File size: 3,212 Bytes
92416a2 | 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 | import OAI.Dynamics.ConditionalShuffle.PhysicalInstrument
namespace OAI
noncomputable section
namespace Revealed
open scoped Classical
open Thorp
lemma imageRank_lt_iff {α β : Type*} [Fintype α] [LinearOrder β]
(f : α → β) (hf : Function.Injective f) (a b : α) :
imageRank f hf a < imageRank f hf b ↔ f a < f b := by
unfold imageRank
exact (Fintype.orderIsoFinOfCardEq (Set.range f)
(Fintype.card_congr (Equiv.ofInjective f hf)).symm).symm.lt_iff_lt
lemma imageRank_comp_perm {α β : Type*} [Fintype α] [LinearOrder β]
(f : α → β) (hf : Function.Injective f) (g : Equiv.Perm α) :
imageRank (f ∘ g) (hf.comp g.injective) = g.trans (imageRank f hf) := by
let e := (imageRank (f ∘ g) (hf.comp g.injective)).symm.trans (g.trans (imageRank f hf))
have he : StrictMono e := by
intro i j hij
change imageRank f hf (g ((imageRank (f ∘ g) (hf.comp g.injective)).symm i)) <
imageRank f hf (g ((imageRank (f ∘ g) (hf.comp g.injective)).symm j))
rw [imageRank_lt_iff]
exact (imageRank_lt_iff (f ∘ g) (hf.comp g.injective) _ _).mp (by simpa using hij)
let o : Fin (Fintype.card α) ≃o Fin (Fintype.card α) :=
{ toEquiv := e, map_rel_iff' := fun {_ _} => he.le_iff_le }
have ho : o = OrderIso.refl _ := Subsingleton.elim _ _
apply Equiv.ext
intro a
have ha := congrArg (fun h : Fin (Fintype.card α) ≃o Fin (Fintype.card α) =>
h (imageRank (f ∘ g) (hf.comp g.injective) a)) ho
simpa [o, e] using ha.symm
def splitLabels (d : ℕ) : Sum (Position d) (Position d) ≃ Position (d+1) where
toFun
| .inl x => outside d x
| .inr x => freeLabel d x
invFun x := if x 0 then .inl (Fin.tail x) else .inr (Fin.tail x)
left_inv x := by cases x <;> simp [outside, freeLabel]
right_inv x := by
cases h : x 0 <;> simpa only [h, Bool.false_eq_true, ite_false, ite_true, outside, freeLabel]
using (Fin.cons_self_tail x)
lemma splitLabels_inl (d : ℕ) (x : Position d) : splitLabels d (.inl x) = outside d x := rfl
lemma splitLabels_inr (d : ℕ) (x : Position d) : splitLabels d (.inr x) = freeLabel d x := rfl
namespace Split
variable {ι : Type} (d : ℕ)
def ranks (L : Sum ι (Position d) ≃ Position (d+1)) : Position d ≃ Fin (2^d) :=
(imageRank (fun a => toLex (L (.inr a))) (by intro a b h; exact Sum.inr_injective (L.injective h))).trans
(finCongr (by simp [Position]))
lemma ranks_pre (L : Sum ι (Position d) ≃ Position (d+1)) (g : State d) :
ranks d ((Equiv.sumCongr (Equiv.refl ι) g).trans L) = g.trans (ranks d L) := by
unfold ranks
have hh := imageRank_comp_perm (fun a => toLex (L (.inr a)))
(by intro a b h; exact Sum.inr_injective (L.injective h)) g
exact congrArg (fun E : Position d ≃ Fin (Fintype.card (Position d)) =>
E.trans (finCongr (show Fintype.card (Position d) = 2^d by simp [Position]))) hh
lemma ranks_assignment (L : Sum ι (Position d) ≃ Position (d+1)) :
ranks d L = (assignment L).trans (ranks d (sectionFrame (outside L))) := by
conv_lhs => rw [← assignment_factor L]
rw [ranks_pre]
lemma ranks_freeRank (g : State (d+1)) : ranks d ((splitLabels d).trans g) = freeRank d g := rfl
end Split
end Revealed
end
end OAI
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