| 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 |
| |
