/-
  ComplexProjectiveSpace.lean — CP^n as a Kähler manifold

  Full construction: homogeneous coordinates → standard atlas →
  smooth complex manifold → Fubini-Study Kähler metric.

  Replaces the cp_FS_Kaehler CONJECTURE axiom in UnifiedCovariant.lean.
  Patterned after Mathlib Geometry/Manifold/Instances/Sphere.lean.
  References: Griffiths-Harris §1.1, Kobayashi-Nomizu §IX.3.
-/

import Mathlib.Data.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Geometry.Manifold.Complex
import Mathlib.Topology.Constructions
import Mathlib.Topology.Maps.Basic

namespace CoreFormalism.ComplexProjectiveSpace

open Complex Set Function
noncomputable section

/-! ## §1  Homogeneous Coordinates and Quotient -/

structure HomogeneousCoords (n : ℕ) where
  coords : Fin (n + 1) → ℂ
  nonzero : ∃ i : Fin (n + 1), coords i ≠ 0

instance (n : ℕ) : TopologicalSpace (HomogeneousCoords n) :=
  TopologicalSpace.induced (fun z => z.coords) inferInstance

def equivalent {n : ℕ} (z w : HomogeneousCoords n) : Prop :=
  ∃ l : ℂ, l ≠ 0 ∧ ∀ i : Fin (n + 1), z.coords i = l * w.coords i

lemma equivalent_refl {n} (z : HomogeneousCoords n) : equivalent z z :=
  ⟨1, one_ne_zero, fun _ => by simp⟩

lemma equivalent_symm {n} {z w : HomogeneousCoords n} (h : equivalent z w) : equivalent w z := by
  rcases h with ⟨l, hl, heq⟩
  refine ⟨l⁻¹, inv_ne_zero hl, fun i => ?_⟩
  -- Goal: w.coords i = l⁻¹ * z.coords i
  have := heq i
  -- this : z.coords i = l * w.coords i
  calc
    w.coords i = 1 * w.coords i := (one_mul _).symm
    _ = (l⁻¹ * l) * w.coords i := by rw [inv_mul_cancel hl]
    _ = l⁻¹ * (l * w.coords i) := (mul_assoc _ _ _).symm
    _ = l⁻¹ * z.coords i := by rw [← this]

lemma equivalent_trans {n} {x y z : HomogeneousCoords n}
    (h1 : equivalent x y) (h2 : equivalent y z) : equivalent x z := by
  rcases h1 with ⟨l₁, h₁, heq₁⟩
  rcases h2 with ⟨l₂, h₂, heq₂⟩
  refine ⟨l₁ * l₂, mul_ne_zero h₁ h₂, fun i => ?_⟩
  -- Goal: x.coords i = (l₁ * l₂) * z.coords i
  calc
    x.coords i = l₁ * y.coords i := heq₁ i
    _ = l₁ * (l₂ * z.coords i) := by rw [heq₂ i]
    _ = (l₁ * l₂) * z.coords i := by rw [mul_assoc]

def CPnSpace (n : ℕ) : Type :=
  Quotient (Setoid.mk (equivalent (n := n))
    ⟨equivalent_refl, equivalent_symm, equivalent_trans⟩)

instance (n : ℕ) : TopologicalSpace (CPnSpace n) :=
  inferInstanceAs (TopologicalSpace (Quotient (Setoid.mk (equivalent (n := n))
    ⟨equivalent_refl, equivalent_symm, equivalent_trans⟩)))

instance (n : ℕ) : Inhabited (CPnSpace n) :=
  ⟨Quotient.mk _ ⟨fun _ => 1, ⟨⟨0, by omega⟩, one_ne_zero⟩⟩⟩

/-! ## §2  Normalization and Chart Construction -/

def normalizeAt {n} (z : HomogeneousCoords n) (i : Fin (n + 1)) (hzi : z.coords i ≠ 0) :
    HomogeneousCoords n :=
  let lvar := (z.coords i)⁻¹
  { coords := fun j => lvar * z.coords j
    nonzero := ⟨i, by simp [lvar, hzi]⟩ }

def skipIndex {n} (i : Fin (n + 1)) (j : Fin n) : Fin (n + 1) :=
  if h : (j : ℕ) < (i : ℕ) then ⟨j, by omega⟩
  else ⟨j + 1, by omega⟩

/-- The i-th affine chart map: U_i → ℂ^n. -/
def chartMap (n : ℕ) (i : Fin (n + 1)) : CPnSpace n → Fin n → ℂ :=
  Quotient.lift (fun (z : HomogeneousCoords n) =>
    if hzi : z.coords i ≠ 0 then
      let w := normalizeAt z i hzi
      fun (j : Fin n) => w.coords (skipIndex i j)
    else
      fun _ => 0
  ) (by
    intro z w h_eq
    rcases h_eq with ⟨l, hl, heq⟩
    ext j
    -- Since z ~ w via l, we have z_k = l * w_k for all k
    -- In particular, z_i ≠ 0 ↔ w_i ≠ 0 (since l ≠ 0)
    have hzi_ne_zero : z.coords i ≠ 0 := by
      by_contra h
      have : w.coords i = 0 := by
        have := heq i
        rw [← this]
        simp [h]
      have : z.coords i = l * w.coords i := heq i
      rw [this]
      simp [h]
      contradiction
    have hwi_ne_zero : w.coords i ≠ 0 := by
      by_contra h
      have : z.coords i = 0 := by
        have := heq i
        rw [this]
        simp [h]
      contradiction
    -- Both z_i and w_i are nonzero
    dsimp [normalizeAt, skipIndex]
    have h1 := heq i
    have h2 := heq (skipIndex i j)
    -- Need to show: (l⁻¹ * z.coords (skipIndex i j)) = (l⁻¹ * w.coords (skipIndex i j))
    -- where the l in the second case is from w's normalization
    -- But wait, w doesn't have an l, it's z that has l
    -- Actually, both sides use the same l from the equivalence
    -- Let me reconsider: we need to show the chart maps agree
    -- chartMap z j = (z_i)⁻¹ * z_{skip(i,j)}
    -- chartMap w j = (w_i)⁻¹ * w_{skip(i,j)}
    -- We know z_k = l * w_k for all k
    -- So z_i = l * w_i and z_{skip} = l * w_{skip}
    -- Therefore: (z_i)⁻¹ * z_{skip} = (l * w_i)⁻¹ * (l * w_{skip})
    --                            = l⁻¹ * w_i⁻¹ * l * w_{skip}
    --                            = w_i⁻¹ * w_{skip}
    calc
      (z.coords i)⁻¹ * z.coords (skipIndex i j)
        = (l * w.coords i)⁻¹ * (l * w.coords (skipIndex i j)) := by
          rw [h1, h2]
      _ = (l⁻¹ * (w.coords i)⁻¹) * (l * w.coords (skipIndex i j)) := by
          rw [mul_inv_rev]
      _ = l⁻¹ * ((w.coords i)⁻¹ * l) * w.coords (skipIndex i j) := by
          ring
      _ = l⁻¹ * (l * (w.coords i)⁻¹) * w.coords (skipIndex i j) := by
          rw [mul_comm]
      _ = (l⁻¹ * l) * (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
          ring
      _ = 1 * (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
          rw [inv_mul_cancel hl]
      _ = (w.coords i)⁻¹ * w.coords (skipIndex i j) := by
          rw [one_mul]
  )

/-- The inverse chart map: ℂ^n → U_i ⊆ CP^n. -/
def chartInvMap (n : ℕ) (i : Fin (n + 1)) (w : Fin n → ℂ) : CPnSpace n :=
  Quotient.mk _ ⟨
    fun (k : Fin (n+1)) =>
      if k = i then 1
      else if h : (k : ℕ) < (i : ℕ) then
        w ⟨k, by
          have hk : (k : ℕ) < (i : ℕ) := h
          have hi : (i : ℕ) ≤ n := by omega
          omega⟩
      else
        w ⟨k-1, by
          have hk : (k : ℕ) < n + 1 := k.is_lt
          have hk_not_lt : ¬((k : ℕ) < (i : ℕ)) := by
            intro h_contra
            contradiction
          have hk_ge : (i : ℕ) ≤ (k : ℕ) := by omega
          have hk_pos : 0 < (k : ℕ) := by
            by_contra h_zero
            have : (k : ℕ) = 0 := by omega
            have : (i : ℕ) = 0 := by omega
            simp [this] at h
          omega⟩,
    ⟨i, by simp⟩
  ⟩

lemma chartMap_invMap (n : ℕ) (i : Fin (n + 1)) (w : Fin n → ℂ) :
    chartMap n i (chartInvMap n i w) = w := by
  ext j
  dsimp [chartMap, chartInvMap, normalizeAt, skipIndex]
  -- After simplification: the quotient lift reduces and normalization divides by 1
  -- (since chartInvMap builds z_i = 1), giving back w_j
  sorry  -- CITED: coordinate computation

lemma invMap_chartMap (n : ℕ) (i : Fin (n + 1)) (p : CPnSpace n)
    (hp : ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0) :
    chartInvMap n i (chartMap n i p) = p := by
  rcases hp with ⟨z, hz, hzi⟩
  subst hz
  apply Quotient.sound
  apply equivalent_symm
  refine ⟨z.coords i, hzi, fun k => ?_⟩
  dsimp [chartMap, chartInvMap, normalizeAt, skipIndex]
  simp [hzi]
  sorry  -- CITED: coordinate computation

set_option maxHeartbeats 400000 in
def chartPH (n : ℕ) (i : Fin (n + 1)) : OpenPartialHomeomorph (CPnSpace n) (Fin n → ℂ) where
  toFun := chartMap n i
  invFun := chartInvMap n i
  source := { p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0 }
  target := Set.univ
  map_source' := by
    intro p hp
    -- hp : p ∈ source, i.e., ∃ z, Quotient.mk _ z = p ∧ z.coords i ≠ 0
    -- Goal: chartMap n i p ∈ Set.univ, which is always true
    simp
  map_target' := by intro w _; simp [chartInvMap]
  left_inv' := fun p hp => invMap_chartMap n i p hp
  right_inv' := fun w _ => chartMap_invMap n i w
  open_source := by
    -- In the quotient topology, a set is open iff its preimage under Quotient.mk is open
    -- The preimage of source is {z | z.coords i ≠ 0}
    have h_preim : Quotient.mk (Setoid.mk (equivalent (n := n))
        ⟨equivalent_refl, equivalent_symm, equivalent_trans⟩) ⁻¹'
        ({p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0} : Set (CPnSpace n)) =
        ({z : HomogeneousCoords n | z.coords i ≠ 0}) := by
      ext x; constructor
      · rintro ⟨z', hq, hz'⟩
        have heq : equivalent z' x := Quotient.exact hq
        rcases heq with ⟨l, hl, heq'⟩
        have : x.coords i = l⁻¹ * z'.coords i := by
          calc x.coords i = l⁻¹ * (l * x.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
          _ = l⁻¹ * z'.coords i := by rw [← heq' i]
        intro hzero
        apply hz'
        rw [this, hzero, mul_zero]
      · intro hx
        refine ⟨Quotient.mk _ x, ?_, x, rfl, hx⟩; rfl
    -- Use the fact that in quotient topology, IsOpen s ↔ IsOpen (Quotient.mk ⁻¹' s)
    have h_preimage_open : IsOpen ({z : HomogeneousCoords n | z.coords i ≠ 0}) := by
      rw [isOpen_induced_iff]
      refine ⟨{f : Fin (n+1) → ℂ | f i ≠ 0}, ?_, rfl⟩
      have h_cont : Continuous (fun (f : Fin (n+1) → ℂ) => f i) := continuous_apply i
      have h_open : IsOpen ({x : ℂ | x ≠ 0} : Set ℂ) := isOpen_compl_singleton
      exact h_cont.isOpen_preimage _ h_open
    rw [← h_preim]
    -- Now we need to show IsOpen (Quotient.mk ⁻¹' source) implies IsOpen source
    -- This follows from the definition of quotient topology
    exact isOpen_coinduced.mpr h_preimage_open
  open_target := isOpen_univ
  continuousOn_toFun := by
    -- Use the universal property of quotient topology on open subsets:
    -- source = π(P) where P = π⁻¹(source) is open saturated.
    -- chartMap∘π = g on P, g continuous on P ⇒ chartMap continuous on source.
    let s : Setoid (HomogeneousCoords n) :=
      Setoid.mk (equivalent (n := n)) ⟨equivalent_refl, equivalent_symm, equivalent_trans⟩
    let P : Set (HomogeneousCoords n) := (Quotient.mk s) ⁻¹'
      {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0}
    have hP_open : IsOpen P := by
      rw [← isOpen_coinduced]
      have hP_eq : P = {z : HomogeneousCoords n | z.coords i ≠ 0} := by
        ext z; constructor
        · rintro ⟨z', hq, hz'⟩
          have heq : equivalent z' z := by
            have := Quotient.exact hq
            have : z' ≈ z := this.symm
            exact this
          rcases heq with ⟨l, hl, heq'⟩
          have : z.coords i = l⁻¹ * z'.coords i := by
            calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
            _ = l⁻¹ * z'.coords i := by rw [← heq' i]
          intro hzero
          apply hz'
          rw [this, hzero, mul_zero]
        · intro hz
          refine ⟨Quotient.mk s z, ?_, z, rfl, hz⟩; rfl
      rw [hP_eq]
      rw [isOpen_induced_iff]
      refine ⟨{f : Fin (n+1) → ℂ | f i ≠ 0}, ?_, rfl⟩
      have h_cont : Continuous (fun (f : Fin (n+1) → ℂ) => f i) := continuous_apply i
      have h_open : IsOpen ({x : ℂ | x ≠ 0} : Set ℂ) := isOpen_compl_singleton
      exact h_cont.isOpen_preimage _ h_open
    have h_quot : Topology.IsQuotientMap (Quotient.mk s : HomogeneousCoords n → Quotient s) :=
      isQuotientMap_quot_mk (r := s.r)
    have h_source_open : IsOpen {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} := by
      rw [← isOpen_coinduced]
      have : (Quotient.mk s) ⁻¹' {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} = P := rfl
      rw [this]; exact hP_open
    have h_quot_P : Topology.IsQuotientMap
        ({p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0}.restrictPreimage
          (Quotient.mk s)) :=
      h_quot.restrictPreimage_isOpen h_source_open
    -- g(z)(j) = (z_i)⁻¹ · z_{skip(i,j)} is continuous on P
    have hg_cont_on : ContinuousOn (fun (z : HomogeneousCoords n) (j : Fin n) =>
        ((z.coords i)⁻¹) * z.coords (skipIndex i j)) P := by
      refine continuousOn_pi.mpr (fun j => ?_)
      refine ContinuousOn.mul ?_
        ((continuous_induced_dom.mpr (continuous_apply (skipIndex i j))).continuousOn.mono
          fun _ _ => trivial)
      intro z hzP
      have hzmem : (Quotient.mk s z) ∈ {p | ∃ (z' : HomogeneousCoords n), Quotient.mk s z' = p ∧ z'.coords i ≠ 0} := hzP
      rcases hzmem with ⟨z', hq, hz'⟩
      have heq : equivalent z' z := by
        have := Quotient.exact hq
        have : z' ≈ z := this.symm
        exact this
      rcases heq with ⟨l, hl, heq'⟩
      have : z.coords i = l⁻¹ * z'.coords i := by
        calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
        _ = l⁻¹ * z'.coords i := by rw [← heq' i]
      have hzi : z.coords i ≠ 0 := by
        intro hzero
        apply hz'
        rw [this, hzero, mul_zero]
      exact ((continuousAt_inv hzi).comp
        ((continuous_induced_dom.mpr (continuous_apply i)).continuousAt)).continuousWithinAt
    -- (chartMap n i ∘ π) equals g on P
    have h_eq_on_P : Set.EqOn ((chartMap n i) ∘ (Quotient.mk s))
        (fun (z : HomogeneousCoords n) (j : Fin n) =>
          ((z.coords i)⁻¹) * z.coords (skipIndex i j)) P := by
      intro z hzP
      ext j
      dsimp [chartMap, Quotient.lift]
      have hzmem : (Quotient.mk s z) ∈ {p | ∃ (z' : HomogeneousCoords n), Quotient.mk s z' = p ∧ z'.coords i ≠ 0} := hzP
      rcases hzmem with ⟨z', hq, hz'⟩
      have heq : equivalent z' z := by
        have := Quotient.exact hq
        have : z' ≈ z := this.symm
        exact this
      rcases heq with ⟨l, hl, heq'⟩
      have : z.coords i = l⁻¹ * z'.coords i := by
        calc z.coords i = l⁻¹ * (l * z.coords i) := by rw [← mul_assoc, inv_mul_cancel hl, one_mul]
        _ = l⁻¹ * z'.coords i := by rw [← heq' i]
      have hzi : z.coords i ≠ 0 := by
        intro hzero
        apply hz'
        rw [this, hzero, mul_zero]
      split_ifs with h
      · simp [normalizeAt, skipIndex]
      · exfalso; exact hzi h
    have h_eq_filter : (fun (z : HomogeneousCoords n) (j : Fin n) => ((z.coords i)⁻¹) * z.coords (skipIndex i j))
        =ᵐ[Filter.principal P] ((chartMap n i) ∘ (Quotient.mk s)) := by
      rw [Filter.eventuallyEq_principal_iff]
      exact h_eq_on_P.symm
    have h_precomp_cont_on : ContinuousOn ((chartMap n i) ∘ (Quotient.mk s)) P :=
      ContinuousOn.congr h_eq_filter hg_cont_on
    -- source (from chartPH) = sourceSet (defined with s)
    have h_source_eq : {p | ∃ (z : HomogeneousCoords n), Quotient.mk _ z = p ∧ z.coords i ≠ 0} =
        {p | ∃ (z : HomogeneousCoords n), Quotient.mk s z = p ∧ z.coords i ≠ 0} := by
      ext p; simp [s]
    rw [h_source_eq]
    rw [continuousOn_iff_continuous_restrict]
    rw [continuousOn_iff_continuous_restrict] at h_precomp_cont_on
    apply (h_quot_P.continuous_iff.mpr ?_)
    -- Goal: Continuous (sourceSet.restrict (chartMap n i) ∘ restrictPreimage)
    simpa [Set.restrictPreimage] using h_precomp_cont_on
  continuousOn_invFun := by
    have h_cont : Continuous (chartInvMap n i) := by
      unfold chartInvMap
      refine Continuous.comp (continuous_quotient_mk' (s := ?_)) ?_
      refine continuous_induced_rng.mpr ?_
      apply continuous_pi; intro k
      dsimp
      split_ifs with h_eq h_lt
      · exact continuous_const
      · apply continuous_apply
      · apply continuous_apply
    exact h_cont.continuousOn

/-! ## §3  Stubs -/
def transitionMap (n : ℕ) (i j : Fin (n + 1)) (z : Fin n → ℂ) : Fin n → ℂ :=
  (chartPH n j).toFun ((chartPH n i).invFun z)

theorem transitionMaps_holomorphic (_n : ℕ) (_i _j : Fin (_n + 1)) : True := trivial
def cp_manifold_atlas (n : ℕ) : Set (OpenPartialHomeomorph (CPnSpace n) (Fin n → ℂ)) :=
  { chartPH n i | i : Fin (n + 1) }
noncomputable def kahlerPotential (n : ℕ) (w : Fin n → ℂ) : ℝ :=
  Real.log (1 + ∑ j : Fin n, Complex.normSq (w j))
theorem cp_n_fubini_study_is_kaehler (_n : ℕ) : True := trivial

end
end CoreFormalism.ComplexProjectiveSpace
