| import OAI.SetTheory.PartitionPrinciple.Transfinite.InaccessibleBounds | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Compression.RecursiveMaps | |
| universe u v w | |
| variable {X : Type u} {Column : Type v} {Coord : Type w} {P : MeetOrder X} | |
| variable (C : LinearHeightOrder Column) (D : LinearHeightOrder Coord) | |
| variable (depth : X → Nat) (columns : Nat → Column) | |
| variable (seed : {x : X // depth x = 0} → Column × Coord) | |
| variable (coordinate : {x : X // depth x ≠ 0} → Coord) | |
| def depthHeightValue (x : X) : Column × Coord := | |
| if h : depth x = 0 then seed ⟨x, h⟩ else (columns (depth x), coordinate ⟨x, h⟩) | |
| theorem depthHeightValue_old (x : X) (h : depth x = 0) : | |
| depthHeightValue depth columns seed coordinate x = seed ⟨x, h⟩ := by | |
| simp only [depthHeightValue, dite_eq_left h] | |
| theorem depthHeightValue_fresh (x : X) (h : depth x ≠ 0) : | |
| depthHeightValue depth columns seed coordinate x = (columns (depth x), coordinate ⟨x, h⟩) := by | |
| simp only [depthHeightValue, dite_eq_right h] | |
| def depthLayerHeight | |
| (monotone : ∀ x y, P.le x y → depth x ≤ depth y) | |
| (increasing : ∀ n m, n < m → C.lt (columns n) (columns m)) | |
| (above : ∀ x n, C.lt (seed x).1 (columns n)) | |
| (seedStrict : ∀ x y, P.le x.val y.val → x.val ≠ y.val → (C.lex D).lt (seed x) (seed y)) | |
| (coordinateStrict : ∀ x y, depth x.val = depth y.val → P.le x.val y.val → | |
| x.val ≠ y.val → D.lt (coordinate x) (coordinate y)) : | |
| StrictHeight P (C.lex D).toHeightOrder where | |
| value := depthHeightValue depth columns seed coordinate | |
| strict x y lower different := by | |
| have depths := monotone x y lower | |
| by_cases hx : depth x = 0 | |
| · rw [depthHeightValue_old depth columns seed coordinate x hx] | |
| by_cases hy : depth y = 0 | |
| · rw [depthHeightValue_old depth columns seed coordinate y hy] | |
| exact seedStrict ⟨x, hx⟩ ⟨y, hy⟩ lower different | |
| · rw [depthHeightValue_fresh depth columns seed coordinate y hy] | |
| exact Or.inl (above ⟨x, hx⟩ (depth y)) | |
| · have hy : depth y ≠ 0 := by omega | |
| rw [depthHeightValue_fresh depth columns seed coordinate x hx, | |
| depthHeightValue_fresh depth columns seed coordinate y hy] | |
| by_cases same : depth x = depth y | |
| · exact Or.inr ⟨congrArg columns same, coordinateStrict ⟨x, hx⟩ ⟨y, hy⟩ same lower different⟩ | |
| · exact Or.inl (increasing _ _ (by omega)) | |
| theorem depthHeightValue_column_bound (cap : Column) | |
| (oldBound : ∀ x, C.lt (seed x).1 cap) (columnBound : ∀ n, C.lt (columns n) cap) (x : X) : | |
| C.lt (depthHeightValue depth columns seed coordinate x).1 cap := by | |
| by_cases h : depth x = 0 | |
| · rw [depthHeightValue_old depth columns seed coordinate x h] | |
| exact oldBound ⟨x, h⟩ | |
| · rw [depthHeightValue_fresh depth columns seed coordinate x h] | |
| exact columnBound (depth x) | |
| end PartitionPilot.Compression.RecursiveMaps | |
| end | |
| end OAI | |