File size: 3,706 Bytes
e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Graphs.AdaptiveStream
namespace OAI
namespace MatchingFPRAS.PhysicalStream.Experiment
variable {α β : Type}
def All : Experiment α → (α → Prop) → Prop
| .done a,P => P a
| .read f,P => All (f false) P ∧ All (f true) P
| .delay _ p,P => All p P
lemma All.mono {p : Experiment α} {P Q : α → Prop} (h : All p P) (hPQ : ∀ a,P a→Q a) : All p Q := by
induction p with
| done a => exact hPQ a h
| read p ih => exact ⟨ih false h.1,ih true h.2⟩
| delay n p ih => exact ih h
lemma all_run (p : Experiment α) (P : α → Prop) (h : All p P) (b : Stream) : P (p.run b).1 := by
induction p generalizing b with
| done a => exact h
| read p ih =>
cases hb : b 0 <;> simp only [run,hb] <;> apply ih <;> first | exact h.1 | exact h.2
| delay n p ih => exact ih h (shift n b)
lemma mean_nonneg (p : Experiment α) (f : α → ℚ) (h : All p (fun a=>0≤f a)) : 0≤p.mean f := by
induction p with
| done a => exact h
| read p ih => exact div_nonneg (add_nonneg (ih false h.1) (ih true h.2)) (by norm_num)
| delay n p ih => exact ih h
lemma all_true (p : Experiment α) : All p (fun _=>True) := by
induction p with
| done a => trivial
| read p ih => exact ⟨ih false,ih true⟩
| delay n p ih => exact ih
lemma all_indicator (p : Experiment α) (P : α → Prop) [DecidablePred P] :
All p P ↔ p.mean (fun a=>if P a then 0 else 1)=0 := by
induction p with
| done a => simp [All,mean]
| delay n p ih => exact ih
| read p ih =>
have hz (b : Bool) : 0≤(p b).mean (fun a=>if P a then 0 else 1) :=
mean_nonneg _ _ ((all_true _).mono (by intro a _; split <;> norm_num))
simp only [All,mean,ih]
constructor
· rintro ⟨h0,h1⟩; rw [h0,h1]; norm_num
· intro h
have h0:=hz false
have h1:=hz true
constructor <;> linarith
lemma all_of_mean_eq {p q : Experiment α} (h : ∀ f : α → ℚ,q.mean f=p.mean f)
(P : α → Prop) (hp : All p P) : All q P := by
classical
rw [all_indicator,h,←all_indicator]
exact hp
lemma cost_bind_le_of_all (p : Experiment α) (q : α → Experiment β) (N : ℕ)
(h : All p (fun a=>(q a).cost≤N)) : (p.bind q).cost≤p.cost+N := by
induction p with
| done a => simpa only [All,bind,cost,Nat.zero_add] using h
| read p ih =>
have h0:=ih false h.1
have h1:=ih true h.2
simp only [cost,bind]
omega
| delay n p ih => simpa only [bind,cost,Nat.add_assoc] using Nat.add_le_add_left (ih h) n
lemma all_bind (p : Experiment α) (q : α → Experiment β) (P : β → Prop) :
All (p.bind q) P ↔ All p (fun a=>All (q a) P) := by
induction p with
| done a => rfl
| read p ih => simp only [bind,All,ih]
| delay n p ih => exact ih
lemma all_map (p : Experiment α) (f : α → β) (P : β → Prop) :
All (p.map f) P ↔ All p (fun a=>P (f a)) := by
rw [map,all_bind]; rfl
variable {α : Type}
def worst : Experiment α → (α → ℕ) → ℕ
| .done a,f => f a
| .read p,f => max (worst (p false) f) (worst (p true) f)
| .delay _ p,f => worst p f
lemma all_worst (p : Experiment α) (f : α → ℕ) : All p (fun a=>f a≤p.worst f) := by
induction p with
| done a => exact le_rfl
| read p ih => exact ⟨(ih false).mono (fun a ha=>ha.trans (le_max_left _ _)),
(ih true).mono (fun a ha=>ha.trans (le_max_right _ _))⟩
| delay n p ih => exact ih
lemma worst_le (p : Experiment α) (f : α → ℕ) (N : ℕ) (h : All p (fun a=>f a≤N)) : p.worst f≤N := by
induction p with
| done a => exact h
| read p ih => exact max_le (ih false h.1) (ih true h.2)
| delay n p ih => exact ih h
end MatchingFPRAS.PhysicalStream.Experiment
end OAI
|