| import OAI.Dynamics.SmoothObstruction.NonlinearMatching |
|
|
| 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.ManifoldCoordinates |
| open Set Metric Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
|
|
| omit [IsManifold I ∞ M] in |
| /-- An exact finite coordinate solution lying in the original chart targets |
| is a genuine manifold orbit, also in a half-space model. -/ |
| theorem realize_array (S : Diffeomorph I I M M ∞) {n : ℕ} |
| (a : Fin (n+1) → M) (u : Fin (n+1) → E (d:=d)) |
| (hu : ∀ t,u t∈(extChartAt I (a t)).target) |
| (hdom : ∀ t : Fin n,u t.castSucc∈domainU I S (a t.castSucc) (a t.succ)) |
| (hstep : ∀ t : Fin n,localMap I S (a t.castSucc) (a t.succ) (u t.castSucc)=u t.succ) : |
| ∃ z : M,∀ t : Fin (n+1),S^[t.val] z∈(chartAt H (a t)).source ∧ |
| (extChartAt I (a t)) (S^[t.val] z)=u t := by |
| let z := (extChartAt I (a 0)).symm (u 0) |
| refine ⟨z,?_⟩ |
| have h (t : ℕ) (ht : t<n+1) : S^[t] z=(extChartAt I (a ⟨t,ht⟩)).symm (u ⟨t,ht⟩) := by |
| induction t with |
| | zero => rfl |
| | succ t ih => |
| have ht' : t<n := by omega |
| have hh := hstep ⟨t,ht'⟩ |
| have htS : S ((extChartAt I (a ⟨t,by omega⟩)).symm (u ⟨t,by omega⟩)) |
| ∈(extChartAt I (a ⟨t+1,ht⟩)).source := by |
| have hm := (hdom ⟨t,ht'⟩).2 |
| change S ((extChartAt I (a ⟨t,by omega⟩)).symm (u ⟨t,by omega⟩)) |
| ∈(chartAt H (a ⟨t+1,ht⟩)).source at hm |
| simpa only [extChartAt_source] using hm |
| rw [Function.iterate_succ_apply',ih (by omega)] |
| exact (extChartAt I (a ⟨t+1,ht⟩)).left_inv htS |>.symm.trans |
| (congrArg (extChartAt I (a ⟨t+1,ht⟩)).symm hh) |
| intro t |
| rw [h t t.isLt] |
| exact ⟨by simpa only [extChartAt_source] using (extChartAt I (a t)).map_target (hu t), |
| (extChartAt I (a t)).right_inv (hu t)⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] |
|
|
| /-- Matching the time-zero coordinate realizes the TWO-sided exact arrays |
| as one actual orbit. In particular no pair of unrelated local branches is |
| substituted for the smooth switch. -/ |
| theorem realize_bilateral_array (S : Diffeomorph I I M M ∞) {n : ℕ} |
| (a b : Fin (n+1) → M) (u v : Fin (n+1) → E (d:=d)) |
| (ha : a 0=b 0) (huv : u 0=v 0) |
| (hu : ∀ t,u t∈(extChartAt I (a t)).target) |
| (hv : ∀ t,v t∈(extChartAt I (b t)).target) |
| (hud : ∀ t : Fin n,u t.castSucc∈domainU I S (a t.castSucc) (a t.succ)) |
| (hvd : ∀ t : Fin n,v t.castSucc∈domainU I S.symm (b t.castSucc) (b t.succ)) |
| (hus : ∀ t : Fin n,localMap I S (a t.castSucc) (a t.succ) (u t.castSucc)=u t.succ) |
| (hvs : ∀ t : Fin n,localMap I S.symm (b t.castSucc) (b t.succ) (v t.castSucc)=v t.succ) : |
| ∃ z : M,(∀ t : Fin (n+1),S^[t.val] z∈(chartAt H (a t)).source ∧ |
| (extChartAt I (a t)) (S^[t.val] z)=u t) ∧ |
| (∀ t : Fin (n+1),S.symm^[t.val] z∈(chartAt H (b t)).source ∧ |
| (extChartAt I (b t)) (S.symm^[t.val] z)=v t) := by |
| obtain ⟨z,hz⟩ := realize_array I S a u hu hud hus |
| obtain ⟨w,hw⟩ := realize_array I S.symm b v hv hvd hvs |
| have hz0 : z=(extChartAt I (a 0)).symm (u 0) := by |
| have hz0 := hz 0 |
| simp only [Fin.val_zero,Function.iterate_zero,id_eq] at hz0 |
| rw [←hz0.2,(extChartAt I (a 0)).left_inv (by simpa only [extChartAt_source] using hz0.1)] |
| have hw0 : w=(extChartAt I (b 0)).symm (v 0) := by |
| have hw0 := hw 0 |
| simp only [Fin.val_zero,Function.iterate_zero,id_eq] at hw0 |
| rw [←hw0.2,(extChartAt I (b 0)).left_inv (by simpa only [extChartAt_source] using hw0.1)] |
| have heq : z=w := by rw [hz0,hw0,ha,huv] |
| exact ⟨z,hz,by simpa only [heq] using hw⟩ |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] |
| {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| open FiniteGreen FiniteGreen.Matching |
|
|
| /-- The authentic local-coordinate nonlinear boundary construction gives |
| ONE actual point of the manifold, with both orbit halves and the fixed |
| terminal equations. The ball-domain conditions keep every inverse chart |
| and every one-step expression inside the ORIGINAL manifold. -/ |
| theorem actual_join_unique (S : Diffeomorph I I M M ∞) {n : ℕ} |
| (a b : Fin (n+1) → M) (x y : M) (hab : a 0=b 0) |
| (hx : ∀ t : Fin (n+1),S^[t.val] x∈(chartAt H (a t)).source) |
| (hy : ∀ t : Fin (n+1),S.symm^[t.val] y∈(chartAt H (b t)).source) |
| (A₁ A₂ : Evolution (E (d:=d))) (B₁ : Splitting A₁) (B₂ : Splitting A₂) |
| (C₁ : B₁.Terminal (H:=F) n) (C₂ : B₂.Terminal (H:=G) n) |
| (L : (B₁.initialSpace×B₂.initialSpace) ≃L[ℝ] E (d:=d)) |
| (hL : ∀ z,L z=(z.1:E (d:=d))-(z.2:E (d:=d))) |
| {U K C r δ : ℝ} (hU : 1≤U) (hK : ‖L.symm.toContinuousLinearMap‖≤K) |
| (hsmall : ‖L.symm.toContinuousLinearMap.comp (correction C₁ C₂)‖≤1/2) |
| (hs₁ : ∀ s u,s≤u → u≤n → ‖(A₁.map u s).comp (B₁.slow s)‖≤U) |
| (hb₁ : ∀ s u,s≤u → u≤n → ‖(A₁.map s u).comp (B₁.fast u)‖≤U) |
| (hs₂ : ∀ s u,s≤u → u≤n → ‖(A₂.map u s).comp (B₂.slow s)‖≤U) |
| (hb₂ : ∀ s u,s≤u → u≤n → ‖(A₂.map s u).comp (B₂.fast u)‖≤U) |
| (hJ₁ : ‖C₁.J‖≤U) (hP₁ : ‖C₁.P‖≤U) |
| (hJ₂ : ‖C₂.J‖≤U) (hP₂ : ‖C₂.P‖≤U) |
| (hC : 0≤C) (hr : 0≤r) (hδ : 0<δ) |
| (hA₁ : ∀ t : Fin n,fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (S^[t.val] x))=A₁.map (t+1) t) |
| (hA₂ : ∀ t : Fin n,fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y))=A₂.map (t+1) t) |
| (hd₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ, |
| DifferentiableAt ℝ (localMap I S (a t.castSucc) (a t.succ)) z) |
| (hd₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ, |
| DifferentiableAt ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) z) |
| (hder₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ, |
| ‖fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) z- |
| fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (S^[t.val] x))‖≤ |
| C*‖z-(extChartAt I (a t.castSucc)) (S^[t.val] x)‖) |
| (hder₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ, |
| ‖fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) z- |
| fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y))‖≤ |
| C*‖z-(extChartAt I (b t.castSucc)) (S.symm^[t.val] y)‖) |
| (ht₁ : ∀ t : Fin (n+1),closedBall ((extChartAt I (a t)) (S^[t.val] x)) δ⊆ |
| (extChartAt I (a t)).target) |
| (ht₂ : ∀ t : Fin (n+1),closedBall ((extChartAt I (b t)) (S.symm^[t.val] y)) δ⊆ |
| (extChartAt I (b t)).target) |
| (hdom₁ : ∀ t : Fin n,closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ⊆ |
| domainU I S (a t.castSucc) (a t.succ)) |
| (hdom₂ : ∀ t : Fin n,closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ⊆ |
| domainU I S.symm (b t.castSucc) (b t.succ)) |
| (hnear : ‖(extChartAt I (b 0)) y-(extChartAt I (a 0)) x‖≤C*r) |
| (hcontract : let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*K)*(1+2*D)+D)*C*δ≤1/4) |
| (hdata : let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*K)*(1+2*D)+D)*C*r≤δ/4) : |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| let Q := (U+U^4)*(2*K)*(1+2*D)+D |
| ∃ z : M, |
| (∀ t : Fin (n+1),S^[t.val] z∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (S^[t.val] z)-(extChartAt I (a t)) (S^[t.val] x)‖≤δ/2) ∧ |
| (∀ t : Fin (n+1),S.symm^[t.val] z∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (S.symm^[t.val] z)-(extChartAt I (b t)) (S.symm^[t.val] y)‖≤δ/2) ∧ |
| C₁.P ((extChartAt I (a (Fin.last n))) (S^[n] z)- |
| (extChartAt I (a (Fin.last n))) (S^[n] x))=0 ∧ |
| C₂.P ((extChartAt I (b (Fin.last n))) (S.symm^[n] z)- |
| (extChartAt I (b (Fin.last n))) (S.symm^[n] y))=0 ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (a t)) (S^[t.val] z)- |
| (extChartAt I (a t)) (S^[t.val] x)‖≤2*C*Q*r) ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (b t)) (S.symm^[t.val] z)- |
| (extChartAt I (b t)) (S.symm^[t.val] y)‖≤2*C*Q*r) ∧ |
| ∀ z' : M, |
| (∀ t : Fin (n+1),S^[t.val] z'∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (S^[t.val] z')-(extChartAt I (a t)) (S^[t.val] x)‖≤δ) → |
| (∀ t : Fin (n+1),S.symm^[t.val] z'∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (S.symm^[t.val] z')-(extChartAt I (b t)) (S.symm^[t.val] y)‖≤δ) → |
| C₁.P ((extChartAt I (a (Fin.last n))) (S^[n] z')- |
| (extChartAt I (a (Fin.last n))) (S^[n] x))=0 → |
| C₂.P ((extChartAt I (b (Fin.last n))) (S.symm^[n] z')- |
| (extChartAt I (b (Fin.last n))) (S.symm^[n] y))=0 → z'=z := by |
| let u : Fin (n+1) → E (d:=d) := fun t => (extChartAt I (a t)) (S^[t.val] x) |
| let v : Fin (n+1) → E (d:=d) := fun t => (extChartAt I (b t)) (S.symm^[t.val] y) |
| let f : Fin n → E (d:=d) → E (d:=d) := fun t => localMap I S (a t.castSucc) (a t.succ) |
| let g : Fin n → E (d:=d) → E (d:=d) := fun t => localMap I S.symm (b t.castSucc) (b t.succ) |
| have hu (t : Fin n) : f t (u t.castSucc)=u t.succ := by |
| simp only [f,u,localMap,Function.comp_apply] |
| rw [(extChartAt I (a t.castSucc)).left_inv (by simpa only [extChartAt_source] using hx t.castSucc)] |
| exact congrArg (extChartAt I (a t.succ)) (Function.iterate_succ_apply' S _ x).symm |
| have hv (t : Fin n) : g t (v t.castSucc)=v t.succ := by |
| simp only [g,v,localMap,Function.comp_apply] |
| rw [(extChartAt I (b t.castSucc)).left_inv (by simpa only [extChartAt_source] using hy t.castSucc)] |
| exact congrArg (extChartAt I (b t.succ)) (Function.iterate_succ_apply' S.symm _ y).symm |
| obtain ⟨w,hw,hwsharp,hf,hg,hmid,hp,hq,hunique⟩ := nonlinear_orbits_unique C₁ C₂ L hL hU hK hsmall |
| hs₁ hb₁ hs₂ hb₂ hJ₁ hP₁ hJ₂ hP₂ f g u v hu hv hA₁ hA₂ hC hr hδ hd₁ hd₂ hder₁ hder₂ |
| (by simpa only [u,v,Fin.val_zero,Function.iterate_zero,id_eq] using hnear) hcontract hdata |
| have hw₁ (t : Fin (n+1)) : ‖w.1 t‖≤δ/2 := |
| (norm_le_pi_norm w.1 t).trans ((norm_fst_le w).trans hw) |
| have hw₂ (t : Fin (n+1)) : ‖w.2 t‖≤δ/2 := |
| (norm_le_pi_norm w.2 t).trans ((norm_snd_le w).trans hw) |
| have hwδ : δ/2≤δ := by linarith |
| have hb₁' (t : Fin (n+1)) : u t+w.1 t∈closedBall (u t) δ := by |
| rw [mem_closedBall,dist_eq_norm,add_sub_cancel_left] |
| exact (hw₁ t).trans hwδ |
| have hb₂' (t : Fin (n+1)) : v t+w.2 t∈closedBall (v t) δ := by |
| rw [mem_closedBall,dist_eq_norm,add_sub_cancel_left] |
| exact (hw₂ t).trans hwδ |
| obtain ⟨z,hz,hz'⟩ := realize_bilateral_array I S a b (fun t => u t+w.1 t) (fun t => v t+w.2 t) |
| hab hmid (fun t => ht₁ t (hb₁' t)) (fun t => ht₂ t (hb₂' t)) |
| (fun t => hdom₁ t (hb₁' t.castSucc)) (fun t => hdom₂ t (hb₂' t.castSucc)) hf hg |
| refine ⟨z,?_,?_,?_,?_,?_,?_,?_⟩ |
| · intro t |
| refine ⟨(hz t).1,?_⟩ |
| rw [(hz t).2] |
| simpa only [u,add_sub_cancel_left] using hw₁ t |
| · intro t |
| refine ⟨(hz' t).1,?_⟩ |
| rw [(hz' t).2] |
| simpa only [v,add_sub_cancel_left] using hw₂ t |
| · have hzL := (hz (Fin.last n)).2 |
| simp only [Fin.val_last] at hzL |
| rw [hzL] |
| simpa only [u,Fin.val_last,add_sub_cancel_left] using hp |
| · have hzL := (hz' (Fin.last n)).2 |
| simp only [Fin.val_last] at hzL |
| rw [hzL] |
| simpa only [v,Fin.val_last,add_sub_cancel_left] using hq |
| · intro t |
| rw [(hz t).2] |
| simpa only [u,add_sub_cancel_left] using |
| (norm_le_pi_norm w.1 t).trans ((norm_fst_le w).trans hwsharp) |
| · intro t |
| rw [(hz' t).2] |
| simpa only [v,add_sub_cancel_left] using |
| (norm_le_pi_norm w.2 t).trans ((norm_snd_le w).trans hwsharp) |
| · intro z' hx' hy' hp' hq' |
| let w' : Paths (E (d:=d)) n n := |
| (fun t => (extChartAt I (a t)) (S^[t.val] z')-u t, |
| fun t => (extChartAt I (b t)) (S.symm^[t.val] z')-v t) |
| have hn : ‖w'‖≤δ := by |
| rw [Prod.norm_def,max_le_iff] |
| constructor |
| · exact (pi_norm_le_iff_of_nonneg hδ.le).mpr (fun t => (hx' t).2) |
| · exact (pi_norm_le_iff_of_nonneg hδ.le).mpr (fun t => (hy' t).2) |
| have huf (t : Fin (n+1)) : u t+w'.1 t=(extChartAt I (a t)) (S^[t.val] z') := by |
| dsimp only [w']; abel |
| have hvf (t : Fin (n+1)) : v t+w'.2 t=(extChartAt I (b t)) (S.symm^[t.val] z') := by |
| dsimp only [w']; abel |
| have he : w'=w := hunique w' hn (fun t => by |
| rw [huf,huf] |
| simp only [f,localMap,Function.comp_apply] |
| rw [(extChartAt I (a t.castSucc)).left_inv |
| (by simpa only [extChartAt_source] using (hx' t.castSucc).1)] |
| exact congrArg (extChartAt I (a t.succ)) (Function.iterate_succ_apply' S _ z').symm) |
| (fun t => by |
| rw [hvf,hvf] |
| simp only [g,localMap,Function.comp_apply] |
| rw [(extChartAt I (b t.castSucc)).left_inv |
| (by simpa only [extChartAt_source] using (hy' t.castSucc).1)] |
| exact congrArg (extChartAt I (b t.succ)) (Function.iterate_succ_apply' S.symm _ z').symm) |
| (by rw [huf,hvf,hab]; rfl) |
| (by simpa only [w',u,Fin.val_last] using hp') |
| (by simpa only [w',v,Fin.val_last] using hq') |
| apply (extChartAt I (a 0)).injOn |
| (by simpa only [extChartAt_source,Fin.val_zero,Function.iterate_zero,id_eq] using (hx' 0).1) |
| (by simpa only [extChartAt_source,Fin.val_zero,Function.iterate_zero,id_eq] using (hz 0).1) |
| have hfirst := congrArg (fun q : Paths (E (d:=d)) n n => q.1 0) he |
| have hc := (hz 0).2 |
| simp only [Fin.val_zero,Function.iterate_zero,id_eq] at hc |
| rw [hc,←hfirst, huf] |
| rfl |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ShortBin |
| open EuclideanSplitting FiniteGreen FiniteGreen.Matching |
| open scoped Matrix.Norms.L2Operator |
| variable {d n : ℕ} {K : ℝ} |
|
|
| lemma Gauge.evolution_step (A : Gauge d n K) (D : EuclideanSpace ℝ (Fin d) →L[ℝ] EuclideanSpace ℝ (Fin d)) |
| (t : ℕ) (h : (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm D*A.U t= |
| A.U (t+1)*rate A.lam 1) : A.evolution.map (t+1) t=D := by |
| change Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ) (transition A.U A.V A.lam t (t+1))=D |
| rw [transition,show ((t+1:ℕ):ℝ)-(t:ℝ)=1 by push_cast; ring,←h,mul_assoc,A.UV t,mul_one] |
| exact (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).apply_symm_apply D |
|
|
| lemma Gauge.unweighted (A : Gauge d n K) (S : Finset (Fin d)) {a : ℝ} |
| (ha : 0≤a) (hs : ∀ j∈S,A.lam j≤a) (hf : ∀ j∈Sᶜ,-a≤A.lam j) : |
| (∀ s t,s≤t → t≤n → ‖(A.evolution.map t s).comp ((A.splitting S).slow s)‖≤Real.exp (2*K+a*n)) ∧ |
| (∀ s t,s≤t → t≤n → ‖(A.evolution.map s t).comp ((A.splitting S).fast t)‖≤Real.exp (2*K+a*n)) := |
| unweighted_bounds A.U A.V A.lam A.UV A.VU S ha A.normU A.normV hs hf |
|
|
| /-- All the finite-interval linear boundary hypotheses are produced from |
| raw common-bin gauge comparisons and fixed endpoint projections. The norm |
| of the matching inverse is a middle-time cost, and the endpoint correction |
| retains threshold cancellation, including a positive slow threshold. -/ |
| theorem gauge_boundary_control (A₀ A Z : Gauge d n K) (S₀ S T : Finset (Fin d)) |
| (V W : Submodule ℝ (EuclideanSpace ℝ (Fin d))) |
| (hdV : Module.finrank ℝ ((A.splitting S).fast n).range=Module.finrank ℝ V) |
| (hdW : Module.finrank ℝ ((Z.splitting T).fast n).range=Module.finrank ℝ W) |
| (hcV : ‖((A.splitting S).fast n).range.starProjection-V.starProjection‖≤1/2) |
| (hcW : ‖((Z.splitting T).fast n).range.starProjection-W.starProjection‖≤1/2) |
| {ε γ g : ℝ} (hε : 0≤ε) (hεhalf : ε≤1/2) |
| (hdS : S₀.card=S.card) (hdT : S₀ᶜ.card=T.card) |
| (hcS : ‖(A₀.space S₀).starProjection-(A.space S).starProjection‖≤ε) |
| (hcT : ‖(A₀.space S₀ᶜ).starProjection-(Z.space T).starProjection‖≤ε) |
| (hsmall : Real.exp (2*K)*(2*ε)≤1/2) |
| (hsA : ∀ j∈S,A.lam j≤γ-g) (hfA : ∀ j∈Sᶜ,γ+g≤A.lam j) |
| (hsZ : ∀ j∈T,Z.lam j≤ -γ-g) (hfZ : ∀ j∈Tᶜ,-γ+g≤Z.lam j) |
| (hgap : (2*Real.exp (2*K))*(2*(2*Real.exp (2*K)^2*Real.exp (-2*g*n)))≤1/2) : |
| let C₁ := (A.splitting S).terminalProjection n V hdV hcV |
| let C₂ := (Z.splitting T).terminalProjection n W hdW hcW |
| ∃ L : ((A.splitting S).initialSpace×(Z.splitting T).initialSpace) ≃L[ℝ] EuclideanSpace ℝ (Fin d), |
| (∀ v,L v=(v.1:EuclideanSpace ℝ (Fin d))-(v.2:EuclideanSpace ℝ (Fin d))) ∧ |
| ‖L.symm.toContinuousLinearMap‖≤2*Real.exp (2*K) ∧ |
| ‖L.symm.toContinuousLinearMap.comp (correction C₁ C₂)‖≤1/2 ∧ |
| ‖C₁.kernel‖≤2*Real.exp (2*K)^2*Real.exp (-2*g*n) ∧ |
| ‖C₂.kernel‖≤2*Real.exp (2*K)^2*Real.exp (-2*g*n) := by |
| intro C₁ C₂ |
| obtain ⟨L,hL,hK⟩ := gauge_matching A₀ A Z S₀ S T hε hεhalf hdS hdT hcS hcT hsmall |
| have hA := endpoint_gap_bounds A.U A.V A.lam A.UV A.VU S n A.normU A.normV hsA hfA |
| have hZ := endpoint_gap_bounds Z.U Z.V Z.lam Z.UV Z.VU T n Z.normU Z.normV hsZ hfZ |
| have heA : ‖C₁.kernel‖≤2*Real.exp (2*K)^2*Real.exp (-2*g*n) := |
| C₁.kernel_norm_gap (Real.exp_nonneg _) hA.2 hA.1 |
| ((A.splitting S).terminalProjection_J_norm n V hdV hcV) |
| ((A.splitting S).terminalProjection_P_norm n V hdV hcV) |
| have heZ : ‖C₂.kernel‖≤2*Real.exp (2*K)^2*Real.exp (-2*g*n) := |
| C₂.kernel_norm_gap (Real.exp_nonneg _) hZ.2 hZ.1 |
| ((Z.splitting T).terminalProjection_J_norm n W hdW hcW) |
| ((Z.splitting T).terminalProjection_P_norm n W hdW hcW) |
| refine ⟨L,hL,hK,?_,heA,heZ⟩ |
| apply (ContinuousLinearMap.opNorm_comp_le _ _).trans |
| apply le_trans (mul_le_mul hK (correction_norm C₁ C₂ heA heZ) (norm_nonneg (correction C₁ C₂)) (by positivity)) |
| simpa only [two_mul] using hgap |
| end SmoothObstruction.ShortBin |
|
|
| namespace SmoothObstruction.ShortBin |
| noncomputable def bvpGrowth (K a : ℝ) (n : ℕ) : ℝ := Real.exp (2*K+a*n)+2 |
| noncomputable def bvpInverseBound (K a : ℝ) (n : ℕ) : ℝ := |
| let U := bvpGrowth K a n |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| (U+U^4)*(4*Real.exp (2*K))*(1+2*D)+D |
| lemma bvpGrowth_ge_two (K a : ℝ) (n : ℕ) : 2≤bvpGrowth K a n := by |
| dsimp [bvpGrowth]; linarith [Real.exp_pos (2*K+a*n)] |
| end SmoothObstruction.ShortBin |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| open FiniteGreen FiniteGreen.Matching ShortBin |
|
|
| omit [IsManifold I ∞ M] in |
| lemma interior_iterate (S : Diffeomorph I I M M ∞) {x : M} (hx : I.IsInteriorPoint x) (t : ℕ) : |
| I.IsInteriorPoint (S^[t] x) := by |
| induction t with |
| | zero => exact hx |
| | succ t ih => |
| rw [Function.iterate_succ_apply'] |
| exact ((S.isLocalDiffeomorph _).isInteriorPoint_iff (by simp)).mp ih |
|
|
| /-- The long gauge equations identify every step of the evolution with |
| an ACTUAL coordinate derivative. -/ |
| lemma gauge_actual_step (S : Diffeomorph I I M M ∞) (a b x : M) |
| {n : ℕ} {K : ℝ} (A : ShortBin.Gauge d n K) (t : ℕ) |
| (hx : x∈(chartAt H a).source) (hSx : S x∈(chartAt H b).source) |
| (hint : I.IsInteriorPoint x) |
| (hEq : (Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm (coord I S a b x)*A.U t= |
| A.U (t+1)*EuclideanSplitting.rate A.lam 1) : |
| fderiv ℝ (localMap I S a b) ((extChartAt I a) x)=A.evolution.map (t+1) t := by |
| rw [A.evolution_step (coord I S a b x) t hEq] |
| exact (hasFDerivAt_localMap I S a b x ⟨hx,hSx⟩ ((interior_coord I a x hx).mpr hint)).fderiv |
|
|
| /-- Actual nonlinear joining, now with ALL linear matching/inverse/kernel |
| hypotheses proved from the genuine gauges and the common-bin comparisons. |
| No boundary inverse, complementary middle space or endpoint graph is assumed. -/ |
| theorem gauged_join_unique (Smap : Diffeomorph I I M M ∞) {n : ℕ} |
| (a b : Fin (n+1) → M) (x y : M) (hab : a 0=b 0) |
| (hx : ∀ t : Fin (n+1),Smap^[t.val] x∈(chartAt H (a t)).source) |
| (hy : ∀ t : Fin (n+1),Smap.symm^[t.val] y∈(chartAt H (b t)).source) |
| (hintx : I.IsInteriorPoint x) (hinty : I.IsInteriorPoint y) |
| {K ε γ g α : ℝ} (A₀ A Z : Gauge d n K) (S₀ S T : Finset (Fin d)) |
| (V W : Submodule ℝ (E (d:=d))) |
| (hdV : Module.finrank ℝ ((A.splitting S).fast n).range=Module.finrank ℝ V) |
| (hdW : Module.finrank ℝ ((Z.splitting T).fast n).range=Module.finrank ℝ W) |
| (hcV : ‖((A.splitting S).fast n).range.starProjection-V.starProjection‖≤1/2) |
| (hcW : ‖((Z.splitting T).fast n).range.starProjection-W.starProjection‖≤1/2) |
| (hε : 0≤ε) (hεhalf : ε≤1/2) |
| (hdS : S₀.card=S.card) (hdT : S₀ᶜ.card=T.card) |
| (hcS : ‖(A₀.space S₀).starProjection-(A.space S).starProjection‖≤ε) |
| (hcT : ‖(A₀.space S₀ᶜ).starProjection-(Z.space T).starProjection‖≤ε) |
| (hsmall : Real.exp (2*K)*(2*ε)≤1/2) |
| (hsA : ∀ j∈S,A.lam j≤γ-g) (hfA : ∀ j∈Sᶜ,γ+g≤A.lam j) |
| (hsZ : ∀ j∈T,Z.lam j≤ -γ-g) (hfZ : ∀ j∈Tᶜ,-γ+g≤Z.lam j) |
| (hgap : (2*Real.exp (2*K))*(2*(2*Real.exp (2*K)^2*Real.exp (-2*g*n)))≤1/2) |
| (hα : |γ|+|g|≤α) |
| (hstep₁ : ∀ t : Fin n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I Smap (a t.castSucc) (a t.succ) (Smap^[t.val] x))*A.U t= |
| A.U (t+1)*EuclideanSplitting.rate A.lam 1) |
| (hstep₂ : ∀ t : Fin n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I Smap.symm (b t.castSucc) (b t.succ) (Smap.symm^[t.val] y))*Z.U t= |
| Z.U (t+1)*EuclideanSplitting.rate Z.lam 1) |
| {C r δ : ℝ} |
| (hC : 0≤C) (hr : 0≤r) (hδ : 0<δ) |
| (hd₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ, |
| DifferentiableAt ℝ (localMap I Smap (a t.castSucc) (a t.succ)) z) |
| (hd₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ, |
| DifferentiableAt ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) z) |
| (hder₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ, |
| ‖fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) z- |
| fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (Smap^[t.val] x))‖≤ |
| C*‖z-(extChartAt I (a t.castSucc)) (Smap^[t.val] x)‖) |
| (hder₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ, |
| ‖fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) z- |
| fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y))‖≤ |
| C*‖z-(extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)‖) |
| (ht₁ : ∀ t : Fin (n+1),closedBall ((extChartAt I (a t)) (Smap^[t.val] x)) δ⊆ |
| (extChartAt I (a t)).target) |
| (ht₂ : ∀ t : Fin (n+1),closedBall ((extChartAt I (b t)) (Smap.symm^[t.val] y)) δ⊆ |
| (extChartAt I (b t)).target) |
| (hdom₁ : ∀ t : Fin n,closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ⊆ |
| domainU I Smap (a t.castSucc) (a t.succ)) |
| (hdom₂ : ∀ t : Fin n,closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ⊆ |
| domainU I Smap.symm (b t.castSucc) (b t.succ)) |
| (hnear : ‖(extChartAt I (b 0)) y-(extChartAt I (a 0)) x‖≤C*r) |
| (hcontract : bvpInverseBound K α n*C*δ≤1/4) |
| (hdata : bvpInverseBound K α n*C*r≤δ/4) : |
| let C₁ := (A.splitting S).terminalProjection n V hdV hcV |
| let C₂ := (Z.splitting T).terminalProjection n W hdW hcW |
| let Q := bvpInverseBound K α n |
| ∃ z : M, |
| (∀ t : Fin (n+1),Smap^[t.val] z∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (Smap^[t.val] z)-(extChartAt I (a t)) (Smap^[t.val] x)‖≤δ/2) ∧ |
| (∀ t : Fin (n+1),Smap.symm^[t.val] z∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (Smap.symm^[t.val] z)-(extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤δ/2) ∧ |
| C₁.P ((extChartAt I (a (Fin.last n))) (Smap^[n] z)- |
| (extChartAt I (a (Fin.last n))) (Smap^[n] x))=0 ∧ |
| C₂.P ((extChartAt I (b (Fin.last n))) (Smap.symm^[n] z)- |
| (extChartAt I (b (Fin.last n))) (Smap.symm^[n] y))=0 ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (a t)) (Smap^[t.val] z)- |
| (extChartAt I (a t)) (Smap^[t.val] x)‖≤2*C*Q*r) ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (b t)) (Smap.symm^[t.val] z)- |
| (extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤2*C*Q*r) ∧ |
| ∀ z' : M, |
| (∀ t : Fin (n+1),Smap^[t.val] z'∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (Smap^[t.val] z')-(extChartAt I (a t)) (Smap^[t.val] x)‖≤δ) → |
| (∀ t : Fin (n+1),Smap.symm^[t.val] z'∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (Smap.symm^[t.val] z')-(extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤δ) → |
| C₁.P ((extChartAt I (a (Fin.last n))) (Smap^[n] z')- |
| (extChartAt I (a (Fin.last n))) (Smap^[n] x))=0 → |
| C₂.P ((extChartAt I (b (Fin.last n))) (Smap.symm^[n] z')- |
| (extChartAt I (b (Fin.last n))) (Smap.symm^[n] y))=0 → z'=z := by |
| intro C₁ C₂ Q |
| obtain ⟨L,hL,hK,hcorr,hκ₁,hκ₂⟩ := gauge_boundary_control A₀ A Z S₀ S T V W |
| hdV hdW hcV hcW hε hεhalf hdS hdT hcS hcT hsmall hsA hfA hsZ hfZ hgap |
| have hnα : 0≤α := (add_nonneg (abs_nonneg γ) (abs_nonneg g)).trans hα |
| have hboundA := A.unweighted S hnα |
| (fun j hj => (hsA j hj).trans (by linarith [le_abs_self γ,neg_le_abs g])) |
| (fun j hj => le_trans (by linarith [neg_le_abs γ,neg_le_abs g]) (hfA j hj)) |
| have hboundZ := Z.unweighted T hnα |
| (fun j hj => (hsZ j hj).trans (by linarith [neg_le_abs γ,neg_le_abs g])) |
| (fun j hj => le_trans (by linarith [le_abs_self γ,neg_le_abs g]) (hfZ j hj)) |
| have hU : 1≤bvpGrowth K α n := (by norm_num : (1:ℝ)≤2).trans (bvpGrowth_ge_two K α n) |
| have hUG : Real.exp (2*K+α*n)≤bvpGrowth K α n := by dsimp [bvpGrowth]; linarith |
| have hA₁ (t : Fin n) : fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (Smap^[t.val] x))=A.evolution.map (t+1) t := |
| gauge_actual_step I Smap _ _ _ A t (hx t.castSucc) |
| (by simpa only [Fin.val_succ,Function.iterate_succ_apply'] using hx t.succ) |
| (interior_iterate I Smap hintx t) (hstep₁ t) |
| have hA₂ (t : Fin n) : fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y))=Z.evolution.map (t+1) t := |
| gauge_actual_step I Smap.symm _ _ _ Z t (hy t.castSucc) |
| (by simpa only [Fin.val_succ,Function.iterate_succ_apply'] using hy t.succ) |
| (interior_iterate I Smap.symm hinty t) (hstep₂ t) |
| have hcontract' : let U := bvpGrowth K α n |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*(2*Real.exp (2*K)))*(1+2*D)+D)*C*δ≤1/4 := by |
| simpa only [bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hcontract |
| have hdata' : let U := bvpGrowth K α n |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*(2*Real.exp (2*K)))*(1+2*D)+D)*C*r≤δ/4 := by |
| simpa only [bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hdata |
| have hjoin := actual_join_unique I Smap a b x y hab hx hy A.evolution Z.evolution |
| (A.splitting S) (Z.splitting T) C₁ C₂ L hL hU hK hcorr |
| (fun s u hsu hun => (hboundA.1 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundA.2 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundZ.1 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundZ.2 s u hsu hun).trans hUG) |
| (((A.splitting S).terminalProjection_J_norm n V hdV hcV).trans (bvpGrowth_ge_two K α n)) |
| (((A.splitting S).terminalProjection_P_norm n V hdV hcV).trans hU) |
| (((Z.splitting T).terminalProjection_J_norm n W hdW hcW).trans (bvpGrowth_ge_two K α n)) |
| (((Z.splitting T).terminalProjection_P_norm n W hdW hcW).trans hU) |
| hC hr hδ hA₁ hA₂ hd₁ hd₂ hder₁ hder₂ ht₁ ht₂ hdom₁ hdom₂ hnear hcontract' hdata' |
| simpa only [Q,bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hjoin |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.FiniteGreen.Matching |
| open Set Metric |
| variable {E H G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [NormedAddCommGroup H] [NormedSpace ℝ H] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| {A₀ A₁ A₂ : Evolution E} (B₀ : Splitting A₀) {B₁ : Splitting A₁} {B₂ : Splitting A₂} |
| {n₁ n₂ : ℕ} (C₁ : B₁.Terminal (H:=H) n₁) (C₂ : B₂.Terminal (H:=G) n₂) |
|
|
| /-- A genuine nonlinear coordinate solution satisfies the exact boundary |
| residual equation, hence the sharp ideal-exchange estimate. -/ |
| theorem genuine_initial_error |
| (L : (B₁.initialSpace×B₂.initialSpace) ≃L[ℝ] E) |
| (hL : ∀ x,L x=(x.1:E)-(x.2:E)) |
| {K P ε κ Q C δ : ℝ} (hK : ‖L.symm.toContinuousLinearMap‖≤K) |
| (hP : 0≤P) (hε : 0≤ε) (hκ : 0≤κ) (hC : 0≤C) |
| (hsmall : ‖L.symm.toContinuousLinearMap.comp (correction C₁ C₂)‖≤1/2) |
| (hc₁ : ‖B₁.initialSpace.starProjection-(B₀.slow 0).range.starProjection‖≤ε) |
| (hc₂ : ‖B₂.initialSpace.starProjection-(B₀.fast 0).range.starProjection‖≤ε) |
| (hk₁ : ‖C₁.kernel‖≤κ) (hk₂ : ‖C₂.kernel‖≤κ) |
| (hs : ‖B₀.slow 0‖≤P) (hf : ‖B₀.fast 0‖≤P) |
| (f₁ : Fin n₁ → E → E) (f₂ : Fin n₂ → E → E) |
| (x₁ : Fin (n₁+1) → E) (x₂ : Fin (n₂+1) → E) |
| (hx₁ : ∀ t,f₁ t (x₁ t.castSucc)=x₁ t.succ) |
| (hx₂ : ∀ t,f₂ t (x₂ t.castSucc)=x₂ t.succ) |
| (hA₁ : ∀ t : Fin n₁,fderiv ℝ (f₁ t) (x₁ t.castSucc)=A₁.map (t+1) t) |
| (hA₂ : ∀ t : Fin n₂,fderiv ℝ (f₂ t) (x₂ t.castSucc)=A₂.map (t+1) t) |
| (hd₁ : ∀ t,∀ y∈closedBall (x₁ t.castSucc) δ,DifferentiableAt ℝ (f₁ t) y) |
| (hd₂ : ∀ t,∀ y∈closedBall (x₂ t.castSucc) δ,DifferentiableAt ℝ (f₂ t) y) |
| (hder₁ : ∀ t,∀ y∈closedBall (x₁ t.castSucc) δ, |
| ‖fderiv ℝ (f₁ t) y-fderiv ℝ (f₁ t) (x₁ t.castSucc)‖≤C*‖y-x₁ t.castSucc‖) |
| (hder₂ : ∀ t,∀ y∈closedBall (x₂ t.castSucc) δ, |
| ‖fderiv ℝ (f₂ t) y-fderiv ℝ (f₂ t) (x₂ t.castSucc)‖≤C*‖y-x₂ t.castSucc‖) |
| (v : Paths E n₁ n₂) (hv : ‖v‖≤δ) |
| (hs₁ : ∀ t,f₁ t (x₁ t.castSucc+v.1 t.castSucc)=x₁ t.succ+v.1 t.succ) |
| (hs₂ : ∀ t,f₂ t (x₂ t.castSucc+v.2 t.castSucc)=x₂ t.succ+v.2 t.succ) |
| (hm : x₁ 0+v.1 0=x₂ 0+v.2 0) |
| (hp₁ : C₁.P (v.1 (Fin.last n₁))=0) (hp₂ : C₂.P (v.2 (Fin.last n₂))=0) |
| (hQ : ‖(boundaryEquiv C₁ C₂ L hL (hsmall.trans_lt (by norm_num))).symm.toContinuousLinearMap‖≤Q) : |
| ‖v.1 0-B₀.slow 0 (x₂ 0-x₁ 0)‖≤4*P*K*(ε+κ)*‖x₂ 0-x₁ 0‖+Q*C*‖v‖^2 := by |
| let R := ArrayRemainder.residual (H:=H) (G:=G) f₁ f₂ |
| (fun t => x₁ t.castSucc) (fun t => x₂ t.castSucc) |
| apply array_nonlinear_initial_error B₀ C₁ C₂ L hL hK hP hε hκ hC hsmall hc₁ hc₂ |
| hk₁ hk₂ hs hf R v (x₂ 0-x₁ 0) |
| (ArrayRemainder.residual_quadratic f₁ f₂ _ _ hC hd₁ hd₂ hder₁ hder₂ hv) _ hQ |
| apply Prod.ext |
| · funext t |
| change v.1 t.succ-A₁.map (t+1) t (v.1 t.castSucc)= |
| 0+(f₁ t (x₁ t.castSucc+v.1 t.castSucc)-f₁ t (x₁ t.castSucc)- |
| (fderiv ℝ (f₁ t) (x₁ t.castSucc)) (v.1 t.castSucc)) |
| rw [hs₁ t,hx₁ t,hA₁ t,zero_add] |
| abel |
| · apply Prod.ext |
| · funext t |
| change v.2 t.succ-A₂.map (t+1) t (v.2 t.castSucc)= |
| 0+(f₂ t (x₂ t.castSucc+v.2 t.castSucc)-f₂ t (x₂ t.castSucc)- |
| (fderiv ℝ (f₂ t) (x₂ t.castSucc)) (v.2 t.castSucc)) |
| rw [hs₂ t,hx₂ t,hA₂ t,zero_add] |
| abel |
| · apply Prod.ext |
| · change v.1 0-v.2 0=(x₂ 0-x₁ 0)+0 |
| have hh := congrArg (fun z => z-x₁ 0-v.2 0) hm |
| simpa only [add_zero] using (by convert hh using 1 <;> abel) |
| · apply Prod.ext |
| · change C₁.P (v.1 (Fin.last n₁))=0+0 |
| simpa only [add_zero] using hp₁ |
| · change C₂.P (v.2 (Fin.last n₂))=0+0 |
| simpa only [add_zero] using hp₂ |
| end SmoothObstruction.FiniteGreen.Matching |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] |
| {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
| open FiniteGreen FiniteGreen.Matching |
|
|
| /-- The authentic local-coordinate nonlinear boundary construction gives |
| ONE actual point of the manifold, with both orbit halves and the fixed |
| terminal equations. The ball-domain conditions keep every inverse chart |
| and every one-step expression inside the ORIGINAL manifold. -/ |
| theorem actual_join_ideal (S : Diffeomorph I I M M ∞) {n : ℕ} |
| (a b : Fin (n+1) → M) (x y : M) (hab : a 0=b 0) |
| (hx : ∀ t : Fin (n+1),S^[t.val] x∈(chartAt H (a t)).source) |
| (hy : ∀ t : Fin (n+1),S.symm^[t.val] y∈(chartAt H (b t)).source) |
| (A₁ A₂ : Evolution (E (d:=d))) (B₁ : Splitting A₁) (B₂ : Splitting A₂) |
| (C₁ : B₁.Terminal (H:=F) n) (C₂ : B₂.Terminal (H:=G) n) |
| (L : (B₁.initialSpace×B₂.initialSpace) ≃L[ℝ] E (d:=d)) |
| (hL : ∀ z,L z=(z.1:E (d:=d))-(z.2:E (d:=d))) |
| {U K C r δ : ℝ} (hU : 1≤U) (hK : ‖L.symm.toContinuousLinearMap‖≤K) |
| (A₀ : Evolution (E (d:=d))) (B₀ : Splitting A₀) |
| {P ε κ : ℝ} (hP : 0≤P) (hε : 0≤ε) (hκ : 0≤κ) |
| (hc₁ : ‖B₁.initialSpace.starProjection-(B₀.slow 0).range.starProjection‖≤ε) |
| (hc₂ : ‖B₂.initialSpace.starProjection-(B₀.fast 0).range.starProjection‖≤ε) |
| (hk₁ : ‖C₁.kernel‖≤κ) (hk₂ : ‖C₂.kernel‖≤κ) |
| (hslow : ‖B₀.slow 0‖≤P) (hfast : ‖B₀.fast 0‖≤P) |
| (hsmall : ‖L.symm.toContinuousLinearMap.comp (correction C₁ C₂)‖≤1/2) |
| (hs₁ : ∀ s u,s≤u → u≤n → ‖(A₁.map u s).comp (B₁.slow s)‖≤U) |
| (hb₁ : ∀ s u,s≤u → u≤n → ‖(A₁.map s u).comp (B₁.fast u)‖≤U) |
| (hs₂ : ∀ s u,s≤u → u≤n → ‖(A₂.map u s).comp (B₂.slow s)‖≤U) |
| (hb₂ : ∀ s u,s≤u → u≤n → ‖(A₂.map s u).comp (B₂.fast u)‖≤U) |
| (hJ₁ : ‖C₁.J‖≤U) (hP₁ : ‖C₁.P‖≤U) |
| (hJ₂ : ‖C₂.J‖≤U) (hP₂ : ‖C₂.P‖≤U) |
| (hC : 0≤C) (hr : 0≤r) (hδ : 0<δ) |
| (hA₁ : ∀ t : Fin n,fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (S^[t.val] x))=A₁.map (t+1) t) |
| (hA₂ : ∀ t : Fin n,fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y))=A₂.map (t+1) t) |
| (hd₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ, |
| DifferentiableAt ℝ (localMap I S (a t.castSucc) (a t.succ)) z) |
| (hd₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ, |
| DifferentiableAt ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) z) |
| (hder₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ, |
| ‖fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) z- |
| fderiv ℝ (localMap I S (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (S^[t.val] x))‖≤ |
| C*‖z-(extChartAt I (a t.castSucc)) (S^[t.val] x)‖) |
| (hder₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ, |
| ‖fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) z- |
| fderiv ℝ (localMap I S.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y))‖≤ |
| C*‖z-(extChartAt I (b t.castSucc)) (S.symm^[t.val] y)‖) |
| (ht₁ : ∀ t : Fin (n+1),closedBall ((extChartAt I (a t)) (S^[t.val] x)) δ⊆ |
| (extChartAt I (a t)).target) |
| (ht₂ : ∀ t : Fin (n+1),closedBall ((extChartAt I (b t)) (S.symm^[t.val] y)) δ⊆ |
| (extChartAt I (b t)).target) |
| (hdom₁ : ∀ t : Fin n,closedBall ((extChartAt I (a t.castSucc)) (S^[t.val] x)) δ⊆ |
| domainU I S (a t.castSucc) (a t.succ)) |
| (hdom₂ : ∀ t : Fin n,closedBall ((extChartAt I (b t.castSucc)) (S.symm^[t.val] y)) δ⊆ |
| domainU I S.symm (b t.castSucc) (b t.succ)) |
| (hnear : ‖(extChartAt I (b 0)) y-(extChartAt I (a 0)) x‖≤C*r) |
| (hcontract : let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*K)*(1+2*D)+D)*C*δ≤1/4) |
| (hdata : let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*K)*(1+2*D)+D)*C*r≤δ/4) : |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| let Q := (U+U^4)*(2*K)*(1+2*D)+D |
| ∃ z : M, |
| (∀ t : Fin (n+1),S^[t.val] z∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (S^[t.val] z)-(extChartAt I (a t)) (S^[t.val] x)‖≤δ/2) ∧ |
| (∀ t : Fin (n+1),S.symm^[t.val] z∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (S.symm^[t.val] z)-(extChartAt I (b t)) (S.symm^[t.val] y)‖≤δ/2) ∧ |
| C₁.P ((extChartAt I (a (Fin.last n))) (S^[n] z)- |
| (extChartAt I (a (Fin.last n))) (S^[n] x))=0 ∧ |
| C₂.P ((extChartAt I (b (Fin.last n))) (S.symm^[n] z)- |
| (extChartAt I (b (Fin.last n))) (S.symm^[n] y))=0 ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (a t)) (S^[t.val] z)- |
| (extChartAt I (a t)) (S^[t.val] x)‖≤2*C*Q*r) ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (b t)) (S.symm^[t.val] z)- |
| (extChartAt I (b t)) (S.symm^[t.val] y)‖≤2*C*Q*r) ∧ |
| ‖(extChartAt I (a 0)) z-(extChartAt I (a 0)) x- |
| B₀.slow 0 ((extChartAt I (b 0)) y-(extChartAt I (a 0)) x)‖≤ |
| 4*P*K*(ε+κ)*(C*r)+Q*C*(2*C*Q*r)^2 ∧ |
| ∀ z' : M, |
| (∀ t : Fin (n+1),S^[t.val] z'∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (S^[t.val] z')-(extChartAt I (a t)) (S^[t.val] x)‖≤δ) → |
| (∀ t : Fin (n+1),S.symm^[t.val] z'∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (S.symm^[t.val] z')-(extChartAt I (b t)) (S.symm^[t.val] y)‖≤δ) → |
| C₁.P ((extChartAt I (a (Fin.last n))) (S^[n] z')- |
| (extChartAt I (a (Fin.last n))) (S^[n] x))=0 → |
| C₂.P ((extChartAt I (b (Fin.last n))) (S.symm^[n] z')- |
| (extChartAt I (b (Fin.last n))) (S.symm^[n] y))=0 → z'=z := by |
| let u : Fin (n+1) → E (d:=d) := fun t => (extChartAt I (a t)) (S^[t.val] x) |
| let v : Fin (n+1) → E (d:=d) := fun t => (extChartAt I (b t)) (S.symm^[t.val] y) |
| let f : Fin n → E (d:=d) → E (d:=d) := fun t => localMap I S (a t.castSucc) (a t.succ) |
| let g : Fin n → E (d:=d) → E (d:=d) := fun t => localMap I S.symm (b t.castSucc) (b t.succ) |
| have hu (t : Fin n) : f t (u t.castSucc)=u t.succ := by |
| simp only [f,u,localMap,Function.comp_apply] |
| rw [(extChartAt I (a t.castSucc)).left_inv (by simpa only [extChartAt_source] using hx t.castSucc)] |
| exact congrArg (extChartAt I (a t.succ)) (Function.iterate_succ_apply' S _ x).symm |
| have hv (t : Fin n) : g t (v t.castSucc)=v t.succ := by |
| simp only [g,v,localMap,Function.comp_apply] |
| rw [(extChartAt I (b t.castSucc)).left_inv (by simpa only [extChartAt_source] using hy t.castSucc)] |
| exact congrArg (extChartAt I (b t.succ)) (Function.iterate_succ_apply' S.symm _ y).symm |
| obtain ⟨w,hw,hwsharp,hf,hg,hmid,hp,hq,hunique⟩ := nonlinear_orbits_unique C₁ C₂ L hL hU hK hsmall |
| hs₁ hb₁ hs₂ hb₂ hJ₁ hP₁ hJ₂ hP₂ f g u v hu hv hA₁ hA₂ hC hr hδ hd₁ hd₂ hder₁ hder₂ |
| (by simpa only [u,v,Fin.val_zero,Function.iterate_zero,id_eq] using hnear) hcontract hdata |
| have hw₁ (t : Fin (n+1)) : ‖w.1 t‖≤δ/2 := |
| (norm_le_pi_norm w.1 t).trans ((norm_fst_le w).trans hw) |
| have hw₂ (t : Fin (n+1)) : ‖w.2 t‖≤δ/2 := |
| (norm_le_pi_norm w.2 t).trans ((norm_snd_le w).trans hw) |
| have hwδ : δ/2≤δ := by linarith |
| have hb₁' (t : Fin (n+1)) : u t+w.1 t∈closedBall (u t) δ := by |
| rw [mem_closedBall,dist_eq_norm,add_sub_cancel_left] |
| exact (hw₁ t).trans hwδ |
| have hb₂' (t : Fin (n+1)) : v t+w.2 t∈closedBall (v t) δ := by |
| rw [mem_closedBall,dist_eq_norm,add_sub_cancel_left] |
| exact (hw₂ t).trans hwδ |
| obtain ⟨z,hz,hz'⟩ := realize_bilateral_array I S a b (fun t => u t+w.1 t) (fun t => v t+w.2 t) |
| hab hmid (fun t => ht₁ t (hb₁' t)) (fun t => ht₂ t (hb₂' t)) |
| (fun t => hdom₁ t (hb₁' t.castSucc)) (fun t => hdom₂ t (hb₂' t.castSucc)) hf hg |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| let Q := (U+U^4)*(2*K)*(1+2*D)+D |
| have hKn : 0≤K := (norm_nonneg L.symm.toContinuousLinearMap).trans hK |
| have hUn : 0≤U := by linarith |
| have hQn : 0≤Q := by dsimp [Q,D]; positivity |
| have hQ := boundary_inverse_norm C₁ C₂ L hL hU hK hsmall |
| hs₁ hb₁ hs₂ hb₂ hJ₁ hP₁ hJ₂ hP₂ |
| have hi := genuine_initial_error B₀ C₁ C₂ L hL hK hP hε hκ hC hsmall hc₁ hc₂ |
| hk₁ hk₂ hslow hfast f g u v hu hv hA₁ hA₂ hd₁ hd₂ hder₁ hder₂ w |
| (hw.trans hwδ) hf hg hmid hp hq hQ |
| have hi' : ‖w.1 0-B₀.slow 0 (v 0-u 0)‖≤ |
| 4*P*K*(ε+κ)*(C*r)+Q*C*(2*C*Q*r)^2 := by |
| apply hi.trans |
| apply add_le_add |
| · apply mul_le_mul_of_nonneg_left _ (by positivity) |
| simpa only [u,v,Fin.val_zero,Function.iterate_zero,id_eq] using hnear |
| · apply mul_le_mul_of_nonneg_left _ (mul_nonneg hQn hC) |
| exact pow_le_pow_left₀ (norm_nonneg w) hwsharp 2 |
| refine ⟨z,?_,?_,?_,?_,?_,?_,?_,?_⟩ |
| · intro t |
| refine ⟨(hz t).1,?_⟩ |
| rw [(hz t).2] |
| simpa only [u,add_sub_cancel_left] using hw₁ t |
| · intro t |
| refine ⟨(hz' t).1,?_⟩ |
| rw [(hz' t).2] |
| simpa only [v,add_sub_cancel_left] using hw₂ t |
| · have hzL := (hz (Fin.last n)).2 |
| simp only [Fin.val_last] at hzL |
| rw [hzL] |
| simpa only [u,Fin.val_last,add_sub_cancel_left] using hp |
| · have hzL := (hz' (Fin.last n)).2 |
| simp only [Fin.val_last] at hzL |
| rw [hzL] |
| simpa only [v,Fin.val_last,add_sub_cancel_left] using hq |
| · intro t |
| rw [(hz t).2] |
| simpa only [u,add_sub_cancel_left] using |
| (norm_le_pi_norm w.1 t).trans ((norm_fst_le w).trans hwsharp) |
| · intro t |
| rw [(hz' t).2] |
| simpa only [v,add_sub_cancel_left] using |
| (norm_le_pi_norm w.2 t).trans ((norm_snd_le w).trans hwsharp) |
| · have hz0 := (hz 0).2 |
| simp only [Fin.val_zero,Function.iterate_zero,id_eq] at hz0 |
| rw [hz0] |
| simpa only [u,v,Fin.val_zero,Function.iterate_zero,id_eq,add_sub_cancel_left] using hi' |
| · intro z' hx' hy' hp' hq' |
| let w' : Paths (E (d:=d)) n n := |
| (fun t => (extChartAt I (a t)) (S^[t.val] z')-u t, |
| fun t => (extChartAt I (b t)) (S.symm^[t.val] z')-v t) |
| have hn : ‖w'‖≤δ := by |
| rw [Prod.norm_def,max_le_iff] |
| constructor |
| · exact (pi_norm_le_iff_of_nonneg hδ.le).mpr (fun t => (hx' t).2) |
| · exact (pi_norm_le_iff_of_nonneg hδ.le).mpr (fun t => (hy' t).2) |
| have huf (t : Fin (n+1)) : u t+w'.1 t=(extChartAt I (a t)) (S^[t.val] z') := by |
| dsimp only [w']; abel |
| have hvf (t : Fin (n+1)) : v t+w'.2 t=(extChartAt I (b t)) (S.symm^[t.val] z') := by |
| dsimp only [w']; abel |
| have he : w'=w := hunique w' hn (fun t => by |
| rw [huf,huf] |
| simp only [f,localMap,Function.comp_apply] |
| rw [(extChartAt I (a t.castSucc)).left_inv |
| (by simpa only [extChartAt_source] using (hx' t.castSucc).1)] |
| exact congrArg (extChartAt I (a t.succ)) (Function.iterate_succ_apply' S _ z').symm) |
| (fun t => by |
| rw [hvf,hvf] |
| simp only [g,localMap,Function.comp_apply] |
| rw [(extChartAt I (b t.castSucc)).left_inv |
| (by simpa only [extChartAt_source] using (hy' t.castSucc).1)] |
| exact congrArg (extChartAt I (b t.succ)) (Function.iterate_succ_apply' S.symm _ z').symm) |
| (by rw [huf,hvf,hab]; rfl) |
| (by simpa only [w',u,Fin.val_last] using hp') |
| (by simpa only [w',v,Fin.val_last] using hq') |
| apply (extChartAt I (a 0)).injOn |
| (by simpa only [extChartAt_source,Fin.val_zero,Function.iterate_zero,id_eq] using (hx' 0).1) |
| (by simpa only [extChartAt_source,Fin.val_zero,Function.iterate_zero,id_eq] using (hz 0).1) |
| have hfirst := congrArg (fun q : Paths (E (d:=d)) n n => q.1 0) he |
| have hc := (hz 0).2 |
| simp only [Fin.val_zero,Function.iterate_zero,id_eq] at hc |
| rw [hc,←hfirst, huf] |
| rfl |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric |
| open scoped Manifold ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| open FiniteGreen FiniteGreen.Matching ShortBin |
| theorem gauged_join_ideal (Smap : Diffeomorph I I M M ∞) {n : ℕ} |
| (a b : Fin (n+1) → M) (x y : M) (hab : a 0=b 0) |
| (hx : ∀ t : Fin (n+1),Smap^[t.val] x∈(chartAt H (a t)).source) |
| (hy : ∀ t : Fin (n+1),Smap.symm^[t.val] y∈(chartAt H (b t)).source) |
| (hintx : I.IsInteriorPoint x) (hinty : I.IsInteriorPoint y) |
| {K ε γ g α : ℝ} (A₀ A Z : Gauge d n K) (S₀ S T : Finset (Fin d)) |
| (V W : Submodule ℝ (E (d:=d))) |
| (hdV : Module.finrank ℝ ((A.splitting S).fast n).range=Module.finrank ℝ V) |
| (hdW : Module.finrank ℝ ((Z.splitting T).fast n).range=Module.finrank ℝ W) |
| (hcV : ‖((A.splitting S).fast n).range.starProjection-V.starProjection‖≤1/2) |
| (hcW : ‖((Z.splitting T).fast n).range.starProjection-W.starProjection‖≤1/2) |
| (hε : 0≤ε) (hεhalf : ε≤1/2) |
| (hdS : S₀.card=S.card) (hdT : S₀ᶜ.card=T.card) |
| (hcS : ‖(A₀.space S₀).starProjection-(A.space S).starProjection‖≤ε) |
| (hcT : ‖(A₀.space S₀ᶜ).starProjection-(Z.space T).starProjection‖≤ε) |
| (hsmall : Real.exp (2*K)*(2*ε)≤1/2) |
| (hsA : ∀ j∈S,A.lam j≤γ-g) (hfA : ∀ j∈Sᶜ,γ+g≤A.lam j) |
| (hsZ : ∀ j∈T,Z.lam j≤ -γ-g) (hfZ : ∀ j∈Tᶜ,-γ+g≤Z.lam j) |
| (hgap : (2*Real.exp (2*K))*(2*(2*Real.exp (2*K)^2*Real.exp (-2*g*n)))≤1/2) |
| (hα : |γ|+|g|≤α) |
| (hstep₁ : ∀ t : Fin n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I Smap (a t.castSucc) (a t.succ) (Smap^[t.val] x))*A.U t= |
| A.U (t+1)*EuclideanSplitting.rate A.lam 1) |
| (hstep₂ : ∀ t : Fin n,(Matrix.toEuclideanCLM (n:=Fin d) (𝕜:=ℝ)).symm |
| (coord I Smap.symm (b t.castSucc) (b t.succ) (Smap.symm^[t.val] y))*Z.U t= |
| Z.U (t+1)*EuclideanSplitting.rate Z.lam 1) |
| {C r δ : ℝ} |
| (hC : 0≤C) (hr : 0≤r) (hδ : 0<δ) |
| (hd₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ, |
| DifferentiableAt ℝ (localMap I Smap (a t.castSucc) (a t.succ)) z) |
| (hd₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ, |
| DifferentiableAt ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) z) |
| (hder₁ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ, |
| ‖fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) z- |
| fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (Smap^[t.val] x))‖≤ |
| C*‖z-(extChartAt I (a t.castSucc)) (Smap^[t.val] x)‖) |
| (hder₂ : ∀ t : Fin n,∀ z∈closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ, |
| ‖fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) z- |
| fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y))‖≤ |
| C*‖z-(extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)‖) |
| (ht₁ : ∀ t : Fin (n+1),closedBall ((extChartAt I (a t)) (Smap^[t.val] x)) δ⊆ |
| (extChartAt I (a t)).target) |
| (ht₂ : ∀ t : Fin (n+1),closedBall ((extChartAt I (b t)) (Smap.symm^[t.val] y)) δ⊆ |
| (extChartAt I (b t)).target) |
| (hdom₁ : ∀ t : Fin n,closedBall ((extChartAt I (a t.castSucc)) (Smap^[t.val] x)) δ⊆ |
| domainU I Smap (a t.castSucc) (a t.succ)) |
| (hdom₂ : ∀ t : Fin n,closedBall ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y)) δ⊆ |
| domainU I Smap.symm (b t.castSucc) (b t.succ)) |
| (hnear : ‖(extChartAt I (b 0)) y-(extChartAt I (a 0)) x‖≤C*r) |
| (hcontract : bvpInverseBound K α n*C*δ≤1/4) |
| (hdata : bvpInverseBound K α n*C*r≤δ/4) : |
| let C₁ := (A.splitting S).terminalProjection n V hdV hcV |
| let C₂ := (Z.splitting T).terminalProjection n W hdW hcW |
| let Q := bvpInverseBound K α n |
| ∃ z : M, |
| (∀ t : Fin (n+1),Smap^[t.val] z∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (Smap^[t.val] z)-(extChartAt I (a t)) (Smap^[t.val] x)‖≤δ/2) ∧ |
| (∀ t : Fin (n+1),Smap.symm^[t.val] z∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (Smap.symm^[t.val] z)-(extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤δ/2) ∧ |
| C₁.P ((extChartAt I (a (Fin.last n))) (Smap^[n] z)- |
| (extChartAt I (a (Fin.last n))) (Smap^[n] x))=0 ∧ |
| C₂.P ((extChartAt I (b (Fin.last n))) (Smap.symm^[n] z)- |
| (extChartAt I (b (Fin.last n))) (Smap.symm^[n] y))=0 ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (a t)) (Smap^[t.val] z)- |
| (extChartAt I (a t)) (Smap^[t.val] x)‖≤2*C*Q*r) ∧ |
| (∀ t : Fin (n+1),‖(extChartAt I (b t)) (Smap.symm^[t.val] z)- |
| (extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤2*C*Q*r) ∧ |
| ‖(extChartAt I (a 0)) z-(extChartAt I (a 0)) x- |
| (A₀.splitting S₀).slow 0 ((extChartAt I (b 0)) y-(extChartAt I (a 0)) x)‖≤ |
| 4*Real.exp (2*K)*(2*Real.exp (2*K))*(ε+2*Real.exp (2*K)^2*Real.exp (-2*g*n))*(C*r)+ |
| Q*C*(2*C*Q*r)^2 ∧ |
| ∀ z' : M, |
| (∀ t : Fin (n+1),Smap^[t.val] z'∈(chartAt H (a t)).source ∧ |
| ‖(extChartAt I (a t)) (Smap^[t.val] z')-(extChartAt I (a t)) (Smap^[t.val] x)‖≤δ) → |
| (∀ t : Fin (n+1),Smap.symm^[t.val] z'∈(chartAt H (b t)).source ∧ |
| ‖(extChartAt I (b t)) (Smap.symm^[t.val] z')-(extChartAt I (b t)) (Smap.symm^[t.val] y)‖≤δ) → |
| C₁.P ((extChartAt I (a (Fin.last n))) (Smap^[n] z')- |
| (extChartAt I (a (Fin.last n))) (Smap^[n] x))=0 → |
| C₂.P ((extChartAt I (b (Fin.last n))) (Smap.symm^[n] z')- |
| (extChartAt I (b (Fin.last n))) (Smap.symm^[n] y))=0 → z'=z := by |
| intro C₁ C₂ Q |
| obtain ⟨L,hL,hK,hcorr,hκ₁,hκ₂⟩ := gauge_boundary_control A₀ A Z S₀ S T V W |
| hdV hdW hcV hcW hε hεhalf hdS hdT hcS hcT hsmall hsA hfA hsZ hfZ hgap |
| have hnα : 0≤α := (add_nonneg (abs_nonneg γ) (abs_nonneg g)).trans hα |
| have hboundA := A.unweighted S hnα |
| (fun j hj => (hsA j hj).trans (by linarith [le_abs_self γ,neg_le_abs g])) |
| (fun j hj => le_trans (by linarith [neg_le_abs γ,neg_le_abs g]) (hfA j hj)) |
| have hboundZ := Z.unweighted T hnα |
| (fun j hj => (hsZ j hj).trans (by linarith [neg_le_abs γ,neg_le_abs g])) |
| (fun j hj => le_trans (by linarith [le_abs_self γ,neg_le_abs g]) (hfZ j hj)) |
| have hU : 1≤bvpGrowth K α n := (by norm_num : (1:ℝ)≤2).trans (bvpGrowth_ge_two K α n) |
| have hUG : Real.exp (2*K+α*n)≤bvpGrowth K α n := by dsimp [bvpGrowth]; linarith |
| have hA₁ (t : Fin n) : fderiv ℝ (localMap I Smap (a t.castSucc) (a t.succ)) |
| ((extChartAt I (a t.castSucc)) (Smap^[t.val] x))=A.evolution.map (t+1) t := |
| gauge_actual_step I Smap _ _ _ A t (hx t.castSucc) |
| (by simpa only [Fin.val_succ,Function.iterate_succ_apply'] using hx t.succ) |
| (interior_iterate I Smap hintx t) (hstep₁ t) |
| have hA₂ (t : Fin n) : fderiv ℝ (localMap I Smap.symm (b t.castSucc) (b t.succ)) |
| ((extChartAt I (b t.castSucc)) (Smap.symm^[t.val] y))=Z.evolution.map (t+1) t := |
| gauge_actual_step I Smap.symm _ _ _ Z t (hy t.castSucc) |
| (by simpa only [Fin.val_succ,Function.iterate_succ_apply'] using hy t.succ) |
| (interior_iterate I Smap.symm hinty t) (hstep₂ t) |
| have hcontract' : let U := bvpGrowth K α n |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*(2*Real.exp (2*K)))*(1+2*D)+D)*C*δ≤1/4 := by |
| simpa only [bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hcontract |
| have hdata' : let U := bvpGrowth K α n |
| let D := (2*U+U^4)*(n+n : ℕ)+U^2 |
| ((U+U^4)*(2*(2*Real.exp (2*K)))*(1+2*D)+D)*C*r≤δ/4 := by |
| simpa only [bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hdata |
| have hs₀ : ‖(A₀.splitting S₀).slow 0‖≤Real.exp (2*K) := |
| EuclideanSplitting.splitting_slow_norm _ _ _ _ _ _ 0 |
| (A₀.normU _ (Nat.zero_le n)) (A₀.normV _ (Nat.zero_le n)) |
| have hf₀ : ‖(A₀.splitting S₀).fast 0‖≤Real.exp (2*K) := |
| EuclideanSplitting.splitting_fast_norm _ _ _ _ _ _ 0 |
| (A₀.normU _ (Nat.zero_le n)) (A₀.normV _ (Nat.zero_le n)) |
| have hc₁ : ‖(A.splitting S).initialSpace.starProjection- |
| ((A₀.splitting S₀).slow 0).range.starProjection‖≤ε := by |
| simpa only [FiniteGreen.Splitting.initialSpace,←A.space_eq_range,←A₀.space_eq_range, |
| norm_sub_rev] using hcS |
| have hc₂ : ‖(Z.splitting T).initialSpace.starProjection- |
| ((A₀.splitting S₀).fast 0).range.starProjection‖≤ε := by |
| have heT : ((A₀.splitting S₀).fast 0).range = A₀.space S₀ᶜ := |
| EuclideanSplitting.splitting_fast_range _ _ _ _ _ _ 0 |
| simpa only [FiniteGreen.Splitting.initialSpace,←Z.space_eq_range, |
| heT,norm_sub_rev] using hcT |
| have hjoin := actual_join_ideal I Smap a b x y hab hx hy A.evolution Z.evolution |
| (A.splitting S) (Z.splitting T) C₁ C₂ L hL hU hK A₀.evolution (A₀.splitting S₀) |
| (Real.exp_nonneg _) hε (by positivity) hc₁ hc₂ hκ₁ hκ₂ hs₀ hf₀ hcorr |
| (fun s u hsu hun => (hboundA.1 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundA.2 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundZ.1 s u hsu hun).trans hUG) |
| (fun s u hsu hun => (hboundZ.2 s u hsu hun).trans hUG) |
| (((A.splitting S).terminalProjection_J_norm n V hdV hcV).trans (bvpGrowth_ge_two K α n)) |
| (((A.splitting S).terminalProjection_P_norm n V hdV hcV).trans hU) |
| (((Z.splitting T).terminalProjection_J_norm n W hdW hcW).trans (bvpGrowth_ge_two K α n)) |
| (((Z.splitting T).terminalProjection_P_norm n W hdW hcW).trans hU) |
| hC hr hδ hA₁ hA₂ hd₁ hd₂ hder₁ hder₂ ht₁ ht₂ hdom₁ hdom₂ hnear hcontract' hdata' |
| simpa only [Q,bvpInverseBound,show 2*(2*Real.exp (2*K))=4*Real.exp (2*K) by ring] using hjoin |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology 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] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| omit [ProperSpace M] in |
| lemma coord_refl_same (a x : M) (hx : x∈(chartAt H a).source) : |
| coord I (Diffeomorph.refl I M ∞) a a x=1 := by |
| rw [←chartChange_coe I a a x hx hx] |
| ext v |
| simp [chartChange] |
|
|
| noncomputable def BilateralGauge.forward (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) {n : ℕ} {B R : ℝ} {x : M} |
| (A : BilateralGauge I S a c n B R x) : ShortBin.Gauge d n B := |
| ShortBin.Gauge.ofFinite A.UF A.VF A.lam (fun t _ => A.invF t) A.normF |
|
|
| noncomputable def BilateralGauge.backward (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) {n : ℕ} {B R : ℝ} {x : M} |
| (A : BilateralGauge I S a c n B R x) : ShortBin.Gauge d n B := |
| ShortBin.Gauge.ofFinite A.UB A.VB (fun j => -A.lam j) (fun t _ => A.invB t) A.normB |
|
|
| /-- The common-bin spaces now belong to the ORIGINAL good bilateral gauges. |
| All carried intermediate charts have disappeared from the conclusion. Only |
| initial coarse-bin coordinates are used, so no orbit-code partition appears. -/ |
| theorem uniform_genuine_bin_splitting (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) : |
| ∃ rF CF rB CB D : ℝ,0 < rF ∧ 1 ≤ CF ∧ 0 < rB ∧ 1 ≤ CB ∧ 1 ≤ D ∧ |
| ∀ (c : M → ι) (J : Type*) (x : J → M) (e : J) |
| (n N : ℕ) (B R h g : ℝ) (A : ∀ i,BilateralGauge I S a c n B R (x i)), |
| (∀ y,y∈K (c y)) → N ≤ n → (∀ i,c (x i)=c (x e)) → |
| (∀ i,I.IsInteriorPoint (x i)) → |
| (∀ i,‖(extChartAt I (a (c (x e)))) (x i)- |
| (extChartAt I (a (c (x e)))) (x e)‖ ≤ h) → |
| CF^N*h ≤ rF → CB^N*h ≤ rB → 0 < g → |
| Real.exp (-2*g*N+4*(B+Real.log D)) ≤ 1/2 → |
| let δ := ((N:ℝ)*CF*CF^(2*N)+N*CB*CB^(2*N))*h |
| δ < Real.exp (-2*(B+Real.log D))/2 → |
| Real.exp (-2*g*N+4*(B+Real.log D))+ |
| δ*Real.exp ((1+8*(d:ℝ))*g*N+2*(B+Real.log D)) ≤ 1/2 → |
| ∃ j : Fin (d+1), let γ := (1+8*(j.val:ℝ))*g |
| (∀ i,(∀ k,(A i).lam k < γ-g ∨ γ+g < (A i).lam k) ∧ |
| (Finset.univ.filter (fun k => γ < (A i).lam k)).card= |
| (Finset.univ.filter (fun k => γ < (A e).lam k)).card) ∧ |
| let ε := Real.exp (-2*g*N+4*(B+Real.log D))+ |
| δ*Real.exp ((1+8*(d:ℝ))*g*N+2*(B+Real.log D)) |
| (∀ i,‖(((A i).forward I S a c).space (Finset.univ.filter (fun k => (A i).lam k ≤ γ))).starProjection- |
| (((A e).forward I S a c).space (Finset.univ.filter (fun k => (A e).lam k ≤ γ))).starProjection‖ ≤ 3*ε) ∧ |
| ∀ i,‖(((A i).forward I S a c).space (Finset.univ.filter (fun k => γ < (A i).lam k))).starProjection- |
| (((A e).forward I S a c).space (Finset.univ.filter (fun k => γ < (A e).lam k))).starProjection‖ ≤ 3*ε := by |
| classical |
| obtain ⟨rF,CF,rB,CB,D,hrF,hCF,hrB,hCB,hD,hh⟩ := |
| uniform_genuine_bin_gauges I hclosed S a K hK hKU |
| refine ⟨rF,CF,rB,CB,D,hrF,hCF,hrB,hCB,hD, |
| fun c J x e n N B R h g A hc hN hcommon hint hdist hsF hsB hg hdecay hnoise hsmall => ?_⟩ |
| let cF (t : ℕ) := c (S^[t] (x e)) |
| let cB (t : ℕ) := c (S.symm^[t] (x e)) |
| have hsource (i : J) : x i∈(chartAt H (a (c (x e)))).source := by |
| rw [←hcommon i] |
| exact hKU _ (hc _) |
| obtain ⟨F,Z,hlam,hcenter,hstepF,hstepB,hprodF,hprodB,hform⟩ := |
| hh c J x e cF cB n N B R h A hc hN rfl |
| (fun _ _ => hc _) (fun _ _ => hc _) hint hsource hdist hsF hsB |
| have hzero (i : J) : (F i).U 0=(A i).UF 0 ∧ (F i).V 0=(A i).VF 0 := by |
| have hh := hform i 0 (Nat.zero_le N) |
| have hid := coord_refl_same I (a (c (x e))) (x i) (hsource i) |
| have hh' := And.intro hh.1 hh.2.1 |
| simpa only [cF,Function.iterate_zero_apply,hcommon i,hid,map_one,one_mul,mul_one] using hh' |
| have hspace (i : J) (T : Finset (Fin d)) : |
| (F i).space T=((A i).forward I S a c).space T := by |
| simp only [ShortBin.Gauge.space,BilateralGauge.forward,ShortBin.Gauge.ofFinite, |
| Nat.zero_min,(hzero i).1,(hzero i).2] |
| have hh0 : 0 ≤ h := (norm_nonneg _).trans (hdist e) |
| let δ := ((N:ℝ)*CF*CF^(2*N)+N*CB*CB^(2*N))*h |
| have hδ : 0 ≤ δ := by dsimp [δ]; positivity |
| have hnF : 0 ≤ (N:ℝ)*CF*CF^(2*N)*h := by positivity |
| have hnB : 0 ≤ (N:ℝ)*CB*CB^(2*N)*h := by positivity |
| have hbF : (N:ℝ)*CF*CF^(2*N)*h ≤ δ := by dsimp [δ]; nlinarith |
| have hbB : (N:ℝ)*CB*CB^(2*N)*h ≤ δ := by dsimp [δ]; nlinarith |
| obtain ⟨j,hgap,hslow,hfast⟩ := ShortBin.common_splittings F Z e hg hδ |
| (fun i => (hprodF i).trans hbF) (fun i => (hprodB i).trans hbB) |
| (fun i => (hcenter i).1) (fun i => (hcenter i).2) |
| (fun i k => by rw [(hlam i k).2,(hlam i k).1]) hdecay hnoise hsmall |
| have hFl (i : J) : (F i).lam=(A i).lam := funext fun k => (hlam i k).1 |
| refine ⟨j,?_,?_,?_⟩ |
| · simpa only [hFl] using hgap |
| · simpa only [hFl,hspace] using hslow |
| · simpa only [hFl,hspace] using hfast |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Bundle Set Filter Topology 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] [ProperSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
|
|
| /-- Spaces of the genuine long-window gauge in its original endpoint chart. -/ |
| noncomputable def BilateralGauge.forwardSpace (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) {n : ℕ} {B R : ℝ} {x : M} |
| (A : BilateralGauge I S a c n B R x) (t : ℕ) (T : Finset (Fin d)) := |
| EuclideanSplitting.groupSpace (A.UF t) (A.VF t) (A.invF t).1 (A.invF t).2 T |
|
|
| noncomputable def BilateralGauge.backwardSpace (S : Diffeomorph I I M M ∞) |
| (a : ι → M) (c : M → ι) {n : ℕ} {B R : ℝ} {x : M} |
| (A : BilateralGauge I S a c n B R x) (t : ℕ) (T : Finset (Fin d)) := |
| EuclideanSplitting.groupSpace (A.UB t) (A.VB t) (A.invB t).1 (A.invB t).2 T |
|
|
| /-- Endpoint-bin comparison with NO carried basis or extra threshold in the |
| conclusion. The reference is chosen among original certificates reaching the |
| same endpoint bin from a fixed starting bin. -/ |
| theorem uniform_original_endpoint_fast_comparison (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) : |
| ∃ r C₁ D : ℝ,0<r ∧ 1≤C₁ ∧ 1≤D ∧ |
| ∀ (c : M → ι) (J : Type*) (x : J → M) (e : J) |
| (n N : ℕ) (B R h γ g : ℝ) (T : J → Finset (Fin d)) |
| (A : ∀ i,BilateralGauge I S a c n B R (x i)), |
| (∀ y,y∈K (c y)) → N≤n → |
| (∀ i,c (S^[n] (x i))=c (S^[n] (x e))) → |
| (∀ i,I.IsInteriorPoint (S^[n] (x i))) → |
| (∀ i,‖(extChartAt I (a (c (S^[n] (x e))))) (S^[n] (x i))- |
| (extChartAt I (a (c (S^[n] (x e))))) (S^[n] (x e))‖≤h) → |
| C₁^N*h≤r → 0≤h → |
| (∀ i,(T e).card=(T i).card) → |
| (∀ i,∀ j∈(T i)ᶜ,(A i).lam j≤γ-g) → |
| (∀ i,∀ j∈T i,γ+g≤(A i).lam j) → |
| (let ε := Real.exp ((γ-g)*N+2*(B+Real.log D))* |
| (Real.exp ((-γ-g)*N+2*(B+Real.log D))+N*C₁*C₁^(2*N)*h) |
| ε≤1/2) → |
| ∀ i,‖(A i |>.forwardSpace I S a c n (T i)).starProjection- |
| (A e |>.forwardSpace I S a c n (T e)).starProjection‖≤ |
| 3*(Real.exp ((γ-g)*N+2*(B+Real.log D))* |
| (Real.exp ((-γ-g)*N+2*(B+Real.log D))+N*C₁*C₁^(2*N)*h)) := by |
| obtain ⟨r,C₁,D,hr,hC,hD,hh⟩ := |
| uniform_endpoint_fast_comparison I hclosed S a K hK hKU |
| refine ⟨r,C₁,D,hr,hC,hD,fun c J x e n N B R h γ g T A hc hN hcommon |
| hint hdist ht hn hdim hslow hfast hsmall => ?_⟩ |
| let cR (t : ℕ) := c (S.symm^[t] (S^[n] (x e))) |
| have hsource (i : J) : S^[n] (x i)∈(chartAt H (a (cR 0))).source := by |
| change S^[n] (x i)∈(chartAt H (a (c (S^[n] (x e))))).source |
| rw [←hcommon i] |
| exact hKU _ (hc _) |
| obtain ⟨F,hlam,hform,hclose⟩ := hh c J x e cR n N B R h γ g T A hc hN |
| (fun _ _ => hc _) hint hsource hdist ht hn hdim hslow hfast hsmall |
| have hzero (i : J) : (F i).U 0=(A i).UF n ∧ (F i).V 0=(A i).VF n := by |
| have hid := coord_refl_same I (a (cR 0)) (S^[n] (x i)) (hsource i) |
| have hc0 : c (S^[n] (x i))=cR 0 := hcommon i |
| simpa only [hc0,hid,map_one,one_mul,mul_one] using hform i |
| have hspace (i : J) : (F i).space (T i)=(A i).forwardSpace I S a c n (T i) := by |
| simp only [ShortBin.Gauge.space,BilateralGauge.forwardSpace,(hzero i).1,(hzero i).2] |
| intro i |
| simpa only [hspace] using hclose i |
| end SmoothObstruction.ManifoldCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
