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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import Mathlib
import OAI.Probability.LatticeSurface.Convexity.KernelRestriction
import OAI.Probability.LatticeSurface.Dynamics.ProjectionTail
import OAI.Probability.LatticeSurface.Dynamics.TwoStageDynamics
namespace OAI
namespace ProjectionComparison
open Set ClusterDisplacement
attribute [local instance] Classical.propDecidable
variable {V E : Type*} [Fintype V] [Fintype E]
noncomputable def meshTail (S : System V E) (b₀ b₁ h₀ h₁ : V → ℝ)
(hh₀ : h₀∈feasible S b₀) (hh₁ : h₁∈feasible S b₁)
(increments : ℕ → V → ℝ) (a : ℝ) (n : ℕ) (x : {x // x∉S.pins}) : ℝ :=
max (recursion S b₁ ⟨h₁,hh₁⟩ h₁ increments n x-
recursion S b₀ ⟨h₀,hh₀⟩ h₀ increments n x-a) 0
end ProjectionComparison
namespace ProjectionComparison
open Set ClusterDisplacement
attribute [local instance] Classical.propDecidable
variable {V E : Type*} [Fintype V] [Fintype E]
omit [Fintype E] in
lemma recursion_feasible (S : System V E) (b h₀ : V → ℝ) (hh₀ : h₀∈feasible S b)
(increments : ℕ → V → ℝ) (n : ℕ) : recursion S b ⟨h₀,hh₀⟩ h₀ increments n∈feasible S b := by
cases n with
| zero => exact hh₀
| succ n => exact (project_spec S b ⟨h₀,hh₀⟩ _).1
lemma mesh_tail_step (S : System V E) (b₀ b₁ h₀ h₁ : V → ℝ)
(hh₀ : h₀∈feasible S b₀) (hh₁ : h₁∈feasible S b₁)
(increments : ℕ → V → ℝ) (a : ℝ) (hb : ∀x∈S.pins,b₁ x-b₀ x≤a) (n : ℕ) :
∃ A B : Kernel {x // x∉S.pins},Firm A ∧ Firm B ∧
meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a (n+1)=
B.apply (A.apply (meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a n)) := by
classical
let f₀ := recursion S b₀ ⟨h₀,hh₀⟩ h₀ increments n
let f₁ := recursion S b₁ ⟨h₁,hh₁⟩ h₁ increments n
let v₀ : V → ℝ := fun x => f₀ x+increments n x
let v₁ : V → ℝ := fun x => f₁ x+increments n x
let g₀ := recursion S b₀ ⟨h₀,hh₀⟩ h₀ increments (n+1)
let g₁ := recursion S b₁ ⟨h₁,hh₁⟩ h₁ increments (n+1)
obtain ⟨A,hA,r,hr,he⟩:=tail_projection_kernel S b₀ b₁ v₀ v₁ g₀ g₁ a
(project_spec S b₀ ⟨h₀,hh₀⟩ v₀) (project_spec S b₁ ⟨h₁,hh₁⟩ v₁) hb
let W : Set V := {x | x∉S.pins}
let u : V → ℝ := fun x => max (v₁ x-v₀ x-a) 0
have hu (x) : u x=max (f₁ x-f₀ x-a) 0 := by dsimp [u,v₁,v₀];rw [add_sub_add_right_eq_sub]
have hup (x) (hx : x∉W) : u x=0 := by
have hp : x∈S.pins := Classical.not_not.mp hx
rw [hu]
have hf₀ : f₀ x=b₀ x := (recursion_feasible S b₀ h₀ hh₀ increments n).2 x hp
have hf₁ : f₁ x=b₁ x := (recursion_feasible S b₁ h₁ hh₁ increments n).2 x hp
rw [hf₀,hf₁,max_eq_right (sub_nonpos.mpr (hb x hp))]
refine ⟨A.restrict W,Kernel.cutoff (fun x : W => r x) (fun x => hr x),A.restrict_firm hA W,
Kernel.cutoff_firm _ _,?_⟩
funext x
rw [Kernel.cutoff_apply]
have harg : meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a n=(fun x : W => u x) := by
funext x
exact (hu x).symm
rw [harg,Kernel.restrict_apply_of_zero A W u hup]
exact he x
end ProjectionComparison
namespace ProjectionComparison
open Set ClusterDisplacement
attribute [local instance] Classical.propDecidable
variable {V E : Type*} [Fintype V] [Fintype E]
theorem projection_mesh_tail_displacement (S : System V E) (b₀ b₁ h₀ h₁ : V → ℝ)
(hh₀ : h₀∈feasible S b₀) (hh₁ : h₁∈feasible S b₁)
(increments : ℕ → V → ℝ) (a : ℝ) (hb : ∀x∈S.pins,b₁ x-b₀ x≤a)
(N k : ℕ) (hk : k≤N) (x : {x // x∉S.pins}) :
|meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a k x-
meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a 0 x|≤
5*(Fintype.card {x // x∉S.pins}+1:ℝ)*Real.sqrt
(energy (meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a 0)-
energy (meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a N)) := by
exact two_stage_displacement _ (mesh_tail_step S b₀ b₁ h₀ h₁ hh₀ hh₁ increments a hb) N k hk x
end ProjectionComparison
end OAI
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