File size: 2,922 Bytes
3277efa | 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 | 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
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