TSTP Solution File: SET677+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET677+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 : n011.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 15:32:57 EDT 2023
% Result : Theorem 0.20s 0.54s
% 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
% 0.12/0.13 % Problem : SET677+3 : TPTP v8.1.2. Released v2.2.0.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n011.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 10:07:39 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.54 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.54
% 0.20/0.54 % SZS status Theorem
% 0.20/0.54
% 0.20/0.55 % SZS output start Proof
% 0.20/0.55 Take the following subset of the input axioms:
% 0.20/0.56 fof(p1, axiom, ![B]: (ilf_type(B, set_type) => ![C]: (ilf_type(C, set_type) => ![D]: (ilf_type(D, relation_type(C, B)) => (subset(identity_relation_of(C), D) => (C=domain(C, B, D) & subset(C, range(C, B, D)))))))).
% 0.20/0.56 fof(p2, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, relation_type(B2, C2)) => (subset(identity_relation_of(C2), D2) => (subset(C2, domain(B2, C2, D2)) & C2=range(B2, C2, D2))))))).
% 0.20/0.56 fof(p34, axiom, ![B2]: ilf_type(B2, set_type)).
% 0.20/0.56 fof(p5, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => (ilf_type(C2, identity_relation_of_type(B2)) <=> ilf_type(C2, relation_type(B2, B2)))))).
% 0.20/0.56 fof(prove_relset_1_44, conjecture, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, identity_relation_of_type(B2)) => (subset(identity_relation_of(B2), C2) => (B2=domain(B2, B2, C2) & B2=range(B2, B2, C2)))))).
% 0.20/0.56
% 0.20/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.56 fresh(y, y, x1...xn) = u
% 0.20/0.56 C => fresh(s, t, x1...xn) = v
% 0.20/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.56 variables of u and v.
% 0.20/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.56 input problem has no model of domain size 1).
% 0.20/0.56
% 0.20/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.56
% 0.20/0.56 Axiom 1 (p34): ilf_type(X, set_type) = true2.
% 0.20/0.56 Axiom 2 (prove_relset_1_44_1): ilf_type(c, identity_relation_of_type(b)) = true2.
% 0.20/0.56 Axiom 3 (prove_relset_1_44_2): subset(identity_relation_of(b), c) = true2.
% 0.20/0.56 Axiom 4 (p5_1): fresh100(X, X, Y, Z) = true2.
% 0.20/0.56 Axiom 5 (p5_1): fresh5(X, X, Y, Z) = ilf_type(Z, relation_type(Y, Y)).
% 0.20/0.56 Axiom 6 (p1): fresh122(X, X, Y, Z, W) = Z.
% 0.20/0.56 Axiom 7 (p1): fresh120(X, X, Y, Z, W) = domain(Z, Y, W).
% 0.20/0.56 Axiom 8 (p2): fresh114(X, X, Y, Z, W) = Z.
% 0.20/0.56 Axiom 9 (p2): fresh112(X, X, Y, Z, W) = range(Y, Z, W).
% 0.20/0.56 Axiom 10 (p5_1): fresh99(X, X, Y, Z) = fresh100(ilf_type(Y, set_type), true2, Y, Z).
% 0.20/0.56 Axiom 11 (p1): fresh121(X, X, Y, Z, W) = fresh122(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.20/0.56 Axiom 12 (p1): fresh119(X, X, Y, Z, W) = fresh120(ilf_type(Z, set_type), true2, Y, Z, W).
% 0.20/0.56 Axiom 13 (p2): fresh113(X, X, Y, Z, W) = fresh114(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.20/0.56 Axiom 14 (p2): fresh111(X, X, Y, Z, W) = fresh112(ilf_type(Z, set_type), true2, Y, Z, W).
% 0.20/0.56 Axiom 15 (p5_1): fresh99(ilf_type(X, identity_relation_of_type(Y)), true2, Y, X) = fresh5(ilf_type(X, set_type), true2, Y, X).
% 0.20/0.56 Axiom 16 (p1): fresh119(subset(identity_relation_of(X), Y), true2, Z, X, Y) = fresh121(ilf_type(Y, relation_type(X, Z)), true2, Z, X, Y).
% 0.20/0.56 Axiom 17 (p2): fresh111(subset(identity_relation_of(X), Y), true2, Z, X, Y) = fresh113(ilf_type(Y, relation_type(Z, X)), true2, Z, X, Y).
% 0.20/0.56
% 0.20/0.56 Lemma 18: ilf_type(c, relation_type(b, b)) = true2.
% 0.20/0.56 Proof:
% 0.20/0.56 ilf_type(c, relation_type(b, b))
% 0.20/0.56 = { by axiom 5 (p5_1) R->L }
% 0.20/0.56 fresh5(true2, true2, b, c)
% 0.20/0.56 = { by axiom 1 (p34) R->L }
% 0.20/0.56 fresh5(ilf_type(c, set_type), true2, b, c)
% 0.20/0.56 = { by axiom 15 (p5_1) R->L }
% 0.20/0.56 fresh99(ilf_type(c, identity_relation_of_type(b)), true2, b, c)
% 0.20/0.56 = { by axiom 2 (prove_relset_1_44_1) }
% 0.20/0.56 fresh99(true2, true2, b, c)
% 0.20/0.56 = { by axiom 10 (p5_1) }
% 0.20/0.56 fresh100(ilf_type(b, set_type), true2, b, c)
% 0.20/0.56 = { by axiom 1 (p34) }
% 0.20/0.56 fresh100(true2, true2, b, c)
% 0.20/0.56 = { by axiom 4 (p5_1) }
% 0.20/0.56 true2
% 0.20/0.56
% 0.20/0.56 Goal 1 (prove_relset_1_44_3): tuple3(b, b) = tuple3(domain(b, b, c), range(b, b, c)).
% 0.20/0.56 Proof:
% 0.20/0.56 tuple3(b, b)
% 0.20/0.56 = { by axiom 6 (p1) R->L }
% 0.20/0.56 tuple3(fresh122(true2, true2, b, b, c), b)
% 0.20/0.56 = { by axiom 1 (p34) R->L }
% 0.20/0.56 tuple3(fresh122(ilf_type(b, set_type), true2, b, b, c), b)
% 0.20/0.56 = { by axiom 11 (p1) R->L }
% 0.20/0.56 tuple3(fresh121(true2, true2, b, b, c), b)
% 0.20/0.56 = { by lemma 18 R->L }
% 0.20/0.56 tuple3(fresh121(ilf_type(c, relation_type(b, b)), true2, b, b, c), b)
% 0.20/0.56 = { by axiom 16 (p1) R->L }
% 0.20/0.56 tuple3(fresh119(subset(identity_relation_of(b), c), true2, b, b, c), b)
% 0.20/0.56 = { by axiom 3 (prove_relset_1_44_2) }
% 0.20/0.56 tuple3(fresh119(true2, true2, b, b, c), b)
% 0.20/0.56 = { by axiom 12 (p1) }
% 0.20/0.56 tuple3(fresh120(ilf_type(b, set_type), true2, b, b, c), b)
% 0.20/0.56 = { by axiom 1 (p34) }
% 0.20/0.56 tuple3(fresh120(true2, true2, b, b, c), b)
% 0.20/0.56 = { by axiom 7 (p1) }
% 0.20/0.56 tuple3(domain(b, b, c), b)
% 0.20/0.56 = { by axiom 8 (p2) R->L }
% 0.20/0.56 tuple3(domain(b, b, c), fresh114(true2, true2, b, b, c))
% 0.20/0.56 = { by axiom 1 (p34) R->L }
% 0.20/0.56 tuple3(domain(b, b, c), fresh114(ilf_type(b, set_type), true2, b, b, c))
% 0.20/0.56 = { by axiom 13 (p2) R->L }
% 0.20/0.56 tuple3(domain(b, b, c), fresh113(true2, true2, b, b, c))
% 0.20/0.56 = { by lemma 18 R->L }
% 0.20/0.56 tuple3(domain(b, b, c), fresh113(ilf_type(c, relation_type(b, b)), true2, b, b, c))
% 0.20/0.56 = { by axiom 17 (p2) R->L }
% 0.20/0.56 tuple3(domain(b, b, c), fresh111(subset(identity_relation_of(b), c), true2, b, b, c))
% 0.20/0.56 = { by axiom 3 (prove_relset_1_44_2) }
% 0.20/0.56 tuple3(domain(b, b, c), fresh111(true2, true2, b, b, c))
% 0.20/0.56 = { by axiom 14 (p2) }
% 0.20/0.56 tuple3(domain(b, b, c), fresh112(ilf_type(b, set_type), true2, b, b, c))
% 0.20/0.56 = { by axiom 1 (p34) }
% 0.20/0.56 tuple3(domain(b, b, c), fresh112(true2, true2, b, b, c))
% 0.20/0.56 = { by axiom 9 (p2) }
% 0.20/0.56 tuple3(domain(b, b, c), range(b, b, c))
% 0.20/0.56 % SZS output end Proof
% 0.20/0.56
% 0.20/0.56 RESULT: Theorem (the conjecture is true).
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