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namespace OAI
section
namespace PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun
open CountableColumns RationalHeights
variable (Label : Type 1)
local instance : Nonempty Column := ⟨natural 0⟩
noncomputable def FinalPoint : Type 1 := (diagram Label).orders.system.Union
noncomputable def FinalActor : Type 1 := (diagram Label).groups.system.Union
noncomputable instance finalGroup : Group (FinalActor Label) :=
inferInstanceAs (Group (diagram Label).groups.system.Union)
noncomputable def finalOrder : MeetOrder (FinalPoint Label) := (diagram Label).orders.unionOrder
noncomputable def finalAction : CosetGeometry.OrderAction (FinalActor Label) (finalOrder Label) :=
(diagram Label).actions.unionAction
noncomputable def finalHeight : StrictHeight (finalOrder Label)
(lexOrder CountableColumns.order).toHeightOrder := (diagram Label).heights.unionHeight
noncomputable def finalBottom : FinalPoint Label := (diagram Label).unionBottom
theorem finalBottom_least (x : FinalPoint Label) : (finalOrder Label).le (finalBottom Label) x :=
(diagram Label).unionBottom_least x
noncomputable def finalInclusion (a : Column) : MeetEmbedding (stage Label a).order (finalOrder Label) :=
(diagram Label).orders.inclusion a
noncomputable def finalGroupInclusion (a : Column) : (stage Label a).Actor →* FinalActor Label :=
(diagram Label).groups.inclusion a
theorem finalGroupInclusion_injective (a : Column) : Function.Injective (finalGroupInclusion Label a) :=
(diagram Label).groups.inclusion_injective a
theorem final_points_coherent {a b : Column} (h : a ≤ b) (x : (stage Label a).Point) :
(finalInclusion Label b).toFun ((transition Label h).points.toFun x) =
(finalInclusion Label a).toFun x := (diagram Label).orders.system.point_map h x
theorem final_actors_coherent {a b : Column} (h : a ≤ b) (g : (stage Label a).Actor) :
finalGroupInclusion Label b ((transition Label h).actors g) = finalGroupInclusion Label a g :=
(diagram Label).groups.system.point_map h g
theorem final_point_cover (x : FinalPoint Label) : ∃ a y, (finalInclusion Label a).toFun y = x :=
(diagram Label).orders.system.cover x
theorem final_actor_cover (g : FinalActor Label) : ∃ a h, finalGroupInclusion Label a h = g :=
(diagram Label).groups.system.cover g
theorem final_height_stage (a : Column) (x : (stage Label a).Point) :
(finalHeight Label).value ((finalInclusion Label a).toFun x) = (stage Label a).height.value x := rfl
theorem final_action_stage (a : Column) (g : (stage Label a).Actor) (x : (stage Label a).Point) :
((finalAction Label).iso (finalGroupInclusion Label a g)).toFun ((finalInclusion Label a).toFun x) =
(finalInclusion Label a).toFun (((stage Label a).action.iso g).toFun x) :=
(diagram Label).actions.act_point a g x
theorem final_agreement (a : Column) (g k : (stage Label a).Actor) (v : (stage Label a).Point)
(agree : ∀ x, (stage Label a).order.le x v →
((stage Label a).action.iso g).toFun x = ((stage Label a).action.iso k).toFun x)
(x : FinalPoint Label) (lower : (finalOrder Label).le x ((finalInclusion Label a).toFun v)) :
((finalAction Label).iso (finalGroupInclusion Label a g)).toFun x =
((finalAction Label).iso (finalGroupInclusion Label a k)).toFun x :=
(diagram Label).actions.union_agreement (diagram Label).agreements a g k v agree x lower
theorem final_no_maximal (x : FinalPoint Label) :
∃ y, (finalOrder Label).le x y ∧ ¬ (finalOrder Label).le y x := by
rcases final_point_cover Label x with ⟨a, x, rfl⟩
let b := start a
have h : a < b := lt_start a
refine ⟨(finalInclusion Label b).toFun (stage Label b).top, ?_, ?_⟩
· rw [← final_points_coherent Label h.le x]
exact ((finalInclusion Label b).le_iff _ _).mpr ((stage Label b).greatest _)
· intro lower
rw [← final_points_coherent Label h.le x] at lower
exact strict_top Label h x (((finalInclusion Label b).le_iff _ _).mp lower)
theorem final_no_greatest : ¬ ∃ top : FinalPoint Label, ∀ x, (finalOrder Label).le x top := by
rintro ⟨top, greatest⟩
rcases final_no_maximal Label top with ⟨y, _, strict⟩
exact strict (greatest y)
end PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun
end
section
namespace PartitionPilot.Compression.RecursiveMaps.MainStage
universe u
theorem Transition.preservesProfiles_transport {s t t' : MainStage.{u}}
(f : Transition s t) (eq : t = t') :
PreservesProfiles (f.transport rfl eq).points ↔ PreservesProfiles f.points := by
cases eq
rfl
open CountableColumns
theorem ordinalStep_preservesProfiles (Label : Type 1) (a : Column) (d : Diagram (Set.Iio a))
(preserves : ∀ {i j} (h : i ≤ j), PreservesProfiles (d.transition h).points)
(i : Set.Iio a) : PreservesProfiles ((ordinalStep Label a d).leg i).points := by
classical
have inhabited : Nonempty (Set.Iio a) := ⟨i⟩
by_cases hasLast : ∃ last : Set.Iio a, ∀ i, i ≤ last
· have eq : ordinalStep Label a d =
(d.lastCone hasLast.choose hasLast.choose_spec).extend (d := d)
((d.stage hasLast.choose).nextTransition Label) :=
(dite_eq_left inhabited).trans (dite_eq_left hasLast)
rw [eq]
exact preservesProfiles_comp (d.transition (hasLast.choose_spec i)).points
((d.stage hasLast.choose).nextTransition Label).points
(preserves (hasLast.choose_spec i)) ((d.stage hasLast.choose).next_preservesProfiles Label)
· let := inhabited
have eq : ordinalStep Label a d = d.limitCone :=
(dite_eq_left inhabited).trans (dite_eq_right hasLast)
rw [eq]
exact d.limit_preservesProfiles preserves i
namespace OrdinalRun
variable (Label : Type 1)
theorem transition_preservesProfiles_at (b : Column) :
∀ a (h : a ≤ b), PreservesProfiles (transition Label h).points := by
refine (show WellFounded ((· < ·) : Column → Column → Prop) from wellFounded_lt).induction
(C := fun b => ∀ a (h : a ≤ b), PreservesProfiles (transition Label h).points) b ?_
intro b ih a h
rcases lt_or_eq_of_le h with lower | rfl
· apply (Transition.preservesProfiles_transport (transition Label h)
(Transfinite.stage_generated (ordinalStep Label) b)).mp
have maps : (transition Label h).transport rfl
(Transfinite.stage_generated (ordinalStep Label) b) =
(ordinalStep Label b (before Label b)).leg ⟨a, lower⟩ :=
Transfinite.transition_generated (ordinalStep Label) b ⟨a, lower⟩
rw [maps]
exact ordinalStep_preservesProfiles Label b (before Label b)
(fun {_ j} hj => ih j.val j.property _ hj) ⟨a, lower⟩
· rw [transition_self]
exact preservesProfiles_identity (stage Label a).order
theorem final_preservesProfiles (a : Column) : PreservesProfiles (finalInclusion Label a) :=
(diagram Label).union_preservesProfiles
(fun {_ j} h => transition_preservesProfiles_at Label j _ h) a
end OrdinalRun
end PartitionPilot.Compression.RecursiveMaps.MainStage
end
end OAI
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