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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 98 99 100 101 102 | import Mathlib
import OAI.Probability.LatticeSurface.Dynamics.ProjectionComparison
namespace OAI
namespace ProjectionComparison
open Set
variable {V E : Type*} [Fintype V]
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
lemma objective_nonneg (v h : V → ℝ) : 0≤objective v h := Finset.sum_nonneg fun _ _ => sq_nonneg _
lemma continuous_objective (v : V → ℝ) : Continuous (objective v) := by
unfold objective
fun_prop
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
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)
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))
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)
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)
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
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)
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)
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)
end ProjectionComparison
end OAI
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