Datasets:

roseking
/

openAI-math-dataset

import OAI.GroupTheory.Thompson.BasicInterval

namespace OAI

/-!
# Nesting and normalization of basic dyadic intervals

The endpoint comparisons are made on a common dyadic grid. In particular,
intersecting open interiors force one basic interval to contain the other.
-/

noncomputable section

namespace ThompsonNonamenability.BasicInterval

open Set

/-- Express a left endpoint on any finer dyadic grid. -/
theorem left_eq_on_finer_grid (I : BasicInterval) {s : ℕ} (h : I.depth ≤ s) :
    I.left = ((I.index.val * 2 ^ (s - I.depth) : ℕ) : ℝ) / (2 : ℝ) ^ s := by
  have hp : (2 : ℝ) ^ s = (2 : ℝ) ^ I.depth * (2 : ℝ) ^ (s - I.depth) := by
    rw [← pow_add, Nat.add_sub_cancel' h]
  have hI : (2 : ℝ) ^ I.depth ≠ 0 := by positivity
  have hD : (2 : ℝ) ^ (s - I.depth) ≠ 0 := by positivity
  simp only [left, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat, hp]
  field_simp [hI, hD]

/-- Express a right endpoint on any finer dyadic grid. -/
theorem right_eq_on_finer_grid (I : BasicInterval) {s : ℕ} (h : I.depth ≤ s) :
    I.right = (((I.index.val + 1) * 2 ^ (s - I.depth) : ℕ) : ℝ) /
      (2 : ℝ) ^ s := by
  have hp : (2 : ℝ) ^ s = (2 : ℝ) ^ I.depth * (2 : ℝ) ^ (s - I.depth) := by
    rw [← pow_add, Nat.add_sub_cancel' h]
  have hI : (2 : ℝ) ^ I.depth ≠ 0 := by positivity
  have hD : (2 : ℝ) ^ (s - I.depth) ≠ 0 := by positivity
  simp only [right, Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_pow,
    Nat.cast_ofNat, hp]
  field_simp [hI, hD]

/-- A contained basic interval has at least the depth of its container. -/
theorem depth_le_of_contained {I K : BasicInterval} (h : K.Contained I) :
    I.depth ≤ K.depth := by
  have hl := length_le_of_contained h
  change ((2 : ℝ) ^ K.depth)⁻¹ ≤ ((2 : ℝ) ^ I.depth)⁻¹ at hl
  have hp : (2 : ℝ) ^ I.depth ≤ (2 : ℝ) ^ K.depth :=
    (inv_le_inv₀ (by positivity) (by positivity)).mp hl
  exact (pow_le_pow_iff_right₀ (by norm_num : (1 : ℝ) < 2)).mp hp

/-- Containment gives the integer endpoint bounds on the finer grid. -/
theorem index_bounds_of_contained {I K : BasicInterval} (h : K.Contained I) :
    I.index.val * 2 ^ (K.depth - I.depth) ≤ K.index.val ∧
      K.index.val + 1 ≤ (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by
  have hd := depth_le_of_contained h
  have hp : 0 < (2 : ℝ) ^ K.depth := by positivity
  have hl := h.1
  have hr := h.2
  rw [left_eq_on_finer_grid I hd, left] at hl
  rw [right_eq_on_finer_grid I hd, right] at hr
  constructor
  · exact_mod_cast (div_le_div_iff_of_pos_right hp).mp hl
  · exact_mod_cast (div_le_div_iff_of_pos_right hp).mp hr

/-- A subinterval is precisely the image of a basic interval under the
container's affine chart. -/
theorem contained_iff_exists_comp {I K : BasicInterval} :
    K.Contained I ↔ ∃ J : BasicInterval, I.comp J = K := by
  constructor
  · intro h
    have hd := depth_le_of_contained h
    obtain ⟨hl, hr⟩ := index_bounds_of_contained h
    have hn : K.index.val - I.index.val * 2 ^ (K.depth - I.depth) <
        2 ^ (K.depth - I.depth) := by
      rw [Nat.add_mul, Nat.one_mul] at hr
      omega
    let J : BasicInterval :=
      ⟨K.depth - I.depth, ⟨K.index.val - I.index.val * 2 ^ (K.depth - I.depth), hn⟩⟩
    refine ⟨J, ext_depth_index ?_ ?_⟩
    · change I.depth + (K.depth - I.depth) = K.depth
      omega
    · change I.index.val * 2 ^ (K.depth - I.depth) +
          (K.index.val - I.index.val * 2 ^ (K.depth - I.depth)) = K.index.val
      omega
  · rintro ⟨J, rfl⟩
    exact comp_contained I J

/-- The affine preimage of a contained basic interval is unique. -/
theorem contained_iff_exists_unique_comp {I K : BasicInterval} :
    K.Contained I ↔ ∃! J : BasicInterval, I.comp J = K := by
  constructor
  · intro h
    obtain ⟨J, hJ⟩ := contained_iff_exists_comp.mp h
    refine ⟨J, hJ, ?_⟩
    intro J' hJ'
    exact comp_injective I (hJ'.trans hJ.symm)
  · rintro ⟨J, hJ, _⟩
    exact contained_iff_exists_comp.mpr ⟨J, hJ⟩

alias containment_iff_exists_unique_comp := contained_iff_exists_unique_comp

/-- The basic interval obtained by applying the inverse chart of `I` to `K`. -/
def normalize (I K : BasicInterval) (h : K.Contained I) : BasicInterval :=
  Classical.choose (contained_iff_exists_comp.mp h)

theorem comp_normalize (I K : BasicInterval) (h : K.Contained I) :
    I.comp (normalize I K h) = K :=
  Classical.choose_spec (contained_iff_exists_comp.mp h)

@[simp] theorem normalize_comp (I J : BasicInterval) (h : (I.comp J).Contained I) :
    normalize I (I.comp J) h = J := by
  apply comp_injective I
  exact comp_normalize I (I.comp J) h

/-- At ordered depths, overlap of open interiors implies containment. -/
theorem contained_of_overlap_of_depth_le {I K : BasicInterval}
    (hd : I.depth ≤ K.depth) (h : (I.interior ∩ K.interior).Nonempty) :
    K.Contained I := by
  obtain ⟨x, hxI, hxK⟩ := h
  have hl : I.left < K.right := lt_trans hxI.1 hxK.2
  have hr : K.left < I.right := lt_trans hxK.1 hxI.2
  have hp : 0 < (2 : ℝ) ^ K.depth := by positivity
  rw [left_eq_on_finer_grid I hd, right] at hl
  rw [right_eq_on_finer_grid I hd, left] at hr
  have hln : I.index.val * 2 ^ (K.depth - I.depth) < K.index.val + 1 := by
    exact_mod_cast (div_lt_div_iff_of_pos_right hp).mp hl
  have hrn : K.index.val < (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by
    exact_mod_cast (div_lt_div_iff_of_pos_right hp).mp hr
  have hl' : I.index.val * 2 ^ (K.depth - I.depth) ≤ K.index.val := by omega
  have hr' : K.index.val + 1 ≤ (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by omega
  constructor
  · rw [left_eq_on_finer_grid I hd, left]
    apply (div_le_div_iff_of_pos_right hp).mpr
    exact_mod_cast hl'
  · rw [right_eq_on_finer_grid I hd, right]
    apply (div_le_div_iff_of_pos_right hp).mpr
    exact_mod_cast hr'

/-- Basic dyadic intervals with intersecting open interiors are nested. -/
theorem overlap_interior_nested {I J : BasicInterval}
    (h : (I.interior ∩ J.interior).Nonempty) : I.Contained J ∨ J.Contained I := by
  rcases le_total I.depth J.depth with hd | hd
  · exact Or.inr (contained_of_overlap_of_depth_le hd h)
  · apply Or.inl
    apply contained_of_overlap_of_depth_le hd
    simpa only [inter_comm] using h

end ThompsonNonamenability.BasicInterval

end

end OAI

	import OAI.GroupTheory.Thompson.BasicInterval

	namespace OAI

	/-!
	# Nesting and normalization of basic dyadic intervals

	The endpoint comparisons are made on a common dyadic grid. In particular,
	intersecting open interiors force one basic interval to contain the other.
	-/

	noncomputable section

	namespace ThompsonNonamenability.BasicInterval

	open Set

	/-- Express a left endpoint on any finer dyadic grid. -/
	theorem left_eq_on_finer_grid (I : BasicInterval) {s : ℕ} (h : I.depth ≤ s) :
	I.left = ((I.index.val * 2 ^ (s - I.depth) : ℕ) : ℝ) / (2 : ℝ) ^ s := by
	have hp : (2 : ℝ) ^ s = (2 : ℝ) ^ I.depth * (2 : ℝ) ^ (s - I.depth) := by
	rw [← pow_add, Nat.add_sub_cancel' h]
	have hI : (2 : ℝ) ^ I.depth ≠ 0 := by positivity
	have hD : (2 : ℝ) ^ (s - I.depth) ≠ 0 := by positivity
	simp only [left, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat, hp]
	field_simp [hI, hD]

	/-- Express a right endpoint on any finer dyadic grid. -/
	theorem right_eq_on_finer_grid (I : BasicInterval) {s : ℕ} (h : I.depth ≤ s) :
	I.right = (((I.index.val + 1) * 2 ^ (s - I.depth) : ℕ) : ℝ) /
	(2 : ℝ) ^ s := by
	have hp : (2 : ℝ) ^ s = (2 : ℝ) ^ I.depth * (2 : ℝ) ^ (s - I.depth) := by
	rw [← pow_add, Nat.add_sub_cancel' h]
	have hI : (2 : ℝ) ^ I.depth ≠ 0 := by positivity
	have hD : (2 : ℝ) ^ (s - I.depth) ≠ 0 := by positivity
	simp only [right, Nat.cast_mul, Nat.cast_add, Nat.cast_one, Nat.cast_pow,
	Nat.cast_ofNat, hp]
	field_simp [hI, hD]

	/-- A contained basic interval has at least the depth of its container. -/
	theorem depth_le_of_contained {I K : BasicInterval} (h : K.Contained I) :
	I.depth ≤ K.depth := by
	have hl := length_le_of_contained h
	change ((2 : ℝ) ^ K.depth)⁻¹ ≤ ((2 : ℝ) ^ I.depth)⁻¹ at hl
	have hp : (2 : ℝ) ^ I.depth ≤ (2 : ℝ) ^ K.depth :=
	(inv_le_inv₀ (by positivity) (by positivity)).mp hl
	exact (pow_le_pow_iff_right₀ (by norm_num : (1 : ℝ) < 2)).mp hp

	/-- Containment gives the integer endpoint bounds on the finer grid. -/
	theorem index_bounds_of_contained {I K : BasicInterval} (h : K.Contained I) :
	I.index.val * 2 ^ (K.depth - I.depth) ≤ K.index.val ∧
	K.index.val + 1 ≤ (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by
	have hd := depth_le_of_contained h
	have hp : 0 < (2 : ℝ) ^ K.depth := by positivity
	have hl := h.1
	have hr := h.2
	rw [left_eq_on_finer_grid I hd, left] at hl
	rw [right_eq_on_finer_grid I hd, right] at hr
	constructor
	· exact_mod_cast (div_le_div_iff_of_pos_right hp).mp hl
	· exact_mod_cast (div_le_div_iff_of_pos_right hp).mp hr

	/-- A subinterval is precisely the image of a basic interval under the
	container's affine chart. -/
	theorem contained_iff_exists_comp {I K : BasicInterval} :
	K.Contained I ↔ ∃ J : BasicInterval, I.comp J = K := by
	constructor
	· intro h
	have hd := depth_le_of_contained h
	obtain ⟨hl, hr⟩ := index_bounds_of_contained h
	have hn : K.index.val - I.index.val * 2 ^ (K.depth - I.depth) <
	2 ^ (K.depth - I.depth) := by
	rw [Nat.add_mul, Nat.one_mul] at hr
	omega
	let J : BasicInterval :=
	⟨K.depth - I.depth, ⟨K.index.val - I.index.val * 2 ^ (K.depth - I.depth), hn⟩⟩
	refine ⟨J, ext_depth_index ?_ ?_⟩
	· change I.depth + (K.depth - I.depth) = K.depth
	omega
	· change I.index.val * 2 ^ (K.depth - I.depth) +
	(K.index.val - I.index.val * 2 ^ (K.depth - I.depth)) = K.index.val
	omega
	· rintro ⟨J, rfl⟩
	exact comp_contained I J

	/-- The affine preimage of a contained basic interval is unique. -/
	theorem contained_iff_exists_unique_comp {I K : BasicInterval} :
	K.Contained I ↔ ∃! J : BasicInterval, I.comp J = K := by
	constructor
	· intro h
	obtain ⟨J, hJ⟩ := contained_iff_exists_comp.mp h
	refine ⟨J, hJ, ?_⟩
	intro J' hJ'
	exact comp_injective I (hJ'.trans hJ.symm)
	· rintro ⟨J, hJ, _⟩
	exact contained_iff_exists_comp.mpr ⟨J, hJ⟩

	alias containment_iff_exists_unique_comp := contained_iff_exists_unique_comp

	/-- The basic interval obtained by applying the inverse chart of `I` to `K`. -/
	def normalize (I K : BasicInterval) (h : K.Contained I) : BasicInterval :=
	Classical.choose (contained_iff_exists_comp.mp h)

	theorem comp_normalize (I K : BasicInterval) (h : K.Contained I) :
	I.comp (normalize I K h) = K :=
	Classical.choose_spec (contained_iff_exists_comp.mp h)

	@[simp] theorem normalize_comp (I J : BasicInterval) (h : (I.comp J).Contained I) :
	normalize I (I.comp J) h = J := by
	apply comp_injective I
	exact comp_normalize I (I.comp J) h

	/-- At ordered depths, overlap of open interiors implies containment. -/
	theorem contained_of_overlap_of_depth_le {I K : BasicInterval}
	(hd : I.depth ≤ K.depth) (h : (I.interior ∩ K.interior).Nonempty) :
	K.Contained I := by
	obtain ⟨x, hxI, hxK⟩ := h
	have hl : I.left < K.right := lt_trans hxI.1 hxK.2
	have hr : K.left < I.right := lt_trans hxK.1 hxI.2
	have hp : 0 < (2 : ℝ) ^ K.depth := by positivity
	rw [left_eq_on_finer_grid I hd, right] at hl
	rw [right_eq_on_finer_grid I hd, left] at hr
	have hln : I.index.val * 2 ^ (K.depth - I.depth) < K.index.val + 1 := by
	exact_mod_cast (div_lt_div_iff_of_pos_right hp).mp hl
	have hrn : K.index.val < (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by
	exact_mod_cast (div_lt_div_iff_of_pos_right hp).mp hr
	have hl' : I.index.val * 2 ^ (K.depth - I.depth) ≤ K.index.val := by omega
	have hr' : K.index.val + 1 ≤ (I.index.val + 1) * 2 ^ (K.depth - I.depth) := by omega
	constructor
	· rw [left_eq_on_finer_grid I hd, left]
	apply (div_le_div_iff_of_pos_right hp).mpr
	exact_mod_cast hl'
	· rw [right_eq_on_finer_grid I hd, right]
	apply (div_le_div_iff_of_pos_right hp).mpr
	exact_mod_cast hr'

	/-- Basic dyadic intervals with intersecting open interiors are nested. -/
	theorem overlap_interior_nested {I J : BasicInterval}
	(h : (I.interior ∩ J.interior).Nonempty) : I.Contained J ∨ J.Contained I := by
	rcases le_total I.depth J.depth with hd \| hd
	· exact Or.inr (contained_of_overlap_of_depth_le hd h)
	· apply Or.inl
	apply contained_of_overlap_of_depth_le hd
	simpa only [inter_comm] using h

	end ThompsonNonamenability.BasicInterval

	end

	end OAI