TSTP Solution File: SET617+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET617+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 : n021.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 15:32:42 EDT 2023
% Result : Theorem 0.21s 0.42s
% 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
% 0.00/0.14 % Problem : SET617+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n021.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Sat Aug 26 11:24:41 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.21/0.42 Command-line arguments: --no-flatten-goal
% 0.21/0.42
% 0.21/0.42 % SZS status Theorem
% 0.21/0.42
% 0.21/0.42 % SZS output start Proof
% 0.21/0.42 Take the following subset of the input axioms:
% 0.21/0.42 fof(commutativity_of_symmetric_difference, axiom, ![B, C]: symmetric_difference(B, C)=symmetric_difference(C, B)).
% 0.21/0.42 fof(no_difference_with_empty_set1, axiom, ![B2]: difference(B2, empty_set)=B2).
% 0.21/0.42 fof(no_difference_with_empty_set2, axiom, ![B2]: difference(empty_set, B2)=empty_set).
% 0.21/0.42 fof(prove_th92, conjecture, ![B2]: (symmetric_difference(B2, empty_set)=B2 & symmetric_difference(empty_set, B2)=B2)).
% 0.21/0.42 fof(symmetric_difference_defn, axiom, ![B2, C2]: symmetric_difference(B2, C2)=union(difference(B2, C2), difference(C2, B2))).
% 0.21/0.42 fof(union_empty_set, axiom, ![B2]: union(B2, empty_set)=B2).
% 0.21/0.42
% 0.21/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.42 fresh(y, y, x1...xn) = u
% 0.21/0.42 C => fresh(s, t, x1...xn) = v
% 0.21/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.42 variables of u and v.
% 0.21/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.42 input problem has no model of domain size 1).
% 0.21/0.42
% 0.21/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.42
% 0.21/0.42 Axiom 1 (union_empty_set): union(X, empty_set) = X.
% 0.21/0.42 Axiom 2 (no_difference_with_empty_set1): difference(X, empty_set) = X.
% 0.21/0.42 Axiom 3 (no_difference_with_empty_set2): difference(empty_set, X) = empty_set.
% 0.21/0.42 Axiom 4 (commutativity_of_symmetric_difference): symmetric_difference(X, Y) = symmetric_difference(Y, X).
% 0.21/0.42 Axiom 5 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.21/0.42
% 0.21/0.42 Lemma 6: symmetric_difference(X, empty_set) = X.
% 0.21/0.42 Proof:
% 0.21/0.42 symmetric_difference(X, empty_set)
% 0.21/0.42 = { by axiom 5 (symmetric_difference_defn) }
% 0.21/0.42 union(difference(X, empty_set), difference(empty_set, X))
% 0.21/0.42 = { by axiom 2 (no_difference_with_empty_set1) }
% 0.21/0.42 union(X, difference(empty_set, X))
% 0.21/0.42 = { by axiom 3 (no_difference_with_empty_set2) }
% 0.21/0.42 union(X, empty_set)
% 0.21/0.42 = { by axiom 1 (union_empty_set) }
% 0.21/0.42 X
% 0.21/0.42
% 0.21/0.42 Goal 1 (prove_th92): tuple2(symmetric_difference(empty_set, b), symmetric_difference(b2, empty_set)) = tuple2(b, b2).
% 0.21/0.42 Proof:
% 0.21/0.42 tuple2(symmetric_difference(empty_set, b), symmetric_difference(b2, empty_set))
% 0.21/0.42 = { by axiom 4 (commutativity_of_symmetric_difference) }
% 0.21/0.42 tuple2(symmetric_difference(b, empty_set), symmetric_difference(b2, empty_set))
% 0.21/0.42 = { by lemma 6 }
% 0.21/0.42 tuple2(b, symmetric_difference(b2, empty_set))
% 0.21/0.42 = { by lemma 6 }
% 0.21/0.42 tuple2(b, b2)
% 0.21/0.42 % SZS output end Proof
% 0.21/0.42
% 0.21/0.42 RESULT: Theorem (the conjecture is true).
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