TSTP Solution File: SET622+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET622+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 : n023.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 0.12s 0.39s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET622+3 : TPTP v8.1.2. Released v2.2.0.
% 0.11/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n023.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sat Aug 26 10:11:26 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.12/0.39 Command-line arguments: --no-flatten-goal
% 0.12/0.39
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 Take the following subset of the input axioms:
% 0.12/0.40 fof(associativity_of_intersection, axiom, ![B, C, D]: intersection(intersection(B, C), D)=intersection(B, intersection(C, D))).
% 0.12/0.40 fof(commutativity_of_intersection, axiom, ![B2, C2]: intersection(B2, C2)=intersection(C2, B2)).
% 0.12/0.40 fof(difference_difference_union2, axiom, ![B2, C2, D2]: difference(B2, difference(C2, D2))=union(difference(B2, C2), intersection(B2, D2))).
% 0.12/0.40 fof(prove_th98, conjecture, ![B2, C2, D2]: difference(B2, symmetric_difference(C2, D2))=union(difference(B2, union(C2, D2)), intersection(intersection(B2, C2), D2))).
% 0.12/0.40 fof(symmetric_difference_and_difference, axiom, ![B2, C2]: symmetric_difference(B2, C2)=difference(union(B2, C2), intersection(B2, C2))).
% 0.12/0.40
% 0.12/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.12/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.12/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.12/0.40 fresh(y, y, x1...xn) = u
% 0.12/0.40 C => fresh(s, t, x1...xn) = v
% 0.12/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.12/0.40 variables of u and v.
% 0.12/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.12/0.40 input problem has no model of domain size 1).
% 0.12/0.40
% 0.12/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.12/0.40
% 0.12/0.40 Axiom 1 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.12/0.40 Axiom 2 (associativity_of_intersection): intersection(intersection(X, Y), Z) = intersection(X, intersection(Y, Z)).
% 0.12/0.40 Axiom 3 (difference_difference_union2): difference(X, difference(Y, Z)) = union(difference(X, Y), intersection(X, Z)).
% 0.12/0.40 Axiom 4 (symmetric_difference_and_difference): symmetric_difference(X, Y) = difference(union(X, Y), intersection(X, Y)).
% 0.12/0.40
% 0.12/0.40 Goal 1 (prove_th98): difference(b, symmetric_difference(c, d)) = union(difference(b, union(c, d)), intersection(intersection(b, c), d)).
% 0.12/0.40 Proof:
% 0.12/0.40 difference(b, symmetric_difference(c, d))
% 0.12/0.40 = { by axiom 4 (symmetric_difference_and_difference) }
% 0.12/0.40 difference(b, difference(union(c, d), intersection(c, d)))
% 0.12/0.40 = { by axiom 1 (commutativity_of_intersection) R->L }
% 0.12/0.40 difference(b, difference(union(c, d), intersection(d, c)))
% 0.12/0.40 = { by axiom 3 (difference_difference_union2) }
% 0.12/0.40 union(difference(b, union(c, d)), intersection(b, intersection(d, c)))
% 0.12/0.40 = { by axiom 1 (commutativity_of_intersection) R->L }
% 0.12/0.40 union(difference(b, union(c, d)), intersection(b, intersection(c, d)))
% 0.12/0.40 = { by axiom 2 (associativity_of_intersection) R->L }
% 0.12/0.40 union(difference(b, union(c, d)), intersection(intersection(b, c), d))
% 0.12/0.40 % SZS output end Proof
% 0.12/0.40
% 0.12/0.40 RESULT: Theorem (the conjecture is true).
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