TSTP Solution File: SET886+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET886+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 : n013.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 15:33:39 EDT 2023
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% 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 : SET886+1 : TPTP v8.1.2. Released v3.2.0.
% 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 : n013.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 : Sat Aug 26 16:08:17 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.38 Command-line arguments: --no-flatten-goal
% 0.19/0.38
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 Take the following subset of the input axioms:
% 0.19/0.38 fof(commutativity_k2_tarski, axiom, ![A, B]: unordered_pair(A, B)=unordered_pair(B, A)).
% 0.19/0.38 fof(t26_zfmisc_1, axiom, ![C, A2, B2]: (subset(unordered_pair(A2, B2), singleton(C)) => A2=C)).
% 0.19/0.38 fof(t27_zfmisc_1, conjecture, ![A3, B2, C2]: (subset(unordered_pair(A3, B2), singleton(C2)) => unordered_pair(A3, B2)=singleton(C2))).
% 0.19/0.38 fof(t69_enumset1, axiom, ![A3]: unordered_pair(A3, A3)=singleton(A3)).
% 0.19/0.38
% 0.19/0.38 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.38 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.38 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.38 fresh(y, y, x1...xn) = u
% 0.19/0.38 C => fresh(s, t, x1...xn) = v
% 0.19/0.38 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.38 variables of u and v.
% 0.19/0.38 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.38 input problem has no model of domain size 1).
% 0.19/0.38
% 0.19/0.38 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.38
% 0.19/0.38 Axiom 1 (t69_enumset1): unordered_pair(X, X) = singleton(X).
% 0.19/0.38 Axiom 2 (commutativity_k2_tarski): unordered_pair(X, Y) = unordered_pair(Y, X).
% 0.19/0.38 Axiom 3 (t26_zfmisc_1): fresh(X, X, Y, Z) = Z.
% 0.19/0.38 Axiom 4 (t27_zfmisc_1): subset(unordered_pair(a, b), singleton(c)) = true.
% 0.19/0.38 Axiom 5 (t26_zfmisc_1): fresh(subset(unordered_pair(X, Y), singleton(Z)), true, X, Z) = X.
% 0.19/0.38
% 0.19/0.38 Lemma 6: c = a.
% 0.19/0.38 Proof:
% 0.19/0.38 c
% 0.19/0.38 = { by axiom 3 (t26_zfmisc_1) R->L }
% 0.19/0.38 fresh(true, true, a, c)
% 0.19/0.38 = { by axiom 4 (t27_zfmisc_1) R->L }
% 0.19/0.38 fresh(subset(unordered_pair(a, b), singleton(c)), true, a, c)
% 0.19/0.38 = { by axiom 5 (t26_zfmisc_1) }
% 0.19/0.38 a
% 0.19/0.38
% 0.19/0.38 Goal 1 (t27_zfmisc_1_1): unordered_pair(a, b) = singleton(c).
% 0.19/0.39 Proof:
% 0.19/0.39 unordered_pair(a, b)
% 0.19/0.39 = { by axiom 5 (t26_zfmisc_1) R->L }
% 0.19/0.39 unordered_pair(a, fresh(subset(unordered_pair(b, a), singleton(c)), true, b, c))
% 0.19/0.39 = { by axiom 2 (commutativity_k2_tarski) }
% 0.19/0.39 unordered_pair(a, fresh(subset(unordered_pair(a, b), singleton(c)), true, b, c))
% 0.19/0.39 = { by axiom 4 (t27_zfmisc_1) }
% 0.19/0.39 unordered_pair(a, fresh(true, true, b, c))
% 0.19/0.39 = { by axiom 3 (t26_zfmisc_1) }
% 0.19/0.39 unordered_pair(a, c)
% 0.19/0.39 = { by lemma 6 }
% 0.19/0.39 unordered_pair(a, a)
% 0.19/0.39 = { by axiom 1 (t69_enumset1) }
% 0.19/0.39 singleton(a)
% 0.19/0.39 = { by lemma 6 R->L }
% 0.19/0.39 singleton(c)
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39
% 0.19/0.39 RESULT: Theorem (the conjecture is true).
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