TSTP Solution File: SET873+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET873+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 : n001.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 15:33:36 EDT 2023
% Result : Theorem 0.20s 0.40s
% 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
% 0.00/0.13 % Problem : SET873+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n001.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat Aug 26 09:35:15 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.20/0.40 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.40
% 0.20/0.40 % SZS status Theorem
% 0.20/0.40
% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Take the following subset of the input axioms:
% 0.20/0.40 fof(commutativity_k2_xboole_0, axiom, ![A, B]: set_union2(A, B)=set_union2(B, A)).
% 0.20/0.40 fof(d1_tarski, axiom, ![A2, B2]: (B2=singleton(A2) <=> ![C]: (in(C, B2) <=> C=A2))).
% 0.20/0.40 fof(l21_zfmisc_1, axiom, ![B2, A2_2]: (subset(set_union2(singleton(A2_2), B2), B2) => in(A2_2, B2))).
% 0.20/0.40 fof(reflexivity_r1_tarski, axiom, ![A3, B2]: subset(A3, A3)).
% 0.20/0.40 fof(t13_zfmisc_1, conjecture, ![A3, B2]: (set_union2(singleton(A3), singleton(B2))=singleton(A3) => A3=B2)).
% 0.20/0.40
% 0.20/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40 fresh(y, y, x1...xn) = u
% 0.20/0.40 C => fresh(s, t, x1...xn) = v
% 0.20/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40 variables of u and v.
% 0.20/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40 input problem has no model of domain size 1).
% 0.20/0.40
% 0.20/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40
% 0.20/0.40 Axiom 1 (reflexivity_r1_tarski): subset(X, X) = true2.
% 0.20/0.40 Axiom 2 (commutativity_k2_xboole_0): set_union2(X, Y) = set_union2(Y, X).
% 0.20/0.40 Axiom 3 (l21_zfmisc_1): fresh4(X, X, Y, Z) = true2.
% 0.20/0.40 Axiom 4 (d1_tarski_3): fresh2(X, X, Y, Z) = Y.
% 0.20/0.41 Axiom 5 (t13_zfmisc_1): set_union2(singleton(a), singleton(b)) = singleton(a).
% 0.20/0.41 Axiom 6 (d1_tarski_3): fresh3(X, X, Y, Z, W) = W.
% 0.20/0.41 Axiom 7 (d1_tarski_3): fresh3(in(X, Y), true2, Z, Y, X) = fresh2(Y, singleton(Z), Z, X).
% 0.20/0.41 Axiom 8 (l21_zfmisc_1): fresh4(subset(set_union2(singleton(X), Y), Y), true2, X, Y) = in(X, Y).
% 0.20/0.41
% 0.20/0.41 Goal 1 (t13_zfmisc_1_1): a = b.
% 0.20/0.41 Proof:
% 0.20/0.41 a
% 0.20/0.41 = { by axiom 4 (d1_tarski_3) R->L }
% 0.20/0.41 fresh2(singleton(a), singleton(a), a, b)
% 0.20/0.41 = { by axiom 7 (d1_tarski_3) R->L }
% 0.20/0.41 fresh3(in(b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 8 (l21_zfmisc_1) R->L }
% 0.20/0.41 fresh3(fresh4(subset(set_union2(singleton(b), singleton(a)), singleton(a)), true2, b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 2 (commutativity_k2_xboole_0) R->L }
% 0.20/0.41 fresh3(fresh4(subset(set_union2(singleton(a), singleton(b)), singleton(a)), true2, b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 5 (t13_zfmisc_1) }
% 0.20/0.41 fresh3(fresh4(subset(singleton(a), singleton(a)), true2, b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 1 (reflexivity_r1_tarski) }
% 0.20/0.41 fresh3(fresh4(true2, true2, b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 3 (l21_zfmisc_1) }
% 0.20/0.41 fresh3(true2, true2, a, singleton(a), b)
% 0.20/0.41 = { by axiom 6 (d1_tarski_3) }
% 0.20/0.41 b
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Theorem (the conjecture is true).
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