| import Mathlib |
| import OAI.Probability.LatticeSurface.Dynamics.ProjectionComparison |
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| namespace OAI |
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| namespace ProjectionComparison |
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| open Set |
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| variable {V E : Type*} [Fintype V] |
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| omit [Fintype V] in |
| lemma isClosed_feasible (S : System V E) (b : V → ℝ) : IsClosed (feasible S b) := by |
| have he : IsClosed {h : V → ℝ | ∀ e,h (S.left e)-h (S.right e)≤S.cost e} := by |
| simp only [ofPred_forall] |
| exact isClosed_iInter fun e => isClosed_le ((continuous_apply (S.left e)).sub (continuous_apply (S.right e))) continuous_const |
| have hp : IsClosed {h : V → ℝ | ∀ x∈S.pins,h x=b x} := by |
| simp only [ofPred_forall] |
| exact isClosed_iInter fun x => isClosed_iInter fun _ => isClosed_eq (continuous_apply _) continuous_const |
| exact he.inter hp |
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|
| lemma objective_nonneg (v h : V → ℝ) : 0≤objective v h := Finset.sum_nonneg fun _ _ => sq_nonneg _ |
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| lemma continuous_objective (v : V → ℝ) : Continuous (objective v) := by |
| unfold objective |
| fun_prop |
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| lemma coordinate_bound (v h : V → ℝ) (B : ℝ) (hB : 0≤B) (hh : objective v h≤B) (x : V) : |
| v x-Real.sqrt B≤h x ∧ h x≤v x+Real.sqrt B := by |
| have hx : (h x-v x)^2≤B := (Finset.single_le_sum (fun y (_ : y∈Finset.univ) => sq_nonneg (h y-v y)) (Finset.mem_univ x)).trans hh |
| have hs := Real.sq_sqrt hB |
| have hn := Real.sqrt_nonneg B |
| constructor <;> nlinarith |
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| lemma exists_projection (S : System V E) (b v : V → ℝ) (hne : (feasible S b).Nonempty) : |
| ∃ h,IsProjection S b v h := by |
| obtain ⟨h₀,hh₀⟩ := hne |
| let B := objective v h₀ |
| let L : V → ℝ := fun x => v x-Real.sqrt B |
| let U : V → ℝ := fun x => v x+Real.sqrt B |
| have hc : IsCompact (feasible S b ∩ Icc L U) := isCompact_Icc.inter_left (isClosed_feasible S b) |
| have hmem (h : V → ℝ) (hh : objective v h≤B) : h∈Icc L U := by |
| constructor |
| · intro x;exact (coordinate_bound v h B (objective_nonneg v h₀) hh x).1 |
| · intro x;exact (coordinate_bound v h B (objective_nonneg v h₀) hh x).2 |
| obtain ⟨h,hh,hmin⟩ := hc.exists_isMinOn ⟨h₀,hh₀,hmem h₀ le_rfl⟩ (continuous_objective v).continuousOn |
| refine ⟨h,hh.1,fun k hk => ?_⟩ |
| by_cases hkb : objective v k≤B |
| · exact hmin ⟨hk,hmem k hkb⟩ |
| · exact (hmin ⟨hh₀,hmem h₀ le_rfl⟩).trans (le_of_not_ge hkb) |
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| lemma projection_unique (S : System V E) (b v h k : V → ℝ) |
| (hh : IsProjection S b v h) (hk : IsProjection S b v k) : h=k := by |
| have h := projection_parameter_stability S b b v v h k 0 le_rfl hh hk |
| (fun x _ => by simp) (fun x _ => by simp) |
| funext x |
| exact sub_eq_zero.mp (abs_nonpos_iff.mp (h x)) |
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| noncomputable def project (S : System V E) (b : V → ℝ) (hne : (feasible S b).Nonempty) (v : V → ℝ) : V → ℝ := |
| Classical.choose (exists_projection S b v hne) |
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| lemma project_spec (S : System V E) (b : V → ℝ) (hne : (feasible S b).Nonempty) (v : V → ℝ) : |
| IsProjection S b v (project S b hne v) := Classical.choose_spec (exists_projection S b v hne) |
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| lemma project_eq_self (S : System V E) (b : V → ℝ) (hne : (feasible S b).Nonempty) |
| (v : V → ℝ) (hv : v∈feasible S b) : project S b hne v=v := by |
| apply projection_unique S b v _ v (project_spec S b hne v) |
| refine ⟨hv,?_⟩ |
| intro k hk |
| simpa [objective] using objective_nonneg v k |
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| noncomputable def recursion (S : System V E) (b : V → ℝ) (hne : (feasible S b).Nonempty) |
| (h₀ : V → ℝ) (increments : ℕ → V → ℝ) : ℕ → V → ℝ |
| | 0 => h₀ |
| | n+1 => project S b hne (fun x => recursion S b hne h₀ increments n x+increments n x) |
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| theorem recursion_parameter_stability (S : System V E) (b₁ b₂ : V → ℝ) |
| (hne₁ : (feasible S b₁).Nonempty) (hne₂ : (feasible S b₂).Nonempty) |
| (h₁ h₂ : V → ℝ) (increments : ℕ → V → ℝ) (δ : ℝ) (hδ : 0≤δ) |
| (hb : ∀ x∈S.pins,|b₁ x-b₂ x|≤δ) (hh : ∀ x,|h₁ x-h₂ x|≤δ) : |
| ∀ n x,|recursion S b₁ hne₁ h₁ increments n x-recursion S b₂ hne₂ h₂ increments n x|≤δ := by |
| intro n |
| induction n with |
| | zero => exact hh |
| | succ n hn => |
| exact projection_parameter_stability S b₁ b₂ _ _ _ _ δ hδ (project_spec S b₁ hne₁ _) (project_spec S b₂ hne₂ _) hb |
| (fun x _ => by simpa only [add_sub_add_right_eq_sub] using hn x) |
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| omit [Fintype V] in |
| theorem limit_parameter_stability {ι : Type*} {l : Filter ι} [l.NeBot] |
| (H₁ H₂ : ι → V → ℝ) (h₁ h₂ : V → ℝ) (δ : ℝ) |
| (hH : ∀ i x,|H₁ i x-H₂ i x|≤δ) |
| (hlim₁ : ∀ x,Filter.Tendsto (fun i => H₁ i x) l (nhds (h₁ x))) |
| (hlim₂ : ∀ x,Filter.Tendsto (fun i => H₂ i x) l (nhds (h₂ x))) : |
| ∀ x,|h₁ x-h₂ x|≤δ := by |
| intro x |
| exact le_of_tendsto ((hlim₁ x).sub (hlim₂ x)).abs (Filter.Eventually.of_forall fun i => hH i x) |
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| end ProjectionComparison |
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| end OAI |
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