TSTP Solution File: SET601+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET601+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 : n028.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 15:32:36 EDT 2023
% Result : Theorem 0.20s 0.72s
% 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 : SET601+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n028.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sat Aug 26 14:27:07 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.72 Command-line arguments: --no-flatten-goal
% 0.20/0.72
% 0.20/0.72 % SZS status Theorem
% 0.20/0.72
% 0.20/0.73 % SZS output start Proof
% 0.20/0.73 Take the following subset of the input axioms:
% 0.20/0.73 fof(associativity_of_intersection, axiom, ![B, C, D]: intersection(intersection(B, C), D)=intersection(B, intersection(C, D))).
% 0.20/0.73 fof(associativity_of_union, axiom, ![B2, C2, D2]: union(union(B2, C2), D2)=union(B2, union(C2, D2))).
% 0.20/0.73 fof(commutativity_of_intersection, axiom, ![B2, C2]: intersection(B2, C2)=intersection(C2, B2)).
% 0.20/0.73 fof(commutativity_of_union, axiom, ![B2, C2]: union(B2, C2)=union(C2, B2)).
% 0.20/0.73 fof(prove_th72, conjecture, ![B2, C2, D2]: union(union(intersection(B2, C2), intersection(C2, D2)), intersection(D2, B2))=intersection(intersection(union(B2, C2), union(C2, D2)), union(D2, B2))).
% 0.20/0.73 fof(union_distributes_over_intersection, axiom, ![B2, C2, D2]: union(B2, intersection(C2, D2))=intersection(union(B2, C2), union(B2, D2))).
% 0.20/0.73 fof(union_intersection, axiom, ![B2, C2]: union(B2, intersection(B2, C2))=B2).
% 0.20/0.73
% 0.20/0.73 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.73 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.73 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.73 fresh(y, y, x1...xn) = u
% 0.20/0.73 C => fresh(s, t, x1...xn) = v
% 0.20/0.73 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.73 variables of u and v.
% 0.20/0.73 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.73 input problem has no model of domain size 1).
% 0.20/0.73
% 0.20/0.73 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.73
% 0.20/0.73 Axiom 1 (commutativity_of_union): union(X, Y) = union(Y, X).
% 0.20/0.73 Axiom 2 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.20/0.73 Axiom 3 (union_intersection): union(X, intersection(X, Y)) = X.
% 0.20/0.73 Axiom 4 (associativity_of_union): union(union(X, Y), Z) = union(X, union(Y, Z)).
% 0.20/0.73 Axiom 5 (associativity_of_intersection): intersection(intersection(X, Y), Z) = intersection(X, intersection(Y, Z)).
% 0.20/0.73 Axiom 6 (union_distributes_over_intersection): union(X, intersection(Y, Z)) = intersection(union(X, Y), union(X, Z)).
% 0.20/0.73
% 0.20/0.73 Lemma 7: union(X, union(Y, Z)) = union(Y, union(X, Z)).
% 0.20/0.73 Proof:
% 0.20/0.73 union(X, union(Y, Z))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 union(union(Y, Z), X)
% 0.20/0.73 = { by axiom 4 (associativity_of_union) }
% 0.20/0.73 union(Y, union(Z, X))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) }
% 0.20/0.73 union(Y, union(X, Z))
% 0.20/0.73
% 0.20/0.73 Lemma 8: intersection(union(X, Y), union(Y, Z)) = union(Y, intersection(X, Z)).
% 0.20/0.73 Proof:
% 0.20/0.73 intersection(union(X, Y), union(Y, Z))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 intersection(union(Y, X), union(Y, Z))
% 0.20/0.73 = { by axiom 6 (union_distributes_over_intersection) R->L }
% 0.20/0.73 union(Y, intersection(X, Z))
% 0.20/0.73
% 0.20/0.73 Lemma 9: intersection(X, union(Y, intersection(X, Z))) = union(intersection(X, Z), intersection(X, Y)).
% 0.20/0.73 Proof:
% 0.20/0.73 intersection(X, union(Y, intersection(X, Z)))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 intersection(X, union(intersection(X, Z), Y))
% 0.20/0.73 = { by axiom 3 (union_intersection) R->L }
% 0.20/0.73 intersection(union(X, intersection(X, Z)), union(intersection(X, Z), Y))
% 0.20/0.73 = { by lemma 8 }
% 0.20/0.73 union(intersection(X, Z), intersection(X, Y))
% 0.20/0.73
% 0.20/0.73 Lemma 10: intersection(union(X, Y), union(Z, Y)) = union(Y, intersection(X, Z)).
% 0.20/0.73 Proof:
% 0.20/0.73 intersection(union(X, Y), union(Z, Y))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 intersection(union(X, Y), union(Y, Z))
% 0.20/0.73 = { by lemma 8 }
% 0.20/0.73 union(Y, intersection(X, Z))
% 0.20/0.73
% 0.20/0.73 Goal 1 (prove_th72): union(union(intersection(b, c), intersection(c, d)), intersection(d, b)) = intersection(intersection(union(b, c), union(c, d)), union(d, b)).
% 0.20/0.73 Proof:
% 0.20/0.73 union(union(intersection(b, c), intersection(c, d)), intersection(d, b))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) }
% 0.20/0.73 union(intersection(d, b), union(intersection(b, c), intersection(c, d)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) }
% 0.20/0.73 union(intersection(b, d), union(intersection(b, c), intersection(c, d)))
% 0.20/0.73 = { by lemma 7 }
% 0.20/0.73 union(intersection(b, c), union(intersection(b, d), intersection(c, d)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), union(intersection(d, b), intersection(c, d)))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 union(intersection(b, c), union(intersection(c, d), intersection(d, b)))
% 0.20/0.73 = { by lemma 10 R->L }
% 0.20/0.73 union(intersection(b, c), intersection(union(d, intersection(c, d)), union(b, intersection(c, d))))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(union(d, intersection(d, c)), union(b, intersection(c, d))))
% 0.20/0.73 = { by axiom 3 (union_intersection) }
% 0.20/0.73 union(intersection(b, c), intersection(d, union(b, intersection(c, d))))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) }
% 0.20/0.73 union(intersection(b, c), intersection(d, union(b, intersection(d, c))))
% 0.20/0.73 = { by lemma 9 }
% 0.20/0.73 union(intersection(b, c), union(intersection(d, c), intersection(d, b)))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 union(intersection(b, c), union(intersection(d, b), intersection(d, c)))
% 0.20/0.73 = { by lemma 9 R->L }
% 0.20/0.73 union(intersection(b, c), intersection(d, union(c, intersection(d, b))))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(d, union(c, intersection(b, d))))
% 0.20/0.73 = { by axiom 6 (union_distributes_over_intersection) }
% 0.20/0.73 union(intersection(b, c), intersection(d, intersection(union(c, b), union(c, d))))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) }
% 0.20/0.73 union(intersection(b, c), intersection(d, intersection(union(c, b), union(d, c))))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(d, intersection(union(d, c), union(c, b))))
% 0.20/0.73 = { by axiom 5 (associativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(intersection(d, union(d, c)), union(c, b)))
% 0.20/0.73 = { by axiom 3 (union_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(intersection(union(d, intersection(d, X)), union(d, c)), union(c, b)))
% 0.20/0.73 = { by axiom 6 (union_distributes_over_intersection) R->L }
% 0.20/0.73 union(intersection(b, c), intersection(union(d, intersection(intersection(d, X), c)), union(c, b)))
% 0.20/0.73 = { by axiom 5 (associativity_of_intersection) }
% 0.20/0.73 union(intersection(b, c), intersection(union(d, intersection(d, intersection(X, c))), union(c, b)))
% 0.20/0.73 = { by axiom 3 (union_intersection) }
% 0.20/0.73 union(intersection(b, c), intersection(d, union(c, b)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(c, b), intersection(d, union(c, b)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 union(intersection(c, b), intersection(union(c, b), d))
% 0.20/0.73 = { by axiom 6 (union_distributes_over_intersection) }
% 0.20/0.73 intersection(union(intersection(c, b), union(c, b)), union(intersection(c, b), d))
% 0.20/0.73 = { by lemma 7 }
% 0.20/0.73 intersection(union(c, union(intersection(c, b), b)), union(intersection(c, b), d))
% 0.20/0.73 = { by axiom 4 (associativity_of_union) R->L }
% 0.20/0.73 intersection(union(union(c, intersection(c, b)), b), union(intersection(c, b), d))
% 0.20/0.73 = { by axiom 3 (union_intersection) }
% 0.20/0.73 intersection(union(c, b), union(intersection(c, b), d))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) }
% 0.20/0.73 intersection(union(b, c), union(intersection(c, b), d))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) }
% 0.20/0.73 intersection(union(b, c), union(d, intersection(c, b)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) }
% 0.20/0.73 intersection(union(b, c), union(d, intersection(b, c)))
% 0.20/0.73 = { by lemma 10 R->L }
% 0.20/0.73 intersection(union(b, c), intersection(union(b, d), union(c, d)))
% 0.20/0.73 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.20/0.73 intersection(union(b, c), intersection(union(c, d), union(b, d)))
% 0.20/0.73 = { by axiom 5 (associativity_of_intersection) R->L }
% 0.20/0.73 intersection(intersection(union(b, c), union(c, d)), union(b, d))
% 0.20/0.73 = { by axiom 1 (commutativity_of_union) R->L }
% 0.20/0.73 intersection(intersection(union(b, c), union(c, d)), union(d, b))
% 0.20/0.73 % SZS output end Proof
% 0.20/0.73
% 0.20/0.73 RESULT: Theorem (the conjecture is true).
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