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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Forcing.ExternalAction
namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
open Attachment CosetGeometry
universe u
variable {A J R : Type u} [Group J] {P : MeetOrder A}
variable (f : Family P R) (a : OrderAction J P)
abbrev FlatLabel := External J f.Letters × A
noncomputable def flatVertex (r : R) : Option (f.Child r) → IdealEmbedding P (f.tile r).order
| none => (f.tile r).root
| some j => (f.tile r).child j
def flatWord (w : External J f.Letters) (r : R) : Option (f.Child r) → External J f.Letters
| none => w
| some j => w * generator J f.Letters ⟨r, j⟩
theorem flatVertex_cover (r : R) (z : (f.tile r).Space) :
∃ i x, (f.flatVertex r i).toFun x = z := by
rcases (f.tile r).cover z with ⟨x, eq⟩ | ⟨j, x, eq⟩
· exact ⟨none, x, eq⟩
· exact ⟨some j, x, eq⟩
theorem flatVertex_copy (w : External J f.Letters) (r : R) (i : Option (f.Child r)) (x : A) :
(f.placedTile a w r).toFun ((f.flatVertex r i).toFun x) =
(f.copy a (f.flatWord w r i)).toFun x := by
cases i with
| none => exact f.root_identification a w r x
| some j => exact f.child_identification a w r j x
inductive FlatStep : f.FlatLabel (J := J) → f.FlatLabel (J := J) → Prop
| coordinate (w : External J f.Letters) (j : J) (x : A) :
FlatStep (w * old J f.Letters j, x) (w, (a.iso j).toFun x)
| tile (w : External J f.Letters) (r : R) (i j : Option (f.Child r)) (x y : A)
(equal : (f.flatVertex r i).toFun x = (f.flatVertex r j).toFun y) :
FlatStep (f.flatWord w r i, x) (f.flatWord w r j, y)
| symm {x y} : FlatStep x y → FlatStep y x
def FlatPath : f.FlatLabel (J := J) → f.FlatLabel (J := J) → Prop := Relation.ReflTransGen (f.FlatStep a)
theorem FlatPath.symm {x y : f.FlatLabel (J := J)} (h : f.FlatPath a x y) : f.FlatPath a y x := by
let : Std.Symm (f.FlatStep a) := ⟨fun _ _ step => FlatStep.symm step⟩
exact (Relation.ReflTransGen.stdSymm (r := f.FlatStep a)).symm x y h
theorem FlatStep.copy_eq {x y : f.FlatLabel (J := J)} (h : f.FlatStep a x y) :
(f.copy a x.1).toFun x.2 = (f.copy a y.1).toFun y.2 := by
induction h with
| coordinate w j x => exact f.copy_coordinate a w j x
| tile w r i j x y equal =>
rw [← f.flatVertex_copy a w r i x, ← f.flatVertex_copy a w r j y, equal]
| symm h ih => exact ih.symm
theorem FlatPath.copy_eq {x y : f.FlatLabel (J := J)} (h : f.FlatPath a x y) :
(f.copy a x.1).toFun x.2 = (f.copy a y.1).toFun y.2 := by
induction h with
| refl => rfl
| tail path step ih => exact ih.trans (FlatStep.copy_eq f a step)
def flatSetoid : Setoid (f.FlatLabel (J := J)) where
r := f.FlatPath a
iseqv := ⟨fun _ => Relation.ReflTransGen.refl,
fun h => FlatPath.symm f a h, fun h k => h.trans k⟩
def FlatQuotient := Quotient (f.flatSetoid a)
def flatClass (w : External J f.Letters) (x : A) : f.FlatQuotient a :=
Quotient.mk (f.flatSetoid a) (w, x)
theorem flatClass_coordinate (w : External J f.Letters) (j : J) (x : A) :
f.flatClass a (w * old J f.Letters j) x = f.flatClass a w ((a.iso j).toFun x) :=
Quotient.sound (Relation.ReflTransGen.single (FlatStep.coordinate w j x))
theorem flatClass_tile (w : External J f.Letters) (r : R) (i j : Option (f.Child r)) (x y : A)
(equal : (f.flatVertex r i).toFun x = (f.flatVertex r j).toFun y) :
f.flatClass a (f.flatWord w r i) x = f.flatClass a (f.flatWord w r j) y :=
Quotient.sound (Relation.ReflTransGen.single (FlatStep.tile w r i j x y equal))
end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
end
end OAI
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