import OAI.NumberTheory.SingleFold.SquareSupport

namespace OAI

namespace SingleFold.LocalFive

lemma mod_five_anisotropic (c : ℤ) (hc : c=2 ∨ c= -2) (a b : ℤ) (h : (5:ℤ) ∣ a^2-c*b^2) :
    (5:ℤ) ∣ a ∧ (5:ℤ) ∣ b := by
  have hh : (a:ZMod 5)^2-(c:ZMod 5)*(b:ZMod 5)^2=0 := by
    exact_mod_cast (ZMod.intCast_zmod_eq_zero_iff_dvd (a^2-c*b^2) 5).mpr h
  have hz : (a:ZMod 5)=0 ∧ (b:ZMod 5)=0 := by
    rcases hc with rfl | rfl
    · exact (by decide : ∀ a b : ZMod 5, a^2-2*b^2=0 → a=0 ∧ b=0) _ _ (by simpa using hh)
    · exact (by decide : ∀ a b : ZMod 5, a^2-(-2)*b^2=0 → a=0 ∧ b=0) _ _ (by simpa using hh)
  exact ⟨(ZMod.intCast_zmod_eq_zero_iff_dvd a 5).mp hz.1,
    (ZMod.intCast_zmod_eq_zero_iff_dvd b 5).mp hz.2⟩

lemma five_dvd_of_sq {a : ℤ} (h : (5:ℤ) ∣ a^2) : (5:ℤ) ∣ a :=
  (show Prime (5:ℤ) by norm_num).dvd_of_dvd_pow h

lemma no_norm_five_int (c : ℤ) (hc : c=2 ∨ c= -2) (N : ℕ) : ∀ R S : ℤ, R^2-c*S^2=5*(N:ℤ)^2 → N=0 := by
  induction N using Nat.strong_induction_on with
  | h N ih =>
    intro R S he
    by_cases hN : N=0
    · exact hN
    obtain ⟨hr,hs⟩ := mod_five_anisotropic c hc R S ⟨(N:ℤ)^2, he⟩
    obtain ⟨r,hr⟩ := hr
    obtain ⟨s,hs⟩ := hs
    have hd : (5:ℤ) ∣ (N:ℤ)^2 := ⟨r^2-c*s^2, by rw [hr,hs] at he; nlinarith only [he]⟩
    have hd' : 5 ∣ N := by exact_mod_cast five_dvd_of_sq hd
    obtain ⟨n,hn⟩ := hd'
    have hnl : n < N := by omega
    have he' : r^2-c*s^2=5*(n:ℤ)^2 := by
      rw [hr,hs,hn] at he
      push_cast at he
      nlinarith only [he]
    have hn0 := ih n hnl r s he'
    omega

lemma no_norm_five (c : ℤ) (hc : c=2 ∨ c= -2) (r s : ℚ) : r^2-(c:ℚ)*s^2 ≠ 5 := by
  intro h
  let N := r.den*s.den
  let R : ℤ := r.num*s.den
  let S : ℤ := s.num*r.den
  have he : R^2-c*S^2=5*(N:ℤ)^2 := by
    have h' : (R:ℚ)^2-(c:ℚ)*(S:ℚ)^2=5*(N:ℚ)^2 := by
      dsimp [R,S,N]
      push_cast
      have hr := r.num_div_den
      have hs := s.num_div_den
      rw [←hr,←hs] at h
      field_simp at h
      nlinarith only [h]
    exact_mod_cast h'
  have hz := no_norm_five_int c hc N R S he
  exact (Nat.mul_ne_zero r.den_ne_zero s.den_ne_zero) hz

lemma no_norm_ten (r s : ℚ) : r^2-10*s^2 ≠ 5 := by
  intro h
  have hr : r ≠ 0 := by intro hh; rw [hh] at h; nlinarith [sq_nonneg s]
  have h' : (5/r)^2-(-2:ℤ)*(5*s/r)^2=5 := by
    push_cast
    field_simp
    nlinarith only [h]
  exact no_norm_five (-2) (Or.inr rfl) (5/r) (5*s/r) h'

end SingleFold.LocalFive

namespace SingleFold.Descent
open WeierstrassCurve WeierstrassCurve.Affine WeierstrassCurve.Affine.Point

lemma alpha_five_when_zero (P : E.Point) (h0 : classOf (alphaVal 0 P)=1) :
    classOf (alphaVal 5 P)=1 ∨ classOf (alphaVal 5 P)=classOf 5 := by
  cases P with
  | zero => exact Or.inl (by simp [←zero_def])
  | some x y h =>
    have hsq := (classOf_eq_one_iff (alphaVal_ne_zero 0 (by norm_num) (some x y h))).mp h0
    by_cases hx : x=0
    · have hs : IsSquare (-25:ℚ) := by
        simp only [alphaVal,adjusted,hx,sub_self] at hsq
        convert hsq using 1; norm_num
      obtain ⟨r,hr⟩ := hs
      nlinarith [sq_nonneg r]
    have hs : IsSquare x := by simpa [alphaVal,adjusted,hx] using hsq
    obtain ⟨r,hr⟩ := hs
    have hxp : 0 < x := lt_of_le_of_ne (by nlinarith [sq_nonneg r]) (Ne.symm hx)
    have hx5 : x ≠ 5 := by
      intro he
      exact not_isSquare_prime 5 (by norm_num) (by change IsSquare (5:ℚ); rw [←he]; exact ⟨r,hr⟩)
    have hx5p : 5 < x := by
      have he := equation h
      by_contra hn
      have hl : x < 5 := lt_of_le_of_ne (le_of_not_gt hn) hx5
      have hm : x*(5-x)*(x+5)>0 := mul_pos (mul_pos hxp (sub_pos.mpr hl)) (by linarith)
      nlinarith [sq_nonneg y]
    obtain ⟨d,s,hd,hs0,he⟩ := supported_five (some x y h)
    have ha : alphaVal 5 (some x y h)=x-5 := by simp [alphaVal,adjusted,sub_ne_zero.mpr hx5]
    rw [ha] at he ⊢
    have hdpos : 0 < d := Nat.pos_of_ne_zero (by intro hh; subst d; norm_num at hd)
    have hnneg : ¬ x-5= -((d:ℚ)*s^2) := by
      have hdq : 0 ≤ (d:ℚ) := Nat.cast_nonneg d
      have hnon := mul_nonneg hdq (sq_nonneg s)
      intro hh
      linarith
    have he := he.resolve_right hnneg
    have hdle := Nat.le_of_dvd (by norm_num : 0 < 10) hd
    interval_cases d <;> norm_num at hd
    · left
      rw [he]
      simpa using classOf_mul_sq (d:=1) (by norm_num) hs0
    · exfalso
      apply LocalFive.no_norm_five 2 (Or.inl rfl) r s
      norm_num at he ⊢
      nlinarith only [hr,he]
    · right
      rw [he]
      exact classOf_mul_sq (by norm_num) hs0
    · exfalso
      apply LocalFive.no_norm_ten r s
      norm_num at he
      nlinarith only [hr,he]

noncomputable def a0 : E.Point →+ Additive SquareClass := alpha 0 (by norm_num) (by norm_num)
noncomputable def a5 : E.Point →+ Additive SquareClass := alpha 5 (by norm_num) (by norm_num)

def T0 : E.Point := some 0 0 (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E]))
def T5 : E.Point := some 5 0 (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E]))
def G : E.Point := some (25/4) (75/8) (E.equation_iff_nonsingular.mp (by norm_num [WeierstrassCurve.Affine.equation_iff,E]))

lemma a0_T0 : a0 T0 = Additive.ofMul (classOf (-1)) := by
  change classOf _ = _
  have hh := classOf_mul_sq (d := -1) (r := 5) (by norm_num) (by norm_num)
  norm_num [T0,alphaVal,adjusted] at *
  exact hh
lemma a0_T5 : a0 T5 = Additive.ofMul (classOf 5) := by
  change classOf (alphaVal 0 T5) = classOf 5
  norm_num [T5,alphaVal,adjusted]
lemma a0_G : a0 G = 0 := by
  change classOf _ = 1
  have hh := classOf_mul_sq (d := 1) (r := 5/2) (by norm_num) (by norm_num)
  norm_num [G,alphaVal,adjusted] at *
  exact hh
lemma a5_G : a5 G = Additive.ofMul (classOf 5) := by
  change classOf _ = _
  have hh := classOf_mul_sq (d := 5) (r := 1/2) (by norm_num) (by norm_num)
  norm_num [G,alphaVal,adjusted] at *
  exact hh

lemma normalize_a0 (P : E.Point) : ∃ S : E.Point,
    (S=0 ∨ S=T0 ∨ S=T5 ∨ S=T0+T5) ∧ a0 (P-S)=0 := by
  obtain h | h | h | h := alpha_zero_classes P
  · refine ⟨0,Or.inl rfl,?_⟩
    change classOf (alphaVal 0 (P-0))=1
    simpa only [sub_zero,classOf_one] using h
  · refine ⟨T0,Or.inr (Or.inl rfl),?_⟩
    rw [map_sub,a0_T0]
    exact sub_eq_zero.mpr h
  · refine ⟨T5,Or.inr (Or.inr (Or.inl rfl)),?_⟩
    rw [map_sub,a0_T5]
    exact sub_eq_zero.mpr h
  · refine ⟨T0+T5,Or.inr (Or.inr (Or.inr rfl)),?_⟩
    rw [map_sub,map_add,a0_T0,a0_T5]
    apply sub_eq_zero.mpr
    change classOf (alphaVal 0 P)=classOf (-1)*classOf 5
    rw [←classOf_mul (by norm_num) (by norm_num)]
    norm_num
    exact h

theorem mod_two_cover (P : E.Point) : ∃ S : E.Point, ∃ b : Bool, ∃ Q : E.Point,
    (S=0 ∨ S=T0 ∨ S=T5 ∨ S=T0+T5) ∧ P = (S + if b then G else 0) + (2:ℕ) • Q := by
  obtain ⟨S,hS,h0⟩ := normalize_a0 P
  have h5 := alpha_five_when_zero (P-S) h0
  rcases h5 with h5 | h5
  · have hk : kummer (P-S)=0 := by
      apply Prod.ext
      · exact h0
      · exact h5
    obtain ⟨Q,hQ⟩ := (kummer_eq_zero_iff _).mp hk
    refine ⟨S,false,Q,hS,?_⟩
    simp only [Bool.false_eq_true, ↓reduceIte, add_zero]
    rw [hQ]
    abel
  · have hk : kummer (P-S-G)=0 := by
      apply Prod.ext
      · change a0 (P-S-G)=0
        rw [map_sub,h0,a0_G,sub_self]
      · change a5 (P-S-G)=0
        rw [map_sub,a5_G]
        exact sub_eq_zero.mpr h5
    obtain ⟨Q,hQ⟩ := (kummer_eq_zero_iff _).mp hk
    refine ⟨S,true,Q,hS,?_⟩
    simp only [↓reduceIte]
    rw [hQ]
    abel

end SingleFold.Descent

end OAI