openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Calculus /MovingIntervalCalculus.lean
| import OAI.Geometry.IsometricImmersion.Energy.WeightPrimitiveParameter | |
| import OAI.Geometry.IsometricImmersion.Energy.HyperbolicEnergyDensity | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory | |
| open scoped ContDiff Topology Interval | |
| namespace SmoothLocal.Hyperbolic | |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE | |
| def movingIntervalIntegral (left right : ℝ → ℝ) (f : Coord → ℝ) (theta : ℝ) : ℝ := | |
| ∫ xi in left theta..right theta, f (coordinatePoint xi theta) | |
| def inwardLeft (x0 theta0 speed : ℝ) (theta : ℝ) : ℝ := | |
| x0 + speed * (theta - theta0) | |
| def inwardRight (x0 theta0 speed : ℝ) (theta : ℝ) : ℝ := | |
| x0 - speed * (theta - theta0) | |
| theorem inwardLeft_hasDerivAt (x0 theta0 speed theta : ℝ) : | |
| HasDerivAt (inwardLeft x0 theta0 speed) speed theta := by | |
| unfold inwardLeft | |
| simpa only [inwardLeft, mul_one, id_eq] using | |
| (((hasDerivAt_id theta).sub_const theta0).const_mul speed).const_add x0 | |
| theorem inwardRight_hasDerivAt (x0 theta0 speed theta : ℝ) : | |
| HasDerivAt (inwardRight x0 theta0 speed) (-speed) theta := by | |
| unfold inwardRight | |
| simpa only [inwardRight, mul_one, zero_sub, id_eq, Pi.sub_def] using | |
| (hasDerivAt_const theta x0).sub | |
| (((hasDerivAt_id theta).sub_const theta0).const_mul speed) | |
| theorem inward_interval_length (xl xr theta0 speed theta : ℝ) : | |
| inwardRight xr theta0 speed theta - inwardLeft xl theta0 speed theta = | |
| xr - xl - 2 * speed * (theta - theta0) := by | |
| unfold inwardRight inwardLeft | |
| ring | |
| theorem movingPoint_hasDerivAt {x : ℝ → ℝ} {dx theta : ℝ} | |
| (hx : HasDerivAt x dx theta) : | |
| HasDerivAt (fun t => coordinatePoint (x t) t) | |
| (dx • Pi.single 0 (1 : ℝ) + Pi.single 1 (1 : ℝ) : Coord) theta := by | |
| simpa only [coordinatePoint, one_smul, id_eq, Pi.add_def] using | |
| (hx.smul_const (Pi.single 0 (1 : ℝ) : Coord)).add | |
| ((hasDerivAt_id theta).smul_const (Pi.single 1 (1 : ℝ) : Coord)) | |
| theorem slice_intervalIntegrable | |
| {f : Coord → ℝ} {radius lo hi xl xr theta : ℝ} | |
| (hf : ContinuousOn f (coordinateRectangle radius lo hi)) | |
| (hxl : xl ∈ Ioo (-radius) radius) (hxr : xr ∈ Ioo (-radius) radius) | |
| (htheta : theta ∈ Ioo lo hi) : | |
| IntervalIntegrable (fun xi => f (coordinatePoint xi theta)) volume xl xr := by | |
| have hc : ContinuousOn (fun xi => f (coordinatePoint xi theta)) (Ioo (-radius) radius) := | |
| hf.comp (by unfold coordinatePoint; fun_prop) (fun xi hxi => point_mem_rectangle hxi htheta) | |
| exact (hc.mono (fun xi hxi => | |
| ⟨lt_of_lt_of_le (lt_min hxl.1 hxr.1) hxi.1, | |
| lt_of_le_of_lt hxi.2 (max_lt hxl.2 hxr.2)⟩)).intervalIntegrable | |
| theorem movingIntervalIntegral_eq_primitive | |
| {f : Coord → ℝ} {left right : ℝ → ℝ} {radius lo hi theta : ℝ} | |
| (hf : ContinuousOn f (coordinateRectangle radius lo hi)) (hradius : 0 < radius) | |
| (hl : left theta ∈ Ioo (-radius) radius) (hr : right theta ∈ Ioo (-radius) radius) | |
| (htheta : theta ∈ Ioo lo hi) : | |
| movingIntervalIntegral left right f theta = | |
| coordinatePrimitive f (coordinatePoint (right theta) theta) - | |
| coordinatePrimitive f (coordinatePoint (left theta) theta) := by | |
| have h0 : (0 : ℝ) ∈ Ioo (-radius) radius := ⟨by linarith, hradius⟩ | |
| have hrI := slice_intervalIntegrable hf h0 hr htheta | |
| have hlI := slice_intervalIntegrable hf h0 hl htheta | |
| simpa [movingIntervalIntegral, coordinatePrimitive, coordinatePoint] using | |
| (intervalIntegral.integral_interval_sub_left hrI hlI).symm | |
| theorem movingPrimitive_hasDerivAt | |
| {f : Coord → ℝ} {x : ℝ → ℝ} {radius lo hi theta dx : ℝ} | |
| (hf : ContDiffOn ℝ ∞ f (coordinateRectangle radius lo hi)) (hradius : 0 < radius) | |
| (hx : HasDerivAt x dx theta) (hxt : x theta ∈ Ioo (-radius) radius) | |
| (htheta : theta ∈ Ioo lo hi) : | |
| HasDerivAt (fun t => coordinatePrimitive f (coordinatePoint (x t) t)) | |
| (dx * f (coordinatePoint (x theta) theta) + | |
| ∫ xi in 0..x theta, coordPartial 1 f (coordinatePoint xi theta)) theta := by | |
| have hp := point_mem_rectangle hxt htheta | |
| have hd : DifferentiableAt ℝ (coordinatePrimitive f) (coordinatePoint (x theta) theta) := | |
| ((coordinatePrimitive_contDiffOn hf).contDiffAt | |
| ((coordinateRectangle_isOpen radius lo hi).mem_nhds hp)).differentiableAt (by simp) | |
| have hchain := HasFDerivAt.comp_hasDerivAt theta | |
| (f := fun t => coordinatePoint (x t) t) hd.hasFDerivAt | |
| (movingPoint_hasDerivAt (x := x) hx) | |
| have ht := coordinatePrimitive_partial_t hf hradius hp | |
| have hs := coordinatePrimitive_partial_s hf hradius hp | |
| simp only [coordPartial] at ht hs | |
| simp only [map_add, map_smul, | |
| smul_eq_mul, ht, hs] at hchain | |
| simpa [coordinatePoint, coordPartial, Function.comp_def] using hchain | |
| theorem movingIntervalIntegral_hasDerivAt | |
| {f : Coord → ℝ} {left right : ℝ → ℝ} {radius lo hi theta dl dr : ℝ} | |
| (hf : ContDiffOn ℝ ∞ f (coordinateRectangle radius lo hi)) (hradius : 0 < radius) | |
| (hl : HasDerivAt left dl theta) (hr : HasDerivAt right dr theta) | |
| (hlt : left theta ∈ Ioo (-radius) radius) | |
| (hrt : right theta ∈ Ioo (-radius) radius) (htheta : theta ∈ Ioo lo hi) : | |
| HasDerivAt (movingIntervalIntegral left right f) | |
| ((∫ xi in left theta..right theta, coordPartial 1 f (coordinatePoint xi theta)) + | |
| dr * f (coordinatePoint (right theta) theta) - | |
| dl * f (coordinatePoint (left theta) theta)) theta := by | |
| have hright := movingPrimitive_hasDerivAt hf hradius hr hrt htheta | |
| have hleft := movingPrimitive_hasDerivAt hf hradius hl hlt htheta | |
| have hdiff := hright.sub hleft | |
| have h0 : (0 : ℝ) ∈ Ioo (-radius) radius := ⟨by linarith, hradius⟩ | |
| have hf1 := (partial_contDiffOn hf (coordinateRectangle_isOpen radius lo hi) 1).continuousOn | |
| have hIsub := intervalIntegral.integral_interval_sub_left | |
| (slice_intervalIntegrable hf1 h0 hrt htheta) | |
| (slice_intervalIntegrable hf1 h0 hlt htheta) | |
| have he : | |
| (dr * f (coordinatePoint (right theta) theta) + | |
| ∫ xi in 0..right theta, coordPartial 1 f (coordinatePoint xi theta)) - | |
| (dl * f (coordinatePoint (left theta) theta) + | |
| ∫ xi in 0..left theta, coordPartial 1 f (coordinatePoint xi theta)) = | |
| (∫ xi in left theta..right theta, coordPartial 1 f (coordinatePoint xi theta)) + | |
| dr * f (coordinatePoint (right theta) theta) - | |
| dl * f (coordinatePoint (left theta) theta) := by linarith | |
| rw [he] at hdiff | |
| apply hdiff.congr_of_eventuallyEq | |
| filter_upwards [hl.continuousAt.preimage_mem_nhds (isOpen_Ioo.mem_nhds hlt), | |
| hr.continuousAt.preimage_mem_nhds (isOpen_Ioo.mem_nhds hrt), | |
| isOpen_Ioo.mem_nhds htheta] with t hlt' hrt' ht' | |
| exact movingIntervalIntegral_eq_primitive hf.continuousOn hradius hlt' hrt' ht' | |
| theorem inwardIntervalIntegral_hasDerivAt | |
| {f : Coord → ℝ} {radius lo hi xl xr theta0 speed theta : ℝ} | |
| (hf : ContDiffOn ℝ ∞ f (coordinateRectangle radius lo hi)) (hradius : 0 < radius) | |
| (hl : inwardLeft xl theta0 speed theta ∈ Ioo (-radius) radius) | |
| (hr : inwardRight xr theta0 speed theta ∈ Ioo (-radius) radius) | |
| (htheta : theta ∈ Ioo lo hi) : | |
| HasDerivAt | |
| (movingIntervalIntegral (inwardLeft xl theta0 speed) (inwardRight xr theta0 speed) f) | |
| ((∫ xi in inwardLeft xl theta0 speed theta..inwardRight xr theta0 speed theta, | |
| coordPartial 1 f (coordinatePoint xi theta)) - | |
| speed * (f (coordinatePoint (inwardRight xr theta0 speed theta) theta) + | |
| f (coordinatePoint (inwardLeft xl theta0 speed theta) theta))) theta := by | |
| convert movingIntervalIntegral_hasDerivAt hf hradius | |
| (inwardLeft_hasDerivAt xl theta0 speed theta) | |
| (inwardRight_hasDerivAt xr theta0 speed theta) hl hr htheta using 1 | |
| ring | |
| theorem slice_integral_partial_xi | |
| {f : Coord → ℝ} {radius lo hi xl xr theta : ℝ} | |
| (hf : ContDiffOn ℝ ∞ f (coordinateRectangle radius lo hi)) | |
| (hxl : xl ∈ Ioo (-radius) radius) (hxr : xr ∈ Ioo (-radius) radius) | |
| (htheta : theta ∈ Ioo lo hi) : | |
| (∫ xi in xl..xr, coordPartial 0 f (coordinatePoint xi theta)) = | |
| f (coordinatePoint xr theta) - f (coordinatePoint xl theta) := by | |
| have hmem {xi : ℝ} (hxi : xi ∈ uIcc xl xr) : | |
| coordinatePoint xi theta ∈ coordinateRectangle radius lo hi := | |
| point_mem_rectangle | |
| ⟨lt_of_lt_of_le (lt_min hxl.1 hxr.1) hxi.1, | |
| lt_of_le_of_lt hxi.2 (max_lt hxl.2 hxr.2)⟩ htheta | |
| apply intervalIntegral.integral_eq_sub_of_hasDerivAt | |
| · intro xi hxi | |
| have hd : DifferentiableAt ℝ f (coordinatePoint xi theta) := | |
| (hf.contDiffAt ((coordinateRectangle_isOpen radius lo hi).mem_nhds | |
| (hmem hxi))).differentiableAt (by simp) | |
| exact hd.hasFDerivAt.comp_hasDerivAt xi (point_hasDerivAt_t xi theta) | |
| · exact slice_intervalIntegrable | |
| (partial_contDiffOn hf (coordinateRectangle_isOpen radius lo hi) 0).continuousOn | |
| hxl hxr htheta | |
| end SmoothLocal.Hyperbolic | |
| end | |
| end OAI | |