TSTP Solution File: GEO085+1 by Twee---2.4.2
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- Process Solution
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% File : Twee---2.4.2
% Problem : GEO085+1 : TPTP v8.1.2. Released v2.4.0.
% 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 : n022.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 : Wed Aug 30 23:27:17 EDT 2023
% Result : Theorem 0.21s 0.47s
% 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 : GEO085+1 : TPTP v8.1.2. Released v2.4.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n022.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Tue Aug 29 22:26:38 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.21/0.47 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.47
% 0.21/0.47 % SZS status Theorem
% 0.21/0.47
% 0.21/0.47 % SZS output start Proof
% 0.21/0.47 Take the following subset of the input axioms:
% 0.21/0.47 fof(c6, axiom, ![C, P]: (end_point(P, C) => ?[Q]: (end_point(Q, C) & P!=Q))).
% 0.21/0.47 fof(closed_defn, axiom, ![C2]: (closed(C2) <=> ~?[P2]: end_point(P2, C2))).
% 0.21/0.47 fof(inner_point_defn, axiom, ![C2, P2]: (inner_point(P2, C2) <=> (incident_c(P2, C2) & ~end_point(P2, C2)))).
% 0.21/0.47 fof(open_defn, axiom, ![C2]: (open(C2) <=> ?[P2]: end_point(P2, C2))).
% 0.21/0.47 fof(theorem_2_7_1, conjecture, ![C2]: (open(C2) => ?[P2, Q2]: (P2!=Q2 & (end_point(P2, C2) & end_point(Q2, 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 (theorem_2_7_1): open(c) = true2.
% 0.21/0.47 Axiom 2 (open_defn_1): fresh17(X, X, Y) = true2.
% 0.21/0.47 Axiom 3 (theorem_2_7_1_1): fresh(X, X, Y, Z) = Z.
% 0.21/0.47 Axiom 4 (c6_1): fresh31(X, X, Y, Z) = true2.
% 0.21/0.47 Axiom 5 (open_defn_1): fresh17(open(X), true2, X) = end_point(p3(X), X).
% 0.21/0.47 Axiom 6 (theorem_2_7_1_1): fresh2(X, X, Y, Z) = Y.
% 0.21/0.47 Axiom 7 (c6_1): fresh31(end_point(X, Y), true2, Y, X) = end_point(q(Y, X), Y).
% 0.21/0.47 Axiom 8 (theorem_2_7_1_1): fresh2(end_point(X, c), true2, Y, X) = fresh(end_point(Y, c), true2, Y, X).
% 0.21/0.47
% 0.21/0.47 Lemma 9: end_point(p3(c), c) = true2.
% 0.21/0.47 Proof:
% 0.21/0.47 end_point(p3(c), c)
% 0.21/0.47 = { by axiom 5 (open_defn_1) R->L }
% 0.21/0.47 fresh17(open(c), true2, c)
% 0.21/0.47 = { by axiom 1 (theorem_2_7_1) }
% 0.21/0.47 fresh17(true2, true2, c)
% 0.21/0.47 = { by axiom 2 (open_defn_1) }
% 0.21/0.47 true2
% 0.21/0.47
% 0.21/0.47 Goal 1 (c6): tuple2(X, end_point(X, Y)) = tuple2(q(Y, X), true2).
% 0.21/0.47 The goal is true when:
% 0.21/0.47 X = p3(c)
% 0.21/0.47 Y = c
% 0.21/0.47
% 0.21/0.47 Proof:
% 0.21/0.47 tuple2(p3(c), end_point(p3(c), c))
% 0.21/0.47 = { by lemma 9 }
% 0.21/0.47 tuple2(p3(c), true2)
% 0.21/0.47 = { by axiom 3 (theorem_2_7_1_1) R->L }
% 0.21/0.47 tuple2(fresh(true2, true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by axiom 4 (c6_1) R->L }
% 0.21/0.47 tuple2(fresh(fresh31(true2, true2, c, p3(c)), true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by lemma 9 R->L }
% 0.21/0.47 tuple2(fresh(fresh31(end_point(p3(c), c), true2, c, p3(c)), true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by axiom 7 (c6_1) }
% 0.21/0.47 tuple2(fresh(end_point(q(c, p3(c)), c), true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by axiom 8 (theorem_2_7_1_1) R->L }
% 0.21/0.47 tuple2(fresh2(end_point(p3(c), c), true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by lemma 9 }
% 0.21/0.47 tuple2(fresh2(true2, true2, q(c, p3(c)), p3(c)), true2)
% 0.21/0.47 = { by axiom 6 (theorem_2_7_1_1) }
% 0.21/0.47 tuple2(q(c, p3(c)), 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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