| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentEnergy |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentLimitSupport |
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| namespace OAI |
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| namespace ProjectionComparison |
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| open Set Filter ClusterDisplacement |
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| attribute [local instance] Classical.propDecidable |
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| variable {V E : Type*} [Fintype V] [Fintype E] |
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| theorem transmission_component_tail (S : System V E) (W : Set V) (hW : S.pins⊆W) |
| (b₀ b₁ h₀ h₁ : V → ℝ) (hh₀ : h₀∈feasible S b₀) (hh₁ : h₁∈feasible S b₁) |
| (w : ℝ → FreeSpace S) (u₀ u₁ : ℝ → V → ℝ) |
| (hlim₀ : TendstoUniformlyOn (fun n => pathMesh S b₀ h₀ hh₀ w (n+1)) u₀ atTop (Icc 0 1)) |
| (hlim₁ : TendstoUniformlyOn (fun n => pathMesh S b₁ h₁ hh₁ w (n+1)) u₁ atTop (Icc 0 1)) |
| (δ : ℝ) (hδ : 0<δ) |
| (hcut₀ : ∀t∈Icc (0:ℝ) 1,∀e,S.cost e-(u₀ t (S.left e)-u₀ t (S.right e))≤δ → |
| S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W)) |
| (hcut₁ : ∀t∈Icc (0:ℝ) 1,∀e,S.cost e-(u₁ t (S.left e)-u₁ t (S.right e))≤δ → |
| S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W)) |
| (a : ℝ) (hb : ∀x∈S.pins,b₁ x-b₀ x≤a) : |
| 0≤energy (componentPathTail S W u₀ u₁ a 0)-energy (componentPathTail S W u₀ u₁ a 1) ∧ |
| ∀t∈Icc (0:ℝ) 1,∀x : {x : W // x.val∉S.pins}, |
| |componentPathTail S W u₀ u₁ a t x-componentPathTail S W u₀ u₁ a 0 x|≤ |
| 5*(Fintype.card {x : W // x.val∉S.pins}+1:ℝ)* |
| Real.sqrt (energy (componentPathTail S W u₀ u₁ a 0)-energy (componentPathTail S W u₀ u₁ a 1)) := by |
| have hc₀ := pathMesh_cut_eventually S W b₀ h₀ hh₀ w u₀ hlim₀ δ hδ hcut₀ |
| have hc₁ := pathMesh_cut_eventually S W b₁ h₁ hh₁ w u₁ hlim₁ δ hδ hcut₁ |
| let F : ℕ → ℝ → {x : W // x.val∉S.pins} → ℝ := fun n t => |
| componentMeshTail S W b₀ b₁ h₀ h₁ hh₀ hh₁ (pathIncrement S w (n+1)) a ⌊t*((n+1:ℕ):ℝ)⌋₊ |
| have hF : ∀t∈Icc (0:ℝ) 1,∀x : {x : W // x.val∉S.pins}, |
| Tendsto (fun n => F n t x) atTop (nhds (componentPathTail S W u₀ u₁ a t x)) := by |
| intro t ht x |
| have h₀ := tendsto_pi_nhds.mp (hlim₀.tendsto_at ht) x.val.val |
| have h₁ := tendsto_pi_nhds.mp (hlim₁.tendsto_at ht) x.val.val |
| simpa only [F,componentMeshTail,meshTail,componentPathTail,pathMesh] using |
| (((h₁.sub h₀).sub_const a).max (tendsto_const_nhds (x:=(0:ℝ)))) |
| have hEt : ∀t∈Icc (0:ℝ) 1, |
| Tendsto (fun n => energy (F n t)) atTop (nhds (energy (componentPathTail S W u₀ u₁ a t))) := by |
| intro t ht |
| exact tendsto_finsetSum _ fun x hx => (hF t ht x).pow 2 |
| have ht₀ : (0:ℝ)∈Icc (0:ℝ) 1 := by constructor <;> norm_num |
| have ht₁ : (1:ℝ)∈Icc (0:ℝ) 1 := by constructor <;> norm_num |
| constructor |
| · apply le_of_tendsto_of_tendsto tendsto_const_nhds ((hEt 0 ht₀).sub (hEt 1 ht₁)) |
| filter_upwards [hc₀,hc₁] with n hn₀ hn₁ |
| have he := component_mesh_loss_nonneg S W hW b₀ b₁ h₀ h₁ hh₀ hh₁ |
| (pathIncrement S w (n+1)) a hb (n+1) hn₀ hn₁ |
| simpa only [F,zero_mul,one_mul,Nat.floor_zero,Nat.floor_natCast] using he |
| · intro t ht x |
| have hleft := ((hF t ht x).sub (hF 0 ht₀ x)).abs |
| have hright := ((hEt 0 ht₀).sub (hEt 1 ht₁)).sqrt.const_mul |
| (5*(Fintype.card {x : W // x.val∉S.pins}+1:ℝ)) |
| apply le_of_tendsto_of_tendsto hleft hright |
| filter_upwards [hc₀,hc₁] with n hn₀ hn₁ |
| have hk : ⌊t*((n+1:ℕ):ℝ)⌋₊ ≤ n+1 := Nat.floor_le_of_le |
| (by simpa only [one_mul] using mul_le_mul_of_nonneg_right ht.2 (Nat.cast_nonneg (n+1))) |
| have he := component_mesh_displacement S W hW b₀ b₁ h₀ h₁ hh₀ hh₁ |
| (pathIncrement S w (n+1)) a hb (n+1) hn₀ hn₁ ⌊t*((n+1:ℕ):ℝ)⌋₊ hk x |
| simpa only [F,zero_mul,one_mul,Nat.floor_zero,Nat.floor_natCast] using he |
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| end ProjectionComparison |
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| end OAI |
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