| 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 |
|
|
