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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import Mathlib
import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentEnergy
import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentLimitSupport
namespace OAI
namespace ProjectionComparison
open Set Filter ClusterDisplacement
attribute [local instance] Classical.propDecidable
variable {V E : Type*} [Fintype V] [Fintype E]
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
end ProjectionComparison
end OAI
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