lean/OAI/Analysis/HardSphere/Allocation/PhysicalVertex.lean

import Mathlib
import OAI.Analysis.HardSphere.Comparison.ParameterTarget
import OAI.Analysis.HardSphere.Allocation.CoarsePhysical
import OAI.Analysis.HardSphere.Allocation.PhysicalTargets

namespace OAI

section

noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal NNReal Topology
namespace HardSphere.Bridge
open Wiener GaussianPacking Comparison

lemma exposedPeriodPair_at {E : Type*} [MeasurableSpace E] {L : Length}
    (Q : Kernel E Point) [IsFiniteKernel Q] {t C : ℝ} (ht : 0<t) (hC : 0≤C)
    (y z y' z' : E→Point) (hy : Measurable y) (hz : Measurable z)
    (hy' : Measurable y') (hz' : Measurable z') (c : Point) (hc : projectPoint L c=0)
    (S T U : Set (E×C(Time,Point))) (hS : MeasurableSet S)
    (hT : MeasurableSet T) (hU : MeasurableSet U)
    (hTU : ∀e f,(e,shiftPath c f)∈U ↔ (e,f)∈T)
    (A : E→Set Point) (hQ : ∀e,Q e=targetUniform (A e)) (e : E)
    (hA : MeasurableSet (A e)) (hAf : volume (A e)≠⊤) {a b : ℝ≥0∞}
    (ha : 0<a) (hb : 0<b) (hvol : a≤volume (A e))
    (hbound : ∀u∈A e,‖u-(fun j ↦ (y e j+z e j)/2)‖≤C)
    (hbound' : ∀u∈A e,‖u-(fun j ↦ ((y' e+c) j+(z' e+c) j)/2)‖≤C)
    (hfirst : ∀ᵐu ∂targetUniform (A e),b≤trialKernel t (y e) (z e) u (Prod.mk e ⁻¹' S))
    (hsecond : ∀ᵐu ∂targetUniform (A e),b≤trialKernel t (y' e+c) (z' e+c) u (Prod.mk e ⁻¹' U)) :
    let ρ := exposedPeriodPair Q t y z y' z' hy hz hy' hz' c S U hS hU
    let P := parameterIndividualReference t y z hy hz S hS
    let P' := parameterIndividualReference t y' z' hy' hz' T hT
    let G := b⁻¹*(a⁻¹*(midpointLower t C)⁻¹)
    IsProbabilityMeasure (ρ e) ∧ IsProbabilityMeasure (P e) ∧ IsProbabilityMeasure (P' e) ∧
      (ρ e).fst≤G • P e ∧ (ρ e).snd≤G • P' e ∧
      (∀ᵐp ∂ρ e,projectPoint L (p.1 midpointTime)=projectPoint L (p.2 midpointTime)) := by
  dsimp only
  let Q₀ := Kernel.const Unit (Q e)
  let S₀ : Set (Unit×C(Time,Point)) := Prod.snd ⁻¹' (Prod.mk e ⁻¹' S)
  let T₀ : Set (Unit×C(Time,Point)) := Prod.snd ⁻¹' (Prod.mk e ⁻¹' T)
  let U₀ : Set (Unit×C(Time,Point)) := Prod.snd ⁻¹' (Prod.mk e ⁻¹' U)
  have hs : MeasurableSet S₀ := (hS.preimage measurable_prodMk_left).preimage measurable_snd
  have ht' : MeasurableSet T₀ := (hT.preimage measurable_prodMk_left).preimage measurable_snd
  have hu : MeasurableSet U₀ := (hU.preimage measurable_prodMk_left).preimage measurable_snd
  have htu : ∀q f,(q,shiftPath c f)∈U₀ ↔ (q,f)∈T₀ := fun _ f ↦ hTU e f
  have hq : ∀q : Unit,Q₀ q=targetUniform (A e) := fun _ ↦ hQ e
  obtain ⟨hp,hm,hn,hdom,hmid⟩ := exposedPeriodPair_physical Q₀ ht hC
    (fun _ ↦ y e) (fun _ ↦ z e) (fun _ ↦ y' e) (fun _ ↦ z' e)
    measurable_const measurable_const measurable_const measurable_const c hc
    S₀ T₀ U₀ hs ht' hu htu (fun _ ↦ A e) hq (fun _ ↦ hA) (fun _ ↦ hAf)
    ha hb (fun _ ↦ hvol) (fun _ ↦ hbound) (fun _ ↦ hbound')
    (fun _ ↦ hfirst) (fun _ ↦ hsecond)
  let ρ₀ := exposedPeriodPair Q₀ t (fun _ ↦ y e) (fun _ ↦ z e)
    (fun _ ↦ y' e) (fun _ ↦ z' e) measurable_const measurable_const
    measurable_const measurable_const c S₀ U₀ hs hu
  have he : ρ₀ ()=exposedPeriodPair Q t y z y' z' hy hz hy' hz' c S U hS hU e := by
    rw [exposedPeriodPair_apply Q t y z y' z' hy hz hy' hz' c S U hS hU A hQ e]
    exact exposedPeriodPair_apply Q₀ t _ _ _ _ _ _ _ _ c S₀ U₀ hs hu (fun _ ↦ A e) hq ()
  have hme : parameterIndividualReference t (fun _ : Unit ↦ y e) (fun _ ↦ z e)
      measurable_const measurable_const S₀ hs ()=parameterIndividualReference t y z hy hz S hS e := by
    rw [parameterIndividualReference_apply,parameterIndividualReference_apply]
    rfl
  have hne : parameterIndividualReference t (fun _ : Unit ↦ y' e) (fun _ ↦ z' e)
      measurable_const measurable_const T₀ ht' ()=parameterIndividualReference t y' z' hy' hz' T hT e := by
    rw [parameterIndividualReference_apply,parameterIndividualReference_apply]
    rfl
  have hf := hdom ()
  have hi := hmid ()
  change (ρ₀ ()).fst≤_ ∧ (ρ₀ ()).snd≤_ at hf
  change ∀ᵐp ∂ρ₀ (),_ at hi
  rw [he,hme,hne] at hf
  rw [he] at hi
  refine ⟨?_,?_,?_,hf.1,hf.2,hi⟩
  · rw [←he]; exact hp.isProbabilityMeasure ()
  · rw [←hme]; exact hm.isProbabilityMeasure ()
  · rw [←hne]; exact hn.isProbabilityMeasure ()

end HardSphere.Bridge

namespace HardSphere.Groups
open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital
attribute [local instance] Classical.propDecidable

lemma datumAccessible_endpoints {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i : Fin N)
    (d : PhysicalDatum N K M G L) (hi : datumAccessible hL g m t r R C S u i d) :
    ‖datumStart hL d i-anchorPoint (datumHome hL d i)‖≤10 ∧
    ‖datumEnd hL d i-anchorPoint (datumHome hL d i)‖≤10 := by
  have ha : 0<volume (Prod.mk d ⁻¹' datumTarget hL g m t r R C S u i 0) :=
    lt_of_lt_of_le (by norm_num) (hi 0)
  obtain ⟨U,hU⟩ := nonempty_of_measure_ne_zero ha.ne'
  have hs : 0<trialKernel t (datumStart hL d i) (datumEnd hL d i) U
      (Prod.mk d ⁻¹' datumPermitted hL g m r R C S u i) :=
    lt_of_lt_of_le (by norm_num : (0:ℝ≥0∞)<ENNReal.ofReal (9/10:ℝ)) hU.2
  obtain ⟨f,hf⟩ := nonempty_of_measure_ne_zero hs.ne'
  exact ⟨by rw [←hf.2.2.1.1]; exact (hf.2.1 0).1,
    by rw [←hf.2.2.1.2]; exact (hf.2.1 1).1⟩

lemma midpoint_bound (v y z U : Point) (hy : ‖y-v‖≤10) (hz : ‖z-v‖≤10)
    (hU : ‖U-v‖≤3) : ‖U-(fun j ↦ (y j+z j)/2)‖≤13 := by
  apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤13)).mpr
  intro j
  have hyj := (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤10)).mp hy j
  have hzj := (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤10)).mp hz j
  have huj := (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤3)).mp hU j
  change |y j-v j|≤10 at hyj
  change |z j-v j|≤10 at hzj
  change |U j-v j|≤3 at huj
  change |U j-(y j+z j)/2|≤13
  rw [abs_le] at hyj hzj huj ⊢
  constructor <;> linarith

lemma unitTarget_sub (v c U : Point) : U-c∈unitTarget v ↔ U∈unitTarget (v+c) := by
  simp only [unitTarget,mem_pi,mem_univ,forall_const,mem_Ico,Pi.sub_apply,Pi.add_apply]
  constructor <;> intro h j <;> specialize h j <;> constructor <;> linarith [h.1,h.2]

def routeCellPeriod {K : ℕ} (v w : Fin 3→Fin K) (j : Fin 7) : Point :=
  anchorPoint v-(anchorPoint w+targetStep j)

lemma routeCellPeriod_chart {K : ℕ} (v w : Fin 3→Fin K) (j : Fin 7) :
    anchorPoint w+targetStep j+routeCellPeriod v w j=anchorPoint v := by
  unfold routeCellPeriod
  abel

def vertexTarget {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) : Set (PhysicalDatum N K M G L×Point) :=
  (univ×ˢunitTarget (anchorPoint v))∩
    (datumTarget hL g m t r R C S u i 0 ∩
      (fun p ↦ (p.1,p.2-routeCellPeriod v w j)) ⁻¹'
        datumTarget hL g m t r R C S u i' j)

lemma vertexTarget_measurable {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) :
    MeasurableSet (vertexTarget (M:=M) hL g m t r R C S u i i' v w j) :=
  (MeasurableSet.univ.prod (unitTarget_measurable _)).inter
    ((datumTarget_measurable hL g m t r R C S u i 0).inter
      ((datumTarget_measurable hL g m t r R C S u i' j).preimage
        (measurable_fst.prodMk (measurable_snd.sub measurable_const))))

lemma vertexTarget_subset {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) (d : PhysicalDatum N K M G L) :
    Prod.mk d ⁻¹' vertexTarget hL g m t r R C S u i i' v w j ⊆ unitTarget (anchorPoint v) :=
  fun _ h ↦ h.1.2

lemma vertexTarget_volume {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) (d : PhysicalDatum N K M G L)
    (hv : datumHome hL d i=v) (hw : datumHome hL d i'=w)
    (hi : datumAccessible hL g m t r R C S u i d)
    (hi' : datumAccessible hL g m t r R C S u i' d) :
    ENNReal.ofReal (49/50:ℝ)≤volume
      (Prod.mk d ⁻¹' vertexTarget hL g m t r R C S u i i' v w j) := by
  let B := unitTarget (anchorPoint v)
  let A := Prod.mk d ⁻¹' datumTarget hL g m t r R C S u i 0
  let T := (fun U : Point ↦ U-routeCellPeriod v w j) ⁻¹'
    (Prod.mk d ⁻¹' datumTarget hL g m t r R C S u i' j)
  have hA : MeasurableSet A := (datumTarget_measurable hL g m t r R C S u i 0).preimage measurable_prodMk_left
  have hT : MeasurableSet T := ((datumTarget_measurable hL g m t r R C S u i' j).preimage
    measurable_prodMk_left).preimage (measurable_id.sub measurable_const)
  have hAB : A⊆B := by
    intro U hU
    have h := hU.1
    simpa [hv,targetStep,B] using h
  have hTB : T⊆B := by
    intro U hU
    have h := hU.1
    rw [hw,unitTarget_sub,routeCellPeriod_chart] at h
    exact h
  have htvol : volume T=volume (Prod.mk d ⁻¹' datumTarget hL g m t r R C S u i' j) := by
    exact (measurePreserving_add_right volume (-routeCellPeriod v w j)).measure_preimage
      ((datumTarget_measurable hL g m t r R C S u i' j).preimage measurable_prodMk_left).nullMeasurableSet
  let : IsProbabilityMeasure (volume.restrict B) := ⟨by
    rw [Measure.restrict_apply_univ,unitTarget_volume]⟩
  have ha : ENNReal.ofReal (99/100:ℝ)≤(volume.restrict B) A := by
    rw [Measure.restrict_apply hA,inter_eq_left.mpr hAB]
    exact hi 0
  have ht : ENNReal.ofReal (99/100:ℝ)≤(volume.restrict B) T := by
    rw [Measure.restrict_apply hT,inter_eq_left.mpr hTB,htvol]
    exact hi' j
  have H := probability_inter_98 (volume.restrict B) hA hT ha ht
  rw [Measure.restrict_apply (hA.inter hT)] at H
  convert H using 1
  congr 1
  ext U
  simp only [vertexTarget,mem_preimage,mem_inter_iff,mem_prod,mem_univ,true_and,A,T,B]
  tauto

def shiftedDatumPermitted {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (r R C S : ℝ) (u : Time) (i : Fin N) (c : Point) :
    Set (PhysicalDatum N K M G L×C(Time,Point)) :=
  (fun p ↦ (p.1,Bridge.shiftPath (-c) p.2)) ⁻¹' datumPermitted hL g m r R C S u i

lemma shiftedDatumPermitted_measurable {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (r R C S : ℝ) (u : Time) (i : Fin N) (c : Point) :
    MeasurableSet (shiftedDatumPermitted (M:=M) hL g m r R C S u i c) :=
  (datumPermitted_measurable hL g m r R C S u i).preimage
    (measurable_fst.prodMk ((Bridge.shiftPath_continuous (-c)).measurable.comp measurable_snd))

lemma shiftedDatumPermitted_equiv {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (r R C S : ℝ) (u : Time) (i : Fin N) (c : Point)
    (d : PhysicalDatum N K M G L) (f : C(Time,Point)) :
    (d,Bridge.shiftPath c f)∈shiftedDatumPermitted hL g m r R C S u i c ↔
      (d,f)∈datumPermitted hL g m r R C S u i := by
  simp only [shiftedDatumPermitted,mem_preimage,Bridge.shiftPath_cancel]

lemma datumSuccess_shift {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (u : Time) (i : Fin N) (c : Point)
    (d : PhysicalDatum N K M G L) (U : Point) :
    trialKernel t (datumStart hL d i+c) (datumEnd hL d i+c) U
      (Prod.mk d ⁻¹' shiftedDatumPermitted hL g m r R C S u i c)=
    datumSuccess hL g m t r R C S u i (d,U-c) := by
  have H := trialKernel_shift t (datumStart hL d i) (datumEnd hL d i) (U-c) c
  rw [sub_add_cancel] at H
  rw [←H,Measure.map_apply (Bridge.shiftPath_continuous c).measurable
    ((shiftedDatumPermitted_measurable hL g m r R C S u i c).preimage measurable_prodMk_left)]
  congr 1
  ext f
  exact shiftedDatumPermitted_equiv hL g m r R C S u i c d f

def vertexPair {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (g : Fin G)
    (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) :
    Kernel (PhysicalDatum N K M G L) (C(Time,Point)×C(Time,Point)) :=
  exposedPeriodPair
    (parameterTargetKernel (B:=unitTarget (anchorPoint v)) (by rw [unitTarget_volume]; exact ENNReal.one_ne_top)
      (vertexTarget hL g m t r R C S u i i' v w j)
      (vertexTarget_measurable hL g m t r R C S u i i' v w j)) t
    (fun d ↦ datumStart hL d i) (fun d ↦ datumEnd hL d i)
    (fun d ↦ datumStart hL d i') (fun d ↦ datumEnd hL d i')
    (datumStart_measurable hL i) (datumEnd_measurable hL i)
    (datumStart_measurable hL i') (datumEnd_measurable hL i') (routeCellPeriod v w j)
    (datumPermitted hL g m r R C S u i)
    (shiftedDatumPermitted hL g m r R C S u i' (routeCellPeriod v w j))
    (datumPermitted_measurable hL g m r R C S u i)
    (shiftedDatumPermitted_measurable hL g m r R C S u i' (routeCellPeriod v w j))

instance vertexPair_finite {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (g : Fin G)
    (m : ℕ) (t r R C S : ℝ) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) :
    IsFiniteKernel (vertexPair (M:=M) hL g m t r R C S u i i' v w j) := by
  unfold vertexPair exposedPeriodPair exposedPairedTrialKernel parameterPairedKernel
  infer_instance

def vertexDensityBound (t : ℝ) : ℝ≥0∞ :=
  (ENNReal.ofReal (9/10:ℝ))⁻¹*((ENNReal.ofReal (49/50:ℝ))⁻¹*(midpointLower t 13)⁻¹)

theorem vertexPair_physical {N K M G : ℕ} {L : Length} (hL : L.val=(K:ℝ))
    (g : Fin G) (m : ℕ) (t r R C S : ℝ) (ht : 0<t) (u : Time) (i i' : Fin N)
    (v w : Fin 3→Fin K) (j : Fin 7) (hc : projectPoint L (routeCellPeriod v w j)=0)
    (d : PhysicalDatum N K M G L) (hv : datumHome hL d i=v) (hw : datumHome hL d i'=w)
    (hi : datumAccessible hL g m t r R C S u i d)
    (hi' : datumAccessible hL g m t r R C S u i' d) :
    let ρ := vertexPair hL g m t r R C S u i i' v w j d
    let P := datumIndividualReference hL g m r R C S t u i d
    let P' := datumIndividualReference hL g m r R C S t u i' d
    IsProbabilityMeasure ρ ∧ IsProbabilityMeasure P ∧ IsProbabilityMeasure P' ∧
      ρ.fst≤vertexDensityBound t • P ∧ ρ.snd≤vertexDensityBound t • P' ∧
      (∀ᵐp ∂ρ,projectPoint L (p.1 midpointTime)=projectPoint L (p.2 midpointTime)) := by
  dsimp only
  let A := fun d : PhysicalDatum N K M G L ↦
    Prod.mk d ⁻¹' vertexTarget hL g m t r R C S u i i' v w j
  have hA : MeasurableSet (A d) :=
    (vertexTarget_measurable hL g m t r R C S u i i' v w j).preimage measurable_prodMk_left
  have hAf : volume (A d)≠⊤ := ne_top_of_le_ne_top
    (by rw [unitTarget_volume]; exact ENNReal.one_ne_top)
    (measure_mono (vertexTarget_subset hL g m t r R C S u i i' v w j d))
  have hmem : ∀ᵐU ∂targetUniform (A d),U∈A d :=
    Measure.ae_smul_measure (ae_restrict_mem hA) _
  have hei := datumAccessible_endpoints hL g m t r R C S u i d hi
  have hei' := datumAccessible_endpoints hL g m t r R C S u i' d hi'
  apply exposedPeriodPair_at _ ht (by norm_num : (0:ℝ)≤13)
    (fun d ↦ datumStart hL d i) (fun d ↦ datumEnd hL d i)
    (fun d ↦ datumStart hL d i') (fun d ↦ datumEnd hL d i')
    (datumStart_measurable hL i) (datumEnd_measurable hL i)
    (datumStart_measurable hL i') (datumEnd_measurable hL i')
    (routeCellPeriod v w j) hc (datumPermitted hL g m r R C S u i)
    (datumPermitted hL g m r R C S u i')
    (shiftedDatumPermitted hL g m r R C S u i' (routeCellPeriod v w j))
    (datumPermitted_measurable hL g m r R C S u i)
    (datumPermitted_measurable hL g m r R C S u i')
    (shiftedDatumPermitted_measurable hL g m r R C S u i' (routeCellPeriod v w j))
    (shiftedDatumPermitted_equiv hL g m r R C S u i' (routeCellPeriod v w j))
    A (parameterTargetKernel_apply _ _ _ (vertexTarget_subset hL g m t r R C S u i i' v w j)) d
    hA hAf (by norm_num : (0:ℝ≥0∞)<ENNReal.ofReal (49/50:ℝ))
    (by norm_num : (0:ℝ≥0∞)<ENNReal.ofReal (9/10:ℝ))
    (vertexTarget_volume hL g m t r R C S u i i' v w j d hv hw hi hi')
  · intro U hU
    exact midpoint_bound _ _ _ U hei.1 hei.2
      (targetCell_radius _ 0 U hU.2.1.1)
  · intro U hU
    have H := midpoint_bound _ _ _ (U-routeCellPeriod v w j) hei'.1 hei'.2
      (targetCell_radius _ j _ hU.2.2.1)
    convert H using 1
    congr 1
    ext k
    change U k-((datumStart hL d i' k+routeCellPeriod v w j k)+
      (datumEnd hL d i' k+routeCellPeriod v w j k))/2=
      (U k-routeCellPeriod v w j k)-(datumStart hL d i' k+datumEnd hL d i' k)/2
    ring
  · filter_upwards [hmem] with U hU
    exact hU.2.1.2
  · filter_upwards [hmem] with U hU
    rw [datumSuccess_shift]
    exact hU.2.2.2

end HardSphere.Groups
end
end

end OAI