| import OAI.Geometry.CylinderCovering.Caps.CapsDefinitions | |
| namespace OAI | |
| namespace CylinderCovering.HalfHeightCaps.CompensationEstimates | |
| lemma square_le_one {x : ℝ} (hx : |x| ≤ 1) : x^2 ≤ 1 := by | |
| nlinarith only [sq_abs x, abs_nonneg x, hx] | |
| lemma alpha_bounds {r : ℝ} (hr : |r| ≤ 1) : 0 ≤ alpha r ∧ alpha r ≤ 1/2 := by | |
| have h := square_le_one hr | |
| unfold alpha | |
| constructor | |
| · positivity | |
| · linarith | |
| lemma profile_bounds {r : ℝ} (hr : |r| ≤ 1) : 0 ≤ F r ∧ F r ≤ 1/8 := by | |
| have h := square_le_one hr | |
| have hprod := mul_nonneg (sq_nonneg r) (sub_nonneg.mpr h) | |
| unfold F | |
| constructor | |
| · positivity | |
| · nlinarith only [h, sq_nonneg r, hprod] | |
| lemma abs_product_le_one {x r : ℝ} (hx : |x| ≤ 1) (hr : |r| ≤ 1) : | |
| |x*r| ≤ 1 := by | |
| rw [abs_mul] | |
| nlinarith only [hx, hr, abs_nonneg x, abs_nonneg r, | |
| mul_nonneg (sub_nonneg.mpr hx) (abs_nonneg r)] | |
| lemma weight_bounds {δ x r : ℝ} (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2) | |
| (hx : |x| ≤ 1) (hr : |r| ≤ 1) : | |
| 1-δ ≤ 1-δ*x*r ∧ 1-δ ≤ 1+δ*x*r ∧ | |
| 1/2 ≤ 1-δ*x*r ∧ 1/2 ≤ 1+δ*x*r := by | |
| have hp := abs_product_le_one hx hr | |
| have hlow := mul_le_mul_of_nonneg_left (abs_le.mp hp).1 hδ | |
| have hhigh := mul_le_mul_of_nonneg_left (abs_le.mp hp).2 hδ | |
| refine ⟨?_, ?_, ?_, ?_⟩ <;> nlinarith only [hlow,hhigh,hδsmall] | |
| lemma weighted_square_le_half {x r : ℝ} (hx : |x| ≤ 1) (hr : |r| ≤ 1) : | |
| x^2 * alpha r ≤ 1/2 := by | |
| have hs := square_le_one hx | |
| obtain ⟨ha,ha'⟩ := alpha_bounds hr | |
| have hm := mul_le_mul_of_nonneg_right hs ha | |
| nlinarith only [hm,ha'] | |
| lemma identity {δ X Y Z r s u v : ℝ} | |
| (hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r) | |
| (hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) : | |
| (u-1/2)*(1-δ*X*r)+(v-1/2)*(1+δ*Y*s) = | |
| δ^2*(X^2*alpha r+Y^2*alpha s) := by | |
| unfold alpha beta at * | |
| linear_combination δ*X*hY-δ*Y*hX-hZ-hZv | |
| theorem radial_excess_bounds {δ X Y Z r s u v : ℝ} | |
| (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2) | |
| (hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1) | |
| (hu : 1/2 ≤ u) (hv : 1/2 ≤ v) | |
| (hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r) | |
| (hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) : | |
| 0 ≤ u-1/2 ∧ u-1/2 ≤ 2*δ^2 ∧ 0 ≤ v-1/2 ∧ v-1/2 ≤ 2*δ^2 := by | |
| have hcomp := identity hY hZ hX hZv | |
| have hwU := (weight_bounds hδ hδsmall hXb hr).2.2.1 | |
| have hwV := (weight_bounds hδ hδsmall hYb hs).2.2.2 | |
| have hA : X^2*alpha r+Y^2*alpha s ≤ 1 := by | |
| linarith only [weighted_square_le_half hXb hr, weighted_square_le_half hYb hs] | |
| have hsum := mul_le_mul_of_nonneg_left hA (sq_nonneg δ) | |
| have hu0 : 0 ≤ u-1/2 := by linarith | |
| have hv0 : 0 ≤ v-1/2 := by linarith | |
| have hU := mul_le_mul_of_nonneg_left hwU hu0 | |
| have hV := mul_le_mul_of_nonneg_left hwV hv0 | |
| exact ⟨hu0, by nlinarith only [hcomp,hsum,hU,hV,hv0], | |
| hv0, by nlinarith only [hcomp,hsum,hU,hV,hu0]⟩ | |
| lemma radial_coordinate_error {δ x r u y : ℝ} | |
| (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2) | |
| (hx : |x| ≤ 1) (hr : |r| ≤ 1) | |
| (hu0 : 0 ≤ u-1/2) (hu : u-1/2 ≤ 2*δ^2) | |
| (hy : y = u*r+δ*x*alpha r) : |y-r/2| ≤ 3*δ/2 := by | |
| obtain ⟨ha,ha'⟩ := alpha_bounds hr | |
| have heq : y-r/2 = (u-1/2)*r+δ*x*alpha r := by rw [hy]; ring | |
| have hfirst : |(u-1/2)*r| ≤ 2*δ^2 := by | |
| rw [abs_mul,abs_of_nonneg hu0] | |
| calc | |
| (u-1/2)*|r| ≤ (u-1/2)*1 := mul_le_mul_of_nonneg_left hr hu0 | |
| _ ≤ 2*δ^2 := by simpa using hu | |
| have hsecond : |δ*x*alpha r| ≤ δ/2 := by | |
| rw [abs_mul,abs_mul,abs_of_nonneg hδ,abs_of_nonneg ha] | |
| calc | |
| δ * |x| * alpha r ≤ δ*1*(1/2) := by gcongr | |
| _ = δ/2 := by ring | |
| rw [heq] | |
| have htriangle := abs_add_le ((u-1/2)*r) (δ*x*alpha r) | |
| have hquad := mul_le_mul_of_nonneg_left hδsmall hδ | |
| nlinarith only [htriangle,hfirst,hsecond,hquad] | |
| theorem angular_errors {δ X Y Z r s u v : ℝ} | |
| (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2) | |
| (hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1) | |
| (hu : 1/2 ≤ u) (hv : 1/2 ≤ v) | |
| (hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r) | |
| (hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) : | |
| |Y-r/2| ≤ 3*δ/2 ∧ |X-s/2| ≤ 3*δ/2 := by | |
| obtain ⟨hu0,hu',hv0,hv'⟩ := radial_excess_bounds hδ hδsmall hXb hYb hr hs hu hv hY hZ hX hZv | |
| refine ⟨radial_coordinate_error hδ hδsmall hXb hr hu0 hu' hY, ?_⟩ | |
| apply radial_coordinate_error hδ hδsmall (x := -Y) (by simpa using hYb) hs hv0 hv' | |
| nlinarith only [hX] | |
| lemma squared_error {δ x s : ℝ} (hδ : 0 ≤ δ) (hx : |x| ≤ 1) (hs : |s| ≤ 1) | |
| (he : |x-s/2| ≤ 3*δ/2) : x^2 ≤ s^2/4+9*δ/4 := by | |
| have hsum : |x+s/2| ≤ 3/2 := by | |
| calc | |
| |x+s/2| ≤ |x|+|s/2| := abs_add_le _ _ | |
| _ ≤ 1+1/2 := by rw [abs_div]; norm_num; linarith | |
| _ = 3/2 := by norm_num | |
| have hprod : |(x-s/2)*(x+s/2)| ≤ (3*δ/2)*(3/2) := by | |
| rw [abs_mul] | |
| exact mul_le_mul he hsum (abs_nonneg _) (by positivity) | |
| have hid : (x-s/2)*(x+s/2)=x^2-s^2/4 := by ring | |
| rw [hid] at hprod | |
| nlinarith only [(abs_le.mp hprod).2] | |
| lemma monotone_profile_comparison (r s : ℝ) : | |
| (s^2*alpha r+r^2*alpha s)/4 ≤ F r+F s := by | |
| unfold alpha F | |
| nlinarith only [sq_nonneg (r^2-s^2)] | |
| theorem weighted_comparison {δ X Y r s : ℝ} (hδ : 0 ≤ δ) | |
| (hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1) | |
| (hYr : |Y-r/2| ≤ 3*δ/2) (hXs : |X-s/2| ≤ 3*δ/2) : | |
| X^2*alpha r+Y^2*alpha s ≤ (s^2*alpha r+r^2*alpha s)/4+9*δ/4 ∧ | |
| X^2*alpha r+Y^2*alpha s ≤ F r+F s+9*δ/4 := by | |
| obtain ⟨har,har'⟩ := alpha_bounds hr | |
| obtain ⟨has,has'⟩ := alpha_bounds hs | |
| have hx := mul_le_mul_of_nonneg_right (squared_error hδ hXb hs hXs) har | |
| have hy := mul_le_mul_of_nonneg_right (squared_error hδ hYb hr hYr) has | |
| have hsum := mul_le_mul_of_nonneg_left (add_le_add har' has') hδ | |
| have hfirst : X^2*alpha r+Y^2*alpha s ≤ | |
| (s^2*alpha r+r^2*alpha s)/4+9*δ/4 := by nlinarith only [hx,hy,hsum] | |
| exact ⟨hfirst, hfirst.trans (by linarith only [monotone_profile_comparison r s])⟩ | |
| theorem cubic_padding_lower {δ r s : ℝ} (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/2) | |
| (hr : |r| ≤ 1) (hs : |s| ≤ 1) : | |
| F r+F s+15*δ/4 ≤ (1-δ)*(F r+F s+8*δ) := by | |
| have hrF := (profile_bounds hr).2 | |
| have hsF := (profile_bounds hs).2 | |
| have hcoef : 0 ≤ 17/4-8*δ-(F r+F s) := by linarith | |
| have hprod := mul_nonneg hδ hcoef | |
| nlinarith only [hprod] | |
| theorem cubic_failure_lower {δ X Y r s u v : ℝ} | |
| (hδ : 0 < δ) (hδsmall : δ ≤ 1/2) | |
| (hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1) | |
| (hu : cubicCap δ r < u) (hv : cubicCap δ s < v) : | |
| δ^2*(F r+F s+15*δ/4) < | |
| (u-1/2)*(1-δ*X*r)+(v-1/2)*(1+δ*Y*s) := by | |
| have hδone : 0 < 1-δ := by linarith | |
| have hFr := (profile_bounds hr).1 | |
| have hFs := (profile_bounds hs).1 | |
| have hu' : δ^2*F r+4*δ^3 < u-1/2 := by unfold cubicCap at hu; linarith | |
| have hv' : δ^2*F s+4*δ^3 < v-1/2 := by unfold cubicCap at hv; linarith | |
| have hcapr : 0 ≤ δ^2*F r+4*δ^3 := by positivity | |
| have hu0 : 0 ≤ u-1/2 := hcapr.trans hu'.le | |
| have hcaps : 0 ≤ δ^2*F s+4*δ^3 := by positivity | |
| have hv0 : 0 ≤ v-1/2 := hcaps.trans hv'.le | |
| have hUstrict := mul_lt_mul_of_pos_right hu' hδone | |
| have hVstrict := mul_lt_mul_of_pos_right hv' hδone | |
| have hUweak := mul_le_mul_of_nonneg_left (weight_bounds hδ.le hδsmall hXb hr).1 hu0 | |
| have hVweak := mul_le_mul_of_nonneg_left (weight_bounds hδ.le hδsmall hYb hs).2.1 hv0 | |
| have hpad := mul_le_mul_of_nonneg_left (cubic_padding_lower hδ.le hδsmall hr hs) | |
| (sq_nonneg δ) | |
| nlinarith only [hUstrict,hVstrict,hUweak,hVweak,hpad] | |
| theorem cubic_failures_incompatible {δ X Y Z r s u v : ℝ} | |
| (hδ : 0 < δ) (hδsmall : δ ≤ 1/2) | |
| (hXb : |X| ≤ 1) (hYb : |Y| ≤ 1) (hr : |r| ≤ 1) (hs : |s| ≤ 1) | |
| (hY : Y = u*r+δ*X*alpha r) (hZ : Z = u+δ*X*beta r) | |
| (hX : X = v*s-δ*Y*alpha s) (hZv : 1-Z = v-δ*Y*beta s) : | |
| ¬ (cubicCap δ r < u ∧ cubicCap δ s < v) := by | |
| rintro ⟨hu,hv⟩ | |
| have hFr := (profile_bounds hr).1 | |
| have hFs := (profile_bounds hs).1 | |
| have huc : 1/2 ≤ cubicCap δ r := by | |
| have hnonneg : 0 ≤ δ^2*F r+4*δ^3 := by positivity | |
| unfold cubicCap | |
| linarith | |
| have hvc : 1/2 ≤ cubicCap δ s := by | |
| have hnonneg : 0 ≤ δ^2*F s+4*δ^3 := by positivity | |
| unfold cubicCap | |
| linarith | |
| obtain ⟨hYr,hXs⟩ := angular_errors hδ.le hδsmall hXb hYb hr hs | |
| (huc.trans hu.le) (hvc.trans hv.le) hY hZ hX hZv | |
| have hupper := (weighted_comparison hδ.le hXb hYb hr hs hYr hXs).2 | |
| have hlower := cubic_failure_lower hδ hδsmall hXb hYb hr hs hu hv | |
| rw [identity hY hZ hX hZv] at hlower | |
| have hupper' := mul_le_mul_of_nonneg_left hupper (sq_nonneg δ) | |
| have hcube : 0 < δ^3 := pow_pos hδ 3 | |
| nlinarith only [hlower,hupper',hcube] | |
| end CylinderCovering.HalfHeightCaps.CompensationEstimates | |
| end OAI | |