import OAI.NumberTheory.Jacobsthal.Harmonic.PolynomialLocalGraph
import OAI.NumberTheory.Jacobsthal.Partitions.InflectionTransport
import OAI.NumberTheory.Jacobsthal.Paths.AffineBranch

namespace OAI

namespace Erdos970

section

namespace ErdosDivisibleInflection

open ErdosCriticalGeometry ErdosImplicitCurvature

noncomputable def swapVars (R : Type*) [CommSemiring R] : MV R ≃ₐ[R] MV R :=
  MvPolynomial.renameEquiv R (Equiv.swap (0 : Fin 2) 1)

theorem swap_twice (R : Type*) [CommSemiring R] (Q : MV R) :
    swapVars R (swapVars R Q) = Q := by
  simpa [swapVars] using (MvPolynomial.renameEquiv R (Equiv.swap (0 : Fin 2) 1)).apply_symm_apply Q

theorem nestedX_swap (R : Type*) [CommSemiring R] (Q : MV R) :
    nestedX R (swapVars R Q) = nestedY R Q := rfl

theorem nestedY_swap (R : Type*) [CommSemiring R] (Q : MV R) :
    nestedY R (swapVars R Q) = nestedX R Q := by
  change nestedX R (swapVars R (swapVars R Q)) = _
  rw [swap_twice]

theorem partial_zero_swap (R : Type*) [CommSemiring R] (Q : MV R) :
    MvPolynomial.pderiv 0 (swapVars R Q) = swapVars R (MvPolynomial.pderiv 1 Q) := by
  apply (nestedX R).injective
  rw [nestedX_partial_zero,nestedX_swap,nestedX_swap,nestedY_partial_one]

theorem partial_one_swap (R : Type*) [CommSemiring R] (Q : MV R) :
    MvPolynomial.pderiv 1 (swapVars R Q) = swapVars R (MvPolynomial.pderiv 0 Q) := by
  apply (nestedY R).injective
  rw [nestedY_partial_one,nestedY_swap,nestedY_swap,nestedX_partial_zero]

theorem eval_swap (R : Type*) [CommSemiring R] (Q : MV R) (x y : R) :
    MvPolynomial.eval ![x,y] (swapVars R Q) = MvPolynomial.eval ![y,x] Q := by
  have h1 := eval_nestedX R (swapVars R Q) x y
  rw [nestedX_swap] at h1
  exact h1.symm.trans (eval_nestedY R Q y x)

theorem peval_swap (Q : MV ℝ) (x y : ℝ) : peval (swapVars ℝ Q) x y = peval Q y x := by
  rw [peval_eq_eval_pair,peval_eq_eval_pair,eval_swap]

theorem complexify_swap (Q : MV ℝ) : complexify (swapVars ℝ Q) = swapVars ℂ (complexify Q) := by
  simp only [complexify,swapVars,MvPolynomial.renameEquiv_apply,MvPolynomial.map_rename]

theorem mixed_partials_complex (Q : MV ℂ) :
    MvPolynomial.pderiv 1 (MvPolynomial.pderiv 0 Q) =
      MvPolynomial.pderiv 0 (MvPolynomial.pderiv 1 Q) := by
  classical
  ext m
  simp [MvPolynomial.coeff_pderiv,add_assoc,add_left_comm,add_comm,mul_left_comm,mul_comm]

theorem inflection_swap (Q : MV ℂ) : inflection (swapVars ℂ Q) = swapVars ℂ (inflection Q) := by
  unfold inflection
  simp only [partial_zero_swap,partial_one_swap,mixed_partials_complex,map_add,map_mul,map_pow]
  have hC : swapVars ℂ (MvPolynomial.C (-2 : ℂ)) = MvPolynomial.C (-2 : ℂ) := by simp [swapVars]
  rw [hC]
  ring

theorem complex_divisibility_swap (Q : MV ℝ)
    (hdiv : complexify Q ∣ inflection (complexify Q)) :
    complexify (swapVars ℝ Q) ∣ inflection (complexify (swapVars ℝ Q)) := by
  rw [complexify_swap,inflection_swap]
  exact map_dvd (swapVars ℂ).toRingHom hdiv

end ErdosDivisibleInflection

end

section

namespace ErdosDivisibleInflection

open ErdosCriticalGeometry ErdosImplicitCurvature

theorem no_regular_y_point (Q : MV ℝ) (hQ : Irreducible (complexify Q))
    (hdeg : 1 < Q.totalDegree) (hdiv : complexify Q ∣ inflection (complexify Q))
    (x y : ℝ) (hzero : peval Q x y = 0) (hy : peval (partialY Q) x y ≠ 0) : False := by
  obtain ⟨a,b,f,hax,hxb,hfx,hf,hcurve,hfy⟩ := exists_local_real_graph Q x y hzero hy
  have hsecond := zero_second_derivative_on_divisible_graph Q f a b hdiv hf hcurve hfy
  obtain ⟨A,B,haffine⟩ := affine_of_second_derivative_zero f a b x ⟨hax,hxb⟩ hf hsecond
  have hdegC : 1 < (complexify Q).totalDegree := by rwa [complexify_totalDegree]
  apply no_affine_interval (complexify Q) hQ hdegC (A : ℂ) (B : ℂ) a b (hax.trans hxb)
  intro t ht
  have hz := hcurve t ht
  rw [haffine t ht] at hz
  have hc := complexify_eval Q t (A*t+B)
  rw [← peval_eq_eval_pair,hz] at hc
  simpa only [Complex.ofReal_mul,Complex.ofReal_add,Complex.ofReal_zero] using hc

theorem no_regular_x_point (Q : MV ℝ) (hQ : Irreducible (complexify Q))
    (hdeg : 1 < Q.totalDegree) (hdiv : complexify Q ∣ inflection (complexify Q))
    (x y : ℝ) (hzero : peval Q x y = 0) (hx : peval (partialX Q) x y ≠ 0) : False := by
  have hirr : Irreducible (complexify (swapVars ℝ Q)) := by
    rw [complexify_swap]
    exact (MulEquiv.irreducible_iff (swapVars ℂ).toMulEquiv).mpr hQ
  have hd : 1 < (swapVars ℝ Q).totalDegree := by
    simpa only [swapVars,MvPolynomial.totalDegree_renameEquiv] using hdeg
  have hz : peval (swapVars ℝ Q) y x = 0 := by rw [peval_swap]; exact hzero
  have hpart : peval (partialY (swapVars ℝ Q)) y x ≠ 0 := by
    change peval (MvPolynomial.pderiv 1 (swapVars ℝ Q)) y x ≠ 0
    rw [partial_one_swap,peval_swap]
    exact hx
  exact no_regular_y_point (swapVars ℝ Q) hirr hd (complex_divisibility_swap Q hdiv) y x hz hpart

theorem no_real_regular_point_of_divisible_inflection (Q : MV ℝ)
    (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree)
    (hdiv : complexify Q ∣ inflection (complexify Q)) (x y : ℝ)
    (hzero : peval Q x y = 0)
    (hregular : peval (partialX Q) x y ≠ 0 ∨ peval (partialY Q) x y ≠ 0) : False := by
  rcases hregular with hx | hy
  · exact no_regular_x_point Q hQ hdeg hdiv x y hzero hx
  · exact no_regular_y_point Q hQ hdeg hdiv x y hzero hy

theorem divisible_inflection_gradient_zero (Q : MV ℝ)
    (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree)
    (hdiv : complexify Q ∣ inflection (complexify Q)) (x y : ℝ)
    (hzero : peval Q x y = 0) :
    peval (partialX Q) x y = 0 ∧ peval (partialY Q) x y = 0 := by
  constructor
  · by_contra hx
    exact no_regular_x_point Q hQ hdeg hdiv x y hzero hx
  · by_contra hy
    exact no_regular_y_point Q hQ hdeg hdiv x y hzero hy

theorem real_divisible_inflection_gradient_zero (Q : MV ℝ)
    (hQ : Irreducible (complexify Q)) (hdeg : 1 < Q.totalDegree)
    (hdiv : Q ∣ inflectionPolynomial Q) (x y : ℝ) (hzero : peval Q x y = 0) :
    peval (partialX Q) x y = 0 ∧ peval (partialY Q) x y = 0 :=
  divisible_inflection_gradient_zero Q hQ hdeg (real_divisibility_to_complex Q hdiv) x y hzero

end ErdosDivisibleInflection

end

end Erdos970

end OAI