openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Energy /WeightPrimitiveParameter.lean
| import OAI.Geometry.IsometricImmersion.Calculus.FinitePrimitive | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter MeasureTheory Function | |
| open scoped ContDiff Topology Interval | |
| namespace SmoothLocal.Weighted | |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Flow | |
| variable {B : Coord → ℝ} {R a b : ℝ} | |
| theorem coefficient_pair_contDiffOn | |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) : | |
| ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (pairRectangle R a b) := by | |
| have hpoint : ContDiff ℝ ∞ (fun p : ℝ × ℝ => coordinatePoint p.1 p.2) := by | |
| unfold coordinatePoint | |
| fun_prop | |
| exact hB.comp hpoint.contDiffOn (fun p hp => point_mem_rectangle hp.1 hp.2) | |
| theorem coefficient_pairPartialY | |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) | |
| {x y : ℝ} (hx : x ∈ Ioo (-R) R) (hy : y ∈ Ioo a b) : | |
| pairPartialY (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y) = | |
| coordPartial 1 B (coordinatePoint x y) := by | |
| have hdB : DifferentiableAt ℝ B (coordinatePoint x y) := | |
| ((hB.contDiffAt ((coordinateRectangle_isOpen R a b).mem_nhds | |
| (point_mem_rectangle hx hy))).differentiableAt (by simp)) | |
| have hdPair : DifferentiableAt ℝ (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y) := | |
| (((coefficient_pair_contDiffOn hB).contDiffAt | |
| ((pairRectangle_isOpen R a b).mem_nhds ⟨hx, hy⟩)).differentiableAt (by simp)) | |
| have hcoord : HasDerivAt (fun t => B (coordinatePoint x t)) | |
| (coordPartial 1 B (coordinatePoint x y)) y := | |
| hdB.hasFDerivAt.comp_hasDerivAt y (coordinatePoint_hasDerivAt_y x y) | |
| have hpair : HasDerivAt (fun t => B (coordinatePoint x t)) | |
| (pairPartialY (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y)) y := | |
| hdPair.hasFDerivAt.comp_hasDerivAt y | |
| ((hasDerivAt_const y x).prodMk (hasDerivAt_id y)) | |
| exact hpair.unique hcoord | |
| theorem coordinatePrimitive_partial_s | |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) | |
| (hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) : | |
| coordPartial 1 (coordinatePrimitive B) p = | |
| ∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1)) := by | |
| have hpairC1 : ContDiffOn ℝ 1 (fun q : ℝ × ℝ => B (coordinatePoint q.1 q.2)) | |
| (pairRectangle R a b) := (coefficient_pair_contDiffOn hB).of_le (by simp) | |
| have hftc := pairPrimitive_hasDerivAt_y hpairC1 hp.1 hp.2 | |
| have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by | |
| have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩ | |
| intro t ht | |
| exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1, | |
| lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩ | |
| have he : pairPrimitive | |
| (pairPartialY (fun q : ℝ × ℝ => B (coordinatePoint q.1 q.2))) (p 0, p 1) = | |
| ∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1)) := by | |
| unfold pairPrimitive | |
| apply intervalIntegral.integral_congr | |
| intro t ht | |
| exact coefficient_pairPartialY hB (hseg ht) hp.2 | |
| rw [he] at hftc | |
| have hftc' : HasDerivAt (fun s => coordinatePrimitive B (coordinatePoint (p 0) s)) | |
| (∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1))) (p 1) := by | |
| simpa [pairPrimitive, coordinatePrimitive, coordinatePoint] using hftc | |
| have hdI : DifferentiableAt ℝ (coordinatePrimitive B) (coordinatePoint (p 0) (p 1)) := by | |
| rw [point_eta] | |
| exact ((coordinatePrimitive_contDiffOn hB).contDiffAt | |
| ((coordinateRectangle_isOpen R a b).mem_nhds hp)).differentiableAt (by simp) | |
| have hchain := hdI.hasFDerivAt.comp_hasDerivAt (p 1) (coordinatePoint_hasDerivAt_y (p 0) (p 1)) | |
| have hcoord : HasDerivAt (fun s => coordinatePrimitive B (coordinatePoint (p 0) s)) | |
| (coordPartial 1 (coordinatePrimitive B) p) (p 1) := by | |
| simpa only [point_eta, Function.comp_def, coordPartial] using hchain | |
| exact hcoord.unique hftc' | |
| theorem coordinatePrimitive_segment_intervalIntegrable | |
| (hB : ContinuousOn B (coordinateRectangle R a b)) | |
| (hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) : | |
| IntervalIntegrable (fun t => B (coordinatePoint t (p 1))) volume 0 (p 0) := by | |
| have hc : ContinuousOn (fun t => B (coordinatePoint t (p 1))) (Ioo (-R) R) := | |
| hB.comp (by unfold coordinatePoint; fun_prop) | |
| (fun t ht => point_mem_rectangle ht hp.2) | |
| have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩ | |
| have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by | |
| intro t ht | |
| exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1, | |
| lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩ | |
| exact (hc.mono hseg).intervalIntegrable | |
| theorem coordinatePrimitive_abs_bound | |
| (hB : ContinuousOn B (coordinateRectangle R a b)) | |
| (hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) | |
| {M : ℝ} (hbound : ∀ q ∈ coordinateRectangle R a b, |B q| ≤ M) : | |
| |coordinatePrimitive B p| ≤ R * M := by | |
| have hM : 0 ≤ M := (abs_nonneg _).trans (hbound p hp) | |
| have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩ | |
| have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by | |
| intro t ht | |
| exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1, | |
| lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩ | |
| have hi := coordinatePrimitive_segment_intervalIntegrable hB hR hp | |
| have hb := intervalIntegral.norm_integral_le_of_norm_le_const (a := 0) (b := p 0) (C := M) | |
| (f := fun t => B (coordinatePoint t (p 1))) | |
| (fun t ht => by | |
| rw [Real.norm_eq_abs] | |
| exact hbound _ (point_mem_rectangle (hseg (uIoc_subset_uIcc ht)) hp.2)) | |
| calc | |
| _ ≤ M * |p 0| := by simpa only [coordinatePrimitive, Real.norm_eq_abs, sub_zero] using hb | |
| _ ≤ M * R := mul_le_mul_of_nonneg_left (abs_lt.mpr hp.1).le hM | |
| _ = R * M := mul_comm _ _ | |
| theorem coordinatePrimitive_partial_s_bound | |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) | |
| (hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) | |
| {M : ℝ} (hBs : ∀ q ∈ coordinateRectangle R a b, |coordPartial 1 B q| ≤ M) : | |
| |coordPartial 1 (coordinatePrimitive B) p| ≤ R * M := by | |
| rw [coordinatePrimitive_partial_s hB hR hp] | |
| exact coordinatePrimitive_abs_bound | |
| (partial_contDiffOn hB (coordinateRectangle_isOpen R a b) 1).continuousOn hR hp hBs | |
| end SmoothLocal.Weighted | |
| end | |
| end OAI | |