Counting transfer definition Erdos9796Counting_Adapter

/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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    https://www.apache.org/licenses/LICENSE-2.0

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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import Definitions.Def_Erdos9796Mission

scoped[EuclideanGeometry] notation "ℝ²" => EuclideanSpace ℝ (Fin 2)

open scoped EuclideanGeometry Finset Real

/-- Oriented angles on the two-dimensional Euclidean space. -/
noncomputable instance Module.orientedEuclideanSpaceFinTwo :
    Module.Oriented ℝ ℝ² (Fin 2) :=
  ⟨Basis.orientation <| PiLp.basisFun 2 _ _⟩

/-- The two-dimensional Euclidean space has real dimension two. -/
instance fact_finrank_euclideanSpace_fin_two :
    Fact (Module.finrank ℝ ℝ² = 2) :=
  ⟨finrank_euclideanSpace_fin⟩

namespace EuclideanGeometry

variable {V P : Type*} {n : ℕ}
variable [NormedAddCommGroup V] [InnerProductSpace ℝ V]
variable [MetricSpace P] [NormedAddTorsor V P]
variable [Module.Oriented ℝ V (Fin 2)]
variable [Fact (Module.finrank ℝ V = 2)] {p : Fin n → P}

/-- Compatibility name for the mission's extreme-point predicate. -/
abbrev ConvexIndep (S : Set ℝ²) : Prop :=
  Erdos9796Mission.ConvexIndep S

/-- A sequence of points forms a counter-clockwise convex polygon. -/
def IsCcwConvexPolygon (p : Fin n → P) : Prop :=
  ∀ ⦃i j k⦄, i < j → j < k → (∡ (p i) (p j) (p k)).sign = 1

theorem IsCcwConvexPolygon.sign_oangle (hp : IsCcwConvexPolygon p)
    {i j k : Fin n} (hij : i < j) (hjk : j < k) :
    (∡ (p i) (p j) (p k)).sign = 1 :=
  hp hij hjk

set_option linter.docPrime false in
theorem IsCcwConvexPolygon.sign_oangle' (hp : IsCcwConvexPolygon p)
    {i j k : Fin n} (hij : i < j) (hjk : j < k) :
    (∡ (p j) (p k) (p i)).sign = 1 := by
  rw [EuclideanGeometry.oangle_rotate_sign]
  exact hp hij hjk

set_option linter.docPrime false in
theorem IsCcwConvexPolygon.sign_oangle'' (hp : IsCcwConvexPolygon p)
    {i j k : Fin n} (hij : i < j) (hjk : j < k) :
    (∡ (p k) (p i) (p j)).sign = 1 := by
  rw [← EuclideanGeometry.oangle_rotate_sign]
  exact hp hij hjk

end EuclideanGeometry

namespace Erdos97

abbrev HasNEquidistantPointsAt :=
  Erdos9796Mission.HasNEquidistantPointsAt

abbrev HasNEquidistantPointsOn :=
  Erdos9796Mission.HasNEquidistantPointsOn

abbrev HasNEquidistantProperty :=
  Erdos9796Mission.HasNEquidistantProperty

end Erdos97

namespace Problem97

/-- A stable namespace anchor used by independently compiled theorem nodes. -/
def prove2meCountingNamespaceAnchor : Unit := ()

namespace ConvexIndep

/-- A stable namespace anchor for lemmas attached to convex independence. -/
def prove2meCountingNamespaceAnchor : Unit := ()

end ConvexIndep

namespace CGN

/-- A stable namespace anchor for the CGN counting subgraph. -/
def prove2meCountingNamespaceAnchor : Unit := ()

end CGN

namespace MEC

/-- A stable namespace anchor for the minimum-enclosing-circle subgraph. -/
def prove2meCountingNamespaceAnchor : Unit := ()

end MEC

namespace Dumitrescu

/-- A stable namespace anchor for the Dumitrescu counting subgraph. -/
def prove2meCountingNamespaceAnchor : Unit := ()

end Dumitrescu

end Problem97