TSTP Solution File: GEO087+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO087+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 : n015.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 23:27:18 EDT 2023
% Result : Theorem 0.18s 0.47s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GEO087+1 : TPTP v8.1.2. Released v2.4.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n015.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 19:32:11 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.18/0.47 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.47
% 0.18/0.47 % SZS status Theorem
% 0.18/0.47
% 0.18/0.48 % SZS output start Proof
% 0.18/0.48 Take the following subset of the input axioms:
% 0.18/0.48 fof(c3, axiom, ![C]: ?[P]: inner_point(P, C)).
% 0.18/0.48 fof(c6, axiom, ![C3, P2]: (end_point(P2, C3) => ?[Q]: (end_point(Q, C3) & P2!=Q))).
% 0.18/0.48 fof(closed_defn, axiom, ![C3]: (closed(C3) <=> ~?[P2]: end_point(P2, C3))).
% 0.18/0.48 fof(corollary_2_9, conjecture, ![C1, C2]: (part_of(C1, C2) => ~?[P2]: meet(P2, C1, C2))).
% 0.18/0.48 fof(inner_point_defn, axiom, ![C3, P2]: (inner_point(P2, C3) <=> (incident_c(P2, C3) & ~end_point(P2, C3)))).
% 0.18/0.48 fof(meet_defn, axiom, ![C3, P2, C1_2]: (meet(P2, C3, C1_2) <=> (incident_c(P2, C3) & (incident_c(P2, C1_2) & ![Q2]: ((incident_c(Q2, C3) & incident_c(Q2, C1_2)) => (end_point(Q2, C3) & end_point(Q2, C1_2))))))).
% 0.18/0.48 fof(part_of_defn, axiom, ![C3, C1_2]: (part_of(C1_2, C3) <=> ![P2]: (incident_c(P2, C1_2) => incident_c(P2, C3)))).
% 0.18/0.48
% 0.18/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.48 fresh(y, y, x1...xn) = u
% 0.18/0.48 C => fresh(s, t, x1...xn) = v
% 0.18/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.48 variables of u and v.
% 0.18/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.48 input problem has no model of domain size 1).
% 0.18/0.48
% 0.18/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.48
% 0.18/0.48 Axiom 1 (corollary_2_9): part_of(c1, c2) = true2.
% 0.18/0.48 Axiom 2 (corollary_2_9_1): meet(p, c1, c2) = true2.
% 0.18/0.48 Axiom 3 (c3): inner_point(p3(X), X) = true2.
% 0.18/0.48 Axiom 4 (meet_defn_3): fresh39(X, X, Y, Z) = true2.
% 0.18/0.48 Axiom 5 (inner_point_defn_2): fresh21(X, X, Y, Z) = true2.
% 0.18/0.48 Axiom 6 (meet_defn_3): fresh20(X, X, Y, Z) = end_point(Z, Y).
% 0.18/0.48 Axiom 7 (part_of_defn_1): fresh13(X, X, Y, Z) = true2.
% 0.18/0.48 Axiom 8 (part_of_defn_1): fresh14(X, X, Y, Z, W) = incident_c(W, Y).
% 0.18/0.48 Axiom 9 (meet_defn_3): fresh38(X, X, Y, Z, W) = fresh39(incident_c(W, Y), true2, Y, W).
% 0.18/0.48 Axiom 10 (inner_point_defn_2): fresh21(inner_point(X, Y), true2, X, Y) = incident_c(X, Y).
% 0.18/0.48 Axiom 11 (part_of_defn_1): fresh14(incident_c(X, Y), true2, Z, Y, X) = fresh13(part_of(Y, Z), true2, Z, X).
% 0.18/0.48 Axiom 12 (meet_defn_3): fresh38(meet(X, Y, Z), true2, Y, Z, W) = fresh20(incident_c(W, Z), true2, Y, W).
% 0.18/0.48
% 0.18/0.48 Lemma 13: incident_c(p3(X), X) = true2.
% 0.18/0.48 Proof:
% 0.18/0.48 incident_c(p3(X), X)
% 0.18/0.48 = { by axiom 10 (inner_point_defn_2) R->L }
% 0.18/0.48 fresh21(inner_point(p3(X), X), true2, p3(X), X)
% 0.18/0.48 = { by axiom 3 (c3) }
% 0.18/0.48 fresh21(true2, true2, p3(X), X)
% 0.18/0.48 = { by axiom 5 (inner_point_defn_2) }
% 0.18/0.48 true2
% 0.18/0.48
% 0.18/0.48 Goal 1 (inner_point_defn_1): tuple(end_point(X, Y), inner_point(X, Y)) = tuple(true2, true2).
% 0.18/0.48 The goal is true when:
% 0.18/0.48 X = p3(c1)
% 0.18/0.48 Y = c1
% 0.18/0.48
% 0.18/0.48 Proof:
% 0.18/0.48 tuple(end_point(p3(c1), c1), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 6 (meet_defn_3) R->L }
% 0.18/0.48 tuple(fresh20(true2, true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 7 (part_of_defn_1) R->L }
% 0.18/0.48 tuple(fresh20(fresh13(true2, true2, c2, p3(c1)), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 1 (corollary_2_9) R->L }
% 0.18/0.48 tuple(fresh20(fresh13(part_of(c1, c2), true2, c2, p3(c1)), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 11 (part_of_defn_1) R->L }
% 0.18/0.48 tuple(fresh20(fresh14(incident_c(p3(c1), c1), true2, c2, c1, p3(c1)), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by lemma 13 }
% 0.18/0.48 tuple(fresh20(fresh14(true2, true2, c2, c1, p3(c1)), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 8 (part_of_defn_1) }
% 0.18/0.48 tuple(fresh20(incident_c(p3(c1), c2), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 12 (meet_defn_3) R->L }
% 0.18/0.48 tuple(fresh38(meet(p, c1, c2), true2, c1, c2, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 2 (corollary_2_9_1) }
% 0.18/0.48 tuple(fresh38(true2, true2, c1, c2, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 9 (meet_defn_3) }
% 0.18/0.48 tuple(fresh39(incident_c(p3(c1), c1), true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by lemma 13 }
% 0.18/0.48 tuple(fresh39(true2, true2, c1, p3(c1)), inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 4 (meet_defn_3) }
% 0.18/0.48 tuple(true2, inner_point(p3(c1), c1))
% 0.18/0.48 = { by axiom 3 (c3) }
% 0.18/0.48 tuple(true2, true2)
% 0.18/0.48 % SZS output end Proof
% 0.18/0.48
% 0.18/0.48 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------