TSTP Solution File: SET908+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET908+1 : TPTP v8.1.2. Released v3.2.0.
% 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 : n002.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 15:33:41 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
% 0.07/0.13 % Problem : SET908+1 : TPTP v8.1.2. Released v3.2.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n002.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Sat Aug 26 13:50:18 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.21/0.42 Command-line arguments: --ground-connectedness --complete-subsets
% 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(antisymmetry_r2_hidden, axiom, ![A, B]: (in(A, B) => ~in(B, A))).
% 0.21/0.42 fof(d1_tarski, axiom, ![A2, B2]: (B2=singleton(A2) <=> ![C]: (in(C, B2) <=> C=A2))).
% 0.21/0.42 fof(d1_xboole_0, axiom, ![A3]: (A3=empty_set <=> ![B2]: ~in(B2, A3))).
% 0.21/0.42 fof(d2_xboole_0, axiom, ![B2, C2, A2_2]: (C2=set_union2(A2_2, B2) <=> ![D]: (in(D, C2) <=> (in(D, A2_2) | in(D, B2))))).
% 0.21/0.42 fof(t49_zfmisc_1, conjecture, ![A3, B2]: set_union2(singleton(A3), B2)!=empty_set).
% 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 (t49_zfmisc_1): set_union2(singleton(a), b) = empty_set.
% 0.21/0.42 Axiom 2 (d1_tarski_2): fresh15(X, X, Y, Z) = true2.
% 0.21/0.42 Axiom 3 (d2_xboole_0_2): fresh10(X, X, Y, Z) = true2.
% 0.21/0.42 Axiom 4 (d1_tarski_2): fresh15(X, singleton(Y), Y, X) = in(Y, X).
% 0.21/0.42 Axiom 5 (d2_xboole_0_3): fresh9(X, X, Y, Z, W) = true2.
% 0.21/0.42 Axiom 6 (d2_xboole_0_2): fresh11(X, X, Y, Z, W, V) = in(V, W).
% 0.21/0.42 Axiom 7 (d2_xboole_0_3): fresh9(in(X, Y), true2, Y, Z, X) = equiv(Y, Z, X).
% 0.21/0.42 Axiom 8 (d2_xboole_0_2): fresh11(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh10(W, set_union2(X, Y), W, Z).
% 0.21/0.42
% 0.21/0.42 Goal 1 (d1_xboole_0_1): tuple2(X, in(Y, X)) = tuple2(empty_set, true2).
% 0.21/0.42 The goal is true when:
% 0.21/0.42 X = empty_set
% 0.21/0.42 Y = a
% 0.21/0.42
% 0.21/0.42 Proof:
% 0.21/0.42 tuple2(empty_set, in(a, empty_set))
% 0.21/0.42 = { by axiom 1 (t49_zfmisc_1) R->L }
% 0.21/0.42 tuple2(empty_set, in(a, set_union2(singleton(a), b)))
% 0.21/0.42 = { by axiom 6 (d2_xboole_0_2) R->L }
% 0.21/0.42 tuple2(empty_set, fresh11(true2, true2, singleton(a), b, set_union2(singleton(a), b), a))
% 0.21/0.42 = { by axiom 5 (d2_xboole_0_3) R->L }
% 0.21/0.42 tuple2(empty_set, fresh11(fresh9(true2, true2, singleton(a), b, a), true2, singleton(a), b, set_union2(singleton(a), b), a))
% 0.21/0.43 = { by axiom 2 (d1_tarski_2) R->L }
% 0.21/0.43 tuple2(empty_set, fresh11(fresh9(fresh15(singleton(a), singleton(a), a, singleton(a)), true2, singleton(a), b, a), true2, singleton(a), b, set_union2(singleton(a), b), a))
% 0.21/0.43 = { by axiom 4 (d1_tarski_2) }
% 0.21/0.43 tuple2(empty_set, fresh11(fresh9(in(a, singleton(a)), true2, singleton(a), b, a), true2, singleton(a), b, set_union2(singleton(a), b), a))
% 0.21/0.43 = { by axiom 7 (d2_xboole_0_3) }
% 0.21/0.43 tuple2(empty_set, fresh11(equiv(singleton(a), b, a), true2, singleton(a), b, set_union2(singleton(a), b), a))
% 0.21/0.43 = { by axiom 8 (d2_xboole_0_2) }
% 0.21/0.43 tuple2(empty_set, fresh10(set_union2(singleton(a), b), set_union2(singleton(a), b), set_union2(singleton(a), b), a))
% 0.21/0.43 = { by axiom 3 (d2_xboole_0_2) }
% 0.21/0.43 tuple2(empty_set, true2)
% 0.21/0.43 % SZS output end Proof
% 0.21/0.43
% 0.21/0.43 RESULT: Theorem (the conjecture is true).
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