Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Geometry.Borsuk.Simplicial.MapChainsEquivSimplexAll
import OAI.Geometry.Borsuk.Simplicial.BarycenterNeg

namespace OAI

namespace BorsukNine.SimplicialChains
open Finset
open scoped RealInnerProductSpace
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [DecidableEq E]

noncomputable def refinement (R : ℝ) : ℕ →
    Chain (hemisphereComplex R : PreAbstractSimplicialComplex E) →ₗ[F₂]
      Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)
  | 0 => LinearMap.id
  | r+1 => (subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter).comp
      (refinement R r)

lemma refinement_boundary (R : ℝ) (r : ℕ) :
    (boundary (hemisphereComplex R : PreAbstractSimplicialComplex E)).comp (refinement R r) =
      (refinement R r).comp (boundary (hemisphereComplex R)) := by
  induction r with
  | zero => simp only [refinement, LinearMap.comp_id, LinearMap.id_comp]
  | succ r ih =>
    rw [refinement, ← LinearMap.comp_assoc, subdivMap_boundary _ _ _ (hemisphere_subdivision_carrier R),
      LinearMap.comp_assoc, ih, ← LinearMap.comp_assoc]

lemma refinement_augmentation (R : ℝ) (r : ℕ) :
    (augmentation (hemisphereComplex R : PreAbstractSimplicialComplex E)).comp
      (refinement R r) = augmentation (hemisphereComplex R) := by
  induction r with
  | zero => simp only [refinement, LinearMap.comp_id]
  | succ r ih =>
    rw [refinement, ← LinearMap.comp_assoc,
      subdivMap_augmentation _ _ _ (hemisphere_subdivision_carrier R), ih]

lemma refinement_norm (R : ℝ) (r : ℕ) :
    (normMap (hemisphereComplex R : PreAbstractSimplicialComplex E) (Equiv.neg E)).comp
      (refinement R r) = (refinement R r).comp (normMap (hemisphereComplex R) (Equiv.neg E)) := by
  induction r with
  | zero => simp only [refinement, LinearMap.comp_id, LinearMap.id_comp]
  | succ r ih =>
    rw [refinement, ← LinearMap.comp_assoc,
      subdivMap_norm _ _ (Equiv.neg E) (Equiv.neg E) (hemisphere_neg R) (hemisphere_neg R)
        barycenter (barycenter_neg), LinearMap.comp_assoc, ih, ← LinearMap.comp_assoc]

lemma refinement_degree (R : ℝ) (r d : ℕ) :
    degreeChains (hemisphereComplex R : PreAbstractSimplicialComplex E) d ≤
      (degreeChains (hemisphereComplex R) d).comap (refinement R r) := by
  induction r with
  | zero => intro c hc; exact hc
  | succ r ih =>
    intro c hc
    change subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter
      (refinement R r c) ∈ degreeChains (hemisphereComplex R) d
    exact subdivMap_degree (hemisphereComplex R : PreAbstractSimplicialComplex E)
      (hemisphereComplex R) barycenter d (ih hc)

lemma refinement_mesh (R : ℝ) (r n : ℕ) {c : Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)}
    (hc : c ∈ degreeChains (hemisphereComplex R) n) {D : ℝ} (hD : 0 ≤ D)
    (hmesh : ∀ s ∈ c.support, ∀ x ∈ s.val, ∀ y ∈ s.val, dist x y ≤ D)
    {t : {s : Finset E // s ∈ hemisphereComplex R}}
    (ht : t ∈ (refinement R r c).support) :
    ∀ x ∈ t.val, ∀ y ∈ t.val, dist x y ≤ (((n:ℝ)-1)/n)^r * D := by
  induction r generalizing t with
  | zero => simpa only [refinement, LinearMap.id_apply, pow_zero, one_mul] using hmesh t ht
  | succ r ih =>
    change t ∈ (subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter
      (refinement R r c)).support at ht
    obtain ⟨s,hs,hts⟩ := support_linearMap _ _ _ _ ht
    rw [subdivMap_simplex _ _ _ s.property] at hts
    have hdeg := refinement_degree R r n hc hs
    change s.val.card = n at hdeg
    have hn : 0 < n := hdeg ▸ s.property.1.card_pos
    have hfac : 0 ≤ ((n:ℝ)-1)/n := by
      have hh : (1:ℝ) ≤ n := by exact_mod_cast hn
      exact div_nonneg (by linarith) (by positivity)
    have hh := subdivSimplex_mesh R s.val n (le_of_eq hdeg)
      ((((n:ℝ)-1)/n)^r * D) (mul_nonneg (pow_nonneg hfac _) hD) (ih hs) hts
    simpa only [pow_succ', mul_assoc] using hh
omit [DecidableEq E] in
lemma hemisphere_mesh_two (R : ℝ) {s : Finset E} (hs : s ∈ hemisphereComplex R)
    {x y : E} (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ 2 := by
  obtain ⟨_, _, _, unit_norm⟩ := hs
  calc
    dist x y ≤ ‖x‖ + ‖y‖ := dist_le_norm_add_norm x y
    _ ≤ 2 := by linarith [(unit_norm x hx).1, (unit_norm y hy).1]
lemma refinement_mesh_uniform (R : ℝ) (r n d : ℕ) (hd : d ≤ n)
    {c : Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)}
    (hc : c ∈ degreeChains (hemisphereComplex R) d)
    {t : {s : Finset E // s ∈ hemisphereComplex R}}
    (ht : t ∈ (refinement R r c).support) :
    ∀ x ∈ t.val, ∀ y ∈ t.val, dist x y ≤ (((n:ℝ)-1)/n)^r * 2 := by
  have hdeg := refinement_degree R r d hc ht
  have htpos := t.property.1.card_pos
  change t.val.card = d at hdeg
  have hdpos : 0 < d := hdeg ▸ htpos
  have hfac : 0 ≤ ((d:ℝ)-1)/d := by
    have hh : (1:ℝ) ≤ d := by exact_mod_cast hdpos
    exact div_nonneg (by linarith) (by positivity)
  have hm := refinement_mesh R r d hc (D := 2) (by norm_num)
    (fun simplex _ _ first_mem _ second_mem =>
      hemisphere_mesh_two R simplex.property first_mem second_mem) ht
  intro x hx y hy
  exact (hm x hx y hy).trans (mul_le_mul_of_nonneg_right
    (pow_le_pow_left₀ hfac (barycenter_mesh_factor hdpos hd) r) (by norm_num))
end BorsukNine.SimplicialChains
end OAI

	import Mathlib
	import OAI.Geometry.Borsuk.Simplicial.MapChainsEquivSimplexAll
	import OAI.Geometry.Borsuk.Simplicial.BarycenterNeg

	namespace OAI

	namespace BorsukNine.SimplicialChains
	open Finset
	open scoped RealInnerProductSpace
	variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [DecidableEq E]

	noncomputable def refinement (R : ℝ) : ℕ →
	Chain (hemisphereComplex R : PreAbstractSimplicialComplex E) →ₗ[F₂]
	Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)
	\| 0 => LinearMap.id
	\| r+1 => (subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter).comp
	(refinement R r)

	lemma refinement_boundary (R : ℝ) (r : ℕ) :
	(boundary (hemisphereComplex R : PreAbstractSimplicialComplex E)).comp (refinement R r) =
	(refinement R r).comp (boundary (hemisphereComplex R)) := by
	induction r with
	\| zero => simp only [refinement, LinearMap.comp_id, LinearMap.id_comp]
	\| succ r ih =>
	rw [refinement, ← LinearMap.comp_assoc, subdivMap_boundary _ _ _ (hemisphere_subdivision_carrier R),
	LinearMap.comp_assoc, ih, ← LinearMap.comp_assoc]

	lemma refinement_augmentation (R : ℝ) (r : ℕ) :
	(augmentation (hemisphereComplex R : PreAbstractSimplicialComplex E)).comp
	(refinement R r) = augmentation (hemisphereComplex R) := by
	induction r with
	\| zero => simp only [refinement, LinearMap.comp_id]
	\| succ r ih =>
	rw [refinement, ← LinearMap.comp_assoc,
	subdivMap_augmentation _ _ _ (hemisphere_subdivision_carrier R), ih]

	lemma refinement_norm (R : ℝ) (r : ℕ) :
	(normMap (hemisphereComplex R : PreAbstractSimplicialComplex E) (Equiv.neg E)).comp
	(refinement R r) = (refinement R r).comp (normMap (hemisphereComplex R) (Equiv.neg E)) := by
	induction r with
	\| zero => simp only [refinement, LinearMap.comp_id, LinearMap.id_comp]
	\| succ r ih =>
	rw [refinement, ← LinearMap.comp_assoc,
	subdivMap_norm _ _ (Equiv.neg E) (Equiv.neg E) (hemisphere_neg R) (hemisphere_neg R)
	barycenter (barycenter_neg), LinearMap.comp_assoc, ih, ← LinearMap.comp_assoc]

	lemma refinement_degree (R : ℝ) (r d : ℕ) :
	degreeChains (hemisphereComplex R : PreAbstractSimplicialComplex E) d ≤
	(degreeChains (hemisphereComplex R) d).comap (refinement R r) := by
	induction r with
	\| zero => intro c hc; exact hc
	\| succ r ih =>
	intro c hc
	change subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter
	(refinement R r c) ∈ degreeChains (hemisphereComplex R) d
	exact subdivMap_degree (hemisphereComplex R : PreAbstractSimplicialComplex E)
	(hemisphereComplex R) barycenter d (ih hc)

	lemma refinement_mesh (R : ℝ) (r n : ℕ) {c : Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)}
	(hc : c ∈ degreeChains (hemisphereComplex R) n) {D : ℝ} (hD : 0 ≤ D)
	(hmesh : ∀ s ∈ c.support, ∀ x ∈ s.val, ∀ y ∈ s.val, dist x y ≤ D)
	{t : {s : Finset E // s ∈ hemisphereComplex R}}
	(ht : t ∈ (refinement R r c).support) :
	∀ x ∈ t.val, ∀ y ∈ t.val, dist x y ≤ (((n:ℝ)-1)/n)^r * D := by
	induction r generalizing t with
	\| zero => simpa only [refinement, LinearMap.id_apply, pow_zero, one_mul] using hmesh t ht
	\| succ r ih =>
	change t ∈ (subdivMap (hemisphereComplex R) (hemisphereComplex R) barycenter
	(refinement R r c)).support at ht
	obtain ⟨s,hs,hts⟩ := support_linearMap _ _ _ _ ht
	rw [subdivMap_simplex _ _ _ s.property] at hts
	have hdeg := refinement_degree R r n hc hs
	change s.val.card = n at hdeg
	have hn : 0 < n := hdeg ▸ s.property.1.card_pos
	have hfac : 0 ≤ ((n:ℝ)-1)/n := by
	have hh : (1:ℝ) ≤ n := by exact_mod_cast hn
	exact div_nonneg (by linarith) (by positivity)
	have hh := subdivSimplex_mesh R s.val n (le_of_eq hdeg)
	((((n:ℝ)-1)/n)^r * D) (mul_nonneg (pow_nonneg hfac _) hD) (ih hs) hts
	simpa only [pow_succ', mul_assoc] using hh

	omit [DecidableEq E] in
	lemma hemisphere_mesh_two (R : ℝ) {s : Finset E} (hs : s ∈ hemisphereComplex R)
	{x y : E} (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ 2 := by
	obtain ⟨_, _, _, unit_norm⟩ := hs
	calc
	dist x y ≤ ‖x‖ + ‖y‖ := dist_le_norm_add_norm x y
	_ ≤ 2 := by linarith [(unit_norm x hx).1, (unit_norm y hy).1]

	lemma refinement_mesh_uniform (R : ℝ) (r n d : ℕ) (hd : d ≤ n)
	{c : Chain (hemisphereComplex R : PreAbstractSimplicialComplex E)}
	(hc : c ∈ degreeChains (hemisphereComplex R) d)
	{t : {s : Finset E // s ∈ hemisphereComplex R}}
	(ht : t ∈ (refinement R r c).support) :
	∀ x ∈ t.val, ∀ y ∈ t.val, dist x y ≤ (((n:ℝ)-1)/n)^r * 2 := by
	have hdeg := refinement_degree R r d hc ht
	have htpos := t.property.1.card_pos
	change t.val.card = d at hdeg
	have hdpos : 0 < d := hdeg ▸ htpos
	have hfac : 0 ≤ ((d:ℝ)-1)/d := by
	have hh : (1:ℝ) ≤ d := by exact_mod_cast hdpos
	exact div_nonneg (by linarith) (by positivity)
	have hm := refinement_mesh R r d hc (D := 2) (by norm_num)
	(fun simplex _ _ first_mem _ second_mem =>
	hemisphere_mesh_two R simplex.property first_mem second_mem) ht
	intro x hx y hy
	exact (hm x hx y hy).trans (mul_le_mul_of_nonneg_right
	(pow_le_pow_left₀ hfac (barycenter_mesh_factor hdpos hd) r) (by norm_num))

	end BorsukNine.SimplicialChains

	end OAI