TSTP Solution File: GEO002-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO002-4 : TPTP v8.1.2. Released v1.1.0.
% 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 : n011.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 23:26:43 EDT 2023
% Result : Unsatisfiable 0.21s 0.58s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO002-4 : TPTP v8.1.2. Released v1.1.0.
% 0.07/0.13 % 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 : n011.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 : Tue Aug 29 22:19:26 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.58 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.58
% 0.21/0.58 % SZS status Unsatisfiable
% 0.21/0.58
% 0.21/0.60 % SZS output start Proof
% 0.21/0.60 Take the following subset of the input axioms:
% 0.21/0.60 fof(between_substitution3, axiom, ![X, Y, Z, W]: (~equalish(X, Y) | (~between(W, Z, X) | between(W, Z, Y)))).
% 0.21/0.60 fof(identity_for_equidistance, axiom, ![X2, Y2, Z2]: (~equidistant(X2, Y2, Z2, Z2) | equalish(X2, Y2))).
% 0.21/0.60 fof(outer_pasch1, axiom, ![V, X2, Y2, Z2, W2]: (~between(X2, W2, V) | (~between(Y2, V, Z2) | between(X2, outer_pasch(W2, X2, Y2, Z2, V), Y2)))).
% 0.21/0.60 fof(outer_pasch2, axiom, ![X2, Y2, V2, Z2, W2]: (~between(X2, W2, V2) | (~between(Y2, V2, Z2) | between(Z2, W2, outer_pasch(W2, X2, Y2, Z2, V2))))).
% 0.21/0.60 fof(prove_a_between_a_and_b, negated_conjecture, ~between(a, a, b)).
% 0.21/0.60 fof(segment_construction1, axiom, ![X2, Y2, V2, W2]: between(X2, Y2, extension(X2, Y2, W2, V2))).
% 0.21/0.60 fof(segment_construction2, axiom, ![X2, Y2, V2, W2]: equidistant(Y2, extension(X2, Y2, W2, V2), W2, V2)).
% 0.21/0.60 fof(transitivity_for_betweeness, axiom, ![X2, Y2, V2, Z2]: (~between(X2, Y2, V2) | (~between(Y2, Z2, V2) | between(X2, Y2, Z2)))).
% 0.21/0.60
% 0.21/0.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.60 fresh(y, y, x1...xn) = u
% 0.21/0.60 C => fresh(s, t, x1...xn) = v
% 0.21/0.60 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.60 variables of u and v.
% 0.21/0.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.60 input problem has no model of domain size 1).
% 0.21/0.60
% 0.21/0.60 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.60
% 0.21/0.60 Axiom 1 (identity_for_equidistance): fresh7(X, X, Y, Z) = true.
% 0.21/0.60 Axiom 2 (transitivity_for_betweeness): fresh(X, X, Y, Z, W) = true.
% 0.21/0.60 Axiom 3 (between_substitution3): fresh9(X, X, Y, Z, W) = true.
% 0.21/0.60 Axiom 4 (between_substitution3): fresh8(X, X, Y, Z, W, V) = between(W, V, Z).
% 0.21/0.60 Axiom 5 (transitivity_for_betweeness): fresh2(X, X, Y, Z, W, V) = between(Y, Z, V).
% 0.21/0.60 Axiom 6 (segment_construction1): between(X, Y, extension(X, Y, Z, W)) = true.
% 0.21/0.60 Axiom 7 (outer_pasch1): fresh5(X, X, Y, Z, W, V, U) = true.
% 0.21/0.60 Axiom 8 (outer_pasch2): fresh3(X, X, Y, Z, W, V, U) = true.
% 0.21/0.60 Axiom 9 (outer_pasch1): fresh6(X, X, Y, Z, W, V, U) = between(Y, outer_pasch(Z, Y, V, U, W), V).
% 0.21/0.60 Axiom 10 (outer_pasch2): fresh4(X, X, Y, Z, W, V, U) = between(U, Z, outer_pasch(Z, Y, V, U, W)).
% 0.21/0.60 Axiom 11 (segment_construction2): equidistant(X, extension(Y, X, Z, W), Z, W) = true.
% 0.21/0.60 Axiom 12 (between_substitution3): fresh8(equalish(X, Y), true, X, Y, Z, W) = fresh9(between(Z, W, X), true, Y, Z, W).
% 0.21/0.60 Axiom 13 (identity_for_equidistance): fresh7(equidistant(X, Y, Z, Z), true, X, Y) = equalish(X, Y).
% 0.21/0.60 Axiom 14 (transitivity_for_betweeness): fresh2(between(X, Y, Z), true, W, X, Z, Y) = fresh(between(W, X, Z), true, W, X, Y).
% 0.21/0.60 Axiom 15 (outer_pasch1): fresh6(between(X, Y, Z), true, W, V, Y, X, Z) = fresh5(between(W, V, Y), true, W, V, Y, X, Z).
% 0.21/0.60 Axiom 16 (outer_pasch2): fresh4(between(X, Y, Z), true, W, V, Y, X, Z) = fresh3(between(W, V, Y), true, W, V, Y, X, Z).
% 0.21/0.60
% 0.21/0.60 Lemma 17: equalish(X, extension(Y, X, Z, Z)) = true.
% 0.21/0.60 Proof:
% 0.21/0.60 equalish(X, extension(Y, X, Z, Z))
% 0.21/0.60 = { by axiom 13 (identity_for_equidistance) R->L }
% 0.21/0.60 fresh7(equidistant(X, extension(Y, X, Z, Z), Z, Z), true, X, extension(Y, X, Z, Z))
% 0.21/0.60 = { by axiom 11 (segment_construction2) }
% 0.21/0.60 fresh7(true, true, X, extension(Y, X, Z, Z))
% 0.21/0.60 = { by axiom 1 (identity_for_equidistance) }
% 0.21/0.60 true
% 0.21/0.60
% 0.21/0.60 Lemma 18: between(X, Y, Y) = true.
% 0.21/0.60 Proof:
% 0.21/0.60 between(X, Y, Y)
% 0.21/0.60 = { by axiom 5 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(true, true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 3 (between_substitution3) R->L }
% 0.21/0.60 fresh2(fresh9(true, true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 2 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(fresh9(fresh(true, true, Y, Y, Y), true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 6 (segment_construction1) R->L }
% 0.21/0.60 fresh2(fresh9(fresh(between(Y, Y, extension(Y, Y, W, V)), true, Y, Y, Y), true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 14 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(fresh9(fresh2(between(Y, Y, extension(Y, Y, W, V)), true, Y, Y, extension(Y, Y, W, V), Y), true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 6 (segment_construction1) }
% 0.21/0.60 fresh2(fresh9(fresh2(true, true, Y, Y, extension(Y, Y, W, V), Y), true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 5 (transitivity_for_betweeness) }
% 0.21/0.60 fresh2(fresh9(between(Y, Y, Y), true, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 12 (between_substitution3) R->L }
% 0.21/0.60 fresh2(fresh8(equalish(Y, extension(X, Y, Z, Z)), true, Y, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by lemma 17 }
% 0.21/0.60 fresh2(fresh8(true, true, Y, extension(X, Y, Z, Z), Y, Y), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 4 (between_substitution3) }
% 0.21/0.60 fresh2(between(Y, Y, extension(X, Y, Z, Z)), true, X, Y, extension(X, Y, Z, Z), Y)
% 0.21/0.60 = { by axiom 14 (transitivity_for_betweeness) }
% 0.21/0.60 fresh(between(X, Y, extension(X, Y, Z, Z)), true, X, Y, Y)
% 0.21/0.60 = { by axiom 6 (segment_construction1) }
% 0.21/0.60 fresh(true, true, X, Y, Y)
% 0.21/0.60 = { by axiom 2 (transitivity_for_betweeness) }
% 0.21/0.60 true
% 0.21/0.60
% 0.21/0.60 Goal 1 (prove_a_between_a_and_b): between(a, a, b) = true.
% 0.21/0.60 Proof:
% 0.21/0.60 between(a, a, b)
% 0.21/0.60 = { by axiom 5 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(true, true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 2 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(fresh(true, true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 3 (between_substitution3) R->L }
% 0.21/0.60 fresh2(fresh(fresh9(true, true, extension(X, b, Y, Y), a, b), true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by lemma 18 R->L }
% 0.21/0.60 fresh2(fresh(fresh9(between(a, b, b), true, extension(X, b, Y, Y), a, b), true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 12 (between_substitution3) R->L }
% 0.21/0.60 fresh2(fresh(fresh8(equalish(b, extension(X, b, Y, Y)), true, b, extension(X, b, Y, Y), a, b), true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by lemma 17 }
% 0.21/0.60 fresh2(fresh(fresh8(true, true, b, extension(X, b, Y, Y), a, b), true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 4 (between_substitution3) }
% 0.21/0.60 fresh2(fresh(between(a, b, extension(X, b, Y, Y)), true, a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 14 (transitivity_for_betweeness) R->L }
% 0.21/0.60 fresh2(fresh2(between(b, outer_pasch(a, b, b, a, a), extension(X, b, Y, Y)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 4 (between_substitution3) R->L }
% 0.21/0.60 fresh2(fresh2(fresh8(true, true, b, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by lemma 17 R->L }
% 0.21/0.60 fresh2(fresh2(fresh8(equalish(b, extension(X, b, Y, Y)), true, b, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 12 (between_substitution3) }
% 0.21/0.60 fresh2(fresh2(fresh9(between(b, outer_pasch(a, b, b, a, a), b), true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 9 (outer_pasch1) R->L }
% 0.21/0.60 fresh2(fresh2(fresh9(fresh6(true, true, b, a, a, b, a), true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by lemma 18 R->L }
% 0.21/0.60 fresh2(fresh2(fresh9(fresh6(between(b, a, a), true, b, a, a, b, a), true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 15 (outer_pasch1) }
% 0.21/0.60 fresh2(fresh2(fresh9(fresh5(between(b, a, a), true, b, a, a, b, a), true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by lemma 18 }
% 0.21/0.60 fresh2(fresh2(fresh9(fresh5(true, true, b, a, a, b, a), true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 7 (outer_pasch1) }
% 0.21/0.60 fresh2(fresh2(fresh9(true, true, extension(X, b, Y, Y), b, outer_pasch(a, b, b, a, a)), true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 3 (between_substitution3) }
% 0.21/0.60 fresh2(fresh2(true, true, a, b, extension(X, b, Y, Y), outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 5 (transitivity_for_betweeness) }
% 0.21/0.60 fresh2(between(a, b, outer_pasch(a, b, b, a, a)), true, a, a, outer_pasch(a, b, b, a, a), b)
% 0.21/0.60 = { by axiom 14 (transitivity_for_betweeness) }
% 0.21/0.60 fresh(between(a, a, outer_pasch(a, b, b, a, a)), true, a, a, b)
% 0.21/0.60 = { by axiom 10 (outer_pasch2) R->L }
% 0.21/0.60 fresh(fresh4(true, true, b, a, a, b, a), true, a, a, b)
% 0.21/0.60 = { by lemma 18 R->L }
% 0.21/0.60 fresh(fresh4(between(b, a, a), true, b, a, a, b, a), true, a, a, b)
% 0.21/0.60 = { by axiom 16 (outer_pasch2) }
% 0.21/0.60 fresh(fresh3(between(b, a, a), true, b, a, a, b, a), true, a, a, b)
% 0.21/0.60 = { by lemma 18 }
% 0.21/0.60 fresh(fresh3(true, true, b, a, a, b, a), true, a, a, b)
% 0.21/0.60 = { by axiom 8 (outer_pasch2) }
% 0.21/0.60 fresh(true, true, a, a, b)
% 0.21/0.60 = { by axiom 2 (transitivity_for_betweeness) }
% 0.21/0.60 true
% 0.21/0.60 % SZS output end Proof
% 0.21/0.60
% 0.21/0.60 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------