TSTP Solution File: SET619+3 by Twee---2.4.2
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- Process Solution
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% File : Twee---2.4.2
% Problem : SET619+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 15:32:43 EDT 2023
% Result : Theorem 0.21s 0.45s
% Output : Proof 0.21s
% 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 : SET619+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
% 0.13/0.35 % Computer : n028.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 14:08:22 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.45 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.45
% 0.21/0.45 % SZS status Theorem
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% 0.21/0.45 % SZS output start Proof
% 0.21/0.45 Take the following subset of the input axioms:
% 0.21/0.45 fof(associativity_of_union, axiom, ![B, C, D]: union(union(B, C), D)=union(B, union(C, D))).
% 0.21/0.45 fof(commutativity_of_intersection, axiom, ![B2, C2]: intersection(B2, C2)=intersection(C2, B2)).
% 0.21/0.45 fof(commutativity_of_symmetric_difference, axiom, ![B2, C2]: symmetric_difference(B2, C2)=symmetric_difference(C2, B2)).
% 0.21/0.45 fof(commutativity_of_union, axiom, ![B2, C2]: union(B2, C2)=union(C2, B2)).
% 0.21/0.45 fof(prove_th95, conjecture, ![B2, C2]: union(B2, C2)=union(symmetric_difference(B2, C2), intersection(B2, C2))).
% 0.21/0.45 fof(symmetric_difference_defn, axiom, ![B2, C2]: symmetric_difference(B2, C2)=union(difference(B2, C2), difference(C2, B2))).
% 0.21/0.45 fof(union_intersection, axiom, ![B2, C2]: union(B2, intersection(B2, C2))=B2).
% 0.21/0.45 fof(union_intersection_difference, axiom, ![B2, C2]: union(intersection(B2, C2), difference(B2, C2))=B2).
% 0.21/0.45
% 0.21/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.45 fresh(y, y, x1...xn) = u
% 0.21/0.45 C => fresh(s, t, x1...xn) = v
% 0.21/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.45 variables of u and v.
% 0.21/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.45 input problem has no model of domain size 1).
% 0.21/0.45
% 0.21/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.45
% 0.21/0.45 Axiom 1 (commutativity_of_union): union(X, Y) = union(Y, X).
% 0.21/0.45 Axiom 2 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.21/0.45 Axiom 3 (commutativity_of_symmetric_difference): symmetric_difference(X, Y) = symmetric_difference(Y, X).
% 0.21/0.45 Axiom 4 (union_intersection): union(X, intersection(X, Y)) = X.
% 0.21/0.45 Axiom 5 (associativity_of_union): union(union(X, Y), Z) = union(X, union(Y, Z)).
% 0.21/0.45 Axiom 6 (union_intersection_difference): union(intersection(X, Y), difference(X, Y)) = X.
% 0.21/0.45 Axiom 7 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.21/0.45
% 0.21/0.45 Goal 1 (prove_th95): union(b, c) = union(symmetric_difference(b, c), intersection(b, c)).
% 0.21/0.45 Proof:
% 0.21/0.45 union(b, c)
% 0.21/0.45 = { by axiom 6 (union_intersection_difference) R->L }
% 0.21/0.45 union(b, union(intersection(c, b), difference(c, b)))
% 0.21/0.45 = { by axiom 2 (commutativity_of_intersection) }
% 0.21/0.45 union(b, union(intersection(b, c), difference(c, b)))
% 0.21/0.45 = { by axiom 5 (associativity_of_union) R->L }
% 0.21/0.45 union(union(b, intersection(b, c)), difference(c, b))
% 0.21/0.45 = { by axiom 4 (union_intersection) }
% 0.21/0.45 union(b, difference(c, b))
% 0.21/0.45 = { by axiom 1 (commutativity_of_union) R->L }
% 0.21/0.45 union(difference(c, b), b)
% 0.21/0.45 = { by axiom 6 (union_intersection_difference) R->L }
% 0.21/0.45 union(difference(c, b), union(intersection(b, c), difference(b, c)))
% 0.21/0.45 = { by axiom 1 (commutativity_of_union) R->L }
% 0.21/0.45 union(difference(c, b), union(difference(b, c), intersection(b, c)))
% 0.21/0.45 = { by axiom 5 (associativity_of_union) R->L }
% 0.21/0.45 union(union(difference(c, b), difference(b, c)), intersection(b, c))
% 0.21/0.45 = { by axiom 7 (symmetric_difference_defn) R->L }
% 0.21/0.45 union(symmetric_difference(c, b), intersection(b, c))
% 0.21/0.45 = { by axiom 1 (commutativity_of_union) }
% 0.21/0.45 union(intersection(b, c), symmetric_difference(c, b))
% 0.21/0.46 = { by axiom 3 (commutativity_of_symmetric_difference) }
% 0.21/0.46 union(intersection(b, c), symmetric_difference(b, c))
% 0.21/0.46 = { by axiom 1 (commutativity_of_union) }
% 0.21/0.46 union(symmetric_difference(b, c), intersection(b, c))
% 0.21/0.46 % SZS output end Proof
% 0.21/0.46
% 0.21/0.46 RESULT: Theorem (the conjecture is true).
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