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namespace OAI
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namespace VertexCover.GridCoupling
open MeasureTheory ProbabilityTheory
open scoped ENNReal
noncomputable def midpoint (p : ℕ) (i : Fin p) : ℝ := -1 + (2*(i:ℝ)+1)/p
noncomputable def index (p : ℕ) (hp : 0 < p) (x : ℝ) : Fin p :=
⟨min ⌊(p:ℝ)*(x+1)/2⌋₊ (p-1), (min_le_right _ _).trans_lt (by omega)⟩
theorem index_measurable (p : ℕ) (hp : 0 < p) : Measurable (index p hp) := by
let g : ℕ → Fin p := fun k => ⟨min k (p-1), (min_le_right _ _).trans_lt (by omega)⟩
exact (measurable_of_countable g).comp
(((measurable_const.mul (measurable_id.add_const 1)).div_const 2).nat_floor)
theorem floor_lt_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
(hx : x ∈ Set.Ico (-1 : ℝ) 1) : ⌊(p:ℝ)*(x+1)/2⌋₊ < p := by
apply (Nat.floor_lt' (Nat.ne_of_gt hp)).mpr
have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
nlinarith [mul_lt_mul_of_pos_left hx.2 hpR]
theorem index_val_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
(hx : x ∈ Set.Ico (-1 : ℝ) 1) : (index p hp x).val = ⌊(p:ℝ)*(x+1)/2⌋₊ := by
exact min_eq_left (by have h := floor_lt_on_interval p hp hx; omega)
theorem index_eq_iff_on_interval (p : ℕ) (hp : 0 < p) {x : ℝ}
(hx : x ∈ Set.Ico (-1 : ℝ) 1) (i : Fin p) :
index p hp x = i ↔ -1 + 2*(i:ℝ)/p ≤ x ∧ x < -1 + 2*((i:ℝ)+1)/p := by
have hz : 0 ≤ (p:ℝ)*(x+1)/2 := div_nonneg (mul_nonneg (Nat.cast_nonneg _) (by linarith [hx.1])) (by norm_num)
have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
rw [Fin.ext_iff, index_val_on_interval p hp hx, Nat.floor_eq_iff hz]
constructor
· rintro ⟨hl, hu⟩
constructor
· have h : 2*(i:ℝ)/p ≤ x+1 := (div_le_iff₀ hpR).mpr (by nlinarith)
linarith
· have h : x+1 < 2*((i:ℝ)+1)/p := (lt_div_iff₀ hpR).mpr (by nlinarith)
linarith
· rintro ⟨hl, hu⟩
have hL : 2*(i:ℝ) ≤ (x+1)*p := (div_le_iff₀ hpR).mp (by linarith)
have hU : (x+1)*p < 2*((i:ℝ)+1) := (lt_div_iff₀ hpR).mp (by linarith)
constructor <;> nlinarith
end VertexCover.GridCoupling
namespace VertexCover.GridCoupling
open MeasureTheory ProbabilityTheory
open scoped ENNReal
theorem cell_subset_interval (p : ℕ) (hp : 0 < p) (i : Fin p) :
Set.Ico (-1 + 2*(i:ℝ)/p) (-1 + 2*((i:ℝ)+1)/p) ⊆ Set.Ico (-1:ℝ) 1 := by
intro x hx
have hpR : (0:ℝ) < p := Nat.cast_pos.mpr hp
have hi : (i:ℝ)+1 ≤ p := by exact_mod_cast i.isLt
have hz : 0 ≤ 2*(i:ℝ)/p := div_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg _)) hpR.le
have hu : 2*((i:ℝ)+1)/p ≤ 2 := (div_le_iff₀ hpR).mpr (by nlinarith)
exact ⟨by linarith [hx.1], by linarith [hx.2]⟩
theorem preimage_cell (p : ℕ) (hp : 0 < p) (i : Fin p) :
(index p hp ⁻¹' {i}) ∩ Set.Ico (-1:ℝ) 1 =
Set.Ico (-1 + 2*(i:ℝ)/p) (-1 + 2*((i:ℝ)+1)/p) := by
ext x
constructor
· rintro ⟨hi, hx⟩
exact (index_eq_iff_on_interval p hp hx i).mp hi
· intro hx
have hx' := cell_subset_interval p hp i hx
exact ⟨(index_eq_iff_on_interval p hp hx' i).mpr hx, hx'⟩
theorem rounding_error (p : ℕ) (hp : 0 < p) {x : ℝ}
(hx : x ∈ Set.Ico (-1 : ℝ) 1) : |x - midpoint p (index p hp x)| ≤ 1/p := by
have h := (index_eq_iff_on_interval p hp hx (index p hp x)).mp rfl
unfold midpoint
rw [abs_le]
simp only [mul_add, mul_one, add_div] at h ⊢
have he : (2:ℝ)/p = 2*(1/p) := by ring
rw [he] at h
constructor <;> linarith
end VertexCover.GridCoupling
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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