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5711324 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.Analysis.CircleDomains.Selection.InitialBlockPiece
import OAI.Analysis.CircleDomains.Selection.PrependOldBlockJump
import OAI.Analysis.CircleDomains.Selection.SinglePieceBlock
import OAI.Analysis.CircleDomains.Selection.ReanchorBlockCode
namespace OAI
/-! Domain-endpoint and disk-origin invariants for normalization of mixed
paid chains into grouped jump codes. -/
open Set MeasureTheory
noncomputable section
namespace CircleDomainRigidity
variable {X Y I K : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [MeasurableSpace C(PathTime, X)] {n : ℕ}
def BlockPathsInDomain (d : PathCodeData X Y) (P : BoundedBlockPattern I n)
(Z : JumpBlockCode d P) : Prop :=
∀b j, (Z.paths b j).1.val ⊥ ∈ d.quotientDomain ∧ (Z.paths b j).1.val ⊤ ∈ d.quotientDomain
def BlockDisksFrom (d : PathCodeData X Y) (W : K → Set X) (P : BoundedBlockPattern I n)
(Z : JumpBlockCode d P) : Prop := ∀b i, ∃k, Z.disks b i = W k
theorem jumpBlockCode_initial_eq
(d : PathCodeData X Y) (ν : Measure X)
(N : ∀n : ℕ, (Fin (n+1) → ℕ) → Set (Fin (n+1) → UnitAddCircle))
(B H : Set X) (O : I → Set X) (P : BoundedBlockPattern I n) (Z : JumpBlockCode d P)
{x : X} (hZ : IsJumpBlockCode d ν N B {x} H O P Z) : (Z.paths 0 0).1.val ⊥ = x := by
have h := (hZ.start 0).1
simpa only [Fin.castSucc_zero, blockAnchors_zero, mem_singleton_iff] using h
theorem jumpBlockCode_terminal_eq
(d : PathCodeData X Y) (ν : Measure X)
(N : ∀n : ℕ, (Fin (n+1) → ℕ) → Set (Fin (n+1) → UnitAddCircle))
(B F : Set X) (O : I → Set X) (P : BoundedBlockPattern I n) (Z : JumpBlockCode d P)
{y : X} (hZ : IsJumpBlockCode d ν N B F {y} O P Z) :
(Z.paths (Fin.last P.1.val) (Fin.last (P.2.2 (Fin.last P.1.val)).val)).1.val ⊤ = y := by
have h := (hZ.finish (Fin.last P.1.val)).1
simpa only [Fin.succ_last, blockAnchors_last, mem_singleton_iff] using h
omit [MeasurableSpace X] in
theorem singlePieceBlock_inDomain (d : PathCodeData X Y) (γ : TaggedPhysicalPath d)
(h0 : γ.1.val ⊥ ∈ d.quotientDomain) (h1 : γ.1.val ⊤ ∈ d.quotientDomain) :
BlockPathsInDomain d (singlePiecePattern I) (singlePieceBlock d γ) := fun _ _ => ⟨h0, h1⟩
omit [MeasurableSpace X] in
theorem singlePieceBlock_disksFrom (d : PathCodeData X Y) (γ : TaggedPhysicalPath d) (W : K → Set X) :
BlockDisksFrom d W (singlePiecePattern I) (singlePieceBlock d γ) := fun _ i => Fin.elim0 i
omit [MeasurableSpace X] in
theorem replaceInitialTruePiece_inDomain (d : PathCodeData X Y) (P : BoundedBlockPattern I n)
(Z : JumpBlockCode d P) (hZ : BlockPathsInDomain d P Z) (γ : TaggedPhysicalPath d)
(h0 : γ.1.val ⊥ ∈ d.quotientDomain) (h1 : γ.1.val ⊤ ∈ d.quotientDomain) :
BlockPathsInDomain d P (replaceInitialTruePiece d P Z γ) := by
intro b
refine Fin.cases ?_ (fun b j => hZ b.succ j) b
intro j
exact Fin.cases ⟨h0, h1⟩ (fun j => hZ 0 j.succ) j
omit [MeasurableSpace X] in
theorem replaceInitialTruePiece_disksFrom (d : PathCodeData X Y) (W : K → Set X)
(P : BoundedBlockPattern I n) (Z : JumpBlockCode d P) (hZ : BlockDisksFrom d W P Z)
(γ : TaggedPhysicalPath d) : BlockDisksFrom d W P (replaceInitialTruePiece d P Z γ) := hZ
omit [MeasurableSpace X] in
theorem prependNewBlockJump_inDomain (d : PathCodeData X Y) (P : BoundedBlockPattern I n)
(Z : JumpBlockCode d P) (hZ : BlockPathsInDomain d P Z) (γ : TaggedPhysicalPath d) (V : Set X)
(h0 : γ.1.val ⊥ ∈ d.quotientDomain) (h1 : γ.1.val ⊤ ∈ d.quotientDomain) :
BlockPathsInDomain d (prependNewJumpPattern P) (prependNewBlockJump d P Z γ V) := by
intro b
refine Fin.cases ?_ (fun b j => hZ b.succ j) b
intro j
exact Fin.cases ⟨h0, h1⟩ (fun j => hZ 0 j) j
omit [MeasurableSpace X] in
theorem prependNewBlockJump_disksFrom (d : PathCodeData X Y) (W : K → Set X)
(P : BoundedBlockPattern I n) (Z : JumpBlockCode d P) (hZ : BlockDisksFrom d W P Z)
(γ : TaggedPhysicalPath d) (k : K) :
BlockDisksFrom d W (prependNewJumpPattern P) (prependNewBlockJump d P Z γ (W k)) := by
intro b
refine Fin.cases ?_ (fun b i => hZ b.succ i) b
intro i
exact Fin.cases ⟨k, rfl⟩ (fun i => hZ 0 i) i
omit [MeasurableSpace X] in
theorem prependOldBlockJump_inDomain (d : PathCodeData X Y) (P : BoundedBlockPattern I n)
(Z : JumpBlockCode d P) (hZ : BlockPathsInDomain d P Z) (γ : TaggedPhysicalPath d) (i : I)
(h0 : γ.1.val ⊥ ∈ d.quotientDomain) (h1 : γ.1.val ⊤ ∈ d.quotientDomain) :
BlockPathsInDomain d (prependOldJumpPattern P i) (prependOldBlockJump d P Z γ i) := by
intro b
exact Fin.cases (fun _ => ⟨h0, h1⟩) (fun b j => hZ b j) b
omit [MeasurableSpace X] in
theorem prependOldBlockJump_disksFrom (d : PathCodeData X Y) (W : K → Set X)
(P : BoundedBlockPattern I n) (Z : JumpBlockCode d P) (hZ : BlockDisksFrom d W P Z)
(γ : TaggedPhysicalPath d) (i : I) :
BlockDisksFrom d W (prependOldJumpPattern P i) (prependOldBlockJump d P Z γ i) := by
intro b
exact Fin.cases (fun j => Fin.elim0 j) (fun b j => hZ b j) b
end CircleDomainRigidity
end
end OAI
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