| import Mathlib |
| import OAI.Computability.MinUncut.Model |
|
|
| namespace OAI |
|
|
| section |
| namespace MinUncut.FiniteGaussian |
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|
| def algebraicNonneg (a b : ℚ) (N : ℕ) : Bool := |
| if 0 ≤ b then |
| if 0 ≤ a then true else decide (a^2 ≤ (N:ℚ)*b^2) |
| else |
| if a < 0 then false else decide ((N:ℚ)*b^2 ≤ a^2) |
|
|
| lemma algebraicNonneg_eq (a b : ℚ) (N : ℕ) : |
| algebraicNonneg a b N = true ↔ 0 ≤ (a:ℝ)+(b:ℝ)*Real.sqrt (N:ℝ) := by |
| have hs := Real.sqrt_nonneg (N:ℝ) |
| have hs2 := Real.sq_sqrt (Nat.cast_nonneg (α := ℝ) N) |
| have hb2 : ((b:ℝ)*Real.sqrt (N:ℝ))^2=(N:ℝ)*(b:ℝ)^2 := by rw [mul_pow,hs2]; ring |
| unfold algebraicNonneg |
| split_ifs with hb ha ha |
| · have hbr : (0:ℝ) ≤ b := by exact_mod_cast hb |
| have har : (0:ℝ) ≤ a := by exact_mod_cast ha |
| simp only [true_iff] |
| positivity |
| · have hbr : (0:ℝ) ≤ b := by exact_mod_cast hb |
| have har : (a:ℝ) < 0 := by exact_mod_cast (lt_of_not_ge ha) |
| have hprod : 0 ≤ (b:ℝ)*Real.sqrt (N:ℝ) := mul_nonneg hbr hs |
| simp only [decide_eq_true_eq] |
| constructor |
| · intro h |
| have h' : (a:ℝ)^2 ≤ (N:ℝ)*(b:ℝ)^2 := by exact_mod_cast h |
| nlinarith |
| · intro h |
| have h' : (a:ℝ)^2 ≤ (N:ℝ)*(b:ℝ)^2 := by nlinarith |
| exact_mod_cast h' |
| · have hbr : (b:ℝ) < 0 := by exact_mod_cast (lt_of_not_ge hb) |
| have har : (a:ℝ) < 0 := by exact_mod_cast ha |
| have hprod : (b:ℝ)*Real.sqrt (N:ℝ) ≤ 0 := mul_nonpos_of_nonpos_of_nonneg hbr.le hs |
| simp only [false_iff,not_le] |
| linarith |
| · have hbr : (b:ℝ) < 0 := by exact_mod_cast (lt_of_not_ge hb) |
| have har : (0:ℝ) ≤ a := by exact_mod_cast (le_of_not_gt ha) |
| have hprod : (b:ℝ)*Real.sqrt (N:ℝ) ≤ 0 := mul_nonpos_of_nonpos_of_nonneg hbr.le hs |
| simp only [decide_eq_true_eq] |
| constructor |
| · intro h |
| have h' : (N:ℝ)*(b:ℝ)^2 ≤ (a:ℝ)^2 := by exact_mod_cast h |
| nlinarith |
| · intro h |
| have h' : (N:ℝ)*(b:ℝ)^2 ≤ (a:ℝ)^2 := by nlinarith |
| exact_mod_cast h' |
|
|
| end MinUncut.FiniteGaussian |
|
|
| end |
|
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| end OAI |
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