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openAI-math-dataset / lean /OAI /Geometry /Cannon /Flows /FlowPatternSetup.lean
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Add lean source and binary files (part 66)
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import Mathlib
import OAI.Geometry.Cannon.Flows.FlowSetup
import OAI.Geometry.Cannon.Flows.IntervalAvoidingShortRange
namespace OAI
section
section
section
noncomputable section
section
section
section
section
namespace CannonApprox
open Set Metric Filter Topology CannonNode CannonMagnification CannonPairs
variable {X Γ : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X]
{action : Γ → X ≃ₜ X} {ι : ℕ → Type*} [∀ n, Fintype (ι n)]
{p : ∀ n, ι n → X} {r : ℕ → ℝ}
omit [CompactSpace X] in
lemma FlowSetup.margins {T : RecordData action ι p r} {ell L CE LE : ℝ} {Q : ℕ}
(F : FlowSetup T ell L CE LE Q) (hL : 1 ≤ L) (_hCE : 0 ≤ CE) (hLE : 0 ≤ LE) :
2 < F.A ∧ 0 ≤ F.B ∧ 0 < F.c₀ ∧ 2*L ≤ F.c₀ ∧
CE*(2*L) ≤ (F.cPat/2)*(F.A-2)^2 ∧
LE*(10*F.cPat)*(F.B+1)*(F.B+2) < F.c₀ ∧ 4*(F.B+1) < F.H := by
have hc : 0 < F.c₀ := by linarith only [F.core,hL]
have hA : 2 < F.A := by rw [F.inner]; nlinarith only [hc,hL]
have hB : 0 ≤ F.B := by rw [F.outer]; linarith only [hA]
have hpat : CE*(2*L) = (F.cPat/2)*(F.A-2)^2 := by
rw [F.pattern]
field_simp [ne_of_gt (sub_pos.mpr hA)]
have hm : LE*(10*F.cPat)*(F.B+1)*(F.B+2) ≤ 12*F.cPat*(F.B+2)^2*LE := by
have hb : 0 ≤ F.B+2 := by positivity
have hnorm : LE*(10*F.cPat)*(F.B+1)*(F.B+2) =
(10*(F.B+1))*((F.cPat*LE)*(F.B+2)) := by ring
rw [hnorm]
calc
_ ≤ (12*(F.B+2))*((F.cPat*LE)*(F.B+2)) :=
mul_le_mul_of_nonneg_right (by linarith only [hB])
(mul_nonneg (mul_nonneg F.pattern_pos.le hLE) hb)
_ = _ := by ring
exact ⟨hA,hB,hc,by linarith only [F.core],hpat.le,hm.trans_lt F.margin,
(le_max_left _ _).trans_lt F.separation⟩
omit [CompactSpace X] in
lemma FlowSetup.center_puncture {T : RecordData action ι p r} {ell L CE LE : ℝ} {Q : ℕ}
(F : FlowSetup T ell L CE LE Q) (hB : 0 ≤ F.B+1) (g : ℕ) (i : Fin (F.S.flow.size g)) :
F.ε ≤ dist (F.S.center g i) T.z₀ := by
by_contra hn
have hi := mem_closedBall.mp (F.S.center_mem hB g i)
apply T.z₀not (F.local_domain ?_)
rw [mem_closedBall]
have ht := dist_triangle T.z₀ (F.S.center g i) T.x₀
rw [dist_comm T.z₀ (F.S.center g i)] at ht
linarith only [ht,hi,lt_of_not_ge hn]
theorem RecordData.pattern_pair (T : RecordData action ι p r)
(hr : ∀ n, 0 < r n) (hrlim : Tendsto r atTop (𝓝 0))
(hpnet : ∀ n x, ∃ i, dist (p n i) x ≤ r n)
(gi gj : Γ) (xi xj pi pj : X) (R U V A B cutoff ε : ℝ)
(hR : 0 < R) (hV : 0 < V) (hA : 0 < A) (hB : 1 ≤ B)
(hwide : 3*U/T.H_diameter < A) (hroom : 2*U/A ≤ ε)
(hΩi : closedBall xi ε ⊆ T.Ω) (hΩj : closedBall xj ε ⊆ T.Ω)
(hui : ∀ y, dist (action gi y) pi*denominator xi R y ≤ U*R)
(huj : ∀ y, dist (action gj y) pj*denominator xj R y ≤ U*R)
(hli : ∀ y, denominator xi R y ≤ cutoff →
R ≤ V*dist (action gi y) pi*denominator xi R y)
(hlj : ∀ y, denominator xj R y ≤ cutoff →
R ≤ V*dist (action gj y) pj*denominator xj R y)
(hsmall : B*R ≤ cutoff) (havoid : A*R < 1/(V*B)) (hfar : 2*A*R ≤ dist pi pj)
(Pat : Bool → Set X)
(hPati : ∀ s, (action gi).symm '' Pat s ⊆ closedBall xi (B*R))
(hPatj : ∀ s, (action gj).symm '' Pat s ⊆ closedBall xj (B*R))
(v₀ v₁ : Bool → X) (γ : ∀ s, Path (v₀ s) (v₁ s))
(hγ : ∀ s, range (γ s) ⊆ Pat s)
(κ li lj : ℝ) (hκ : 0 < κ) (hli_pos : 0 < li) (hlj_pos : 0 < lj)
(hgap : ∀ s, κ*li ≤ T.f ((action gi).symm (v₁ s))-T.f ((action gi).symm (v₀ s)))
(hsep : ∀ x ∈ Pat false, ∀ y ∈ Pat true,
T.f ((action gj).symm x)+lj/4 ≤ T.f ((action gj).symm y)) :
(min (κ/4) (1/16))*li ≤ CannonCrossEnergy.oscillation T.f
((action gi).symm '' closedBall pj (A*R)) ∨
(min (κ/4) (1/16))*lj ≤ CannonCrossEnergy.oscillation T.f
((action gj).symm '' closedBall pi (A*R)) := by
have hP : ∀ s z, z ∈ range (γ s) → A*R < dist z pi ∧ A*R < dist z pj := by
intro s z hz
exact ⟨havoid.trans_le (pattern_puncture_distance (action gi) xi pi R V B cutoff
hR hV hB hsmall hli (Pat s) (hPati s) z (hγ s hz)),
havoid.trans_le (pattern_puncture_distance (action gj) xj pj R V B cutoff
hR hV hB hsmall hlj (Pat s) (hPatj s) z (hγ s hz))⟩
have henergy (g : Γ) (K : Set X) (hK : IsCompact K) (hKΩ : (action g).symm '' K ⊆ T.Ω) :
Tendsto (fun n => meshEnergy (p n) (r n) 2 (T.H n ∘ (action g).symm) K) atTop (𝓝 0) := by
apply T.energy g K hK _ 2 (by norm_num)
intro z hz
exact ⟨(action g).symm z,hKΩ (mem_image_of_mem _ hz),(action g).apply_symm_apply z⟩
have hh := fixed_source_pair ι p r hr hrlim hpnet T.Ω T.H T.f T.continuous T.H_continuous
T.unit T.H_limit T.H_diameter T.H_diameter_pos T.H_level
(action gi) (action gj) xi xj pi pj R U A ε hR hA hwide hroom hΩi hΩj hui huj hfar
v₀ v₁ γ hP (κ*li) (lj/4) (mul_pos hκ hli_pos) (by positivity) hgap
(fun x hx y hy => hsep x (hγ false hx) y (hγ true hy)) (henergy gi) (henergy gj)
rcases hh with hh | hh
· left
calc
_ ≤ (κ/4)*li := mul_le_mul_of_nonneg_right (min_le_left _ _) hli_pos.le
_ = (κ*li)/4 := by ring
_ ≤ _ := hh
· right
calc
_ ≤ (1/16)*lj := mul_le_mul_of_nonneg_right (min_le_right _ _) hlj_pos.le
_ = (lj/4)/4 := by ring
_ ≤ _ := hh
end CannonApprox
end
section
section
section
section
section
section
section
section
section
namespace CannonCapacity
open Finset Set
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
section
open Finset
namespace CannonCapacity
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
namespace CannonCapacity
open Set Metric Finset
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
section
namespace CannonCapacity
open Set Metric Finset CannonAnnular
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
section
namespace CannonCapacity
open Set Metric Finset CannonAnnular CannonPacking
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
namespace CannonCapacity
open Set Metric Finset CannonAnnular
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
section
namespace CannonCapacity
open Set Metric Finset CannonAnnular CannonPacking
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
section
open Set Metric Finset
namespace CannonCapacity
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
namespace CannonCapacity
open Set Metric Finset CannonPacking CannonAnnular Filter Topology
attribute [local instance] Classical.propDecidable
end CannonCapacity
end
namespace CannonCapacity
open Set Finset
attribute [local instance] Classical.propDecidable
end CannonCapacity
namespace CannonCapacity
open Set Metric Finset
attribute [local instance] Classical.propDecidable
end CannonCapacity
namespace CannonNode
open Set Metric Finset CannonCapacity CannonFlow
attribute [local instance] Classical.propDecidable
end CannonNode
namespace CannonNode
open Set Metric Finset CannonCapacity CannonFlow CannonPacking CannonAnnular Filter Topology
attribute [local instance] Classical.propDecidable
end CannonNode
end
namespace CannonNode
open Set Metric Finset Filter Topology CannonPacking CannonAnnular CannonMagnification
attribute [local instance] Classical.propDecidable
end CannonNode
end
section
section
open Set Filter Finset
namespace CannonFlow
namespace FiniteFlow
attribute [local instance] Classical.propDecidable
variable (T : FiniteFlow)
end FiniteFlow
namespace FiniteFlow
attribute [local instance] Classical.propDecidable
variable (T : FiniteFlow)
end FiniteFlow
end CannonFlow
namespace CannonFlow
open Finset
attribute [local instance] Classical.propDecidable
end CannonFlow
end
namespace CannonNode
open Set Metric Finset CannonFlow
attribute [local instance] Classical.propDecidable
end CannonNode
end
namespace CannonNode
open Set Metric Finset Filter Topology CannonPacking CannonAnnular CannonMagnification CannonFlow
attribute [local instance] Classical.propDecidable
end CannonNode
end
section
open Set Filter Finset
namespace CannonFlow
namespace FiniteFlow
attribute [local instance] Classical.propDecidable
variable (T : FiniteFlow)
end FiniteFlow
namespace FiniteFlow
attribute [local instance] Classical.propDecidable
variable (T : FiniteFlow)
end FiniteFlow
end CannonFlow
namespace CannonFlow
open Finset
attribute [local instance] Classical.propDecidable
end CannonFlow
end
section
open Set Finset MeasureTheory Filter
namespace CannonPairCost
attribute [local instance] Classical.propDecidable
end CannonPairCost
end
namespace CannonNode.SourceConstruction
open Set Metric Finset Filter Topology CannonFlow
attribute [local instance] Classical.propDecidable
variable {X : Type*} [MetricSpace X] {f : X → ℝ} {x₀ : X}
{ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ}
variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n)
end CannonNode.SourceConstruction
namespace CannonNode.SourceConstruction
open Set Metric Finset MeasureTheory CannonFlow
attribute [local instance] Classical.propDecidable
variable {X : Type*} [MetricSpace X] {f : X → ℝ} {x₀ : X}
{ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ}
variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n)
end CannonNode.SourceConstruction
end
namespace CannonNode
open Set Metric CannonCapacity CannonMagnification
namespace SourceConstruction
attribute [local instance] Classical.propDecidable
variable {X : Type*} [MetricSpace X] [ProperSpace X] {f : X → ℝ} {x₀ : X}
{ε ell c₀ A B H ρ : ℝ} {qdim n : ℕ}
variable (S : SourceConstruction f x₀ ε ell c₀ A B H ρ qdim n)
end SourceConstruction
end CannonNode
end
namespace CannonApprox
open Set Metric CannonNode CannonMagnification
variable {X Γ : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X]
{action : Γ → X ≃ₜ X} {ι : ℕ → Type*} [∀ n, Fintype (ι n)]
{p : ∀ n, ι n → X} {r : ℕ → ℝ}
structure FlowPatternSetup (T : RecordData action ι p r) {ell L CE LE : ℝ} {Q : ℕ}
(F : FlowSetup T ell L CE LE Q)
(h : ∀ g, Fin (F.S.flow.size g) → X ≃ₜ X) where
net : Finset X
net_nonempty : net.Nonempty
pat : ∀ g, Fin (F.S.flow.size g) → Bool → net
data : ∀ g i, F.S.good (F.S.state g i) →
∃ v₀ v₁ : Bool → X, ∃ γ : ∀ s, Path (v₀ s) (v₁ s),
(∀ s, range (γ s) ⊆ closedBall ((pat g i s : net) : X)
(3*F.cPat*radius ell (160*(Q+1)*F.n))) ∧
(∀ s, F.S.κ*F.S.len g i ≤ T.f ((h g i).symm (v₁ s))-T.f ((h g i).symm (v₀ s))) ∧
(∀ s, (h g i).symm '' closedBall ((pat g i s : net) : X)
(3*F.cPat*radius ell (160*(Q+1)*F.n)) ⊆ closedBall (F.S.center g i) ((F.B+2)*F.S.scale g)) ∧
∀ z ∈ closedBall ((pat g i false : net) : X) (3*F.cPat*radius ell (160*(Q+1)*F.n)),
∀ w ∈ closedBall ((pat g i true : net) : X) (3*F.cPat*radius ell (160*(Q+1)*F.n)),
T.f ((h g i).symm z)+F.S.len g i/4 ≤ T.f ((h g i).symm w)
theorem exists_flow_patterns (T : RecordData action ι p r) {ell L CE LE : ℝ} {Q : ℕ}
(F : FlowSetup T ell L CE LE Q) (hL : 1 ≤ L) (hCE : 0 ≤ CE) (hLE : 0 ≤ LE)
(hpaths : ∀ x u w : X, ∀ r : ℝ, 0 < r → dist u x ≤ r → dist w x ≤ r →
JoinedIn (ball x (L*r)) u w)
(h : ∀ g, Fin (F.S.flow.size g) → X ≃ₜ X)
(hexp : ∀ g i, Expansion (h g i) (F.S.center g i) (F.S.scale g) CE LE) :
Nonempty (FlowPatternSetup T F h) := by
classical
obtain ⟨hA,hB,hc,hcore,hcoef₁,hcoef₂,hH⟩ := F.margins hL hCE hLE
obtain ⟨Z,_,hnet,_⟩ := CannonNet.exists_compact_net (X:=X)
(F.cPat*radius ell (160*(Q+1)*F.n)) (mul_pos F.pattern_pos (radius_pos _ _))
obtain ⟨z,hz,_⟩ := hnet T.x₀
have hZ : Z.Nonempty := ⟨z,hz⟩
let ballOf : (Bool → Z) → Bool → Set X := fun pat s =>
closedBall (pat s : X) (3*F.cPat*radius ell (160*(Q+1)*F.n))
let HasPattern (g : ℕ) (i : Fin (F.S.flow.size g)) (pat : Bool → Z) : Prop :=
∃ v₀ v₁ : Bool → X, ∃ γ : ∀ s, Path (v₀ s) (v₁ s),
(∀ s, range (γ s) ⊆ ballOf pat s) ∧
(∀ s, F.S.κ*F.S.len g i ≤ T.f ((h g i).symm (v₁ s))-T.f ((h g i).symm (v₀ s))) ∧
(∀ s, (h g i).symm '' ballOf pat s ⊆ closedBall (F.S.center g i) ((F.B+2)*F.S.scale g)) ∧
∀ z ∈ ballOf pat false, ∀ w ∈ ballOf pat true,
T.f ((h g i).symm z)+F.S.len g i/4 ≤ T.f ((h g i).symm w)
have hp : ∀ g i, F.S.good (F.S.state g i) → ∃ pat, HasPattern g i pat := by
intro g i hi
have hexp' : Expansion (h g i) (F.S.state g i).center
(radius ell (F.S.state g i).depth) CE LE := by
rw [F.S.state_depth g i]
exact hexp g i
obtain ⟨pat,v₀,v₁,γ,hγ,hgap,hball,hsep⟩ := F.S.node_pattern (fun z : Z => (z:X))
T.Ω T.continuous F.local_domain L CE LE F.cPat hL hCE hLE F.pattern_pos
hA hB hcore hcoef₁ hcoef₂ F.small
(fun x => by obtain ⟨z,hz,hd⟩ := hnet x; exact ⟨⟨z,hz⟩,hd⟩) hpaths
(F.S.state g i) hi (h g i) hexp'
refine ⟨pat,v₀,v₁,γ,hγ,hgap,?_,hsep⟩
intro s y hy
obtain ⟨w,hw,hball⟩ := hball s
obtain ⟨v,hv,rfl⟩ := hy
obtain ⟨u,hu,huv⟩ := hball hv
rw [← huv,(h g i).symm_apply_apply]
have hloc := (F.S.Yloc (F.S.state g i) w hw).2
have hsm := mul_le_mul_of_nonneg_right F.S.pattern_small (radius_pos ell (F.S.state g i).depth).le
have ht := dist_triangle u w (F.S.center g i)
change dist u w < F.c₀*radius ell ((F.S.state g i).depth+160*(Q+1)*F.n) at hu
rw [radius_add] at hu
rw [mem_closedBall]
rw [F.S.state_depth g i] at hu hloc hsm
change F.c₀*radius ell (160*(Q+1)*F.n)*F.S.scale g ≤ 1*F.S.scale g at hsm
change dist w (F.S.center g i) ≤ (F.B+1)*F.S.scale g at hloc
change dist u w < F.c₀*(radius ell (160*(Q+1)*F.n)*F.S.scale g) at hu
nlinarith only [ht,hu,hloc,hsm]
let pat : ∀ g, Fin (F.S.flow.size g) → Bool → Z := fun g i =>
if hi : F.S.good (F.S.state g i) then (hp g i hi).choose else fun _ => ⟨z,hz⟩
refine ⟨⟨Z,hZ,pat,?_⟩⟩
intro g i hi
dsimp only [pat]
rw [dite_eq_left hi]
exact (hp g i hi).choose_spec
end CannonApprox
end
end
end
end
end
end
end
end OAI