| import OAI.Dynamics.SmoothObstruction.TubeProjection |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory Filter Topology TopologicalSpace Set Finset |
| open scoped ENNReal NNReal BoundedContinuousFunction |
| open Finset MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Finset Matrix |
| open scoped Matrix.Norms.L2Operator |
| open Matrix |
|
|
| namespace SmoothObstruction.CanonicalRelation |
| open Set MeasureTheory Filter Topology |
|
|
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| [StandardBorelSpace X] [StandardBorelSpace Y] |
|
|
| /-- A Borel relation with exactly one output at every input has a genuine |
| measurable solution map. Its definition does not depend on any auxiliary |
| frame certificate used to obtain existence and uniqueness. -/ |
| theorem measurable_solution (R : Set (X × Y)) (hR : MeasurableSet R) |
| (hex : ∀ x,∃! y,(x,y)∈R) : |
| ∃ f : X → Y,Measurable f ∧ (∀ x,(x,f x)∈R) ∧ |
| ∀ x y,(x,y)∈R → y=f x := by |
| classical |
| let f : X → Y := fun x => (hex x).choose |
| have hf (x : X) : (x,f x)∈R := (hex x).choose_spec.1 |
| have hu (x : X) (y : Y) (hy : (x,y)∈R) : y=f x := |
| (hex x).choose_spec.2 y hy |
| let : StandardBorelSpace R := hR.standardBorel |
| let π : R → X := fun z => z.val.1 |
| have hπ : Measurable π := measurable_fst.comp measurable_subtype_coe |
| have hsurj : Function.Surjective π := fun x => ⟨⟨(x,f x),hf x⟩,rfl⟩ |
| have he : f ∘ π=(fun z : R => z.val.2) := by |
| funext z |
| exact (hu z.val.1 z.val.2 z.property).symm |
| have hm : Measurable (f ∘ π) := by |
| rw [he] |
| exact measurable_snd.comp measurable_subtype_coe |
| exact ⟨f,(Measurable.measurable_comp_iff_of_surjective hπ hsurj).mp hm,hf,hu⟩ |
|
|
| /-- The retained set has its original Borel subspace, not a conditioned |
| probability space. The measurable total extension is only for convenient |
| pushforward notation; it is equal to the canonical solution on the carrier. -/ |
| theorem retained_solution [Nonempty Y] (G : Set X) (hG : MeasurableSet G) |
| (R : X → Y → Prop) (hR : MeasurableSet {z : X×Y | z.1∈G ∧ R z.1 z.2}) |
| (hex : ∀ x∈G,∃! y,R x y) : |
| ∃ f : X → Y,Measurable f ∧ (∀ x∈G,R x (f x)) ∧ |
| ∀ x∈G,∀ y,R x y → y=f x := by |
| classical |
| let : StandardBorelSpace G := hG.standardBorel |
| let R' : Set (G×Y) := {z | R z.1.val z.2} |
| have hR' : MeasurableSet R' := by |
| have hm : Measurable (fun z : G×Y => (z.1.val,z.2)) := |
| (measurable_subtype_coe.comp measurable_fst).prodMk measurable_snd |
| have he : R'= (fun z : G×Y => (z.1.val,z.2)) ⁻¹' {z : X×Y | z.1∈G ∧ R z.1 z.2} := by |
| ext z |
| simp only [R',mem_ofPred_eq,Set.mem_preimage] |
| exact ⟨fun h => ⟨z.1.property,h⟩,fun h => h.2⟩ |
| rw [he] |
| exact hR.preimage hm |
| obtain ⟨f,hf,hfr,hfu⟩ := measurable_solution R' hR' (fun x => hex x x.property) |
| obtain ⟨F,hF,hFf⟩ := (MeasurableEmbedding.subtype_coe hG).exists_measurable_extend hf |
| (fun _ => inferInstance) |
| have he (x : G) : F x=f x := congrFun hFf x |
| refine ⟨F,hF,?_,?_⟩ |
| · intro x hx |
| rw [he ⟨x,hx⟩] |
| exact hfr ⟨x,hx⟩ |
| · intro x hx y hy |
| rw [he ⟨x,hx⟩] |
| exact hfu ⟨x,hx⟩ y hy |
|
|
| end SmoothObstruction.CanonicalRelation |
|
|
| namespace SmoothObstruction.CanonicalRelation |
| open Set Filter Topology |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| /-- Local analytic branches agree with the canonical Borel solution merely |
| by satisfying its boundary relation inside the uniqueness tube. The frame |
| witnesses and chart codes need not be continuous functions of the inputs. -/ |
| theorem local_derivative (G : Set E) (x : E) (hx : x∈G) |
| (R : E → F → Prop) (f g : E → F) (L : E →L[ℝ] F) |
| (hu : ∀ u∈G,∀ v,R u v → v=f u) |
| (hg : HasFDerivAt g L x) (hrel : ∀ᶠ u in 𝓝[G] x,R u (g u)) : |
| HasFDerivWithinAt f L G x := by |
| have he : g =ᶠ[𝓝[G] x] f := by |
| filter_upwards [hrel,self_mem_nhdsWithin] with u huR huG |
| exact hu u huG (g u) huR |
| exact hg.hasFDerivWithinAt.congr_of_eventuallyEq he.symm (he.eq_of_nhdsWithin hx).symm |
| end SmoothObstruction.CanonicalRelation |
|
|
| namespace SmoothObstruction.SelectedTube |
| open Set MeasureTheory |
| variable {X C E : Type*} [MeasurableSpace X] [Fintype C] |
| [MeasurableSpace C] [MeasurableSingletonClass C] |
| [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] |
|
|
| /-- The literal fixed-atlas coordinate tube, with chart sources checked. |
| Only the base point chooses the chart; the second point need not be good. -/ |
| def tube (q : X → C) (U : C → Set X) (φ : C → X → E) |
| (f : ℕ → X → X) (n : ℕ) (δ : ℝ) (x z : X) : Prop := |
| ∀ t : Fin (n+1),f t z∈U (q (f t x)) ∧ |
| ‖φ (q (f t x)) (f t z)-φ (q (f t x)) (f t x)‖≤δ |
|
|
| omit [NormedAddCommGroup E] [BorelSpace E] [SecondCountableTopology E] in |
| lemma measurable_evaluate (φ : C → X → E) (hφ : ∀ c,Measurable (φ c)) : |
| Measurable (fun p : C×X => φ p.1 p.2) := |
| measurable_from_prod_countable_right hφ |
|
|
| lemma measurable_tube (q : X → C) (hq : Measurable q) |
| (U : C → Set X) (hU : ∀ c,MeasurableSet (U c)) |
| (φ : C → X → E) (hφ : ∀ c,Measurable (φ c)) |
| (f : ℕ → X → X) (hf : ∀ t,Measurable (f t)) (n : ℕ) (δ : ℝ) : |
| MeasurableSet {p : X×X | tube q U φ f n δ p.1 p.2} := by |
| have hu : MeasurableSet {p : C×X | p.2∈U p.1} := by |
| have he : {p : C×X | p.2∈U p.1}=⋃ c,({c} ×ˢ U c) := by |
| ext p |
| simp only [mem_ofPred_eq,mem_iUnion,mem_prod,mem_singleton_iff] |
| exact ⟨fun h => ⟨p.1,rfl,h⟩,fun ⟨c,hc,h⟩ => hc ▸ h⟩ |
| rw [he] |
| exact MeasurableSet.iUnion (fun c => (measurableSet_singleton c).prod (hU c)) |
| simp only [tube,Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro t |
| have hx : Measurable (fun p : X×X => f t p.1) := (hf t).comp measurable_fst |
| have hz : Measurable (fun p : X×X => f t p.2) := (hf t).comp measurable_snd |
| have hc := hq.comp hx |
| exact (hu.preimage (hc.prodMk hz)).inter |
| (measurableSet_le (((measurable_evaluate φ hφ).comp (hc.prodMk hz)).sub |
| ((measurable_evaluate φ hφ).comp (hc.prodMk hx))).norm measurable_const) |
| end SmoothObstruction.SelectedTube |
|
|
| namespace SmoothObstruction.MeasurableSwitch |
| open Set MeasureTheory |
| variable {X C E : Type*} [MeasurableSpace X] [Fintype C] |
| [MeasurableSpace C] [MeasurableSingletonClass C] |
| [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] |
|
|
| /-- Fixed-bin future/past observations and the two old coordinate tubes. |
| All auxiliary frames have disappeared from this boundary relation. -/ |
| def relation (F P : X → X → E) (TF TB : X → X → Prop) (u : X×X) (z : X) : Prop := |
| F u.1 z=F u.1 u.1 ∧ P u.2 z=P u.2 u.2 ∧ TF u.1 z ∧ TB u.2 z |
|
|
| lemma measurable_relation (F P : X → X → E) |
| (hF : Measurable (Function.uncurry F)) (hP : Measurable (Function.uncurry P)) |
| (TF TB : X → X → Prop) (hTF : MeasurableSet {p : X×X | TF p.1 p.2}) |
| (hTB : MeasurableSet {p : X×X | TB p.1 p.2}) : |
| MeasurableSet {p : (X×X)×X | relation F P TF TB p.1 p.2} := by |
| have h₁ : Measurable (fun p : (X×X)×X => (p.1.1,p.2)) := |
| (measurable_fst.comp measurable_fst).prodMk measurable_snd |
| have h₂ : Measurable (fun p : (X×X)×X => (p.1.2,p.2)) := |
| (measurable_snd.comp measurable_fst).prodMk measurable_snd |
| have hx : Measurable (fun p : (X×X)×X => p.1.1) := |
| measurable_fst.comp measurable_fst |
| have hy : Measurable (fun p : (X×X)×X => p.1.2) := |
| measurable_snd.comp measurable_fst |
| exact (measurableSet_eq_fun (hF.comp h₁) (hF.comp (hx.prodMk hx))).inter |
| ((measurableSet_eq_fun (hP.comp h₂) (hP.comp (hy.prodMk hy))).inter |
| ((hTF.preimage h₁).inter (hTB.preimage h₂))) |
|
|
| /-- The original retained carrier gives one globally measurable pair map. |
| There is no measurable moving-frame selection and no conditioning of a law. |
| Existence and uniqueness are supplied by the actual finite-tube solve. -/ |
| theorem canonical_pair [StandardBorelSpace X] [Nonempty X] |
| (G : Set (X×X)) (hG : MeasurableSet G) |
| (F P : X → X → E) (hF : Measurable (Function.uncurry F)) |
| (hP : Measurable (Function.uncurry P)) |
| (TF TB : X → X → Prop) (hTF : MeasurableSet {p : X×X | TF p.1 p.2}) |
| (hTB : MeasurableSet {p : X×X | TB p.1 p.2}) |
| (hex : ∀ u∈G,∃! z,relation F P TF TB u z) |
| (hex' : ∀ u∈G,∃! w,relation F P TF TB u.swap w) : |
| ∃ Φ : X×X → X×X,Measurable Φ ∧ |
| (∀ u∈G,relation F P TF TB u (Φ u).1 ∧ relation F P TF TB u.swap (Φ u).2) ∧ |
| (∀ u∈G,∀ z,relation F P TF TB u z → z=(Φ u).1) ∧ |
| (∀ u∈G,∀ w,relation F P TF TB u.swap w → w=(Φ u).2) := by |
| have hR := measurable_relation F P hF hP TF TB hTF hTB |
| have hRG : MeasurableSet {p : (X×X)×X | p.1∈G ∧ relation F P TF TB p.1 p.2} := |
| (hG.preimage measurable_fst).inter hR |
| have hRG' : MeasurableSet {p : (X×X)×X | p.1∈G ∧ relation F P TF TB p.1.swap p.2} := |
| (hG.preimage measurable_fst).inter (hR.preimage |
| ((measurable_swap.comp measurable_fst).prodMk measurable_snd)) |
| obtain ⟨z,hz,hzr,hzu⟩ := CanonicalRelation.retained_solution G hG |
| (relation F P TF TB) hRG hex |
| obtain ⟨w,hw,hwr,hwu⟩ := CanonicalRelation.retained_solution G hG |
| (fun u => relation F P TF TB u.swap) hRG' hex' |
| exact ⟨fun u => (z u,w u),hz.prodMk hw,fun u hu => ⟨hzr u hu,hwr u hu⟩,hzu,hwu⟩ |
| end SmoothObstruction.MeasurableSwitch |
|
|
| namespace SmoothObstruction.EndpointObservation |
| open Set MeasureTheory |
| variable {X C B E : Type*} [MeasurableSpace X] |
| [Fintype C] [Fintype B] [MeasurableSpace C] [MeasurableSingletonClass C] |
| [MeasurableSpace B] [MeasurableSingletonClass B] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| /-- The source endpoint observation in the ambient coordinate space. Its |
| finite index consists ONLY of the starting coarse bin and endpoint coarse |
| bin; an internal frame witness is not part of this index. -/ |
| def observation (c : X → C) (b : X → B) (f : X → X) |
| (φ : B → X → E) (P : C → B → E →L[ℝ] E) (x z : X) : E := |
| P (c x) (b (f x)) (φ (b (f x)) (f z)) |
|
|
| lemma measurable_observation (c : X → C) (hc : Measurable c) |
| (b : X → B) (hb : Measurable b) (f : X → X) (hf : Measurable f) |
| (φ : B → X → E) (hφ : ∀ j,Measurable (φ j)) (P : C → B → E →L[ℝ] E) : |
| Measurable (Function.uncurry (observation c b f φ P)) := by |
| have hh : Measurable (fun u : (C×B)×X => P u.1.1 u.1.2 (φ u.1.2 (f u.2))) := by |
| apply measurable_from_prod_countable_right |
| intro ij |
| exact (P ij.1 ij.2).continuous.measurable.comp ((hφ ij.2).comp hf) |
| exact hh.comp (((hc.comp measurable_fst).prodMk |
| (hb.comp (hf.comp measurable_fst))).prodMk measurable_snd) |
|
|
| omit [MeasurableSpace X] [Fintype C] [Fintype B] |
| [MeasurableSpace C] [MeasurableSingletonClass C] |
| [MeasurableSpace B] [MeasurableSingletonClass B] [MeasurableSpace E] [BorelSpace E] |
|
|
| /-- Positive bin margins turn the ambient observations into the actual fixed |
| smooth endpoint maps on a neighborhood; there is no differentiability claim |
| for the discontinuously selected observation itself. -/ |
| lemma observation_fixed (c : X → C) (b : X → B) (f : X → X) |
| (φ : B → X → E) (P : C → B → E →L[ℝ] E) |
| (x y z : X) (hc : c x=c y) (hb : b (f x)=b (f y)) : |
| observation c b f φ P x z=observation c b f φ P y z := by |
| simp only [observation,hc,hb] |
|
|
| lemma observation_eq_iff (c : X → C) (b : X → B) (f : X → X) |
| (φ : B → X → E) (P : C → B → E →L[ℝ] E) (x y z : X) : |
| observation c b f φ P x z=observation c b f φ P x y ↔ |
| P (c x) (b (f x)) (φ (b (f x)) (f z)-φ (b (f x)) (f y))=0 := by |
| simp only [observation,map_sub,sub_eq_zero] |
| end SmoothObstruction.EndpointObservation |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set MeasureTheory |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] |
|
|
| /-- A chart has no prescribed behaviour outside its source. This Borel |
| extension, used only to state global relations, agrees with it on the source. -/ |
| noncomputable def borelChart (a : M) : M → EuclideanSpace ℝ (Fin d) := by |
| classical |
| exact (chartAt H a).source.piecewise (extChartAt I a) (fun _ => 0) |
|
|
| lemma borelChart_eq (a z : M) (hz : z∈(chartAt H a).source) : |
| borelChart I a z=(extChartAt I a) z := by |
| classical |
| exact Set.piecewise_eq_of_mem _ _ _ hz |
|
|
| variable [MeasurableSpace M] [BorelSpace M] |
|
|
| lemma measurable_borelChart (a : M) : Measurable (borelChart I a) := by |
| classical |
| apply ContinuousOn.measurable_piecewise _ continuousOn_const (chartAt H a).open_source.measurableSet |
| simpa only [extChartAt_source] using continuousOn_extChartAt (I:=I) a |
|
|
| variable {ι C : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| [Fintype C] [MeasurableSpace C] [MeasurableSingletonClass C] |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (c : M → ι) (n : ℕ) |
|
|
| lemma measurable_originalTube (hc : Measurable c) |
| (hsource : ∀ z,z∈(chartAt H (a (c z))).source) (δ : ℝ) : |
| MeasurableSet {p : M×M | originalTube I S a c (n:=n) δ p.1 p.2} := by |
| have he : {p : M×M | originalTube I S a c (n:=n) δ p.1 p.2}= |
| {p | SelectedTube.tube c (fun j => (chartAt H (a j)).source) |
| (fun j => borelChart I (a j)) (fun t => S^[t]) n δ p.1 p.2} := by |
| ext p |
| constructor <;> intro hp t <;> obtain ⟨hs,ht⟩ := hp t |
| · exact ⟨hs,by simpa only [borelChart_eq I _ _ hs, |
| borelChart_eq I _ _ (hsource _)] using ht⟩ |
| · exact ⟨hs,by simpa only [borelChart_eq I _ _ hs, |
| borelChart_eq I _ _ (hsource _)] using ht⟩ |
| rw [he] |
| exact SelectedTube.measurable_tube c hc (fun j => (chartAt H (a j)).source) |
| (fun j => (chartAt H (a j)).open_source.measurableSet) |
| (fun j => borelChart I (a j)) (fun j => measurable_borelChart I (a j)) |
| (fun t => S^[t]) (fun t => S.continuous.measurable.iterate t) n δ |
|
|
| noncomputable def originalObservation (P : M → C) |
| (W : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))) (x z : M) := |
| (W (P x,P (S^[n] x))).starProjection |
| (borelChart I (a (c (S^[n] x))) (S^[n] z)) |
|
|
| lemma measurable_originalObservation (hc : Measurable c) (P : M → C) (hP : Measurable P) |
| (W : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))) : |
| Measurable (Function.uncurry (originalObservation I S a c n P W)) := by |
| exact EndpointObservation.measurable_observation P hP (fun x => (P x,c x)) |
| (hP.prodMk hc) (S^[n]) (S.continuous.measurable.iterate n) |
| (fun j : C×ι => borelChart I (a j.2)) (fun j => measurable_borelChart I (a j.2)) |
| (fun q j => (W (q,j.1)).starProjection) |
|
|
| omit [MeasurableSpace M] [BorelSpace M] |
| [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| [Fintype C] [MeasurableSpace C] [MeasurableSingletonClass C] |
|
|
| lemma originalObservation_eq_iff (P : M → C) |
| (W : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))) (x z : M) |
| (hx : S^[n] x∈(chartAt H (a (c (S^[n] x)))).source) |
| (hz : S^[n] z∈(chartAt H (a (c (S^[n] x)))).source) : |
| originalObservation I S a c n P W x z=originalObservation I S a c n P W x x ↔ |
| originalEndpoint I S a c (n:=n) (W (P x,P (S^[n] x))) x z := by |
| simp only [originalObservation,originalEndpoint,borelChart_eq I _ _ hx,borelChart_eq I _ _ hz] |
|
|
| lemma originalTube_mono {δ ε : ℝ} {x z : M} (h : δ≤ε) |
| (hx : originalTube I S a c (n:=n) δ x z) : |
| originalTube I S a c (n:=n) ε x z := fun t => ⟨(hx t).1,(hx t).2.trans h⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] |
| {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| open FiniteGreen ShortBin |
|
|
| /-- A measurable, witness-independent ORIGINAL pair switch. Its Borel relation |
| uses zero-extended charts, but all asserted equations and tube estimates are in |
| the original atlas. The retained set has not been normalized or repartitioned. -/ |
| theorem canonical_original_switch (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r₀ C₀ : ℝ,0<r₀ ∧ 1≤C₀ ∧ |
| ∀ (C J : Type*) [Fintype C] [MeasurableSpace C] [MeasurableSingletonClass C] |
| [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] |
| (c : M → ι) (P : M → C) (L : M → J) (G R : Set M) |
| (n : ℕ) (B R₀ ε g α r δ : ℝ) (γ : C → ℝ) |
| (A : ∀ x:G,BilateralGauge I S a c n B R₀ x) |
| (T : G → Finset (Fin d)) (ref : G → G) |
| (Wf Wb : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))), |
| Measurable c → Measurable P → Measurable L → MeasurableSet R → R⊆G → |
| (∀ v,v∈K (c v)) → |
| (∀ x y,L x=L y → P x=P y) → |
| (∀ x y,P x=P y → c x=c y) → |
| (∀ x∈R,I.IsInteriorPoint x) → |
| (∀ x:G,P (ref x)=P x) → |
| (∀ x y:G,P x=P y → (T x).card=(T y).card) → |
| (∀ x y:G,P x=P y → ref x=ref y) → |
| (∀ x:G, |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))ᶜ).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)ᶜ).starProjection‖≤ε ∧ |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)).starProjection‖≤ε) → |
| (∀ x:G, |
| Module.finrank ℝ ((A x).forwardSpace I S a c n (T x))= |
| Module.finrank ℝ (Wf (P x,P (S^[n] x))) ∧ |
| Module.finrank ℝ ((A x).backwardSpace I S a c n (T x)ᶜ)= |
| Module.finrank ℝ (Wb (P x,P (S.symm^[n] x))) ∧ |
| ‖((A x).forwardSpace I S a c n (T x)).starProjection- |
| (Wf (P x,P (S^[n] x))).starProjection‖≤1/2 ∧ |
| ‖((A x).backwardSpace I S a c n (T x)ᶜ).starProjection- |
| (Wb (P x,P (S.symm^[n] x))).starProjection‖≤1/2) → |
| 0≤ε → ε≤1/2 → Real.exp (2*B)*(2*ε)≤1/2 → |
| (∀ x:G,∀ j∈(T x)ᶜ,(A x).lam j≤γ (P x)-g) → |
| (∀ x:G,∀ j∈T x,γ (P x)+g≤(A x).lam j) → |
| (2*Real.exp (2*B))*(2*(2*Real.exp (2*B)^2*Real.exp (-2*g*n)))≤1/2 → |
| (∀ x:G,|γ (P x)|+|g|≤α) → 0≤r → 0<δ → δ≤r₀ → |
| (∀ x∈R,∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) δ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) δ⊆interior (range I)) → |
| (∀ x∈R,∀ y∈R,L x=L y → |
| ‖(extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x‖≤C₀*r) → |
| bvpInverseBound B α n*C₀*δ≤1/4 → bvpInverseBound B α n*C₀*r≤δ/4 → |
| ∃ Φ : M×M → M×M,Measurable Φ ∧ |
| (∀ x y:G,x.val∈R → y.val∈R → L x=L y → |
| let z := (Φ (x.val,y.val)).1 |
| let w := (Φ (x.val,y.val)).2 |
| originalTube I S a c (n:=n) (δ/2) x z ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) y z ∧ |
| originalTube I S a c (n:=n) (δ/2) y w ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) x w ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x z ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y z ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y w ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x w ∧ |
| ‖(extChartAt I (a (c x))) z-(extChartAt I (a (c x))) x- |
| (((A (ref x)).forward I S a c).splitting (T (ref x))ᶜ).slow 0 |
| ((extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x)‖≤ |
| 4*Real.exp (2*B)*(2*Real.exp (2*B))*(ε+2*Real.exp (2*B)^2*Real.exp (-2*g*n))*(C₀*r)+ |
| bvpInverseBound B α n*C₀*(2*C₀*bvpInverseBound B α n*r)^2) ∧ |
| ∀ x∈R,∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ x z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z → z=x := by |
| obtain ⟨r₀,C₀,hr₀,hC₀,hjoin⟩ := original_gauged_join I hclosed S a K hK hKU |
| refine ⟨r₀,C₀,hr₀,hC₀,?_⟩ |
| intro C J _ _ _ _ _ _ c P L G R n B R₀ ε g α r δ γ A T ref Wf Wb |
| hc hP hL hR hRG hcover hLP hPc hint href hcard hrefeq hstart hend hε hεhalf |
| hsmall hslow hfast hgap hα hr hδ hδr hsafe hnear hcontract hdata |
| let F := originalObservation I S a c n P Wf |
| let V := originalObservation I S.symm a c n P Wb |
| let TF := originalTube I S a c (n:=n) δ |
| let TB := originalTube I S.symm a c (n:=n) δ |
| let Gpair : Set (M×M) := {u | u.1∈R ∧ u.2∈R ∧ L u.1=L u.2} |
| have hGpair : MeasurableSet Gpair := |
| (hR.preimage measurable_fst).inter ((hR.preimage measurable_snd).inter |
| (measurableSet_eq_fun (hL.comp measurable_fst) (hL.comp measurable_snd))) |
| have hsrc (x : M) : x∈(chartAt H (a (c x))).source := hKU _ (hcover x) |
| have hspec (x y : G) (hx : x.val∈R) (hy : y.val∈R) (hxy : L x=L y) := by |
| have hp := hLP x y hxy |
| have hcst : |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A y).forwardSpace I S a c 0 (T y)).starProjection‖≤ε := by |
| rw [hrefeq x y hp] |
| exact (hstart y).2 |
| exact hjoin c n B R₀ ε (γ (P x)) g α r δ (ref x) x y |
| (A (ref x)) (A x) (A y) (T (ref x)) (T x) (T y) |
| (Wf (P x,P (S^[n] x))) (Wb (P y,P (S.symm^[n] y))) hcover (hPc x y hp) |
| (hint x hx) (hint y hy) (hend x).1 (hend y).2.1 (hend x).2.2.1 (hend y).2.2.2 |
| hε hεhalf (hcard _ _ (href x)) (hcard _ _ ((href x).trans hp)) (hstart x).1 hcst hsmall |
| (hslow x) (hfast x) (by simpa only [hp] using hslow y) (by simpa only [hp] using hfast y) |
| hgap (hα x) hr hδ hδr (fun t => (hsafe x hx t).1) (fun t => (hsafe y hy t).2) |
| (hnear x hx y hy hxy) hcontract hdata |
| have hrel (x y : M) (z : M) : |
| MeasurableSwitch.relation F V TF TB (x,y) z ↔ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| TF x z ∧ TB y z := by |
| constructor |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mp hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mp hb,htf,htb⟩ |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mpr hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mpr hb,htf,htb⟩ |
| have hex (u : M×M) (hu : u∈Gpair) : ∃! z,MeasurableSwitch.relation F V TF TB u z := by |
| let x : G := ⟨u.1,hRG hu.1⟩ |
| let y : G := ⟨u.2,hRG hu.2.1⟩ |
| obtain ⟨z,hzf,hzb,hze,hze',hsharp,hsharp',hideal,hunique⟩ := hspec x y hu.1 hu.2.1 hu.2.2 |
| refine ⟨z,(hrel _ _ _).mpr ⟨hze,hze',?_,?_⟩,?_⟩ |
| · exact originalTube_mono I S a c n (by linarith) hzf |
| · exact originalTube_mono I S.symm a c n (by linarith) hzb |
| · intro z' hz' |
| obtain ⟨hf,hb,htf,htb⟩ := (hrel _ _ _).mp hz' |
| exact hunique z' htf htb hf hb |
| have hswap (u : M×M) (hu : u∈Gpair) : u.swap∈Gpair := ⟨hu.2.1,hu.1,hu.2.2.symm⟩ |
| obtain ⟨Φ,hΦ,hΦr,hΦu,hΦu'⟩ := MeasurableSwitch.canonical_pair Gpair hGpair F V |
| (measurable_originalObservation I S a c n hc P hP Wf) |
| (measurable_originalObservation I S.symm a c n hc P hP Wb) TF TB |
| (measurable_originalTube I S a c n hc hsrc δ) |
| (measurable_originalTube I S.symm a c n hc hsrc δ) hex (fun u hu => hex u.swap (hswap u hu)) |
| refine ⟨Φ,hΦ,?_,?_⟩ |
| · intro x y hx hy hxy |
| obtain ⟨z,hzf,hzb,hze,hze',hsf,hsb,hideal,hunique⟩ := hspec x y hx hy hxy |
| obtain ⟨w,hwf,hwb,hwe,hwe',htf,htb,_,_⟩ := hspec y x hy hx hxy.symm |
| have hz : z=(Φ (x.val,y.val)).1 := hΦu _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hze,hze',originalTube_mono I S a c n (by linarith) hzf, |
| originalTube_mono I S.symm a c n (by linarith) hzb⟩) |
| have hw : w=(Φ (x.val,y.val)).2 := hΦu' _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hwe,hwe',originalTube_mono I S a c n (by linarith) hwf, |
| originalTube_mono I S.symm a c n (by linarith) hwb⟩) |
| dsimp only |
| rw [←hz,←hw] |
| exact ⟨hzf,hzb,hwf,hwb,hze,hze',hwe,hwe',hsf,hsb,htf,htb,hideal⟩ |
| · intro x hx z htf htb hef heb |
| obtain ⟨w,_,_,_,_,_,_,_,hunique⟩ := hspec (⟨x,hRG hx⟩ : G) ⟨x,hRG hx⟩ hx hx rfl |
| have hselfF : originalTube I S a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| have hselfB : originalTube I S.symm a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| exact (hunique z htf htb hef heb).trans (hunique x hselfF hselfB rfl rfl).symm |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] |
| {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| open FiniteGreen ShortBin |
| |
| /-- A measurable, witness-independent ORIGINAL pair switch. Its Borel relation |
| uses zero-extended charts, but all asserted equations and tube estimates are in |
| the original atlas. The retained set has not been normalized or repartitioned. -/ |
| theorem canonical_original_switch_unique (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r₀ C₀ : ℝ,0<r₀ ∧ 1≤C₀ ∧ |
| ∀ (C J : Type*) [Fintype C] [MeasurableSpace C] [MeasurableSingletonClass C] |
| [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] |
| (c : M → ι) (P : M → C) (L : M → J) (G R : Set M) |
| (n : ℕ) (B R₀ ε g α r δ : ℝ) (γ : C → ℝ) |
| (A : ∀ x:G,BilateralGauge I S a c n B R₀ x) |
| (T : G → Finset (Fin d)) (ref : G → G) |
| (Wf Wb : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))), |
| Measurable c → Measurable P → Measurable L → MeasurableSet R → R⊆G → |
| (∀ v,v∈K (c v)) → |
| (∀ x y,L x=L y → P x=P y) → |
| (∀ x y,P x=P y → c x=c y) → |
| (∀ x∈R,I.IsInteriorPoint x) → |
| (∀ x:G,P (ref x)=P x) → |
| (∀ x y:G,P x=P y → (T x).card=(T y).card) → |
| (∀ x y:G,P x=P y → ref x=ref y) → |
| (∀ x:G, |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))ᶜ).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)ᶜ).starProjection‖≤ε ∧ |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)).starProjection‖≤ε) → |
| (∀ x:G, |
| Module.finrank ℝ ((A x).forwardSpace I S a c n (T x))= |
| Module.finrank ℝ (Wf (P x,P (S^[n] x))) ∧ |
| Module.finrank ℝ ((A x).backwardSpace I S a c n (T x)ᶜ)= |
| Module.finrank ℝ (Wb (P x,P (S.symm^[n] x))) ∧ |
| ‖((A x).forwardSpace I S a c n (T x)).starProjection- |
| (Wf (P x,P (S^[n] x))).starProjection‖≤1/2 ∧ |
| ‖((A x).backwardSpace I S a c n (T x)ᶜ).starProjection- |
| (Wb (P x,P (S.symm^[n] x))).starProjection‖≤1/2) → |
| 0≤ε → ε≤1/2 → Real.exp (2*B)*(2*ε)≤1/2 → |
| (∀ x:G,∀ j∈(T x)ᶜ,(A x).lam j≤γ (P x)-g) → |
| (∀ x:G,∀ j∈T x,γ (P x)+g≤(A x).lam j) → |
| (2*Real.exp (2*B))*(2*(2*Real.exp (2*B)^2*Real.exp (-2*g*n)))≤1/2 → |
| (∀ x:G,|γ (P x)|+|g|≤α) → 0≤r → 0<δ → δ≤r₀ → |
| (∀ x∈R,∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) δ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) δ⊆interior (range I)) → |
| (∀ x∈R,∀ y∈R,L x=L y → |
| ‖(extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x‖≤C₀*r) → |
| bvpInverseBound B α n*C₀*δ≤1/4 → bvpInverseBound B α n*C₀*r≤δ/4 → |
| ∃ Φ : M×M → M×M,Measurable Φ ∧ |
| (∀ x y:G,x.val∈R → y.val∈R → L x=L y → |
| let z := (Φ (x.val,y.val)).1 |
| let w := (Φ (x.val,y.val)).2 |
| originalTube I S a c (n:=n) (δ/2) x z ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) y z ∧ |
| originalTube I S a c (n:=n) (δ/2) y w ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) x w ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x z ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y z ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y w ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x w ∧ |
| ‖(extChartAt I (a (c x))) z-(extChartAt I (a (c x))) x- |
| (((A (ref x)).forward I S a c).splitting (T (ref x))ᶜ).slow 0 |
| ((extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x)‖≤ |
| 4*Real.exp (2*B)*(2*Real.exp (2*B))*(ε+2*Real.exp (2*B)^2*Real.exp (-2*g*n))*(C₀*r)+ |
| bvpInverseBound B α n*C₀*(2*C₀*bvpInverseBound B α n*r)^2) ∧ |
| (∀ x∈R,∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ x z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z → z=x) ∧ |
| ∀ x∈R,∀ y∈R,L x=L y → ∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ y z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z → |
| z=(Φ (x,y)).1 := by |
| obtain ⟨r₀,C₀,hr₀,hC₀,hjoin⟩ := original_gauged_join I hclosed S a K hK hKU |
| refine ⟨r₀,C₀,hr₀,hC₀,?_⟩ |
| intro C J _ _ _ _ _ _ c P L G R n B R₀ ε g α r δ γ A T ref Wf Wb |
| hc hP hL hR hRG hcover hLP hPc hint href hcard hrefeq hstart hend hε hεhalf |
| hsmall hslow hfast hgap hα hr hδ hδr hsafe hnear hcontract hdata |
| let F := originalObservation I S a c n P Wf |
| let V := originalObservation I S.symm a c n P Wb |
| let TF := originalTube I S a c (n:=n) δ |
| let TB := originalTube I S.symm a c (n:=n) δ |
| let Gpair : Set (M×M) := {u | u.1∈R ∧ u.2∈R ∧ L u.1=L u.2} |
| have hGpair : MeasurableSet Gpair := |
| (hR.preimage measurable_fst).inter ((hR.preimage measurable_snd).inter |
| (measurableSet_eq_fun (hL.comp measurable_fst) (hL.comp measurable_snd))) |
| have hsrc (x : M) : x∈(chartAt H (a (c x))).source := hKU _ (hcover x) |
| have hspec (x y : G) (hx : x.val∈R) (hy : y.val∈R) (hxy : L x=L y) := by |
| have hp := hLP x y hxy |
| have hcst : |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A y).forwardSpace I S a c 0 (T y)).starProjection‖≤ε := by |
| rw [hrefeq x y hp] |
| exact (hstart y).2 |
| exact hjoin c n B R₀ ε (γ (P x)) g α r δ (ref x) x y |
| (A (ref x)) (A x) (A y) (T (ref x)) (T x) (T y) |
| (Wf (P x,P (S^[n] x))) (Wb (P y,P (S.symm^[n] y))) hcover (hPc x y hp) |
| (hint x hx) (hint y hy) (hend x).1 (hend y).2.1 (hend x).2.2.1 (hend y).2.2.2 |
| hε hεhalf (hcard _ _ (href x)) (hcard _ _ ((href x).trans hp)) (hstart x).1 hcst hsmall |
| (hslow x) (hfast x) (by simpa only [hp] using hslow y) (by simpa only [hp] using hfast y) |
| hgap (hα x) hr hδ hδr (fun t => (hsafe x hx t).1) (fun t => (hsafe y hy t).2) |
| (hnear x hx y hy hxy) hcontract hdata |
| have hrel (x y : M) (z : M) : |
| MeasurableSwitch.relation F V TF TB (x,y) z ↔ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| TF x z ∧ TB y z := by |
| constructor |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mp hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mp hb,htf,htb⟩ |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mpr hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mpr hb,htf,htb⟩ |
| have hex (u : M×M) (hu : u∈Gpair) : ∃! z,MeasurableSwitch.relation F V TF TB u z := by |
| let x : G := ⟨u.1,hRG hu.1⟩ |
| let y : G := ⟨u.2,hRG hu.2.1⟩ |
| obtain ⟨z,hzf,hzb,hze,hze',hsharp,hsharp',hideal,hunique⟩ := hspec x y hu.1 hu.2.1 hu.2.2 |
| refine ⟨z,(hrel _ _ _).mpr ⟨hze,hze',?_,?_⟩,?_⟩ |
| · exact originalTube_mono I S a c n (by linarith) hzf |
| · exact originalTube_mono I S.symm a c n (by linarith) hzb |
| · intro z' hz' |
| obtain ⟨hf,hb,htf,htb⟩ := (hrel _ _ _).mp hz' |
| exact hunique z' htf htb hf hb |
| have hswap (u : M×M) (hu : u∈Gpair) : u.swap∈Gpair := ⟨hu.2.1,hu.1,hu.2.2.symm⟩ |
| obtain ⟨Φ,hΦ,hΦr,hΦu,hΦu'⟩ := MeasurableSwitch.canonical_pair Gpair hGpair F V |
| (measurable_originalObservation I S a c n hc P hP Wf) |
| (measurable_originalObservation I S.symm a c n hc P hP Wb) TF TB |
| (measurable_originalTube I S a c n hc hsrc δ) |
| (measurable_originalTube I S.symm a c n hc hsrc δ) hex (fun u hu => hex u.swap (hswap u hu)) |
| refine ⟨Φ,hΦ,?_,?_,?_⟩ |
| · intro x y hx hy hxy |
| obtain ⟨z,hzf,hzb,hze,hze',hsf,hsb,hideal,hunique⟩ := hspec x y hx hy hxy |
| obtain ⟨w,hwf,hwb,hwe,hwe',htf,htb,_,_⟩ := hspec y x hy hx hxy.symm |
| have hz : z=(Φ (x.val,y.val)).1 := hΦu _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hze,hze',originalTube_mono I S a c n (by linarith) hzf, |
| originalTube_mono I S.symm a c n (by linarith) hzb⟩) |
| have hw : w=(Φ (x.val,y.val)).2 := hΦu' _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hwe,hwe',originalTube_mono I S a c n (by linarith) hwf, |
| originalTube_mono I S.symm a c n (by linarith) hwb⟩) |
| dsimp only |
| rw [←hz,←hw] |
| exact ⟨hzf,hzb,hwf,hwb,hze,hze',hwe,hwe',hsf,hsb,htf,htb,hideal⟩ |
| · intro x hx z htf htb hef heb |
| obtain ⟨w,_,_,_,_,_,_,_,hunique⟩ := hspec (⟨x,hRG hx⟩ : G) ⟨x,hRG hx⟩ hx hx rfl |
| have hselfF : originalTube I S a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| have hselfB : originalTube I S.symm a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| exact (hunique z htf htb hef heb).trans (hunique x hselfF hselfB rfl rfl).symm |
| · intro x hx y hy hxy z htf htb hef heb |
| exact hΦu _ ⟨hx,hy,hxy⟩ z ((hrel x y z).mpr ⟨hef,heb,htf,htb⟩) |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric MeasureTheory |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [MeasurableSpace M] [BorelSpace M] [StandardBorelSpace M] [Nonempty M] |
| {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| open FiniteGreen ShortBin |
|
|
| /-- A measurable, witness-independent ORIGINAL pair switch. Its Borel relation |
| uses zero-extended charts, but all asserted equations and tube estimates are in |
| the original atlas. The retained set has not been normalized or repartitioned. -/ |
| theorem canonical_original_switch_complete (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r₀ C₀ : ℝ,0<r₀ ∧ 1≤C₀ ∧ |
| ∀ (C J : Type*) [Fintype C] [MeasurableSpace C] [MeasurableSingletonClass C] |
| [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] |
| (c : M → ι) (P : M → C) (L : M → J) (G R : Set M) |
| (n : ℕ) (B R₀ ε g α r δ : ℝ) (γ : C → ℝ) |
| (A : ∀ x:G,BilateralGauge I S a c n B R₀ x) |
| (T : G → Finset (Fin d)) (ref : G → G) |
| (Wf Wb : C×C → Submodule ℝ (EuclideanSpace ℝ (Fin d))), |
| Measurable c → Measurable P → Measurable L → MeasurableSet R → R⊆G → |
| (∀ v,v∈K (c v)) → |
| (∀ x y,L x=L y → P x=P y) → |
| (∀ x y,P x=P y → c x=c y) → |
| (∀ x∈R,I.IsInteriorPoint x) → |
| (∀ x:G,P (ref x)=P x) → |
| (∀ x y:G,P x=P y → (T x).card=(T y).card) → |
| (∀ x y:G,P x=P y → ref x=ref y) → |
| (∀ x:G, |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))ᶜ).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)ᶜ).starProjection‖≤ε ∧ |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A x).forwardSpace I S a c 0 (T x)).starProjection‖≤ε) → |
| (∀ x:G, |
| Module.finrank ℝ ((A x).forwardSpace I S a c n (T x))= |
| Module.finrank ℝ (Wf (P x,P (S^[n] x))) ∧ |
| Module.finrank ℝ ((A x).backwardSpace I S a c n (T x)ᶜ)= |
| Module.finrank ℝ (Wb (P x,P (S.symm^[n] x))) ∧ |
| ‖((A x).forwardSpace I S a c n (T x)).starProjection- |
| (Wf (P x,P (S^[n] x))).starProjection‖≤1/2 ∧ |
| ‖((A x).backwardSpace I S a c n (T x)ᶜ).starProjection- |
| (Wb (P x,P (S.symm^[n] x))).starProjection‖≤1/2) → |
| 0≤ε → ε≤1/2 → Real.exp (2*B)*(2*ε)≤1/2 → |
| (∀ x:G,∀ j∈(T x)ᶜ,(A x).lam j≤γ (P x)-g) → |
| (∀ x:G,∀ j∈T x,γ (P x)+g≤(A x).lam j) → |
| (2*Real.exp (2*B))*(2*(2*Real.exp (2*B)^2*Real.exp (-2*g*n)))≤1/2 → |
| (∀ x:G,|γ (P x)|+|g|≤α) → 0≤r → 0<δ → δ≤r₀ → |
| (∀ x∈R,∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) δ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) δ⊆interior (range I)) → |
| (∀ x∈R,∀ y∈R,L x=L y → |
| ‖(extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x‖≤C₀*r) → |
| bvpInverseBound B α n*C₀*δ≤1/4 → bvpInverseBound B α n*C₀*r≤δ/4 → |
| ∃ Φ : M×M → M×M,Measurable Φ ∧ |
| (∀ x y:G,x.val∈R → y.val∈R → L x=L y → |
| let z := (Φ (x.val,y.val)).1 |
| let w := (Φ (x.val,y.val)).2 |
| originalTube I S a c (n:=n) (δ/2) x z ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) y z ∧ |
| originalTube I S a c (n:=n) (δ/2) y w ∧ |
| originalTube I S.symm a c (n:=n) (δ/2) x w ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x z ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y z ∧ |
| originalTube I S a c (n:=n) (2*C₀*bvpInverseBound B α n*r) y w ∧ |
| originalTube I S.symm a c (n:=n) (2*C₀*bvpInverseBound B α n*r) x w ∧ |
| ‖(extChartAt I (a (c x))) z-(extChartAt I (a (c x))) x- |
| (((A (ref x)).forward I S a c).splitting (T (ref x))ᶜ).slow 0 |
| ((extChartAt I (a (c y))) y-(extChartAt I (a (c x))) x)‖≤ |
| 4*Real.exp (2*B)*(2*Real.exp (2*B))*(ε+2*Real.exp (2*B)^2*Real.exp (-2*g*n))*(C₀*r)+ |
| bvpInverseBound B α n*C₀*(2*C₀*bvpInverseBound B α n*r)^2 ∧ |
| ‖(extChartAt I (a (c y))) w-(extChartAt I (a (c y))) y- |
| (((A (ref y)).forward I S a c).splitting (T (ref y))ᶜ).slow 0 |
| ((extChartAt I (a (c x))) x-(extChartAt I (a (c y))) y)‖≤ |
| 4*Real.exp (2*B)*(2*Real.exp (2*B))*(ε+2*Real.exp (2*B)^2*Real.exp (-2*g*n))*(C₀*r)+ |
| bvpInverseBound B α n*C₀*(2*C₀*bvpInverseBound B α n*r)^2) ∧ |
| (∀ x∈R,∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ x z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x z → z=x) ∧ |
| ∀ x∈R,∀ y∈R,L x=L y → ∀ z, |
| originalTube I S a c (n:=n) δ x z → originalTube I S.symm a c (n:=n) δ y z → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z → |
| z=(Φ (x,y)).1 := by |
| obtain ⟨r₀,C₀,hr₀,hC₀,hjoin⟩ := original_gauged_join I hclosed S a K hK hKU |
| refine ⟨r₀,C₀,hr₀,hC₀,?_⟩ |
| intro C J _ _ _ _ _ _ c P L G R n B R₀ ε g α r δ γ A T ref Wf Wb |
| hc hP hL hR hRG hcover hLP hPc hint href hcard hrefeq hstart hend hε hεhalf |
| hsmall hslow hfast hgap hα hr hδ hδr hsafe hnear hcontract hdata |
| let F := originalObservation I S a c n P Wf |
| let V := originalObservation I S.symm a c n P Wb |
| let TF := originalTube I S a c (n:=n) δ |
| let TB := originalTube I S.symm a c (n:=n) δ |
| let Gpair : Set (M×M) := {u | u.1∈R ∧ u.2∈R ∧ L u.1=L u.2} |
| have hGpair : MeasurableSet Gpair := |
| (hR.preimage measurable_fst).inter ((hR.preimage measurable_snd).inter |
| (measurableSet_eq_fun (hL.comp measurable_fst) (hL.comp measurable_snd))) |
| have hsrc (x : M) : x∈(chartAt H (a (c x))).source := hKU _ (hcover x) |
| have hspec (x y : G) (hx : x.val∈R) (hy : y.val∈R) (hxy : L x=L y) := by |
| have hp := hLP x y hxy |
| have hcst : |
| ‖((A (ref x)).forwardSpace I S a c 0 (T (ref x))).starProjection- |
| ((A y).forwardSpace I S a c 0 (T y)).starProjection‖≤ε := by |
| rw [hrefeq x y hp] |
| exact (hstart y).2 |
| exact hjoin c n B R₀ ε (γ (P x)) g α r δ (ref x) x y |
| (A (ref x)) (A x) (A y) (T (ref x)) (T x) (T y) |
| (Wf (P x,P (S^[n] x))) (Wb (P y,P (S.symm^[n] y))) hcover (hPc x y hp) |
| (hint x hx) (hint y hy) (hend x).1 (hend y).2.1 (hend x).2.2.1 (hend y).2.2.2 |
| hε hεhalf (hcard _ _ (href x)) (hcard _ _ ((href x).trans hp)) (hstart x).1 hcst hsmall |
| (hslow x) (hfast x) (by simpa only [hp] using hslow y) (by simpa only [hp] using hfast y) |
| hgap (hα x) hr hδ hδr (fun t => (hsafe x hx t).1) (fun t => (hsafe y hy t).2) |
| (hnear x hx y hy hxy) hcontract hdata |
| have hrel (x y : M) (z : M) : |
| MeasurableSwitch.relation F V TF TB (x,y) z ↔ |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z ∧ |
| TF x z ∧ TB y z := by |
| constructor |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mp hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mp hb,htf,htb⟩ |
| · rintro ⟨hf,hb,htf,htb⟩ |
| exact ⟨(originalObservation_eq_iff I S a c n P Wf x z (hsrc _) (htf ⟨n,by omega⟩).1).mpr hf, |
| (originalObservation_eq_iff I S.symm a c n P Wb y z (hsrc _) (htb ⟨n,by omega⟩).1).mpr hb,htf,htb⟩ |
| have hex (u : M×M) (hu : u∈Gpair) : ∃! z,MeasurableSwitch.relation F V TF TB u z := by |
| let x : G := ⟨u.1,hRG hu.1⟩ |
| let y : G := ⟨u.2,hRG hu.2.1⟩ |
| obtain ⟨z,hzf,hzb,hze,hze',hsharp,hsharp',hideal,hunique⟩ := hspec x y hu.1 hu.2.1 hu.2.2 |
| refine ⟨z,(hrel _ _ _).mpr ⟨hze,hze',?_,?_⟩,?_⟩ |
| · exact originalTube_mono I S a c n (by linarith) hzf |
| · exact originalTube_mono I S.symm a c n (by linarith) hzb |
| · intro z' hz' |
| obtain ⟨hf,hb,htf,htb⟩ := (hrel _ _ _).mp hz' |
| exact hunique z' htf htb hf hb |
| have hswap (u : M×M) (hu : u∈Gpair) : u.swap∈Gpair := ⟨hu.2.1,hu.1,hu.2.2.symm⟩ |
| obtain ⟨Φ,hΦ,hΦr,hΦu,hΦu'⟩ := MeasurableSwitch.canonical_pair Gpair hGpair F V |
| (measurable_originalObservation I S a c n hc P hP Wf) |
| (measurable_originalObservation I S.symm a c n hc P hP Wb) TF TB |
| (measurable_originalTube I S a c n hc hsrc δ) |
| (measurable_originalTube I S.symm a c n hc hsrc δ) hex (fun u hu => hex u.swap (hswap u hu)) |
| refine ⟨Φ,hΦ,?_,?_,?_⟩ |
| · intro x y hx hy hxy |
| obtain ⟨z,hzf,hzb,hze,hze',hsf,hsb,hideal,hunique⟩ := hspec x y hx hy hxy |
| obtain ⟨w,hwf,hwb,hwe,hwe',htf,htb,hideal',_⟩ := hspec y x hy hx hxy.symm |
| have hz : z=(Φ (x.val,y.val)).1 := hΦu _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hze,hze',originalTube_mono I S a c n (by linarith) hzf, |
| originalTube_mono I S.symm a c n (by linarith) hzb⟩) |
| have hw : w=(Φ (x.val,y.val)).2 := hΦu' _ ⟨hx,hy,hxy⟩ _ ((hrel _ _ _).mpr |
| ⟨hwe,hwe',originalTube_mono I S a c n (by linarith) hwf, |
| originalTube_mono I S.symm a c n (by linarith) hwb⟩) |
| dsimp only |
| rw [←hz,←hw] |
| exact ⟨hzf,hzb,hwf,hwb,hze,hze',hwe,hwe',hsf,hsb,htf,htb,hideal,hideal'⟩ |
| · intro x hx z htf htb hef heb |
| obtain ⟨w,_,_,_,_,_,_,_,hunique⟩ := hspec (⟨x,hRG hx⟩ : G) ⟨x,hRG hx⟩ hx hx rfl |
| have hselfF : originalTube I S a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| have hselfB : originalTube I S.symm a c (n:=n) δ x x := by |
| intro t |
| exact ⟨hsrc _,by simpa only [sub_self,norm_zero] using hδ.le⟩ |
| exact (hunique z htf htb hef heb).trans (hunique x hselfF hselfB rfl rfl).symm |
| · intro x hx y hy hxy z htf htb hef heb |
| exact hΦu _ ⟨hx,hy,hxy⟩ z ((hrel x y z).mpr ⟨hef,heb,htf,htb⟩) |
| end SmoothObstruction.ManifoldCoordinates |
| |
| end |
| |
| end OAI |
| |
