import Mathlib
import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear
import OAI.Geometry.CAT0Fillings.Gradient.Multiplier
import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain

namespace OAI

section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
  [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
  {hT : IsMetricCurrent T} (q : ChartGeometry hT)

lemma sobolev_tendsto {P : ℕ → q.Sobolev} {Q : q.Sobolev}
    (hv : Tendsto (fun j => q.inclusion (P j)) atTop (𝓝 (q.inclusion Q)))
    (hg : Tendsto (fun j => q.closedGradient (P j)) atTop (𝓝 (q.closedGradient Q))) :
    Tendsto P atTop (𝓝 Q) := by
  apply tendsto_subtype_rng.mpr
  exact ((WithLp.prodContinuousLinearEquiv 2 ℝ
    (Lp ℝ 2 (MassMeasure.currentMassMeasure hT)) (Lp (Euc (k+1)) 2 q.atlasMeasure)).symm.continuous.tendsto
      (q.inclusion Q,q.closedGradient Q)).comp (hv.prodMk_nhds hg)

lemma closed_chain_in_closed {F F' : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
    (hdF : ∀ t, HasDerivAt F (F' t) t) (hcF : Continuous F')
    {C : ℝ} (hC : 0 ≤ C) (hbF : ∀ t, |F' t| ≤ C) (P : q.Sobolev)
    {S : Set q.Sobolev} (hS : IsClosed S)
    (hmem : ∀ (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev (hF.comp hu) ∈ S) :
    ∃ Q : q.Sobolev, Q ∈ S ∧
      (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
        (fun x => F ((q.inclusion P) x)) ∧
      (q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
        (fun w => F' ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
  obtain ⟨u,K,hu,hl⟩ := q.exists_lipschitz_approximants P
  have hv := (q.valueProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl
  have hg := (q.gradientProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl
  change Tendsto (fun j => value (hu j)) atTop (𝓝 (q.inclusion P)) at hv
  change Tendsto (fun j => q.gradient (hu j)) atTop (𝓝 (q.closedGradient P)) at hg
  obtain ⟨s,hs,hae⟩ := (tendstoInMeasure_of_tendsto_Lp hv).exists_seq_tendsto_ae
  have hnames : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, ∀ j,
      (value (hT := hT) (hu (s j))) x = u (s j) x := by
    apply ae_all_iff.mpr
    intro j
    exact ((Foundations.boundedLip_of_lipschitz (hu (s j))).memLp 2).coeFn_toLp
  have hae' : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT,
      Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x)) := by
    filter_upwards [hae,hnames] with x hx he
    simpa only [he] using hx
  let b (j : ℕ) (w : ℕ × Euc (k+1)) := F' (u (s j) (q.atlasParam w))
  let c (w : ℕ × Euc (k+1)) := F' ((q.inclusion P) (q.atlasParam w))
  have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure :=
    (hcF.measurable.comp ((hu (s j)).continuous.measurable.comp q.measurable_atlasParam)).aestronglyMeasurable
  have hc : AEStronglyMeasurable c q.atlasMeasure :=
    hcF.comp_aestronglyMeasurable ((Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving)
  have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C :=
    Eventually.of_forall fun point => hbF (u (s j) (q.atlasParam point))
  have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C :=
    Eventually.of_forall fun point => hbF ((q.inclusion P) (q.atlasParam point))
  have hbc : ∀ᵐ w ∂q.atlasMeasure, Tendsto (fun j => b j w) atTop (𝓝 (c w)) := by
    have ha : ∀ᵐ w ∂q.atlasMeasure,
        Tendsto (fun j => u (s j) (q.atlasParam w)) atTop (𝓝 ((q.inclusion P) (q.atlasParam w))) := by
      apply ae_of_ae_map (μ := q.atlasMeasure) (p := fun x =>
        Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x))) q.measurable_atlasParam.aemeasurable
      rwa [q.map_atlasMeasure]
    filter_upwards [ha] with w hw
    exact (hcF.tendsto _).comp hw
  have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC hbc (hg.comp hs.tendsto_atTop)
  have he (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.gradient (hu (s j))) =
      q.gradient (hF.comp (hu (s j))) := by
    apply Lp.ext
    exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (q.gradient_chain_ae (hu (s j)) hF hdF).symm
  simp only [Function.comp_apply,he] at hgl
  have hvl := (q.continuous_composeValue hF).tendsto (q.inclusion P) |>.comp (hv.comp hs.tendsto_atTop)
  simp only [Function.comp_def,q.composeValue_value] at hvl
  obtain ⟨Q,hQ,hG⟩ := q.closedGradient_closed (q.composeValue hF (q.inclusion P))
    (ClosedCalculus.mulLp hc hcC (q.closedGradient P))
    (fun j => q.lipSobolev (hF.comp (hu (s j)))) hvl hgl
  refine ⟨Q,?_,?_,?_⟩
  · apply hS.mem_of_tendsto (q.sobolev_tendsto (P := fun j => q.lipSobolev (hF.comp (hu (s j)))) (by simpa only [q.inclusion_lipSobolev,hQ] using hvl) (by simpa only [q.closedGradient_lipSobolev,hG] using hgl))
    exact Eventually.of_forall fun j => hmem _ _ (hu (s j))
  · rw [hQ]
    exact q.composeValue_ae hF (q.inclusion P)
  · rw [hG]
    exact ClosedCalculus.mulLp_ae hc hcC _

end CAT0Fillings.ChartGeometry
end

section

open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
  [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
  {hT : IsMetricCurrent T} (q : ChartGeometry hT)

lemma closed_chain_uniform_in_closed {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
    (hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, |D t| ≤ C)
    (Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0)
    (hFj : ∀ j, LipschitzWith (K j) (Fj j))
    (hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t)
    (hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, |Dj j t| ≤ C)
    (hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t)))
    (a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0))
    (he : ∀ j t, |Fj j t-F t| ≤ a j) (P : q.Sobolev)
    {S : Set q.Sobolev} (hclosed : IsClosed S)
    (hmem : ∀ j (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev ((hFj j).comp hu) ∈ S) :
    ∃ Q : q.Sobolev, Q ∈ S ∧
      (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
        (fun x => F ((q.inclusion P) x)) ∧
      (q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
        (fun w => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
  choose Q hQS hQ hG using (fun j => q.closed_chain_in_closed (hFj j) (hdj j) (hcj j) hC (hbj j) P hclosed (hmem j))
  let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w))
  let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w))
  have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving
  have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm
  have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable
  have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun _ => hbj j _
  have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun _ => hbD _
  have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC
    (Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point)))
    (tendsto_const_nhds (x := q.closedGradient P))
  have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by
    apply Lp.ext
    exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm
  simp only [heG] at hgl
  let V := q.composeValue hF (q.inclusion P)
  have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by
    apply tendsto_iff_norm_sub_tendsto_zero.mpr
    apply squeeze_zero (fun _ => norm_nonneg _)
      (g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖)
    · intro j
      apply Lp.norm_le_mul_norm_of_ae_le_mul
      filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P),
        ((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp
          (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1
      change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1
      rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one]
      exact he j _
    · simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖
  obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl
  refine ⟨R,?_,?_,?_⟩
  · apply hclosed.mem_of_tendsto (q.sobolev_tendsto (P := Q) (by simpa only [hR] using hv) (by simpa only [hRG] using hgl))
    exact Eventually.of_forall hQS
  · rw [hR]
    exact q.composeValue_ae hF (q.inclusion P)
  · rw [hRG]
    exact ClosedCalculus.mulLp_ae hc hcC _

end CAT0Fillings.ChartGeometry
end

end OAI