TSTP Solution File: SET906+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET906+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 15:33:41 EDT 2023
% Result : Theorem 0.21s 0.46s
% 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 : SET906+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
% 0.14/0.35 % Computer : n024.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 14:14:54 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.46 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.46
% 0.21/0.46 % SZS status Theorem
% 0.21/0.46
% 0.21/0.46 % SZS output start Proof
% 0.21/0.46 Take the following subset of the input axioms:
% 0.21/0.47 fof(commutativity_k2_xboole_0, axiom, ![A, B]: set_union2(A, B)=set_union2(B, A)).
% 0.21/0.47 fof(d2_tarski, axiom, ![C, A2, B2]: (C=unordered_pair(A2, B2) <=> ![D]: (in(D, C) <=> (D=A2 | D=B2)))).
% 0.21/0.47 fof(d2_xboole_0, axiom, ![B2, C2, A2_2]: (C2=set_union2(A2_2, B2) <=> ![D2]: (in(D2, C2) <=> (in(D2, A2_2) | in(D2, B2))))).
% 0.21/0.47 fof(d3_tarski, axiom, ![B2, A2_2]: (subset(A2_2, B2) <=> ![C2]: (in(C2, A2_2) => in(C2, B2)))).
% 0.21/0.47 fof(t47_zfmisc_1, conjecture, ![A3, B2, C2]: (subset(set_union2(unordered_pair(A3, B2), C2), C2) => in(A3, C2))).
% 0.21/0.47
% 0.21/0.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.47 fresh(y, y, x1...xn) = u
% 0.21/0.47 C => fresh(s, t, x1...xn) = v
% 0.21/0.47 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.47 variables of u and v.
% 0.21/0.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.47 input problem has no model of domain size 1).
% 0.21/0.47
% 0.21/0.47 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.47
% 0.21/0.47 Axiom 1 (commutativity_k2_xboole_0): set_union2(X, Y) = set_union2(Y, X).
% 0.21/0.47 Axiom 2 (d2_tarski_1): equiv2(X, Y, X) = true2.
% 0.21/0.47 Axiom 3 (d2_tarski_4): fresh16(X, X, Y, Z) = true2.
% 0.21/0.47 Axiom 4 (d2_xboole_0_2): fresh11(X, X, Y, Z) = true2.
% 0.21/0.47 Axiom 5 (d3_tarski_1): fresh6(X, X, Y, Z) = true2.
% 0.21/0.47 Axiom 6 (d2_xboole_0_3): fresh10(X, X, Y, Z, W) = true2.
% 0.21/0.47 Axiom 7 (d3_tarski_1): fresh7(X, X, Y, Z, W) = in(W, Z).
% 0.21/0.47 Axiom 8 (t47_zfmisc_1): subset(set_union2(unordered_pair(a, b), c), c) = true2.
% 0.21/0.47 Axiom 9 (d2_tarski_4): fresh17(X, X, Y, Z, W, V) = in(V, W).
% 0.21/0.47 Axiom 10 (d2_xboole_0_2): fresh12(X, X, Y, Z, W, V) = in(V, W).
% 0.21/0.47 Axiom 11 (d2_xboole_0_3): fresh10(in(X, Y), true2, Y, Z, X) = equiv(Y, Z, X).
% 0.21/0.47 Axiom 12 (d3_tarski_1): fresh7(subset(X, Y), true2, X, Y, Z) = fresh6(in(Z, X), true2, Y, Z).
% 0.21/0.47 Axiom 13 (d2_tarski_4): fresh17(equiv2(X, Y, Z), true2, X, Y, W, Z) = fresh16(W, unordered_pair(X, Y), W, Z).
% 0.21/0.47 Axiom 14 (d2_xboole_0_2): fresh12(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh11(W, set_union2(X, Y), W, Z).
% 0.21/0.47
% 0.21/0.47 Goal 1 (t47_zfmisc_1_1): in(a, c) = true2.
% 0.21/0.47 Proof:
% 0.21/0.47 in(a, c)
% 0.21/0.47 = { by axiom 7 (d3_tarski_1) R->L }
% 0.21/0.47 fresh7(true2, true2, set_union2(c, unordered_pair(a, b)), c, a)
% 0.21/0.47 = { by axiom 8 (t47_zfmisc_1) R->L }
% 0.21/0.47 fresh7(subset(set_union2(unordered_pair(a, b), c), c), true2, set_union2(c, unordered_pair(a, b)), c, a)
% 0.21/0.47 = { by axiom 1 (commutativity_k2_xboole_0) }
% 0.21/0.47 fresh7(subset(set_union2(c, unordered_pair(a, b)), c), true2, set_union2(c, unordered_pair(a, b)), c, a)
% 0.21/0.47 = { by axiom 12 (d3_tarski_1) }
% 0.21/0.47 fresh6(in(a, set_union2(c, unordered_pair(a, b))), true2, c, a)
% 0.21/0.47 = { by axiom 1 (commutativity_k2_xboole_0) R->L }
% 0.21/0.47 fresh6(in(a, set_union2(unordered_pair(a, b), c)), true2, c, a)
% 0.21/0.47 = { by axiom 10 (d2_xboole_0_2) R->L }
% 0.21/0.47 fresh6(fresh12(true2, true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 6 (d2_xboole_0_3) R->L }
% 0.21/0.47 fresh6(fresh12(fresh10(true2, true2, unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 3 (d2_tarski_4) R->L }
% 0.21/0.47 fresh6(fresh12(fresh10(fresh16(unordered_pair(a, b), unordered_pair(a, b), unordered_pair(a, b), a), true2, unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 13 (d2_tarski_4) R->L }
% 0.21/0.47 fresh6(fresh12(fresh10(fresh17(equiv2(a, b, a), true2, a, b, unordered_pair(a, b), a), true2, unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 2 (d2_tarski_1) }
% 0.21/0.47 fresh6(fresh12(fresh10(fresh17(true2, true2, a, b, unordered_pair(a, b), a), true2, unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 9 (d2_tarski_4) }
% 0.21/0.47 fresh6(fresh12(fresh10(in(a, unordered_pair(a, b)), true2, unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 11 (d2_xboole_0_3) }
% 0.21/0.47 fresh6(fresh12(equiv(unordered_pair(a, b), c, a), true2, unordered_pair(a, b), c, set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 14 (d2_xboole_0_2) }
% 0.21/0.47 fresh6(fresh11(set_union2(unordered_pair(a, b), c), set_union2(unordered_pair(a, b), c), set_union2(unordered_pair(a, b), c), a), true2, c, a)
% 0.21/0.47 = { by axiom 4 (d2_xboole_0_2) }
% 0.21/0.47 fresh6(true2, true2, c, a)
% 0.21/0.47 = { by axiom 5 (d3_tarski_1) }
% 0.21/0.47 true2
% 0.21/0.47 % SZS output end Proof
% 0.21/0.47
% 0.21/0.47 RESULT: Theorem (the conjecture is true).
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