import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetBounds
import OAI.Geometry.IsometricImmersion.Metrics.MetricSmoothness

namespace OAI

noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix

namespace SmoothLocal.Geometry

def metricQuotientNumeratorBound (G : ℝ) : ℝ := G + 16 * G^2
def metricInverseJetBound (G d : ℝ) : ℝ :=
  LowQuotient.boundThroughThree (metricQuotientNumeratorBound G) d

theorem metricInverseJetBound_nonneg {G d : ℝ} (hG : 0 ≤ G) (hd : 0 < d) :
    0 ≤ metricInverseJetBound G d := by
  have hN : 0 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; positivity
  obtain ⟨h0, h1, h2, h3⟩ := LowQuotient.bounds_nonneg hN hd
  dsimp [metricInverseJetBound, LowQuotient.boundThroughThree]
  positivity

variable {g : MetricField} {U : Set Coord} {G d : ℝ}

theorem metricDet_coordinate_bound
    (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hG : 0 ≤ G)
    (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G) :
    CoordinateBound (fun p => (g p).det) U 3 (16 * G^2) := by
  have hdiag := CoordinateBound.mul_through_three (hg.1 0 0) (hg.1 1 1) hU hG hG (hgB 0 0) (hgB 1 1)
  have hoff := CoordinateBound.mul_through_three (hg.1 0 1) (hg.1 1 0) hU hG hG (hgB 0 1) (hgB 1 0)
  have hdet := CoordinateBound.sub ((hg.1 0 0).mul (hg.1 1 1))
    ((hg.1 0 1).mul (hg.1 1 0)) hU hdiag hoff
  have hc : 8 * G * G + 8 * G * G = 16 * G^2 := by ring
  simpa only [Matrix.det_fin_two, hc] using hdet

theorem metricAdjugate_coordinate_bound
    (hg : SmoothPositiveOn g U) (hU : IsOpen U)
    (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G) (i j : Fin 2) :
    CoordinateBound (fun p => (g p).adjugate i j) U 3 G := by
  fin_cases i <;> fin_cases j
  · simpa [Matrix.adjugate_fin_two] using hgB 1 1
  · simpa [Matrix.adjugate_fin_two] using CoordinateBound.neg (hg.1 0 1) hU (hgB 0 1)
  · simpa [Matrix.adjugate_fin_two] using CoordinateBound.neg (hg.1 1 0) hU (hgB 1 0)
  · simpa [Matrix.adjugate_fin_two] using hgB 0 0

theorem inverseMetric_eq_adjugate_div (g : MetricField) (i j : Fin 2) :
    (fun p => inverseMetric g p i j) = (fun p => (g p).adjugate i j / (g p).det) := by
  funext p
  simp [inverseMetric, Matrix.inv_def, Ring.inverse_eq_inv, div_eq_mul_inv, smul_eq_mul, mul_comm]

theorem inverseMetric_coordinate_bound
    (hg : SmoothPositiveOn g U) (hU : IsOpen U) (hG : 0 ≤ G) (hd : 0 < d)
    (hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) U 3 G)
    (hdet : ∀ p ∈ U, d ≤ |(g p).det|) (i j : Fin 2) :
    CoordinateBound (fun p => inverseMetric g p i j) U 3 (metricInverseJetBound G d) := by
  have hN : 0 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; positivity
  have hGN : G ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; nlinarith [sq_nonneg G]
  have hDN : 16 * G^2 ≤ metricQuotientNumeratorBound G := by dsimp [metricQuotientNumeratorBound]; linarith
  have hadj := (metricAdjugate_coordinate_bound hg hU hgB i j).mono le_rfl hGN
  have hdetB := (metricDet_coordinate_bound hg hU hG hgB).mono le_rfl hDN
  rw [inverseMetric_eq_adjugate_div]
  intro ds hds p hp
  exact LowQuotient.quotient_bound_through_three (metricAdjugate_contDiffOn hg i j)
    (metricDet_contDiffOn hg) hU hN hd hadj hdetB hdet ds hds hp

end SmoothLocal.Geometry

end

end OAI