File size: 5,006 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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import Mathlib
import OAI.Combinatorics.MatchingCount.Sampling.Endpoints
import OAI.Combinatorics.MatchingCount.Tapes.RunBits
namespace OAI
namespace MatchingFPRAS.Counting
open Finset
variable {n : ℕ}
lemma rowMaximum_cast (hn : 2≤n) (G : Fin n → Fin n → ℚ) (v : Fin n → ℚ) (i : Fin n) :
(rowMaximum hn G v i:ℝ)=RealStage.rowMaximum hn (fun i j=>(G i j:ℝ)) (fun i=>(v i:ℝ)) i := by
unfold rowMaximum RealStage.rowMaximum
rw [Finset.apply_sup'_eq_sup'_comp _ (fun q : ℚ=>(q:ℝ)) (by intro a b; exact Rat.cast_max a b)]
simp only [Function.comp_apply,Rat.cast_div]
lemma scaleUpdate_cast (hn : 2≤n) (G : Fin n → Fin n → ℚ) (v : Fin n → ℚ) (i j : Fin n) :
(scaleUpdate hn G v i j:ℝ)=RealStage.scaleUpdate hn (fun i j=>(G i j:ℝ)) (fun i=>(v i:ℝ)) i j := by
by_cases h : j=i
· subst j
simp only [scaleUpdate,RealStage.scaleUpdate,Function.update_self,rowMaximum_cast]
· simp only [scaleUpdate,RealStage.scaleUpdate,Function.update_of_ne h]
lemma scalingTrace_cast (hn : 2≤n) (G : Fin n → Fin n → ℚ) (k : ℕ) (i : Fin n) :
(scalingTrace hn G k i:ℝ)=RealStage.scalingTrace hn (fun i j=>(G i j:ℝ)) k i := by
induction k generalizing i with
| zero => norm_num [scalingTrace,RealStage.scalingTrace]
| succ k ih =>
simp only [scalingTrace,RealStage.scalingTrace]
split_ifs with h
· rw [scaleUpdate_cast]
have hi : (fun j=>(Counting.scalingTrace hn G k j:ℝ))=RealStage.scalingTrace hn (fun i j=>(G i j:ℝ)) k := by
funext j
exact ih j
rw [hi]
· exact ih i
lemma ratioEstimate_cast (a d : ℚ) :
(ratioEstimate a d:ℝ)=RealStage.ratioEstimate a d := by
simp only [ratioEstimate,RealStage.ratioEstimate]
have hp : (0:ℝ)<a ↔ (0:ℚ)<a := by exact_mod_cast Iff.rfl
simp only [hp]
split_ifs <;> push_cast <;> rfl
lemma upperEstimate_cast (D : ℕ) (d z : ℚ) :
(upperEstimate D d z:ℝ)=RealStage.upperEstimate D d z := by
simp only [upperEstimate,RealStage.upperEstimate]
have hp : (0:ℝ)<d ↔ (0:ℚ)<d := by exact_mod_cast Iff.rfl
simp only [hp]
split_ifs <;> push_cast <;> norm_num [alpha,RealStage.alpha]
lemma sigma_cast (D K : ℕ) (e : ℚ) : (sigma D K e:ℝ)=RealStage.sigma D K e := by
simp only [sigma,RealStage.sigma,alpha,RealStage.alpha]
push_cast
rfl
end MatchingFPRAS.Counting
namespace MatchingFPRAS.FiniteTrial
open Finset
def cooling (n : ℕ) : ℚ := (1+1/(n:ℚ))⁻¹
lemma cooling_cast (n : ℕ) : (cooling n:ℝ)=Trial.cooling n := by
simp [cooling,Trial.cooling]
def raw (G : GraphInput) (j : ℕ) (a b : Fin G.n) : ℚ :=
if (min a b,max a b)∈G.edges then 1 else cooling G.n^j
lemma raw_cast (G : GraphInput) (j : ℕ) (a b : Fin G.n) :
(raw G j a b:ℝ)=Trial.raw G j a b := by
unfold raw Trial.raw GraphBridge.rawActivity
split_ifs <;> simp [cooling_cast]
structure ScaleState (G : GraphInput) (j : ℕ) where
scale : Fin G.n → ℚ
positive : ∀ i,0<scale i
upper : ∀ i,scale i≤2^j
def ScaleState.toReal {G : GraphInput} {j : ℕ} (s : ScaleState G j) : Trial.ScaleState G j where
scale i:=(s.scale i:ℝ)
positive i:=by exact_mod_cast s.positive i
upper i:=by exact_mod_cast s.upper i
def ScaleState.activity {G : GraphInput} {j : ℕ} (s : ScaleState G j) : Fin G.n → Fin G.n → ℚ :=
fun a b=>raw G j a b*s.scale a*s.scale b
def ScaleState.nextActivity {G : GraphInput} {j : ℕ} (s : ScaleState G j) : Fin G.n → Fin G.n → ℚ :=
fun a b=>raw G (j+1) a b*s.scale a*s.scale b
lemma ScaleState.activity_cast {G : GraphInput} {j : ℕ} (s : ScaleState G j) :
(fun a b=>(s.activity a b:ℝ))=s.toReal.activity := by
funext a b
simp only [ScaleState.activity,Trial.ScaleState.activity,Rat.cast_mul,raw_cast,ScaleState.toReal]
lemma ScaleState.nextActivity_cast {G : GraphInput} {j : ℕ} (s : ScaleState G j) :
(fun a b=>(s.nextActivity a b:ℝ))=s.toReal.nextActivity := by
funext a b
simp only [ScaleState.nextActivity,Trial.ScaleState.nextActivity,Rat.cast_mul,raw_cast,ScaleState.toReal]
lemma ScaleState.activity_pos {G : GraphInput} {j : ℕ} (s : ScaleState G j) (a b : Fin G.n) :
0<s.activity a b := by
have h:=s.toReal.activity_pos a b
rw [←s.activity_cast] at h
dsimp only at h
exact_mod_cast h
lemma ScaleState.nextActivity_pos {G : GraphInput} {j : ℕ} (s : ScaleState G j) (a b : Fin G.n) :
0<s.nextActivity a b := by
have h:=s.toReal.nextActivity_pos a b
rw [←s.nextActivity_cast] at h
dsimp only at h
exact_mod_cast h
def initialScaleState (G : GraphInput) (hn : 2≤G.n) : ScaleState G 0 where
scale _:=Initializer.initialScale G.n
positive _:=Initializer.initialScale_pos G.n hn
upper i:=by
have h:=(Trial.initialScaleState G hn).upper i
change (Initializer.initialScale G.n:ℝ)≤2^0 at h
exact_mod_cast h
lemma initialScaleState_cast (G : GraphInput) (hn : 2≤G.n) :
(initialScaleState G hn).toReal=Trial.initialScaleState G hn := rfl
end MatchingFPRAS.FiniteTrial
end OAI
|