TSTP Solution File: SET950+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET950+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 : n020.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 15:33:49 EDT 2023
% Result : Theorem 0.21s 0.43s
% 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
% 0.00/0.13 % Problem : SET950+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
% 0.14/0.35 % Computer : n020.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 10:24:45 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.43 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.21/0.43
% 0.21/0.43 % SZS status Theorem
% 0.21/0.43
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Take the following subset of the input axioms:
% 0.21/0.44 fof(antisymmetry_r2_hidden, axiom, ![A, B]: (in(A, B) => ~in(B, A))).
% 0.21/0.44 fof(d2_zfmisc_1, axiom, ![C, A2, B2]: (C=cartesian_product2(A2, B2) <=> ![D]: (in(D, C) <=> ?[E, F]: (in(E, A2) & (in(F, B2) & D=ordered_pair(E, F)))))).
% 0.21/0.44 fof(d3_tarski, axiom, ![B2, A2_2]: (subset(A2_2, B2) <=> ![C2]: (in(C2, A2_2) => in(C2, B2)))).
% 0.21/0.44 fof(fc1_zfmisc_1, axiom, ![A3, B2]: ~empty(ordered_pair(A3, B2))).
% 0.21/0.44 fof(t103_zfmisc_1, conjecture, ![A3, B2, C2, D2]: ~(subset(A3, cartesian_product2(B2, C2)) & (in(D2, A3) & ![E2, F2]: ~(in(E2, B2) & (in(F2, C2) & D2=ordered_pair(E2, F2)))))).
% 0.21/0.44
% 0.21/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.44 fresh(y, y, x1...xn) = u
% 0.21/0.44 C => fresh(s, t, x1...xn) = v
% 0.21/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.44 variables of u and v.
% 0.21/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.44 input problem has no model of domain size 1).
% 0.21/0.44
% 0.21/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.44
% 0.21/0.44 Axiom 1 (t103_zfmisc_1): in(d, a) = true2.
% 0.21/0.44 Axiom 2 (t103_zfmisc_1_1): subset(a, cartesian_product2(b, c)) = true2.
% 0.21/0.44 Axiom 3 (d3_tarski_1): fresh4(X, X, Y, Z) = true2.
% 0.21/0.44 Axiom 4 (d2_zfmisc_1_1): fresh13(X, X, Y, Z, W) = true2.
% 0.21/0.44 Axiom 5 (d2_zfmisc_1_6): fresh7(X, X, Y, Z, W) = true2.
% 0.21/0.44 Axiom 6 (d2_zfmisc_1_7): fresh6(X, X, Y, Z, W) = true2.
% 0.21/0.44 Axiom 7 (d3_tarski_1): fresh5(X, X, Y, Z, W) = in(W, Z).
% 0.21/0.44 Axiom 8 (d2_zfmisc_1_5): fresh2(X, X, Y, Z, W) = W.
% 0.21/0.44 Axiom 9 (d2_zfmisc_1_1): fresh12(X, X, Y, Z, W, V) = equiv(Y, Z, V).
% 0.21/0.44 Axiom 10 (d3_tarski_1): fresh5(subset(X, Y), true2, X, Y, Z) = fresh4(in(Z, X), true2, Y, Z).
% 0.21/0.44 Axiom 11 (d2_zfmisc_1_1): fresh12(in(X, Y), true2, Z, W, Y, X) = fresh13(Y, cartesian_product2(Z, W), Z, W, X).
% 0.21/0.44 Axiom 12 (d2_zfmisc_1_6): fresh7(equiv(X, Y, Z), true2, X, Y, Z) = in(f(X, Y, Z), Y).
% 0.21/0.44 Axiom 13 (d2_zfmisc_1_7): fresh6(equiv(X, Y, Z), true2, X, Y, Z) = in(e(X, Y, Z), X).
% 0.21/0.44 Axiom 14 (d2_zfmisc_1_5): fresh2(equiv(X, Y, Z), true2, X, Y, Z) = ordered_pair(e(X, Y, Z), f(X, Y, Z)).
% 0.21/0.44
% 0.21/0.44 Lemma 15: equiv(b, c, d) = true2.
% 0.21/0.44 Proof:
% 0.21/0.44 equiv(b, c, d)
% 0.21/0.44 = { by axiom 9 (d2_zfmisc_1_1) R->L }
% 0.21/0.44 fresh12(true2, true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 3 (d3_tarski_1) R->L }
% 0.21/0.44 fresh12(fresh4(true2, true2, cartesian_product2(b, c), d), true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 1 (t103_zfmisc_1) R->L }
% 0.21/0.44 fresh12(fresh4(in(d, a), true2, cartesian_product2(b, c), d), true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 10 (d3_tarski_1) R->L }
% 0.21/0.44 fresh12(fresh5(subset(a, cartesian_product2(b, c)), true2, a, cartesian_product2(b, c), d), true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 2 (t103_zfmisc_1_1) }
% 0.21/0.44 fresh12(fresh5(true2, true2, a, cartesian_product2(b, c), d), true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 7 (d3_tarski_1) }
% 0.21/0.44 fresh12(in(d, cartesian_product2(b, c)), true2, b, c, cartesian_product2(b, c), d)
% 0.21/0.44 = { by axiom 11 (d2_zfmisc_1_1) }
% 0.21/0.44 fresh13(cartesian_product2(b, c), cartesian_product2(b, c), b, c, d)
% 0.21/0.44 = { by axiom 4 (d2_zfmisc_1_1) }
% 0.21/0.44 true2
% 0.21/0.44
% 0.21/0.44 Goal 1 (t103_zfmisc_1_2): tuple2(d, in(X, b), in(Y, c)) = tuple2(ordered_pair(X, Y), true2, true2).
% 0.21/0.44 The goal is true when:
% 0.21/0.44 X = e(b, c, d)
% 0.21/0.44 Y = f(b, c, d)
% 0.21/0.44
% 0.21/0.44 Proof:
% 0.21/0.44 tuple2(d, in(e(b, c, d), b), in(f(b, c, d), c))
% 0.21/0.44 = { by axiom 13 (d2_zfmisc_1_7) R->L }
% 0.21/0.44 tuple2(d, fresh6(equiv(b, c, d), true2, b, c, d), in(f(b, c, d), c))
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 tuple2(d, fresh6(true2, true2, b, c, d), in(f(b, c, d), c))
% 0.21/0.44 = { by axiom 6 (d2_zfmisc_1_7) }
% 0.21/0.44 tuple2(d, true2, in(f(b, c, d), c))
% 0.21/0.44 = { by axiom 12 (d2_zfmisc_1_6) R->L }
% 0.21/0.44 tuple2(d, true2, fresh7(equiv(b, c, d), true2, b, c, d))
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 tuple2(d, true2, fresh7(true2, true2, b, c, d))
% 0.21/0.44 = { by axiom 5 (d2_zfmisc_1_6) }
% 0.21/0.44 tuple2(d, true2, true2)
% 0.21/0.44 = { by axiom 8 (d2_zfmisc_1_5) R->L }
% 0.21/0.44 tuple2(fresh2(true2, true2, b, c, d), true2, true2)
% 0.21/0.44 = { by lemma 15 R->L }
% 0.21/0.44 tuple2(fresh2(equiv(b, c, d), true2, b, c, d), true2, true2)
% 0.21/0.44 = { by axiom 14 (d2_zfmisc_1_5) }
% 0.21/0.44 tuple2(ordered_pair(e(b, c, d), f(b, c, d)), true2, true2)
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44
% 0.21/0.44 RESULT: Theorem (the conjecture is true).
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