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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 | import Mathlib
import OAI.Probability.LatticeSurface.Dynamics.ProjectionRestriction
namespace OAI
namespace ProjectionComparison
open Set Filter
attribute [local instance] Classical.propDecidable
variable {V E : Type*} [Fintype V] [Fintype E]
lemma recursion_restrict (S : System V E) (W : Set V) (hW : S.pins⊆W)
(b h : V → ℝ) (hh : h∈feasible S b) (inc : ℕ → V → ℝ) (N : ℕ)
(hcut : ∀i<N,∀e,
recursion S b ⟨h,hh⟩ h inc (i+1) (S.left e)-recursion S b ⟨h,hh⟩ h inc (i+1) (S.right e)=S.cost e →
S.left e∉S.pins → S.right e∉S.pins → (S.left e∈W ↔ S.right e∈W)) :
∀i≤N,(fun x : W => recursion S b ⟨h,hh⟩ h inc i x)=
recursion (restrictedSystem S W) (fun x => b x)
⟨fun x => h x,feasible_restrict S W b h hh⟩ (fun x => h x) (fun i x => inc i x) i := by
intro i
induction i with
| zero => intro hi;rfl
| succ i ih =>
intro hi
have he := ih (Nat.le_of_succ_le hi)
have hp := projection_restrict S W hW b (fun x => recursion S b ⟨h,hh⟩ h inc i x+inc i x)
(recursion S b ⟨h,hh⟩ h inc (i+1)) (project_spec S b ⟨h,hh⟩ _) (hcut i (Nat.lt_of_succ_le hi))
apply projection_unique (restrictedSystem S W) (fun x => b x)
(fun x => recursion S b ⟨h,hh⟩ h inc i x+inc i x) _ _ hp
change IsProjection (restrictedSystem S W) (fun x => b x)
(fun x => (fun x : W => recursion S b ⟨h,hh⟩ h inc i x) x+inc i x) _
rw [he]
exact project_spec _ _ _ _
end ProjectionComparison
end OAI
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