TSTP Solution File: SEU149+3 by Twee---2.4.2
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- Process Solution
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% File : Twee---2.4.2
% Problem : SEU149+3 : 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 17:51:13 EDT 2023
% Result : Theorem 0.22s 0.41s
% Output : Proof 0.22s
% 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 : SEU149+3 : 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 : n013.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 : Wed Aug 23 21:55:47 EDT 2023
% 0.14/0.36 % CPUTime :
% 0.22/0.41 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.22/0.41
% 0.22/0.41 % SZS status Theorem
% 0.22/0.41
% 0.22/0.42 % SZS output start Proof
% 0.22/0.42 Take the following subset of the input axioms:
% 0.22/0.42 fof(commutativity_k2_tarski, axiom, ![A, B]: unordered_pair(A, B)=unordered_pair(B, A)).
% 0.22/0.42 fof(d1_tarski, axiom, ![A2, B2]: (B2=singleton(A2) <=> ![C]: (in(C, B2) <=> C=A2))).
% 0.22/0.42 fof(d2_tarski, axiom, ![B2, C2, A2_2]: (C2=unordered_pair(A2_2, B2) <=> ![D]: (in(D, C2) <=> (D=A2_2 | D=B2)))).
% 0.22/0.42 fof(t8_zfmisc_1, conjecture, ![A3, B2, C2]: (singleton(A3)=unordered_pair(B2, C2) => A3=B2)).
% 0.22/0.42
% 0.22/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.42 fresh(y, y, x1...xn) = u
% 0.22/0.42 C => fresh(s, t, x1...xn) = v
% 0.22/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.42 variables of u and v.
% 0.22/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.42 input problem has no model of domain size 1).
% 0.22/0.42
% 0.22/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.42
% 0.22/0.42 Axiom 1 (commutativity_k2_tarski): unordered_pair(X, Y) = unordered_pair(Y, X).
% 0.22/0.42 Axiom 2 (t8_zfmisc_1): singleton(a) = unordered_pair(b, c).
% 0.22/0.42 Axiom 3 (d2_tarski_1): equiv(X, Y, X) = true2.
% 0.22/0.42 Axiom 4 (d2_tarski_4): fresh6(X, X, Y, Z) = true2.
% 0.22/0.42 Axiom 5 (d1_tarski_3): fresh3(X, X, Y, Z) = Y.
% 0.22/0.42 Axiom 6 (d1_tarski_3): fresh4(X, X, Y, Z, W) = W.
% 0.22/0.42 Axiom 7 (d2_tarski_4): fresh7(X, X, Y, Z, W, V) = in(V, W).
% 0.22/0.42 Axiom 8 (d1_tarski_3): fresh4(in(X, Y), true2, Z, Y, X) = fresh3(Y, singleton(Z), Z, X).
% 0.22/0.42 Axiom 9 (d2_tarski_4): fresh7(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh6(W, unordered_pair(X, Y), W, Z).
% 0.22/0.42
% 0.22/0.42 Lemma 10: unordered_pair(c, b) = singleton(a).
% 0.22/0.42 Proof:
% 0.22/0.42 unordered_pair(c, b)
% 0.22/0.42 = { by axiom 1 (commutativity_k2_tarski) R->L }
% 0.22/0.42 unordered_pair(b, c)
% 0.22/0.42 = { by axiom 2 (t8_zfmisc_1) R->L }
% 0.22/0.42 singleton(a)
% 0.22/0.42
% 0.22/0.42 Lemma 11: in(X, unordered_pair(X, Y)) = true2.
% 0.22/0.42 Proof:
% 0.22/0.42 in(X, unordered_pair(X, Y))
% 0.22/0.42 = { by axiom 7 (d2_tarski_4) R->L }
% 0.22/0.42 fresh7(true2, true2, X, Y, unordered_pair(X, Y), X)
% 0.22/0.42 = { by axiom 3 (d2_tarski_1) R->L }
% 0.22/0.42 fresh7(equiv(X, Y, X), true2, X, Y, unordered_pair(X, Y), X)
% 0.22/0.42 = { by axiom 9 (d2_tarski_4) }
% 0.22/0.42 fresh6(unordered_pair(X, Y), unordered_pair(X, Y), unordered_pair(X, Y), X)
% 0.22/0.42 = { by axiom 4 (d2_tarski_4) }
% 0.22/0.42 true2
% 0.22/0.42
% 0.22/0.42 Lemma 12: a = c.
% 0.22/0.42 Proof:
% 0.22/0.42 a
% 0.22/0.42 = { by axiom 5 (d1_tarski_3) R->L }
% 0.22/0.42 fresh3(singleton(a), singleton(a), a, c)
% 0.22/0.42 = { by axiom 8 (d1_tarski_3) R->L }
% 0.22/0.42 fresh4(in(c, singleton(a)), true2, a, singleton(a), c)
% 0.22/0.42 = { by lemma 10 R->L }
% 0.22/0.42 fresh4(in(c, unordered_pair(c, b)), true2, a, singleton(a), c)
% 0.22/0.42 = { by lemma 11 }
% 0.22/0.42 fresh4(true2, true2, a, singleton(a), c)
% 0.22/0.42 = { by axiom 6 (d1_tarski_3) }
% 0.22/0.42 c
% 0.22/0.42
% 0.22/0.42 Goal 1 (t8_zfmisc_1_1): a = b.
% 0.22/0.42 Proof:
% 0.22/0.42 a
% 0.22/0.42 = { by lemma 12 }
% 0.22/0.42 c
% 0.22/0.42 = { by axiom 5 (d1_tarski_3) R->L }
% 0.22/0.42 fresh3(singleton(c), singleton(c), c, b)
% 0.22/0.42 = { by lemma 12 R->L }
% 0.22/0.42 fresh3(singleton(a), singleton(c), c, b)
% 0.22/0.42 = { by axiom 8 (d1_tarski_3) R->L }
% 0.22/0.42 fresh4(in(b, singleton(a)), true2, c, singleton(a), b)
% 0.22/0.42 = { by lemma 10 R->L }
% 0.22/0.42 fresh4(in(b, unordered_pair(c, b)), true2, c, singleton(a), b)
% 0.22/0.42 = { by axiom 1 (commutativity_k2_tarski) R->L }
% 0.22/0.42 fresh4(in(b, unordered_pair(b, c)), true2, c, singleton(a), b)
% 0.22/0.42 = { by lemma 11 }
% 0.22/0.42 fresh4(true2, true2, c, singleton(a), b)
% 0.22/0.42 = { by axiom 6 (d1_tarski_3) }
% 0.22/0.42 b
% 0.22/0.42 % SZS output end Proof
% 0.22/0.42
% 0.22/0.42 RESULT: Theorem (the conjecture is true).
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