openAI-math-dataset / lean /OAI /Probability /LatticeSurface /Dynamics /ProjectionRestrictionMesh.lean
| 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 | |