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import Mathlib.Data.Prod.Lex
import Mathlib.Order.WithBot
namespace OAI
namespace MatroidProphet.Pivots
open Set Finset
variable {α A T : Type*} [Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T]
abbrev BlockOld (F : T → Set α) :=
{p : T ×ₗ α // (ofLex p).2 ∈ F (ofLex p).1}
noncomputable instance (F : T → Set α) : Fintype (BlockOld F) := Fintype.ofFinite _
def blockOldStage {F : T → Set α} (o : BlockOld F) : T := (ofLex o.val).1
def blockOldLabel {F : T → Set α} (o : BlockOld F) : α := (ofLex o.val).2
def blockStage (F : T → Set α) (τ : A → T) : BlockOld F ⊕ A → T :=
Sum.elim (blockOldStage (F := F)) τ
noncomputable def blockKey (F : T → Set α) (τ : A → T) :
BlockOld F ⊕ A → T ×ₗ (WithTop α ×ₗ ℕ)
| Sum.inl o => toLex (blockOldStage (F := F) o, toLex ((blockOldLabel (α := α) (F := F) o : WithTop α), 0))
| Sum.inr a => toLex (τ a, toLex (⊤, ((Fintype.equivFin A) a).val))
lemma blockKey_injective
{α : Type u_1} {A : Type u_2} {T : Type u_3}
[Fintype α] [LinearOrder α] [Fintype A] [Fintype T] [LinearOrder T]
(F : T → Set α) (τ : A → T) :
Function.Injective (blockKey F τ) := by
intro x y h
cases x with
| inl x =>
cases y with
| inl y =>
have hs : blockOldStage (F := F) x = blockOldStage (F := F) y := congrArg (fun z => (ofLex z).1) h
have hl : (blockOldLabel (α := α) (F := F) x : WithTop α) = blockOldLabel (α := α) (F := F) y :=
congrArg (fun z => (ofLex (ofLex z).2).1) h
have hl' : blockOldLabel (α := α) (F := F) x = blockOldLabel (α := α) (F := F) y := WithTop.coe_inj.mp hl
have heq : x = y := Subtype.ext (congrArg toLex (Prod.ext hs hl'))
exact congrArg Sum.inl heq
| inr y =>
have hh : (blockOldLabel (α := α) (F := F) x : WithTop α) = ⊤ :=
congrArg (fun z => (ofLex (ofLex z).2).1) h
exact (WithTop.coe_ne_top hh).elim
| inr x =>
cases y with
| inl y =>
have hh : (blockOldLabel (α := α) (F := F) y : WithTop α) = ⊤ :=
(congrArg (fun z => (ofLex (ofLex z).2).1) h).symm
exact (WithTop.coe_ne_top hh).elim
| inr y =>
have hi : ((Fintype.equivFin A) x).val = ((Fintype.equivFin A) y).val :=
congrArg (fun z => (ofLex (ofLex z).2).2) h
exact congrArg Sum.inr ((Fintype.equivFin A).injective (Fin.ext hi))
noncomputable def blockTime (F : T → Set α) (τ : A → T) : BlockOld F ⊕ A → ℕ := by
letI := LinearOrder.lift' (blockKey F τ) (blockKey_injective F τ)
exact fun o => ((Fintype.orderIsoFinOfCardEq (BlockOld F ⊕ A) rfl).symm o).val
lemma blockTime_lt_iff (F : T → Set α) (τ : A → T) (x y : BlockOld F ⊕ A) :
blockTime F τ x < blockTime F τ y ↔ blockKey F τ x < blockKey F τ y := by
let := LinearOrder.lift' (blockKey F τ) (blockKey_injective F τ)
change (Fintype.orderIsoFinOfCardEq (BlockOld F ⊕ A) rfl).symm x <
(Fintype.orderIsoFinOfCardEq (BlockOld F ⊕ A) rfl).symm y ↔ _
exact OrderIso.lt_iff_lt _
lemma blockTime_injective (F : T → Set α) (τ : A → T) :
Function.Injective (blockTime F τ) := by
intro x y h
apply blockKey_injective F τ
apply le_antisymm <;> apply le_of_not_gt <;> intro hh
· exact (lt_irrefl _ (h ▸ (blockTime_lt_iff F τ y x).2 hh))
· exact (lt_irrefl _ (h ▸ (blockTime_lt_iff F τ x y).2 hh))
lemma blockTime_old_strictMono (F : T → Set α) (τ : A → T) :
StrictMono (fun o : BlockOld F => blockTime F τ (Sum.inl o)) := by
intro x y hxy
apply (blockTime_lt_iff F τ _ _).2
change toLex (blockOldStage (F := F) x, toLex ((blockOldLabel (α := α) (F := F) x : WithTop α), 0)) <
toLex (blockOldStage (F := F) y, toLex ((blockOldLabel (α := α) (F := F) y : WithTop α), 0))
have hh : x.val < y.val := hxy
rcases Prod.Lex.toLex_lt_toLex.mp hh with hs | ⟨hs, hl⟩
· exact Prod.Lex.toLex_lt_toLex.mpr (Or.inl hs)
· exact Prod.Lex.toLex_lt_toLex.mpr (Or.inr ⟨hs,
Prod.Lex.toLex_lt_toLex.mpr (Or.inl (WithTop.coe_lt_coe.mpr hl))⟩)
lemma stage_lt_implies_blockTime_lt (F : T → Set α) (τ : A → T)
{x y : BlockOld F ⊕ A} (h : blockStage F τ x < blockStage F τ y) :
blockTime F τ x < blockTime F τ y := by
apply (blockTime_lt_iff F τ _ _).2
apply Prod.Lex.toLex_lt_toLex.mpr
left
cases x <;> cases y <;> exact h
lemma blockTime_le_implies_stage_le (F : T → Set α) (τ : A → T)
{x y : BlockOld F ⊕ A} (h : blockTime F τ x ≤ blockTime F τ y) :
blockStage F τ x ≤ blockStage F τ y := by
by_contra hh
exact (not_lt_of_ge h) (stage_lt_implies_blockTime_lt F τ (lt_of_not_ge hh))
end MatroidProphet.Pivots
end OAI
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