| import Mathlib |
| import OAI.Probability.LatticeSurface.Coercivity.GridTrace |
|
|
| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators |
|
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| namespace GridTrace |
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| namespace NatGrid |
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| end NatGrid |
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| end GridTrace |
|
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| namespace GridTrace.NatGrid |
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| def lo (x : Vertex) (r : ℕ) : Vertex := (x.1-r,x.2-r) |
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| def hi (N : ℕ) (x : Vertex) (r : ℕ) : Vertex := (min N (x.1+r),min N (x.2+r)) |
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| def box (N : ℕ) (x : Vertex) (r : ℕ) : Finset Vertex := rect (lo x r) (hi N x r) |
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| def square (N : ℕ) : Finset Vertex := rect (0,0) (N,N) |
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|
| abbrev Edge := Vertex ⊕ Vertex |
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| def edges (l u : Vertex) : Finset Edge := |
| ((Finset.Ico l.1 u.1) ×ˢ (Finset.Icc l.2 u.2)).disjSum |
| ((Finset.Icc l.1 u.1) ×ˢ (Finset.Ico l.2 u.2)) |
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| def edgeGrad (v : Vertex → ℝ) : Edge → ℝ |
| | .inl (i,j) => |v (i+1,j)-v (i,j)| |
| | .inr (i,j) => |v (i,j+1)-v (i,j)| |
|
|
| def anchor : Edge → Vertex |
| | .inl x => x |
| | .inr x => x |
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| lemma square_mem {N : ℕ} {x : Vertex} : x ∈ square N ↔ x.1 ≤ N ∧ x.2 ≤ N := by |
| simp [square,rect] |
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| lemma mem_box_iff {N r : ℕ} {x y : Vertex} : |
| y ∈ box N x r ↔ y.1 ≤ N ∧ y.2 ≤ N ∧ |
| y.1 ≤ x.1+r ∧ x.1 ≤ y.1+r ∧ y.2 ≤ x.2+r ∧ x.2 ≤ y.2+r := by |
| simp only [box,rect,lo,hi,Finset.mem_product,Finset.mem_Icc,le_min_iff] |
| omega |
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|
| lemma box_self {N : ℕ} {x : Vertex} (hx : x ∈ square N) (r : ℕ) : x ∈ box N x r := by |
| rw [mem_box_iff] |
| have hx := square_mem.mp hx |
| omega |
|
|
| lemma box_subset {N r s : ℕ} {x : Vertex} (hrs : r ≤ s) : box N x r ⊆ box N x s := by |
| intro y hy |
| rw [mem_box_iff] at hy ⊢ |
| omega |
|
|
| lemma box_subset_square (N : ℕ) (x : Vertex) (r : ℕ) : box N x r ⊆ square N := by |
| intro y hy |
| exact square_mem.mpr ⟨(mem_box_iff.mp hy).1,(mem_box_iff.mp hy).2.1⟩ |
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|
| lemma box_zero {N : ℕ} {x : Vertex} (hx : x ∈ square N) : box N x 0 = {x} := by |
| ext y |
| rw [mem_box_iff,Finset.mem_singleton] |
| have hx := square_mem.mp hx |
| constructor |
| · intro h |
| exact Prod.ext (by omega) (by omega) |
| · rintro rfl;omega |
|
|
| lemma box_full {N r : ℕ} {x : Vertex} (hx : x ∈ square N) (hr : N ≤ r) : |
| box N x r = square N := by |
| ext y |
| rw [mem_box_iff,square_mem] |
| have hx := square_mem.mp hx |
| omega |
|
|
| lemma side_lower {N x r : ℕ} (hx : x ≤ N) (hr : r ≤ N) : |
| r+1 ≤ (Finset.Icc (x-r) (min N (x+r))).card := by |
| rw [Nat.card_Icc] |
| omega |
|
|
| lemma side_upper (N x r : ℕ) : |
| (Finset.Icc (x-r) (min N (x+r))).card ≤ 2*r+1 := by |
| rw [Nat.card_Icc] |
| omega |
|
|
| lemma box_card_lower {N r : ℕ} {x : Vertex} (hx : x ∈ square N) (hr : r ≤ N) : |
| (r+1)^2 ≤ (box N x r).card := by |
| have hx := square_mem.mp hx |
| simpa [box,rect,lo,hi,pow_two] using Nat.mul_le_mul (side_lower hx.1 hr) (side_lower hx.2 hr) |
|
|
| lemma edges_sum (v : Vertex → ℝ) (l u : Vertex) : |
| ∑ e ∈ edges l u, edgeGrad v e = grad₁ v l u+grad₂ v l u := by |
| simp [edges,Finset.sum_disjSum,Finset.sum_product,edgeGrad,grad₁,grad₂] |
|
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| lemma edges_subset {l u l' u' : Vertex} (hl1 : l'.1 ≤ l.1) (hl2 : l'.2 ≤ l.2) |
| (hu1 : u.1 ≤ u'.1) (hu2 : u.2 ≤ u'.2) : edges l u ⊆ edges l' u' := by |
| intro e he |
| cases e with |
| | inl e => |
| simp only [edges,Finset.inl_mem_disjSum,Finset.mem_product,Finset.mem_Ico,Finset.mem_Icc] at he ⊢ |
| omega |
| | inr e => |
| simp only [edges,Finset.inr_mem_disjSum,Finset.mem_product,Finset.mem_Ico,Finset.mem_Icc] at he ⊢ |
| omega |
|
|
| lemma box_edges_subset (N : ℕ) (x : Vertex) (r : ℕ) : |
| edges (lo x r) (hi N x r) ⊆ edges (0,0) (N,N) := by |
| apply edges_subset <;> simp [lo,hi] |
|
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| lemma anchor_mem_rect {l u : Vertex} {e : Edge} (he : e ∈ edges l u) : anchor e ∈ rect l u := by |
| cases e with |
| | inl e => |
| simp only [edges,Finset.inl_mem_disjSum,Finset.mem_product,Finset.mem_Ico,Finset.mem_Icc] at he |
| exact Finset.mem_product.mpr ⟨Finset.mem_Icc.mpr ⟨he.1.1,he.1.2.le⟩,by simpa [anchor] using he.2⟩ |
| | inr e => |
| simp only [edges,Finset.inr_mem_disjSum,Finset.mem_product,Finset.mem_Ico,Finset.mem_Icc] at he |
| exact Finset.mem_product.mpr ⟨by simpa [anchor] using he.1,Finset.mem_Icc.mpr ⟨he.2.1,he.2.2.le⟩⟩ |
|
|
| lemma edges_card_bound (N : ℕ) (x : Vertex) (r : ℕ) : |
| (edges (lo x r) (hi N x r)).card ≤ 2*(2*r+1)^2 := by |
| have hx := side_upper N x.1 r |
| have hy := side_upper N x.2 r |
| have hx' : (Finset.Ico (x.1-r) (min N (x.1+r))).card ≤ 2*r+1 := by |
| exact (Finset.card_le_card (by intro i hi;exact Finset.mem_Icc.mpr ⟨(Finset.mem_Ico.mp hi).1,(Finset.mem_Ico.mp hi).2.le⟩)).trans hx |
| have hy' : (Finset.Ico (x.2-r) (min N (x.2+r))).card ≤ 2*r+1 := by |
| exact (Finset.card_le_card (by intro i hi;exact Finset.mem_Icc.mpr ⟨(Finset.mem_Ico.mp hi).1,(Finset.mem_Ico.mp hi).2.le⟩)).trans hy |
| simp only [edges,lo,hi,Finset.card_disjSum,Finset.card_product] |
| nlinarith [Nat.mul_le_mul hx' hy, Nat.mul_le_mul hx hy'] |
|
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| lemma dyadic_step (v : Vertex → ℝ) {N : ℕ} {x : Vertex} (hx : x ∈ square N) |
| (s : ℕ) (hs : 1 ≤ s) (hsN : s ≤ N) : |
| |avg (box N x (s-1)) v-avg (box N x (2*s-1)) v| ≤ |
| (4/(s:ℝ))*(∑ e ∈ edges (lo x (2*s-1)) (hi N x (2*s-1)), edgeGrad v e) := by |
| have hA : (box N x (s-1)).Nonempty := ⟨x,box_self hx _⟩ |
| have hcard : (s:ℝ)^2 ≤ ((box N x (s-1)).card:ℝ) := by |
| have h := box_card_lower hx (show s-1 ≤ N by omega) |
| have he : s-1+1=s := by omega |
| rw [he] at h |
| exact_mod_cast h |
| have hside (i : ℕ) : ((Finset.Icc (i-(2*s-1)) (min N (i+(2*s-1)))).card:ℝ) ≤ 4*s := by |
| have h := side_upper N i (2*s-1) |
| have hh : 2*(2*s-1)+1 ≤ 4*s := by omega |
| exact_mod_cast h.trans hh |
| have hsp : (0:ℝ) < s := by exact_mod_cast hs |
| have hc : (0:ℝ) < (box N x (s-1)).card := Nat.cast_pos.mpr (Finset.card_pos.mpr hA) |
| have h := rect_avg_bound v (lo x (2*s-1)) (hi N x (2*s-1)) |
| (box N x (s-1)) hA (box_subset (by omega)) (4*s) (hside x.1) (hside x.2) |
| rw [← edges_sum] at h |
| exact h.trans (mul_le_mul_of_nonneg_right (by |
| apply (div_le_div_iff₀ hc hsp).mpr |
| nlinarith) (Finset.sum_nonneg (fun e _ => by cases e <;> exact abs_nonneg _))) |
|
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| end GridTrace.NatGrid |
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| end |
|
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| end OAI |
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|
