TSTP Solution File: SET918+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET918+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 : n029.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 15:33:44 EDT 2023
% Result : Theorem 0.20s 0.45s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET918+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
% 0.12/0.34 % Computer : n029.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Aug 26 14:36:10 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.20/0.45 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.45
% 0.20/0.45 % SZS status Theorem
% 0.20/0.45
% 0.20/0.46 % SZS output start Proof
% 0.20/0.46 Take the following subset of the input axioms:
% 0.20/0.46 fof(commutativity_k2_tarski, axiom, ![A, B]: unordered_pair(A, B)=unordered_pair(B, A)).
% 0.20/0.46 fof(commutativity_k3_xboole_0, axiom, ![A3, B2]: set_intersection2(A3, B2)=set_intersection2(B2, A3)).
% 0.20/0.46 fof(d1_tarski, axiom, ![A2, B2]: (B2=singleton(A2) <=> ![C]: (in(C, B2) <=> C=A2))).
% 0.20/0.46 fof(d2_tarski, axiom, ![B2, C2, A2_2]: (C2=unordered_pair(A2_2, B2) <=> ![D]: (in(D, C2) <=> (D=A2_2 | D=B2)))).
% 0.20/0.46 fof(d3_xboole_0, axiom, ![B2, C2, A2_2]: (C2=set_intersection2(A2_2, B2) <=> ![D2]: (in(D2, C2) <=> (in(D2, A2_2) & in(D2, B2))))).
% 0.20/0.46 fof(t59_zfmisc_1, conjecture, ![A3, B2, C2]: ~(set_intersection2(unordered_pair(A3, B2), C2)=singleton(A3) & (in(B2, C2) & A3!=B2))).
% 0.20/0.46
% 0.20/0.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.46 fresh(y, y, x1...xn) = u
% 0.20/0.46 C => fresh(s, t, x1...xn) = v
% 0.20/0.46 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.46 variables of u and v.
% 0.20/0.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.46 input problem has no model of domain size 1).
% 0.20/0.46
% 0.20/0.46 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.46
% 0.20/0.46 Axiom 1 (t59_zfmisc_1_1): in(b, c) = true2.
% 0.20/0.46 Axiom 2 (commutativity_k3_xboole_0): set_intersection2(X, Y) = set_intersection2(Y, X).
% 0.20/0.46 Axiom 3 (commutativity_k2_tarski): unordered_pair(X, Y) = unordered_pair(Y, X).
% 0.20/0.46 Axiom 4 (d2_tarski_1): equiv2(X, Y, X) = true2.
% 0.20/0.46 Axiom 5 (t59_zfmisc_1): set_intersection2(unordered_pair(a, b), c) = singleton(a).
% 0.20/0.46 Axiom 6 (d2_tarski_4): fresh16(X, X, Y, Z) = true2.
% 0.20/0.46 Axiom 7 (d3_xboole_0_2): fresh11(X, X, Y, Z) = true2.
% 0.20/0.46 Axiom 8 (d1_tarski_3): fresh4(X, X, Y, Z) = Y.
% 0.20/0.46 Axiom 9 (d3_xboole_0_3): fresh10(X, X, Y, Z, W) = equiv(Y, Z, W).
% 0.20/0.46 Axiom 10 (d3_xboole_0_3): fresh9(X, X, Y, Z, W) = true2.
% 0.20/0.46 Axiom 11 (d1_tarski_3): fresh5(X, X, Y, Z, W) = W.
% 0.20/0.46 Axiom 12 (d2_tarski_4): fresh17(X, X, Y, Z, W, V) = in(V, W).
% 0.20/0.46 Axiom 13 (d3_xboole_0_2): fresh12(X, X, Y, Z, W, V) = in(V, W).
% 0.20/0.46 Axiom 14 (d3_xboole_0_3): fresh10(in(X, Y), true2, Z, Y, X) = fresh9(in(X, Z), true2, Z, Y, X).
% 0.20/0.46 Axiom 15 (d1_tarski_3): fresh5(in(X, Y), true2, Z, Y, X) = fresh4(Y, singleton(Z), Z, X).
% 0.20/0.46 Axiom 16 (d2_tarski_4): fresh17(equiv2(X, Y, Z), true2, X, Y, W, Z) = fresh16(W, unordered_pair(X, Y), W, Z).
% 0.20/0.46 Axiom 17 (d3_xboole_0_2): fresh12(equiv(X, Y, Z), true2, X, Y, W, Z) = fresh11(W, set_intersection2(X, Y), W, Z).
% 0.20/0.46
% 0.20/0.46 Goal 1 (t59_zfmisc_1_2): a = b.
% 0.20/0.46 Proof:
% 0.20/0.46 a
% 0.20/0.46 = { by axiom 8 (d1_tarski_3) R->L }
% 0.20/0.46 fresh4(singleton(a), singleton(a), a, b)
% 0.20/0.46 = { by axiom 15 (d1_tarski_3) R->L }
% 0.20/0.46 fresh5(in(b, singleton(a)), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 5 (t59_zfmisc_1) R->L }
% 0.20/0.46 fresh5(in(b, set_intersection2(unordered_pair(a, b), c)), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 2 (commutativity_k3_xboole_0) }
% 0.20/0.46 fresh5(in(b, set_intersection2(c, unordered_pair(a, b))), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 13 (d3_xboole_0_2) R->L }
% 0.20/0.46 fresh5(fresh12(true2, true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 10 (d3_xboole_0_3) R->L }
% 0.20/0.46 fresh5(fresh12(fresh9(true2, true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 1 (t59_zfmisc_1_1) R->L }
% 0.20/0.46 fresh5(fresh12(fresh9(in(b, c), true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 14 (d3_xboole_0_3) R->L }
% 0.20/0.46 fresh5(fresh12(fresh10(in(b, unordered_pair(b, a)), true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 12 (d2_tarski_4) R->L }
% 0.20/0.46 fresh5(fresh12(fresh10(fresh17(true2, true2, b, a, unordered_pair(b, a), b), true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 4 (d2_tarski_1) R->L }
% 0.20/0.46 fresh5(fresh12(fresh10(fresh17(equiv2(b, a, b), true2, b, a, unordered_pair(b, a), b), true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 16 (d2_tarski_4) }
% 0.20/0.46 fresh5(fresh12(fresh10(fresh16(unordered_pair(b, a), unordered_pair(b, a), unordered_pair(b, a), b), true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 6 (d2_tarski_4) }
% 0.20/0.46 fresh5(fresh12(fresh10(true2, true2, c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 9 (d3_xboole_0_3) }
% 0.20/0.46 fresh5(fresh12(equiv(c, unordered_pair(b, a), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 3 (commutativity_k2_tarski) }
% 0.20/0.46 fresh5(fresh12(equiv(c, unordered_pair(a, b), b), true2, c, unordered_pair(a, b), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 17 (d3_xboole_0_2) }
% 0.20/0.46 fresh5(fresh11(set_intersection2(c, unordered_pair(a, b)), set_intersection2(c, unordered_pair(a, b)), set_intersection2(c, unordered_pair(a, b)), b), true2, a, singleton(a), b)
% 0.20/0.46 = { by axiom 7 (d3_xboole_0_2) }
% 0.20/0.47 fresh5(true2, true2, a, singleton(a), b)
% 0.20/0.47 = { by axiom 11 (d1_tarski_3) }
% 0.20/0.47 b
% 0.20/0.47 % SZS output end Proof
% 0.20/0.47
% 0.20/0.47 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------