TSTP Solution File: SEU056+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SEU056+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 : n032.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 17:50:54 EDT 2023
% Result : Theorem 0.16s 0.41s
% Output : Proof 0.16s
% 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 : SEU056+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.32 % Computer : n032.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Wed Aug 23 18:32:26 EDT 2023
% 0.12/0.32 % CPUTime :
% 0.16/0.41 Command-line arguments: --ground-connectedness --complete-subsets
% 0.16/0.41
% 0.16/0.41 % SZS status Theorem
% 0.16/0.41
% 0.16/0.42 % SZS output start Proof
% 0.16/0.42 Take the following subset of the input axioms:
% 0.16/0.42 fof(commutativity_k3_xboole_0, axiom, ![A, B]: set_intersection2(A, B)=set_intersection2(B, A)).
% 0.16/0.42 fof(d7_xboole_0, axiom, ![A2, B2]: (disjoint(A2, B2) <=> set_intersection2(A2, B2)=empty_set)).
% 0.16/0.42 fof(rc1_xboole_0, axiom, ?[A3]: empty(A3)).
% 0.16/0.42 fof(t121_funct_1, axiom, ![C, A2_2, B2]: ((relation(C) & function(C)) => (one_to_one(C) => relation_image(C, set_intersection2(A2_2, B2))=set_intersection2(relation_image(C, A2_2), relation_image(C, B2))))).
% 0.16/0.42 fof(t125_funct_1, conjecture, ![A3, B2, C2]: ((relation(C2) & function(C2)) => ((disjoint(A3, B2) & one_to_one(C2)) => disjoint(relation_image(C2, A3), relation_image(C2, B2))))).
% 0.16/0.42 fof(t149_relat_1, axiom, ![A2_2]: (relation(A2_2) => relation_image(A2_2, empty_set)=empty_set)).
% 0.16/0.42 fof(t6_boole, axiom, ![A2_2]: (empty(A2_2) => A2_2=empty_set)).
% 0.16/0.42
% 0.16/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.16/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.16/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.16/0.42 fresh(y, y, x1...xn) = u
% 0.16/0.42 C => fresh(s, t, x1...xn) = v
% 0.16/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.16/0.42 variables of u and v.
% 0.16/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.16/0.42 input problem has no model of domain size 1).
% 0.16/0.42
% 0.16/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.16/0.42
% 0.16/0.42 Axiom 1 (t125_funct_1_2): one_to_one(c) = true2.
% 0.16/0.42 Axiom 2 (t125_funct_1): function(c) = true2.
% 0.16/0.42 Axiom 3 (t125_funct_1_1): relation(c) = true2.
% 0.16/0.42 Axiom 4 (rc1_xboole_0): empty(a7) = true2.
% 0.16/0.42 Axiom 5 (t125_funct_1_3): disjoint(a, b) = true2.
% 0.16/0.42 Axiom 6 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.16/0.42 Axiom 7 (t149_relat_1): fresh9(X, X, Y) = empty_set.
% 0.16/0.42 Axiom 8 (t6_boole): fresh3(X, X, Y) = empty_set.
% 0.16/0.42 Axiom 9 (d7_xboole_0): fresh16(X, X, Y, Z) = true2.
% 0.16/0.42 Axiom 10 (d7_xboole_0_1): fresh15(X, X, Y, Z) = empty_set.
% 0.16/0.42 Axiom 11 (t149_relat_1): fresh9(relation(X), true2, X) = relation_image(X, empty_set).
% 0.16/0.42 Axiom 12 (t6_boole): fresh3(empty(X), true2, X) = X.
% 0.16/0.42 Axiom 13 (t121_funct_1): fresh21(X, X, Y, Z, W) = relation_image(Z, set_intersection2(Y, W)).
% 0.16/0.42 Axiom 14 (t121_funct_1): fresh20(X, X, Y, Z, W) = fresh21(function(Z), true2, Y, Z, W).
% 0.16/0.42 Axiom 15 (d7_xboole_0): fresh16(set_intersection2(X, Y), empty_set, X, Y) = disjoint(X, Y).
% 0.16/0.42 Axiom 16 (d7_xboole_0_1): fresh15(disjoint(X, Y), true2, X, Y) = set_intersection2(X, Y).
% 0.16/0.42 Axiom 17 (t121_funct_1): fresh20(one_to_one(X), true2, Y, X, Z) = fresh10(relation(X), true2, Y, X, Z).
% 0.16/0.42 Axiom 18 (t121_funct_1): fresh10(X, X, Y, Z, W) = set_intersection2(relation_image(Z, Y), relation_image(Z, W)).
% 0.16/0.42
% 0.16/0.42 Lemma 19: empty_set = a7.
% 0.16/0.42 Proof:
% 0.16/0.42 empty_set
% 0.16/0.42 = { by axiom 8 (t6_boole) R->L }
% 0.16/0.42 fresh3(true2, true2, a7)
% 0.16/0.42 = { by axiom 4 (rc1_xboole_0) R->L }
% 0.16/0.42 fresh3(empty(a7), true2, a7)
% 0.16/0.42 = { by axiom 12 (t6_boole) }
% 0.16/0.42 a7
% 0.16/0.42
% 0.16/0.42 Goal 1 (t125_funct_1_4): disjoint(relation_image(c, a), relation_image(c, b)) = true2.
% 0.16/0.42 Proof:
% 0.16/0.42 disjoint(relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 15 (d7_xboole_0) R->L }
% 0.16/0.42 fresh16(set_intersection2(relation_image(c, a), relation_image(c, b)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 6 (commutativity_k3_xboole_0) R->L }
% 0.16/0.42 fresh16(set_intersection2(relation_image(c, b), relation_image(c, a)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 18 (t121_funct_1) R->L }
% 0.16/0.42 fresh16(fresh10(true2, true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 3 (t125_funct_1_1) R->L }
% 0.16/0.42 fresh16(fresh10(relation(c), true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 17 (t121_funct_1) R->L }
% 0.16/0.42 fresh16(fresh20(one_to_one(c), true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 1 (t125_funct_1_2) }
% 0.16/0.42 fresh16(fresh20(true2, true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 14 (t121_funct_1) }
% 0.16/0.42 fresh16(fresh21(function(c), true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 2 (t125_funct_1) }
% 0.16/0.42 fresh16(fresh21(true2, true2, b, c, a), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 13 (t121_funct_1) }
% 0.16/0.42 fresh16(relation_image(c, set_intersection2(b, a)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 6 (commutativity_k3_xboole_0) }
% 0.16/0.42 fresh16(relation_image(c, set_intersection2(a, b)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 16 (d7_xboole_0_1) R->L }
% 0.16/0.42 fresh16(relation_image(c, fresh15(disjoint(a, b), true2, a, b)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 5 (t125_funct_1_3) }
% 0.16/0.42 fresh16(relation_image(c, fresh15(true2, true2, a, b)), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 10 (d7_xboole_0_1) }
% 0.16/0.42 fresh16(relation_image(c, empty_set), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 11 (t149_relat_1) R->L }
% 0.16/0.42 fresh16(fresh9(relation(c), true2, c), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 3 (t125_funct_1_1) }
% 0.16/0.42 fresh16(fresh9(true2, true2, c), empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 7 (t149_relat_1) }
% 0.16/0.42 fresh16(empty_set, empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by lemma 19 }
% 0.16/0.42 fresh16(a7, empty_set, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by lemma 19 }
% 0.16/0.42 fresh16(a7, a7, relation_image(c, a), relation_image(c, b))
% 0.16/0.42 = { by axiom 9 (d7_xboole_0) }
% 0.16/0.42 true2
% 0.16/0.42 % SZS output end Proof
% 0.16/0.42
% 0.16/0.42 RESULT: Theorem (the conjecture is true).
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