TSTP Solution File: SET620+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET620+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 : n007.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 0.40s
% 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
% 0.06/0.12 % Problem : SET620+3 : TPTP v8.1.2. Released v2.2.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n007.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Sat Aug 26 15:26:26 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.21/0.40 Command-line arguments: --flatten
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% 0.21/0.40 % SZS status Theorem
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% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 Take the following subset of the input axioms:
% 0.21/0.41 fof(commutativity_of_intersection, axiom, ![B, C]: intersection(B, C)=intersection(C, B)).
% 0.21/0.41 fof(difference_distributes_over_union, axiom, ![D, B2, C2]: difference(union(B2, C2), D)=union(difference(B2, D), difference(C2, D))).
% 0.21/0.41 fof(difference_into_intersection, axiom, ![B2, C2]: difference(B2, intersection(B2, C2))=difference(B2, C2)).
% 0.21/0.41 fof(prove_th96, conjecture, ![B2, C2]: symmetric_difference(B2, C2)=difference(union(B2, C2), intersection(B2, C2))).
% 0.21/0.41 fof(symmetric_difference_defn, axiom, ![B2, C2]: symmetric_difference(B2, C2)=union(difference(B2, C2), difference(C2, B2))).
% 0.21/0.41
% 0.21/0.41 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.41 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.41 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.41 fresh(y, y, x1...xn) = u
% 0.21/0.41 C => fresh(s, t, x1...xn) = v
% 0.21/0.41 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.41 variables of u and v.
% 0.21/0.41 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.41 input problem has no model of domain size 1).
% 0.21/0.41
% 0.21/0.41 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.41
% 0.21/0.41 Axiom 1 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.21/0.41 Axiom 2 (difference_into_intersection): difference(X, intersection(X, Y)) = difference(X, Y).
% 0.21/0.41 Axiom 3 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.21/0.41 Axiom 4 (difference_distributes_over_union): difference(union(X, Y), Z) = union(difference(X, Z), difference(Y, Z)).
% 0.21/0.41
% 0.21/0.41 Goal 1 (prove_th96): symmetric_difference(b, c) = difference(union(b, c), intersection(b, c)).
% 0.21/0.41 Proof:
% 0.21/0.41 symmetric_difference(b, c)
% 0.21/0.41 = { by axiom 3 (symmetric_difference_defn) }
% 0.21/0.41 union(difference(b, c), difference(c, b))
% 0.21/0.41 = { by axiom 2 (difference_into_intersection) R->L }
% 0.21/0.41 union(difference(b, c), difference(c, intersection(c, b)))
% 0.21/0.41 = { by axiom 1 (commutativity_of_intersection) }
% 0.21/0.41 union(difference(b, c), difference(c, intersection(b, c)))
% 0.21/0.41 = { by axiom 2 (difference_into_intersection) R->L }
% 0.21/0.41 union(difference(b, intersection(b, c)), difference(c, intersection(b, c)))
% 0.21/0.41 = { by axiom 4 (difference_distributes_over_union) R->L }
% 0.21/0.41 difference(union(b, c), intersection(b, c))
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41
% 0.21/0.41 RESULT: Theorem (the conjecture is true).
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