| import Mathlib |
|
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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 |
| |
