0.50 fresh87(fresh18(ilf_type(cross_product(b, c), set_type), true2, cross_product(b, c)), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 5 (p10) }
% 0.20/0.50 fresh87(subset(cross_product(b, c), cross_product(b, c)), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 8 (p1) }
% 0.20/0.50 fresh89(ilf_type(c, set_type), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 1 (p19) }
% 0.20/0.50 fresh89(true2, true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 6 (p1) }
% 0.20/0.50 fresh90(ilf_type(cross_product(b, c), set_type), true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 1 (p19) }
% 0.20/0.50 fresh90(true2, true2, cross_product(b, c), b, c)
% 0.20/0.50 = { by axiom 3 (p1) }
% 0.20/0.50 true2
% 0.20/0.50 % SZS output end Proof
% 0.20/0.50
% 0.20/0.50 RESULT: Theorem (the conjecture is true).
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