TSTP Solution File: SEU137+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SEU137+2 : TPTP v8.1.2. Released v3.3.0.
% 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 : n014.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 17:51:09 EDT 2023
% Result : Theorem 0.20s 0.49s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU137+2 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % 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 : n014.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 : Wed Aug 23 23:07:19 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.49 Command-line arguments: --ground-connectedness --complete-subsets
% 0.20/0.49
% 0.20/0.49 % SZS status Theorem
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% 0.20/0.49 % SZS output start Proof
% 0.20/0.49 Take the following subset of the input axioms:
% 0.20/0.49 fof(commutativity_k2_xboole_0, axiom, ![A, B]: set_union2(A, B)=set_union2(B, A)).
% 0.20/0.49 fof(l32_xboole_1, lemma, ![A2, B2]: (set_difference(A2, B2)=empty_set <=> subset(A2, B2))).
% 0.20/0.49 fof(t1_boole, axiom, ![A3]: set_union2(A3, empty_set)=A3).
% 0.20/0.49 fof(t39_xboole_1, lemma, ![A3, B2]: set_union2(A3, set_difference(B2, A3))=set_union2(A3, B2)).
% 0.20/0.49 fof(t45_xboole_1, conjecture, ![A3, B2]: (subset(A3, B2) => B2=set_union2(A3, set_difference(B2, A3)))).
% 0.20/0.49
% 0.20/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.49 fresh(y, y, x1...xn) = u
% 0.20/0.49 C => fresh(s, t, x1...xn) = v
% 0.20/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.49 variables of u and v.
% 0.20/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.49 input problem has no model of domain size 1).
% 0.20/0.49
% 0.20/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.49
% 0.20/0.49 Axiom 1 (commutativity_k2_xboole_0): set_union2(X, Y) = set_union2(Y, X).
% 0.20/0.49 Axiom 2 (t1_boole): set_union2(X, empty_set) = X.
% 0.20/0.49 Axiom 3 (t45_xboole_1): subset(a, b) = true2.
% 0.20/0.49 Axiom 4 (l32_xboole_1_1): fresh25(X, X, Y, Z) = empty_set.
% 0.20/0.49 Axiom 5 (t39_xboole_1): set_union2(X, set_difference(Y, X)) = set_union2(X, Y).
% 0.20/0.49 Axiom 6 (l32_xboole_1_1): fresh25(subset(X, Y), true2, X, Y) = set_difference(X, Y).
% 0.20/0.49
% 0.20/0.49 Goal 1 (t45_xboole_1_1): b = set_union2(a, set_difference(b, a)).
% 0.20/0.49 Proof:
% 0.20/0.49 b
% 0.20/0.49 = { by axiom 2 (t1_boole) R->L }
% 0.20/0.49 set_union2(b, empty_set)
% 0.20/0.49 = { by axiom 4 (l32_xboole_1_1) R->L }
% 0.20/0.49 set_union2(b, fresh25(true2, true2, a, b))
% 0.20/0.49 = { by axiom 3 (t45_xboole_1) R->L }
% 0.20/0.49 set_union2(b, fresh25(subset(a, b), true2, a, b))
% 0.20/0.49 = { by axiom 6 (l32_xboole_1_1) }
% 0.20/0.49 set_union2(b, set_difference(a, b))
% 0.20/0.49 = { by axiom 5 (t39_xboole_1) }
% 0.20/0.49 set_union2(b, a)
% 0.20/0.49 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.20/0.49 set_union2(a, b)
% 0.20/0.49 = { by axiom 5 (t39_xboole_1) R->L }
% 0.20/0.49 set_union2(a, set_difference(b, a))
% 0.20/0.49 % SZS output end Proof
% 0.20/0.49
% 0.20/0.49 RESULT: Theorem (the conjecture is true).
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