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namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.FixedTop
open CosetGeometry
universe u
variable {A J : Type u} (P : MeetOrder A)
def Le : Option A → Option A → Prop
| _, none => True
| none, some _ => False
| some x, some y => P.le x y
def meet : Option A → Option A → Option A
| none, y => y
| x, none => x
| some x, some y => some (P.meet x y)
def order : MeetOrder (Option A) where
le := Le P
meet := meet P
refl x := by
cases x with
| none => trivial
| some x => exact P.refl x
trans {x y z} hxy hyz := by
cases x <;> cases y <;> cases z <;> simp only [Le] at *
exact P.trans hxy hyz
antisymm {x y} hxy hyx := by
cases x <;> cases y <;> simp only [Le] at *
exact congrArg some (P.antisymm hxy hyx)
meet_le_left x y := by
cases x <;> cases y <;> simp only [meet, Le]
· exact P.refl _
· exact P.meet_le_left _ _
meet_le_right x y := by
cases x <;> cases y <;> simp only [meet, Le]
· exact P.refl _
· exact P.meet_le_right _ _
le_meet {x y z} hxy hxz := by
cases x <;> cases y <;> cases z <;> simp only [meet, Le] at *
all_goals first | exact hxy | exact hxz | exact P.le_meet hxy hxz
def embedding : IdealEmbedding P (order P) where
toFun := some
le_iff _ _ := Iff.rfl
map_meet _ _ := rfl
ideal x z h := by
cases z with
| none => exact False.elim h
| some z => exact ⟨z, rfl⟩
theorem strict_top (x : A) : (order P).le ((embedding P).toFun x) none ∧
¬ (order P).le none ((embedding P).toFun x) := ⟨True.intro, id⟩
theorem greatest (x : Option A) : (order P).le x none := by
cases x <;> exact True.intro
theorem no_new_below (x : A) (z : Option A) (h : (order P).le z ((embedding P).toFun x)) :
∃ y, (embedding P).toFun y = z := (embedding P).ideal x z h
variable [Group J] (a : OrderAction J P)
def act (g : J) : Option A → Option A
| none => none
| some x => some ((a.iso g).toFun x)
def actionIso (g : J) : MeetIso (order P) (order P) where
toFun := act P a g
le_iff x y := by
cases x <;> cases y <;> simp only [act, order, Le]
exact (a.iso g).le_iff _ _
map_meet x y := by
cases x <;> cases y <;> simp only [act, order, meet]
exact congrArg some ((a.iso g).map_meet _ _)
inverse
| none => none
| some x => some ((a.iso g).inverse x)
left_inverse x := by
cases x with
| none => rfl
| some x => exact congrArg some ((a.iso g).left_inverse x)
right_inverse x := by
cases x with
| none => rfl
| some x => exact congrArg some ((a.iso g).right_inverse x)
def action : OrderAction J (order P) where
iso := actionIso P a
one x := by
cases x with
| none => rfl
| some x => exact congrArg some (a.one x)
mul g h x := by
cases x with
| none => rfl
| some x => exact congrArg some (a.mul g h x)
theorem fixes_top (g : J) : ((action P a).iso g).toFun none = none := rfl
theorem extends_action (g : J) (x : A) :
((action P a).iso g).toFun ((embedding P).toFun x) =
(embedding P).toFun ((a.iso g).toFun x) := rfl
end PartitionPilot.Compression.RecursiveMaps.FixedTop
namespace PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
open CosetGeometry
universe u
variable {A J R : Type u} [Group J] {P : MeetOrder A}
variable (f : Family P R) (a : OrderAction J P)
def Extension := Option (f.Node a)
noncomputable def extensionOrder : MeetOrder (f.Extension a) := FixedTop.order (f.gluedOrder a)
noncomputable def extensionAction : OrderAction (External J f.Letters) (f.extensionOrder a) :=
FixedTop.action (f.gluedOrder a) (f.externalAction a)
noncomputable def extensionCopy (w : External J f.Letters) : IdealEmbedding P (f.extensionOrder a) :=
(f.copy a w).comp (FixedTop.embedding (f.gluedOrder a))
theorem extension_greatest (x : f.Extension a) : (f.extensionOrder a).le x none :=
FixedTop.greatest (f.gluedOrder a) x
theorem extension_strict_top (w : External J f.Letters) (x : A) :
(f.extensionOrder a).le ((f.extensionCopy a w).toFun x) none ∧
¬ (f.extensionOrder a).le none ((f.extensionCopy a w).toFun x) :=
FixedTop.strict_top (f.gluedOrder a) ((f.copy a w).toFun x)
theorem extension_fixes_top (g : External J f.Letters) :
((f.extensionAction a).iso g).toFun none = none := rfl
theorem extensionAction_copy (g w : External J f.Letters) (x : A) :
((f.extensionAction a).iso g).toFun ((f.extensionCopy a w).toFun x) =
(f.extensionCopy a (g * w)).toFun x :=
congrArg some (f.externalAction_copy a g w x)
theorem extension_extends_old (j : J) (x : A) :
((f.extensionAction a).iso (old J f.Letters j)).toFun ((f.extensionCopy a 1).toFun x) =
(f.extensionCopy a 1).toFun ((a.iso j).toFun x) :=
congrArg some (f.extends_old_action a j x)
theorem extensionCopy_principal (w : External J f.Letters) (z : f.Extension a) :
(f.extensionOrder a).le z ((f.extensionCopy a w).toFun f.top) ↔
∃ x, (f.extensionCopy a w).toFun x = z :=
(f.extensionCopy a w).principal f.top f.greatest z
end PartitionPilot.Compression.RecursiveMaps.TileRequests.Family
end
end OAI
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