TSTP Solution File: SET912+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET912+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 : n024.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 15:33:42 EDT 2023
% Result : Theorem 0.13s 0.40s
% 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 : SET912+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
% 0.13/0.35 % Computer : n024.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 10:48:09 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.13/0.40 Command-line arguments: --no-flatten-goal
% 0.13/0.40
% 0.13/0.40 % SZS status Theorem
% 0.13/0.40
% 0.13/0.40 % SZS output start Proof
% 0.13/0.40 Take the following subset of the input axioms:
% 0.21/0.40 fof(commutativity_k2_tarski, axiom, ![A, B]: unordered_pair(A, B)=unordered_pair(B, A)).
% 0.21/0.40 fof(commutativity_k3_xboole_0, axiom, ![A3, B2]: set_intersection2(A3, B2)=set_intersection2(B2, A3)).
% 0.21/0.40 fof(t28_xboole_1, axiom, ![A2, B2]: (subset(A2, B2) => set_intersection2(A2, B2)=A2)).
% 0.21/0.40 fof(t38_zfmisc_1, axiom, ![C, B2, A2_2]: (subset(unordered_pair(A2_2, B2), C) <=> (in(A2_2, C) & in(B2, C)))).
% 0.21/0.40 fof(t53_zfmisc_1, conjecture, ![A3, B2, C2]: ((in(A3, B2) & in(C2, B2)) => set_intersection2(unordered_pair(A3, C2), B2)=unordered_pair(A3, C2))).
% 0.21/0.40
% 0.21/0.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.40 fresh(y, y, x1...xn) = u
% 0.21/0.40 C => fresh(s, t, x1...xn) = v
% 0.21/0.40 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.40 variables of u and v.
% 0.21/0.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.40 input problem has no model of domain size 1).
% 0.21/0.40
% 0.21/0.40 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.40
% 0.21/0.40 Axiom 1 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.21/0.40 Axiom 2 (commutativity_k2_tarski): unordered_pair(X, Y) = unordered_pair(Y, X).
% 0.21/0.40 Axiom 3 (t53_zfmisc_1): in(c, b) = true2.
% 0.21/0.40 Axiom 4 (t53_zfmisc_1_1): in(a, b) = true2.
% 0.21/0.40 Axiom 5 (t28_xboole_1): fresh(X, X, Y, Z) = Y.
% 0.21/0.40 Axiom 6 (t38_zfmisc_1): fresh5(X, X, Y, Z, W) = true2.
% 0.21/0.40 Axiom 7 (t38_zfmisc_1): fresh4(X, X, Y, Z, W) = subset(unordered_pair(Y, Z), W).
% 0.21/0.40 Axiom 8 (t28_xboole_1): fresh(subset(X, Y), true2, X, Y) = set_intersection2(X, Y).
% 0.21/0.41 Axiom 9 (t38_zfmisc_1): fresh4(in(X, Y), true2, Z, X, Y) = fresh5(in(Z, Y), true2, Z, X, Y).
% 0.21/0.41
% 0.21/0.41 Goal 1 (t53_zfmisc_1_2): set_intersection2(unordered_pair(a, c), b) = unordered_pair(a, c).
% 0.21/0.41 Proof:
% 0.21/0.41 set_intersection2(unordered_pair(a, c), b)
% 0.21/0.41 = { by axiom 1 (commutativity_k3_xboole_0) }
% 0.21/0.41 set_intersection2(b, unordered_pair(a, c))
% 0.21/0.41 = { by axiom 2 (commutativity_k2_tarski) }
% 0.21/0.41 set_intersection2(b, unordered_pair(c, a))
% 0.21/0.41 = { by axiom 1 (commutativity_k3_xboole_0) R->L }
% 0.21/0.41 set_intersection2(unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 8 (t28_xboole_1) R->L }
% 0.21/0.41 fresh(subset(unordered_pair(c, a), b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 2 (commutativity_k2_tarski) R->L }
% 0.21/0.41 fresh(subset(unordered_pair(a, c), b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 7 (t38_zfmisc_1) R->L }
% 0.21/0.41 fresh(fresh4(true2, true2, a, c, b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 3 (t53_zfmisc_1) R->L }
% 0.21/0.41 fresh(fresh4(in(c, b), true2, a, c, b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 9 (t38_zfmisc_1) }
% 0.21/0.41 fresh(fresh5(in(a, b), true2, a, c, b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 4 (t53_zfmisc_1_1) }
% 0.21/0.41 fresh(fresh5(true2, true2, a, c, b), true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 6 (t38_zfmisc_1) }
% 0.21/0.41 fresh(true2, true2, unordered_pair(c, a), b)
% 0.21/0.41 = { by axiom 5 (t28_xboole_1) }
% 0.21/0.41 unordered_pair(c, a)
% 0.21/0.41 = { by axiom 2 (commutativity_k2_tarski) R->L }
% 0.21/0.41 unordered_pair(a, c)
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41
% 0.21/0.41 RESULT: Theorem (the conjecture is true).
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