TSTP Solution File: SET635+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET635+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 : n024.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 15:32:48 EDT 2023
% Result : Theorem 0.11s 0.38s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : SET635+3 : TPTP v8.1.2. Released v2.2.0.
% 0.10/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n024.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Sat Aug 26 12:25:53 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.11/0.38 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.11/0.38
% 0.11/0.38 % SZS status Theorem
% 0.11/0.38
% 0.11/0.39 % SZS output start Proof
% 0.11/0.39 Take the following subset of the input axioms:
% 0.11/0.39 fof(commutativity_of_intersection, axiom, ![B, C]: intersection(B, C)=intersection(C, B)).
% 0.11/0.39 fof(difference_and_intersection, axiom, ![D, B2, C2]: intersection(B2, difference(C2, D))=difference(intersection(B2, C2), D)).
% 0.11/0.39 fof(difference_and_intersection_and_union, axiom, ![B2, C2, D2]: difference(B2, intersection(C2, D2))=union(difference(B2, C2), difference(B2, D2))).
% 0.11/0.39 fof(difference_empty_set, axiom, ![B2, C2]: (difference(B2, C2)=empty_set <=> subset(B2, C2))).
% 0.11/0.39 fof(intersection_is_subset, axiom, ![B2, C2]: subset(intersection(B2, C2), B2)).
% 0.11/0.39 fof(prove_th117, conjecture, ![B2, C2, D2]: intersection(B2, difference(C2, D2))=difference(intersection(B2, C2), intersection(B2, D2))).
% 0.11/0.39 fof(union_empty_set, axiom, ![B2]: union(B2, empty_set)=B2).
% 0.11/0.39
% 0.11/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.11/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.11/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.11/0.39 fresh(y, y, x1...xn) = u
% 0.11/0.39 C => fresh(s, t, x1...xn) = v
% 0.11/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.11/0.39 variables of u and v.
% 0.11/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.11/0.39 input problem has no model of domain size 1).
% 0.11/0.39
% 0.11/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.11/0.39
% 0.11/0.39 Axiom 1 (union_empty_set): union(X, empty_set) = X.
% 0.11/0.39 Axiom 2 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.17/0.39 Axiom 3 (difference_empty_set_1): fresh12(X, X, Y, Z) = empty_set.
% 0.17/0.39 Axiom 4 (intersection_is_subset): subset(intersection(X, Y), X) = true2.
% 0.17/0.39 Axiom 5 (difference_and_intersection): intersection(X, difference(Y, Z)) = difference(intersection(X, Y), Z).
% 0.17/0.39 Axiom 6 (difference_empty_set_1): fresh12(subset(X, Y), true2, X, Y) = difference(X, Y).
% 0.17/0.39 Axiom 7 (difference_and_intersection_and_union): difference(X, intersection(Y, Z)) = union(difference(X, Y), difference(X, Z)).
% 0.17/0.39
% 0.17/0.39 Goal 1 (prove_th117): intersection(b, difference(c, d)) = difference(intersection(b, c), intersection(b, d)).
% 0.17/0.39 Proof:
% 0.17/0.39 intersection(b, difference(c, d))
% 0.17/0.39 = { by axiom 5 (difference_and_intersection) }
% 0.17/0.39 difference(intersection(b, c), d)
% 0.17/0.39 = { by axiom 2 (commutativity_of_intersection) }
% 0.17/0.39 difference(intersection(c, b), d)
% 0.17/0.39 = { by axiom 1 (union_empty_set) R->L }
% 0.17/0.39 union(difference(intersection(c, b), d), empty_set)
% 0.17/0.39 = { by axiom 3 (difference_empty_set_1) R->L }
% 0.17/0.39 union(difference(intersection(c, b), d), fresh12(true2, true2, intersection(b, c), b))
% 0.17/0.39 = { by axiom 4 (intersection_is_subset) R->L }
% 0.17/0.39 union(difference(intersection(c, b), d), fresh12(subset(intersection(b, c), b), true2, intersection(b, c), b))
% 0.17/0.39 = { by axiom 6 (difference_empty_set_1) }
% 0.17/0.39 union(difference(intersection(c, b), d), difference(intersection(b, c), b))
% 0.17/0.39 = { by axiom 2 (commutativity_of_intersection) }
% 0.17/0.39 union(difference(intersection(c, b), d), difference(intersection(c, b), b))
% 0.17/0.39 = { by axiom 7 (difference_and_intersection_and_union) R->L }
% 0.17/0.39 difference(intersection(c, b), intersection(d, b))
% 0.17/0.39 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.17/0.39 difference(intersection(c, b), intersection(b, d))
% 0.17/0.39 = { by axiom 2 (commutativity_of_intersection) R->L }
% 0.17/0.39 difference(intersection(b, c), intersection(b, d))
% 0.17/0.39 % SZS output end Proof
% 0.17/0.39
% 0.17/0.39 RESULT: Theorem (the conjecture is true).
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