File size: 4,360 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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import OAI.Dynamics.ConditionalShuffle.Fibers
namespace OAI
noncomputable section
namespace Revealed.Split
open scoped Classical
variable {ι α β : Type*} [decidableEq_α : DecidableEq α] [DecidableEq β]
lemma trans_free_swap (L : Sum ι α ≃ β) (a b : α) :
L.trans (Equiv.swap (L (.inr a)) (L (.inr b))) =
(Equiv.sumCongr (Equiv.refl ι) (Equiv.swap a b)).trans L := by
ext x
cases x with
| inl i => simp [Equiv.swap_apply_def, L.injective.eq_iff]
| inr x =>
by_cases hxa : x = a
· subst x
simp
· by_cases hxb : x = b
· subst x
simp
· simp [Equiv.swap_apply_def, L.injective.eq_iff, hxa, hxb]
end Revealed.Split
namespace Thorp.Conditional
open scoped Classical
open Revealed.Split
variable {ι α : Type} [fintype_ι : Fintype ι] [fintype_α : Fintype α] [decidableEq_α : DecidableEq α]
lemma step_same_at_iff (d : ℕ) (c c' : Coins (d + 1)) (x : Position (d + 1)) :
step (d + 1) c x = step (d + 1) c' x ↔ c (Fin.tail x) = c' (Fin.tail x) := by
obtain ⟨⟨b,s⟩, rfl⟩ := (splitPosition d).symm.surjective x
change rotate (d + 1) (Fin.cons (b ^^ c s) s) =
rotate (d + 1) (Fin.cons (b ^^ c' s) s) ↔ c s = c' s
rw [(rotate (d + 1)).injective.eq_iff]
constructor
· intro h
have hh := congrFun h 0
simp only [Fin.cons_zero] at hh
cases b <;> simpa using hh
· intro h
rw [h]
lemma outside_step_injective (d : ℕ) (L : Sum ι α ≃ Position (d + 1))
(h : ∀ s : Position d, ∃ i : ι, Fin.tail (L (.inl i)) = s) :
Function.Injective (fun c : Coins (d + 1) => outside (L.trans (step (d + 1) c))) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
intro c c' hc
funext s
obtain ⟨i, hi⟩ := h s
have hh := congrFun (congrArg Subtype.val hc) i
change step (d + 1) c (L (.inl i)) = step (d + 1) c' (L (.inl i)) at hh
have he := (step_same_at_iff d c c' _).mp hh
rwa [hi] at he
lemma pair_free_labels (d : ℕ) (L : Sum ι α ≃ Position (d + 1)) (s : Position d)
(h : ∀ i : ι, Fin.tail (L (.inl i)) ≠ s) :
∃ a b : α, a ≠ b ∧ L (.inr a) = Fin.cons false s ∧ L (.inr b) = Fin.cons true s := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
let retained_decidableEq_α := decidableEq_α
have ex (bit : Bool) : ∃ a : α, L (.inr a) = Fin.cons bit s := by
obtain ⟨x,hx⟩ := L.surjective (Fin.cons bit s)
cases x with
| inl i =>
have hh := congrArg Fin.tail hx
simp only [Fin.tail_cons] at hh
exact False.elim (h i hh)
| inr a => exact ⟨a,hx⟩
obtain ⟨a,ha⟩ := ex false
obtain ⟨b,hb⟩ := ex true
refine ⟨a,b,?_,ha,hb⟩
intro hh
subst b
have hc := congrFun (ha.symm.trans hb) 0
simp at hc
lemma free_pair_toggle (d : ℕ) (L : Sum ι α ≃ Position (d + 1)) (s : Position d)
(a b : α) (ha : L (.inr a) = Fin.cons false s) (hb : L (.inr b) = Fin.cons true s)
(c : Coins (d + 1)) :
L.trans (step (d + 1) (toggleCoin s c)) =
(Equiv.sumCongr (Equiv.refl ι) (Equiv.swap a b)).trans (L.trans (step (d + 1) c)) := by
let retained_fintype_ι := fintype_ι
let retained_fintype_α := fintype_α
rw [Parity.step_toggle]
have hs := trans_free_swap L a b
rw [ha,hb] at hs
rw [← Equiv.trans_assoc, ← hs]
rfl
lemma physical_step_signDichotomy [Nontrivial α] (d : ℕ) (L : Sum ι α ≃ Position (d + 1))
(o : Outside ι α (Position (d + 1))) :
Trim.signDichotomy (Revealed.Disintegration.conditional
(fairMass (fun c : Coins (d + 1) =>
(outside (L.trans (step (d + 1) c)), assignment (L.trans (step (d + 1) c))))) o) := by
by_cases h : ∀ s : Position d, ∃ i : ι, Fin.tail (L (.inl i)) = s
· exact Trim.signDichotomy_conditional_injective _ _ (outside_step_injective d L h) o
· push Not at h
obtain ⟨s,hs⟩ := h
obtain ⟨a,b,hab,ha,hb⟩ := pair_free_labels d L s hs
apply Trim.signDichotomy_conditional_symmetry _ _ (toggleCoin s) (Equiv.swap a b)
· intro c
rw [free_pair_toggle d L s a b ha hb c, outside_pre]
· intro c
rw [free_pair_toggle d L s a b ha hb c, assignment_pre]
· rw [Specht.complexSign_eq, Equiv.Perm.sign_swap hab]
simp
end Thorp.Conditional
end
end OAI
|