openAI-math-dataset / lean /OAI /Probability /LatticeSurface /Dynamics /ProjectionComponentEnergy.lean
| import Mathlib | |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionComponentMesh | |
| namespace OAI | |
| namespace ProjectionComparison | |
| open Set Filter ClusterDisplacement | |
| attribute [local instance] Classical.propDecidable | |
| variable {V E : Type*} [Fintype V] [Fintype E] | |
| lemma mesh_tail_energy_antitone (S : System V E) (b₀ b₁ h₀ h₁ : V → ℝ) | |
| (hh₀ : h₀∈feasible S b₀) (hh₁ : h₁∈feasible S b₁) | |
| (inc : ℕ → V → ℝ) (a : ℝ) (hb : ∀x∈S.pins,b₁ x-b₀ x≤a) : | |
| Antitone (fun i => energy (meshTail S b₀ b₁ h₀ h₁ hh₀ hh₁ inc a i)) := by | |
| apply antitone_nat_of_succ_le | |
| intro i | |
| obtain ⟨A,B,hA,hB,he⟩ := mesh_tail_step S b₀ b₁ h₀ h₁ hh₀ hh₁ inc a hb i | |
| rw [he] | |
| exact (B.energy_bound _).trans (A.energy_bound _) | |
| lemma component_mesh_loss_nonneg (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) | |
| (inc : ℕ → V → ℝ) (a : ℝ) (hb : ∀x∈S.pins,b₁ x-b₀ x≤a) (N : ℕ) | |
| (hcut₀ : ∀i<N,∀e, | |
| recursion S b₀ ⟨h₀,hh₀⟩ h₀ inc (i+1) (S.left e)-recursion S b₀ ⟨h₀,hh₀⟩ h₀ 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)) | |
| (hcut₁ : ∀i<N,∀e, | |
| recursion S b₁ ⟨h₁,hh₁⟩ h₁ inc (i+1) (S.left e)-recursion S b₁ ⟨h₁,hh₁⟩ h₁ 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)) | |
| : 0≤energy (componentMeshTail S W b₀ b₁ h₀ h₁ hh₀ hh₁ inc a 0)- | |
| energy (componentMeshTail S W b₀ b₁ h₀ h₁ hh₀ hh₁ inc a N) := by | |
| have hr₀ := recursion_restrict S W hW b₀ h₀ hh₀ inc N hcut₀ | |
| have hr₁ := recursion_restrict S W hW b₁ h₁ hh₁ inc N hcut₁ | |
| have he (i : ℕ) (hi : i≤N) : | |
| meshTail (restrictedSystem S W) (fun x => b₀ x) (fun x => b₁ x) | |
| (fun x => h₀ x) (fun x => h₁ x) (feasible_restrict S W b₀ h₀ hh₀) | |
| (feasible_restrict S W b₁ h₁ hh₁) (fun i x => inc i x) a i= | |
| componentMeshTail S W b₀ b₁ h₀ h₁ hh₀ hh₁ inc a i := by | |
| funext x | |
| change max (_-_-a) 0=max (_-_-a) 0 | |
| rw [←congrFun (hr₀ i hi) x.val,←congrFun (hr₁ i hi) x.val] | |
| have ht := mesh_tail_energy_antitone (restrictedSystem S W) (fun x => b₀ x) (fun x => b₁ x) | |
| (fun x => h₀ x) (fun x => h₁ x) (feasible_restrict S W b₀ h₀ hh₀) | |
| (feasible_restrict S W b₁ h₁ hh₁) (fun i x => inc i x) a (fun x hx => hb x hx) (Nat.zero_le N) | |
| dsimp only at ht | |
| rw [he 0 (Nat.zero_le N),he N le_rfl] at ht | |
| exact sub_nonneg.mpr ht | |
| end ProjectionComparison | |
| end OAI | |