File size: 2,800 Bytes
37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.InstructionRectangles
namespace OAI
namespace ForcedComputation.Radix
open ShearFlows
def InUnit (R : RationalBox 2) : Prop := ∀ j, 0 ≤ R.lower j ∧ R.upper j ≤ 1
theorem sourceBox_inUnit {B : ℚ} (hB : 0 < B) {l a : ℚ}
(hl : 0 ≤ l ∧ l + 1 ≤ B) (ha : 0 ≤ a ∧ a + 1 ≤ B) :
InUnit (sourceBox B l a) := by
intro j
fin_cases j
· exact ⟨div_nonneg hl.1 hB.le, (div_le_one hB).mpr hl.2⟩
· exact ⟨div_nonneg ha.1 hB.le, (div_le_one hB).mpr ha.2⟩
theorem targetBox_inUnit {B : ℚ} (hB : 0 < B) {l b : ℚ}
(hl : 0 ≤ l ∧ l + 1 ≤ B) (hb : 0 ≤ b ∧ b + 1 ≤ B) (m : HeadMove) :
InUnit (targetBox B l b m) := by
have hp : 0 < B ^ 2 := pow_pos hB 2
have nested {a c : ℚ} (ha : 0 ≤ a ∧ a + 1 ≤ B) (hc : 0 ≤ c ∧ c + 1 ≤ B) :
0 ≤ a / B + c / B ^ 2 ∧ a / B + (c + 1) / B ^ 2 ≤ 1 := by
refine ⟨add_nonneg (div_nonneg ha.1 hB.le) (div_nonneg hc.1 hp.le), ?_⟩
calc
a / B + (c + 1) / B ^ 2 ≤ a / B + B / B ^ 2 := by
exact add_le_add (le_refl _) ((div_le_div_iff_of_pos_right hp).mpr hc.2)
_ = (a + 1) / B := by field_simp
_ ≤ 1 := (div_le_one hB).mpr ha.2
intro j
cases m <;> fin_cases j
· exact ⟨le_rfl, le_rfl⟩
· exact nested hl hb
· exact ⟨div_nonneg hl.1 hB.le, (div_le_one hB).mpr hl.2⟩
· exact ⟨div_nonneg hb.1 hB.le, (div_le_one hB).mpr hb.2⟩
· exact nested hb hl
· exact ⟨le_rfl, le_rfl⟩
theorem center_mem {R : RationalBox 2} (hR : R.positive) : R.center ∈ R.carrier := by
intro j
have h : (R.lower j : ℝ) < R.upper j := by exact_mod_cast hR j
dsimp [RationalBox.center]
constructor <;> linarith
theorem halfWidth_le_half {R : RationalBox 2} (hR : InUnit R) (j : Fin 2) :
R.halfWidth j ≤ 1 / 2 := by
have hl : (0 : ℝ) ≤ R.lower j := by exact_mod_cast (hR j).1
have hu : (R.upper j : ℝ) ≤ 1 := by exact_mod_cast (hR j).2
dsimp [RationalBox.halfWidth]
linarith
theorem scaledBox_between {κ : ℚ} (hκ : 0 ≤ κ) (offset : Fin 2 → ℚ)
{R : RationalBox 2} (hR : InUnit R) {x : Plane}
(hx : x ∈ (scaledBox κ offset R).carrier) (j : Fin 2) :
(offset j : ℝ) ≤ x j ∧ x j ≤ (offset j : ℝ) + (κ : ℝ) := by
have hp : (0 : ℝ) ≤ κ := by exact_mod_cast hκ
have hl : (0 : ℝ) ≤ R.lower j := by exact_mod_cast (hR j).1
have hu : (R.upper j : ℝ) ≤ 1 := by exact_mod_cast (hR j).2
have hj := hx j
simp only [scaledBox, Rat.cast_add, Rat.cast_mul] at hj
exact ⟨(le_add_of_nonneg_right (mul_nonneg hp hl)).trans hj.1,
hj.2.trans (by simpa only [mul_one] using (add_le_add (le_refl (offset j : ℝ)) (mul_le_mul_of_nonneg_left hu hp)))⟩
end ForcedComputation.Radix
end OAI
|