Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear
import OAI.Geometry.CAT0Fillings.Gradient.Multiplier
import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain

namespace OAI

section

open Set Filter
open scoped Topology NNReal

namespace CAT0Fillings.ClosedCalculus

noncomputable def levelApprox (a ε t : ℝ) := ε*Real.arctan ((t-a)/ε)
noncomputable def levelApproxDeriv (a ε t : ℝ) := ε^2/(ε^2+(t-a)^2)

lemma levelApprox_hasDerivAt {ε : ℝ} (hε : 0 < ε) (a t : ℝ) :
    HasDerivAt (levelApprox a ε) (levelApproxDeriv a ε t) t := by
  have hd := ((((hasDerivAt_id t).sub_const a).div_const ε).arctan).const_mul ε
  convert hd using 1 <;> try rfl
  dsimp [levelApproxDeriv]
  field_simp [hε.ne']

lemma levelApproxDeriv_bound {ε : ℝ} (hε : 0 < ε) (a t : ℝ) : |levelApproxDeriv a ε t| ≤ 1 := by
  have hp := add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))
  dsimp [levelApproxDeriv]
  rw [abs_of_nonneg (div_nonneg (sq_nonneg _) hp.le)]
  exact (div_le_one hp).mpr (le_add_of_nonneg_right (sq_nonneg _))

lemma levelApprox_lipschitz {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
    LipschitzWith 1 (levelApprox a ε) := by
  apply lipschitzWith_of_nnnorm_deriv_le (fun t => (levelApprox_hasDerivAt hε a t).differentiableAt)
  intro t
  rw [←NNReal.coe_le_coe,coe_nnnorm,NNReal.coe_one,(levelApprox_hasDerivAt hε a t).deriv,Real.norm_eq_abs]
  exact levelApproxDeriv_bound hε a t

lemma continuous_levelApproxDeriv {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
    Continuous (levelApproxDeriv a ε) := by
  change Continuous (fun t : ℝ => ε^2/(ε^2+(t-a)^2))
  exact continuous_const.div (continuous_const.add ((continuous_id.sub continuous_const).pow 2))
    (fun t => (add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))).ne')

lemma levelApprox_error {ε : ℝ} (hε : 0 ≤ ε) (a t : ℝ) :
    |levelApprox a ε t| ≤ ε*(Real.pi/2) := by
  dsimp [levelApprox]
  rw [abs_mul,abs_of_nonneg hε]
  exact mul_le_mul_of_nonneg_left (abs_le.mpr
    ⟨(Real.neg_pi_div_two_lt_arctan _).le,(Real.arctan_lt_pi_div_two _).le⟩) hε

lemma levelApproxDeriv_tendsto {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0))
    (hep : ∀ j, 0 < e j) (a t : ℝ) :
    Tendsto (fun j => levelApproxDeriv a (e j) t) atTop (𝓝 (if t = a then 1 else 0)) := by
  by_cases ht : t = a
  · subst t
    simp only [levelApproxDeriv,sub_self,zero_pow (by decide : 2 ≠ 0),add_zero,
      div_self (pow_ne_zero 2 (hep _).ne'),ite_true]
    exact tendsto_const_nhds
  · rw [ite_eq_right ht]
    have hta : t-a ≠ 0 := sub_ne_zero.mpr ht
    have hh := (he.pow 2).div ((he.pow 2).add_const ((t-a)^2))
      (by simpa using pow_ne_zero 2 hta)
    convert hh using 1 <;> simp only [levelApproxDeriv,zero_pow (by decide : 2 ≠ 0),zero_add,zero_div]
    rfl

end CAT0Fillings.ClosedCalculus
end

section

open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings.ChartGeometry
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
  [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
  {hT : IsMetricCurrent T} (q : ChartGeometry hT)

lemma closed_chain_uniform {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
    (hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, |D t| ≤ C)
    (Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0)
    (hFj : ∀ j, LipschitzWith (K j) (Fj j))
    (hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t)
    (hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, |Dj j t| ≤ C)
    (hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t)))
    (a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0))
    (he : ∀ j t, |Fj j t-F t| ≤ a j) (P : q.Sobolev) :
    ∃ Q : q.Sobolev,
      (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
        (fun x => F ((q.inclusion P) x)) ∧
      (q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
        (fun w => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
  choose Q hQ hG using (fun j => q.closed_chain (hFj j) (hdj j) (hcj j) hC (hbj j) P)
  let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w))
  let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w))
  have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving
  have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm
  have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable
  have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun _ => hbj j _
  have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun _ => hbD _
  have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC
    (Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point)))
    (tendsto_const_nhds (x := q.closedGradient P))
  have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by
    apply Lp.ext
    exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm
  simp only [heG] at hgl
  let V := q.composeValue hF (q.inclusion P)
  have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by
    apply tendsto_iff_norm_sub_tendsto_zero.mpr
    apply squeeze_zero (fun _ => norm_nonneg _)
      (g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖)
    · intro j
      apply Lp.norm_le_mul_norm_of_ae_le_mul
      filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P),
        ((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp
          (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1
      change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1
      rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one]
      exact he j _
    · simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖
  obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl
  refine ⟨R,?_,?_⟩
  · rw [hR]
    exact q.composeValue_ae hF (q.inclusion P)
  · rw [hRG]
    exact ClosedCalculus.mulLp_ae hc hcC _

end CAT0Fillings.ChartGeometry
end

section

open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal

namespace CAT0Fillings.ChartGeometry
open ClosedCalculus

variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
  [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
  {hT : IsMetricCurrent T} (q : ChartGeometry hT)

lemma closedGradient_level_zero (hz : IsCycle T) (P : q.Sobolev) (a : ℝ) :
    ∀ᵐ w ∂q.atlasMeasure, (q.inclusion P) (q.atlasParam w) = a →
      (q.closedGradient P) w = 0 := by
  let e (j : ℕ) : ℝ := 1/((j:ℝ)+1)
  have hep (j : ℕ) : 0 < e j := by dsimp [e]; positivity
  have he : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
  have hD : Measurable (fun t : ℝ => if t = a then (1:ℝ) else 0) := by
    convert (show Measurable (({a} : Set ℝ).indicator (fun _ : ℝ => (1:ℝ))) from
      measurable_const.indicator (measurableSet_singleton a)) using 1
    funext t
    simp only [Set.indicator_apply,Set.mem_singleton_iff]

  have hb (t : ℝ) : |if t = a then (1:ℝ) else 0| ≤ 1 := by split_ifs <;> norm_num
  obtain ⟨R,hR,hG⟩ := q.closed_chain_uniform (F := fun _ => (0:ℝ))
    (D := fun t => if t = a then (1:ℝ) else 0) (LipschitzWith.const (0:ℝ))
    hD (by norm_num : (0:ℝ) ≤ 1) hb
    (fun j => levelApprox a (e j)) (fun j => levelApproxDeriv a (e j)) (fun _ => 1)
    (fun j => levelApprox_lipschitz (hep j) a) (fun j => levelApprox_hasDerivAt (hep j) a)
    (fun j => continuous_levelApproxDeriv (hep j) a) (fun j => levelApproxDeriv_bound (hep j) a)
    (levelApproxDeriv_tendsto he hep a) (fun j => e j*(Real.pi/2))
    (by simpa only [zero_mul] using he.mul_const (Real.pi/2))
    (fun j t => by simpa only [sub_zero] using levelApprox_error (hep j).le a t) P
  have hR0 : q.inclusion R = 0 := Lp.ext (hR.trans (Lp.coeFn_zero ℝ 2 (MassMeasure.currentMassMeasure hT)).symm)
  have hRz : R = 0 := q.inclusion_injective hz (hR0.trans (map_zero q.inclusion).symm)
  rw [hRz,map_zero] at hG
  filter_upwards [hG,(Lp.coeFn_zero (Euc (k+1)) 2 q.atlasMeasure)] with w hw hz
  intro ha
  simpa only [hz,ha,ite_true,one_smul,Pi.zero_apply] using hw.symm

end CAT0Fillings.ChartGeometry
end

end OAI

	import Mathlib
	import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear
	import OAI.Geometry.CAT0Fillings.Gradient.Multiplier
	import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain

	namespace OAI

	section

	open Set Filter
	open scoped Topology NNReal

	namespace CAT0Fillings.ClosedCalculus

	noncomputable def levelApprox (a ε t : ℝ) := ε*Real.arctan ((t-a)/ε)
	noncomputable def levelApproxDeriv (a ε t : ℝ) := ε^2/(ε^2+(t-a)^2)

	lemma levelApprox_hasDerivAt {ε : ℝ} (hε : 0 < ε) (a t : ℝ) :
	HasDerivAt (levelApprox a ε) (levelApproxDeriv a ε t) t := by
	have hd := ((((hasDerivAt_id t).sub_const a).div_const ε).arctan).const_mul ε
	convert hd using 1 <;> try rfl
	dsimp [levelApproxDeriv]
	field_simp [hε.ne']

	lemma levelApproxDeriv_bound {ε : ℝ} (hε : 0 < ε) (a t : ℝ) : \|levelApproxDeriv a ε t\| ≤ 1 := by
	have hp := add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))
	dsimp [levelApproxDeriv]
	rw [abs_of_nonneg (div_nonneg (sq_nonneg _) hp.le)]
	exact (div_le_one hp).mpr (le_add_of_nonneg_right (sq_nonneg _))

	lemma levelApprox_lipschitz {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
	LipschitzWith 1 (levelApprox a ε) := by
	apply lipschitzWith_of_nnnorm_deriv_le (fun t => (levelApprox_hasDerivAt hε a t).differentiableAt)
	intro t
	rw [←NNReal.coe_le_coe,coe_nnnorm,NNReal.coe_one,(levelApprox_hasDerivAt hε a t).deriv,Real.norm_eq_abs]
	exact levelApproxDeriv_bound hε a t

	lemma continuous_levelApproxDeriv {ε : ℝ} (hε : 0 < ε) (a : ℝ) :
	Continuous (levelApproxDeriv a ε) := by
	change Continuous (fun t : ℝ => ε^2/(ε^2+(t-a)^2))
	exact continuous_const.div (continuous_const.add ((continuous_id.sub continuous_const).pow 2))
	(fun t => (add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg (t-a))).ne')

	lemma levelApprox_error {ε : ℝ} (hε : 0 ≤ ε) (a t : ℝ) :
	\|levelApprox a ε t\| ≤ ε*(Real.pi/2) := by
	dsimp [levelApprox]
	rw [abs_mul,abs_of_nonneg hε]
	exact mul_le_mul_of_nonneg_left (abs_le.mpr
	⟨(Real.neg_pi_div_two_lt_arctan _).le,(Real.arctan_lt_pi_div_two _).le⟩) hε

	lemma levelApproxDeriv_tendsto {e : ℕ → ℝ} (he : Tendsto e atTop (𝓝 0))
	(hep : ∀ j, 0 < e j) (a t : ℝ) :
	Tendsto (fun j => levelApproxDeriv a (e j) t) atTop (𝓝 (if t = a then 1 else 0)) := by
	by_cases ht : t = a
	· subst t
	simp only [levelApproxDeriv,sub_self,zero_pow (by decide : 2 ≠ 0),add_zero,
	div_self (pow_ne_zero 2 (hep _).ne'),ite_true]
	exact tendsto_const_nhds
	· rw [ite_eq_right ht]
	have hta : t-a ≠ 0 := sub_ne_zero.mpr ht
	have hh := (he.pow 2).div ((he.pow 2).add_const ((t-a)^2))
	(by simpa using pow_ne_zero 2 hta)
	convert hh using 1 <;> simp only [levelApproxDeriv,zero_pow (by decide : 2 ≠ 0),zero_add,zero_div]
	rfl

	end CAT0Fillings.ClosedCalculus
	end

	section

	open Set Filter MeasureTheory
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings.ChartGeometry
	variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
	[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
	{hT : IsMetricCurrent T} (q : ChartGeometry hT)

	lemma closed_chain_uniform {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F)
	(hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, \|D t\| ≤ C)
	(Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0)
	(hFj : ∀ j, LipschitzWith (K j) (Fj j))
	(hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t)
	(hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, \|Dj j t\| ≤ C)
	(hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t)))
	(a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0))
	(he : ∀ j t, \|Fj j t-F t\| ≤ a j) (P : q.Sobolev) :
	∃ Q : q.Sobolev,
	(q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT]
	(fun x => F ((q.inclusion P) x)) ∧
	(q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure]
	(fun w => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by
	choose Q hQ hG using (fun j => q.closed_chain (hFj j) (hdj j) (hcj j) hC (hbj j) P)
	let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w))
	let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w))
	have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving
	have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm
	have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable
	have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, \|b j w\| ≤ C := Eventually.of_forall fun _ => hbj j _
	have hcC : ∀ᵐ w ∂q.atlasMeasure, \|c w\| ≤ C := Eventually.of_forall fun _ => hbD _
	have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC
	(Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point)))
	(tendsto_const_nhds (x := q.closedGradient P))
	have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by
	apply Lp.ext
	exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm
	simp only [heG] at hgl
	let V := q.composeValue hF (q.inclusion P)
	have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by
	apply tendsto_iff_norm_sub_tendsto_zero.mpr
	apply squeeze_zero (fun _ => norm_nonneg _)
	(g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖)
	· intro j
	apply Lp.norm_le_mul_norm_of_ae_le_mul
	filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P),
	((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp
	(μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1
	change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1
	rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one]
	exact he j _
	· simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖
	obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl
	refine ⟨R,?_,?_⟩
	· rw [hR]
	exact q.composeValue_ae hF (q.inclusion P)
	· rw [hRG]
	exact ClosedCalculus.mulLp_ae hc hcC _

	end CAT0Fillings.ChartGeometry
	end

	section

	open Set Filter MeasureTheory
	open scoped Topology ENNReal NNReal

	namespace CAT0Fillings.ChartGeometry
	open ClosedCalculus

	variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
	[CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)}
	{hT : IsMetricCurrent T} (q : ChartGeometry hT)

	lemma closedGradient_level_zero (hz : IsCycle T) (P : q.Sobolev) (a : ℝ) :
	∀ᵐ w ∂q.atlasMeasure, (q.inclusion P) (q.atlasParam w) = a →
	(q.closedGradient P) w = 0 := by
	let e (j : ℕ) : ℝ := 1/((j:ℝ)+1)
	have hep (j : ℕ) : 0 < e j := by dsimp [e]; positivity
	have he : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
	have hD : Measurable (fun t : ℝ => if t = a then (1:ℝ) else 0) := by
	convert (show Measurable (({a} : Set ℝ).indicator (fun _ : ℝ => (1:ℝ))) from
	measurable_const.indicator (measurableSet_singleton a)) using 1
	funext t
	simp only [Set.indicator_apply,Set.mem_singleton_iff]

	have hb (t : ℝ) : \|if t = a then (1:ℝ) else 0\| ≤ 1 := by split_ifs <;> norm_num
	obtain ⟨R,hR,hG⟩ := q.closed_chain_uniform (F := fun _ => (0:ℝ))
	(D := fun t => if t = a then (1:ℝ) else 0) (LipschitzWith.const (0:ℝ))
	hD (by norm_num : (0:ℝ) ≤ 1) hb
	(fun j => levelApprox a (e j)) (fun j => levelApproxDeriv a (e j)) (fun _ => 1)
	(fun j => levelApprox_lipschitz (hep j) a) (fun j => levelApprox_hasDerivAt (hep j) a)
	(fun j => continuous_levelApproxDeriv (hep j) a) (fun j => levelApproxDeriv_bound (hep j) a)
	(levelApproxDeriv_tendsto he hep a) (fun j => e j*(Real.pi/2))
	(by simpa only [zero_mul] using he.mul_const (Real.pi/2))
	(fun j t => by simpa only [sub_zero] using levelApprox_error (hep j).le a t) P
	have hR0 : q.inclusion R = 0 := Lp.ext (hR.trans (Lp.coeFn_zero ℝ 2 (MassMeasure.currentMassMeasure hT)).symm)
	have hRz : R = 0 := q.inclusion_injective hz (hR0.trans (map_zero q.inclusion).symm)
	rw [hRz,map_zero] at hG
	filter_upwards [hG,(Lp.coeFn_zero (Euc (k+1)) 2 q.atlasMeasure)] with w hw hz
	intro ha
	simpa only [hz,ha,ite_true,one_smul,Pi.zero_apply] using hw.symm

	end CAT0Fillings.ChartGeometry
	end

	end OAI