| 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 |
|
|
