openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Programs /BoxBounds.lean
| 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 | |