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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 | import OAI.Dynamics.ConditionalShuffle.ResetPartial
namespace OAI
noncomputable section
open scoped BigOperators Classical
open Filter Topology
namespace Thorp.Conditional
def reversePosition (d : ℕ) : State d where
toFun x := fun i => x i.rev
invFun x := fun i => x i.rev
left_inv x := by funext i; simp
right_inv x := by funext i; simp
@[simp] lemma reversePosition_apply (d : ℕ) (x : Position d) (i : Fin d) :
reversePosition d x i = x i.rev := rfl
@[simp] lemma reversePosition_involutive (d : ℕ) (x : Position d) :
reversePosition d (reversePosition d x) = x := by ext i; simp
lemma reversePosition_cons (d : ℕ) (b : Bool) (x : Position d) :
reversePosition (d+1) (Fin.cons b x) = Fin.snoc (reversePosition d x) b := by
ext i; refine Fin.lastCases ?_ (fun j => ?_) i
· simp
· simp [Fin.rev_castSucc]
lemma reversePosition_snoc (d : ℕ) (b : Bool) (x : Position d) :
reversePosition (d+1) (Fin.snoc x b) = Fin.cons b (reversePosition d x) := by
have := congrArg (reversePosition (d+1)) (reversePosition_cons d b (reversePosition d x))
simpa only [reversePosition_involutive] using this.symm
def reverseCoins (d : ℕ) : Coins (d+1) ≃ Coins (d+1) where
toFun c := c ∘ reversePosition d
invFun c := c ∘ reversePosition d
left_inv c := by funext x; change c (reversePosition d (reversePosition d x)) = c x; rw [reversePosition_involutive]
right_inv c := by funext x; change c (reversePosition d (reversePosition d x)) = c x; rw [reversePosition_involutive]
lemma reverse_step_cancel (d : ℕ) (c : Coins (d+1)) (x : Position (d+1)) :
step (d+1) (reverseCoins d c) (reversePosition (d+1) (step (d+1) c x)) =
reversePosition (d+1) x := by
obtain ⟨⟨b,x⟩,rfl⟩ := (Fin.consEquiv (fun _ : Fin (d+1) => Bool)).surjective x
change step (d+1) (reverseCoins d c) (reversePosition (d+1) (step (d+1) c (Fin.cons b x))) = reversePosition (d+1) (Fin.cons b x)
rw [step_cons d c b x, scatterPosition_eq_snoc, reversePosition_snoc, step_cons d (reverseCoins d c),
scatterPosition_eq_snoc, reversePosition_cons]
have he : reverseCoins d c (reversePosition d x) = c x := by change c (reversePosition d (reversePosition d x)) = c x; rw [reversePosition_involutive]
rw [he, Bool.xor_assoc, Bool.xor_self, Bool.xor_false]
lemma reverse_step (d : ℕ) (c : Coins (d+1)) :
step (d+1) (reverseCoins d c) =
reversePosition (d+1) * (step (d+1) c)⁻¹ * reversePosition (d+1) := by
apply Equiv.ext; intro x
have hh := reverse_step_cancel d c ((step (d+1) c).symm (reversePosition (d+1) x))
change step (d+1) (reverseCoins d c) x =
reversePosition (d+1) ((step (d+1) c).symm (reversePosition (d+1) x))
simpa only [Equiv.apply_symm_apply, reversePosition_involutive] using hh
def reverseHistory (d t : ℕ) : History (d+1) t ≃ History (d+1) t where
toFun ω := fun i => reverseCoins d (ω i.rev)
invFun ω := fun i => reverseCoins d (ω i.rev)
left_inv ω := by
funext i x
change ω i.rev.rev (reversePosition d (reversePosition d x)) = ω i x
rw [Fin.rev_rev, reversePosition_involutive]
right_inv ω := by
funext i x
change ω i.rev.rev (reversePosition d (reversePosition d x)) = ω i x
rw [Fin.rev_rev, reversePosition_involutive]
lemma run_cons (d t : ℕ) (c : Coins d) (ω : History d t) :
run d (t+1) (Fin.cons c ω) = run d t ω * step d c := by
simp [run, List.ofFn_succ]
lemma reverseHistory_cons (d t : ℕ) (c : Coins (d+1)) (ω : History (d+1) t) :
reverseHistory d (t+1) (Fin.cons c ω) = Fin.snoc (reverseHistory d t ω) (reverseCoins d c) := by
ext i; refine Fin.lastCases ?_ (fun j => ?_) i
· simp [reverseHistory]
· simp [reverseHistory, Fin.rev_castSucc]
lemma reverse_run (d t : ℕ) (ω : History (d+1) t) :
run (d+1) t (reverseHistory d t ω) =
reversePosition (d+1) * (run (d+1) t ω)⁻¹ * reversePosition (d+1) := by
have hr : reversePosition (d+1) * reversePosition (d+1) = 1 := by
apply Equiv.ext; intro x; exact reversePosition_involutive (d+1) x
induction t with
| zero => simp [run_zero, hr]
| succ t ih =>
obtain ⟨⟨c,ω⟩,rfl⟩ := (Fin.consEquiv (fun _ : Fin (t+1) => Coins (d+1))).surjective ω
change run (d+1) (t+1) (reverseHistory d (t+1) (Fin.cons c ω)) =
reversePosition (d+1) * (run (d+1) (t+1) (Fin.cons c ω))⁻¹ * reversePosition (d+1)
rw [reverseHistory_cons, run_succ]
simp only [Fin.snoc_last, Fin.snoc_castSucc]
rw [ih, reverse_step, run_cons, mul_inv_rev]
calc
_ = reversePosition (d+1) * (step (d+1) c)⁻¹ *
(reversePosition (d+1) * reversePosition (d+1)) *
(run (d+1) t ω)⁻¹ * reversePosition (d+1) := by group
_ = _ := by rw [hr]; group
end Thorp.Conditional
end
end OAI
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