import Mathlib

namespace OAI

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namespace VertexCover.Parameters

noncomputable section

def a (m : ℕ) : ℝ := 1 / (4*m)
def b₀ (m : ℕ) : ℝ := 1 / (2*m)
def ν (m : ℕ) : ℝ := 1 / (2^7 * (m:ℝ)^3)
def β (m : ℕ) : ℝ := 1 / (2^10 * (m:ℝ)^3)
def ell (m : ℕ) : ℕ := 2^11*m^3
def h (m : ℕ) : ℕ := 2^12*m^3
def d (m : ℕ) : ℕ := 2^38*m^10
def σ (m : ℕ) : ℝ := 1 / (2^46 * (m:ℝ)^12)
def p (m : ℕ) : ℕ := 2*d m+1
def t (m : ℕ) : ℕ := 2^17*m^4

theorem d_square (m : ℕ) : d m = (2^19*m^5)^2 := by
  unfold d
  ring

theorem sqrt_d (m : ℕ) : Real.sqrt (d m) = 2^19*(m:ℝ)^5 := by
  rw [d_square]
  simp only [Nat.cast_pow, Nat.cast_mul, Nat.cast_ofNat, Real.sqrt_sq_eq_abs]
  exact abs_of_nonneg (by positivity)

theorem t_eq (m : ℕ) (hm : 0 < m) : (t m : ℝ) = a m * Real.sqrt (d m) := by
  rw [sqrt_d]
  have hp : (m:ℝ) ≠ 0 := by positivity
  simp only [t, a, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  field_simp
  ring

theorem positive (m : ℕ) (hm : 0 < m) :
    0 < ν m ∧ 0 < β m ∧ 0 < ell m ∧ 0 < h m ∧ 0 < d m ∧ 0 < σ m := by
  dsimp [ν, β, ell, h, d, σ]
  constructor <;> try positivity
  constructor <;> try positivity
  constructor <;> try positivity
  constructor <;> try positivity
  constructor <;> positivity

theorem dimensions (m : ℕ) (hm : 4 ≤ m) :
    4 ≤ d m ∧ 2 ≤ h m ∧ h m < d m ∧ Even (d m) := by
  have h1 : 1 ≤ m := by omega
  have h3 : 1 ≤ m^3 := one_le_pow₀ h1
  have h310 : m^3 ≤ m^10 := Nat.pow_le_pow_right (by omega) (by decide)
  have h10 : 1 ≤ m^10 := one_le_pow₀ h1
  dsimp [d, h]
  refine ⟨by omega, by omega, ?_, ?_⟩
  · omega
  · exact ⟨137438953472*m^10, by omega⟩

theorem restriction_budget (m : ℕ) (hm : 4 ≤ m) :
    16*(h m : ℝ)+32*(h m : ℝ)^2 ≤ ν m*(d m : ℝ)/2 := by
  have hm1 : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega)
  have hp : (m:ℝ) ≠ 0 := by exact_mod_cast (show m ≠ 0 by omega)
  have h36 : (m:ℝ)^3 ≤ (m:ℝ)^6 := pow_le_pow_right₀ hm1 (by decide)
  have h67 : (m:ℝ)^6 ≤ (m:ℝ)^7 := pow_le_pow_right₀ hm1 (by decide)
  have h6 : (0:ℝ) ≤ (m:ℝ)^6 := by positivity
  have heq : ν m*(d m : ℝ)/2 = 2^30*(m:ℝ)^7 := by
    simp only [ν, d, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
    field_simp
  rw [heq]
  simp only [h, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  nlinarith [sq_nonneg ((m:ℝ)^3)]

theorem β_h (m : ℕ) (hm : 0 < m) : β m*(h m : ℝ)/2 = 2 := by
  have hp : (m:ℝ) ≠ 0 := by positivity
  simp only [β, h, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  field_simp

theorem ell_eq (m : ℕ) (hm : 0 < m) : (ell m : ℝ) = 2/β m := by
  have hp : (m:ℝ) ≠ 0 := by positivity
  simp only [ell, β, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
  field_simp

theorem σ_eq (m : ℕ) : σ m = 1 / ((h m : ℝ)^2*(ell m : ℝ)^2) := by
  unfold σ h ell
  push_cast
  congr 1
  ring

theorem grid_budget (m : ℕ) (hm : 4 ≤ m) :
    (d m : ℝ)/(p m : ℝ) < 1/2 ∧ (1:ℝ)/2 < (t m : ℝ)/2 ∧ 2*m < p m := by
  have hd := (dimensions m hm).1
  have hm1 : 1 ≤ m := by omega
  have hm4 : 1 ≤ m^4 := one_le_pow₀ hm1
  have hmd : m ≤ d m := by
    have hpow : m^1 ≤ m^10 := Nat.pow_le_pow_right (by omega) (by decide)
    simp only [pow_one] at hpow
    dsimp [d]
    omega
  refine ⟨?_, ?_, ?_⟩
  · have hp : (0:ℝ) < p m := by unfold p; positivity
    apply (div_lt_iff₀ hp).mpr
    simp only [p, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one]
    linarith
  · have ht : 1 < t m := by dsimp [t]; omega
    have ht' : (1:ℝ) < t m := by exact_mod_cast ht
    linarith
  · dsimp [p]; omega

end

end VertexCover.Parameters

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end OAI