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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
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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