import OAI.Geometry.IsometricImmersion.Caps.BoundedPatchCauchy
import OAI.Geometry.IsometricImmersion.Calculus.ShearedMetricFiniteJets
import OAI.Geometry.IsometricImmersion.Estimates.FixedRadiusCutBounds

namespace OAI

noncomputable section
open Set Filter Function
open scoped ContDiff Topology Matrix NNReal

namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow
open SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor

theorem exists_actual_off_pulse_cauchy_bounds_at_radius
    {gStar : MetricField} {V : Set Coord}
    (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V)
    {G d kappa q0 r : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d)
    (hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20)
    (hr : 0 < r) (hrhalf : r < 1/2) (hLr : boundedClassWidth kappa M*r ≤ 1/20)
    (hrsmall : heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))*
      (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M))
    (K : ℕ) :
    ∃ B : ℝ, 0 ≤ B ∧ ∀ N : ℕ, ∀ delta : ℝ, 0 < delta →
        ∀ᶠ tau : ℕ in atTop,
          ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U W : Set Coord)
            (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ),
            SmoothPositiveOn (perturbedMetric g0 eta.val) U → IsOpen U →
            CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z →
            CapInductionFlow (perturbedMetric g0 eta.val) U G (M : ℝ) d (1/(M : ℝ)) (1/(M : ℝ)) kappa z Y W →
            BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
            (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare,
              ‖iteratedFDeriv ℝ k (fun a => perturbedMetric g0 eta.val a i j) p‖ ≤ G) →
            (∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) →
            |hessianQuotient (perturbedMetric g0 eta.val) z 0-q0| ≤ 1/(100*boundedClassWidth kappa M) →
            (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare,
              ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j-
                testMetric gStar q0 (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) q i j) p‖ ≤
                  metricApproximationAccuracy tau) →
            ∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ n ≤ K,
              ‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0)
                (-delta/(tau : ℝ))) x‖ ≤ B ∧
              ‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0)
                (-delta/(tau : ℝ))) x‖ ≤ B := by
  let L := boundedClassWidth kappa M
  have hL : 0 < L := boundedClassWidth_pos kappa M
  have hMR : (0 : ℝ) < M := Nat.cast_pos.mpr hM
  obtain ⟨A0,hA0,hcutK⟩ := exists_actual_finite_sheared_lower_cut_bound_at_radius
    (G := G) (Z := (M : ℝ)) (d := d) (c := 1/(M : ℝ)) (e0 := 1/(M : ℝ))
    (kappa := kappa) (q0 := q0) (L := L) hgStar hV hSV hG (Nat.cast_nonneg M) hd
      (one_div_pos.mpr hMR) hL hq0 hr hrhalf hLr hrsmall (max 8 (K+2))
  let A : ℝ≥0 := ⟨A0,hA0⟩
  obtain ⟨Cmetric,hCmetric,hmetricK⟩ := exists_uniform_off_pulse_sheared_metric_bounds
    hgStar hV hSV hL hr hrhalf hLr hq0 (max 9 (K+3))
  obtain ⟨B,hB,hcauchy⟩ := exists_bounded_patch_cauchy_bound G d kappa q0 r Cmetric M A
    hG hd hkappa hM hr hrhalf hCmetric hq0 hLr hrsmall K
  refine ⟨B,hB,?_⟩
  intro N delta hdelt
  have hwidth : ∀ᶠ tau : ℕ in atTop, 1 ≤ (tau : ℝ) ∧ delta/(2*(tau : ℝ)) ≤ r/4 :=
    (tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually
      (pulse_width_eventually hr delta)
  filter_upwards [hcutK N delta hdelt,hwidth,
    eventually_ge_atTop (max 1 (max 9 (K+3)))] with tau hcutTau hwidthTau htau
  intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ x hx n hn
  have htau1 : 1 ≤ tau := (le_max_left _ _).trans htau
  have htauk : max 9 (K+3) ≤ tau := (le_max_right _ _).trans htau
  have htr : (0 : ℝ) < tau := zero_lt_one.trans_le (by exact_mod_cast htau1)
  have hdw : delta/(tau : ℝ) ≤ r := by
    have he : delta/(tau : ℝ) = 2*(delta/(2*(tau : ℝ))) := by ring
    rw [he]
    linarith [hwidthTau.2]
  have hb0 : -delta/(tau : ℝ) ≤ 0 := div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hdelt.le) htr.le
  have hab : -r ≤ -delta/(tau : ℝ) := by simpa only [neg_div] using neg_le_neg hdw
  have hSU : modelSquare ⊆ U := hf.squareSubset.trans hf.domainSubset
  have hfullMetric := hmetricK N delta hdelt tau htau1 htauk
    (perturbedMetric g0 eta.val) U hg hU hSU happ
  have hmetric : ∀ i j, CoordinateBound
      (fun p => metricInShearCoordinates (perturbedMetric g0 eta.val) q0 p i j)
      (shrinkingSlab (-(L*r)) (L*r) (-r) (-delta/(tau : ℝ)) (L/4)) (max 9 (K+3)) Cmetric := by
    intro i j ds hds p hp
    have haxes := shrinking_class_slab_axis_bounds hr.le hb0 hp
    exact hfullMetric i j ds hds p ⟨haxes.1,hp.1⟩
  have hactualCut : ∀ ds : List (Fin 2), ds.length ≤ max 8 (K+2) →
      ∀ x : ℝ, |x| ≤ L*r →
        |iteratedCoordPartial ds (heightInShearCoordinates z q0) (coordinatePoint x (-r))| ≤ (A : ℝ) :=
    hcutTau (perturbedMetric g0 eta.val) U W z Y hg hU hgB hdet hh hf hcenter happ
  exact hcauchy g0 eta z (-delta/(tau : ℝ)) hclass hab hb0 hgB hdet hcenter hmetric hactualCut x hx n hn

end SmoothLocal.Perturbation

end

end OAI