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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.ProductBits
namespace OAI
namespace MatchingFPRAS.TableMarking
open Finset Transport Variation
variable {V E I : Type*} [Fintype V] [Fintype E] [DecidableEq E] [Fintype I] [DecidableEq I]
variable (inc : E → V → Prop)
abbrev Marked := Σ M : PM inc, (↥M.val → I)
def reveal (M : PM inc) (marks : E → I) : Marked (I:=I) inc := ⟨M,fun e=>marks e.val⟩
variable {A : Type*} [Fintype A] [Nonempty A]
def draw (f : A → PM inc) (c : E → WeightedOptions I) (k : ℕ)
(bits : A×(E → Fin k → Bool)) : Marked (I:=I) inc :=
reveal inc (f bits.1) (fun e=>(c e).draw k (bits.2 e))
noncomputable def portion (c : E → WeightedOptions I) (e : E) (i : I) : ℝ := (c e).law i
noncomputable def law (w : E → ℝ) (c : E → WeightedOptions I) : Marked (I:=I) inc → ℝ :=
sigmaLift (gibbs w) (fun M=>product (fun e : M.val=>portion c e.val))
lemma reveal_law (p : PM inc → ℝ) (c : E → WeightedOptions I) :
push (tensor p (product (portion c))) (fun x : PM inc×(E → I)=>reveal inc x.1 x.2)=
sigmaLift p (fun M=>product (fun e : M.val=>portion c e.val)) := by
calc
_ = sigmaLift p (fun M=>push (product (portion c)) (fun m=>fun e : M.val=>m e.val)) := by
exact push_tensor_sigma p (product (portion c)) (fun M m=>fun e : M.val=>m e.val)
_ = _ := ?_
congr 1
funext M
apply product_restrict
intro e
exact WeightedOptions.real_law_sum (c e)
lemma draw_tv (f : A → PM inc) (w : E → ℝ) (c : E → WeightedOptions I) (k : ℕ) :
tv (imageLaw (draw inc f c k)) (law inc w c)≤
tv (imageLaw f) (gibbs w)+(∑ e : E,(2:ℝ)*(c e).size)/(2:ℝ)^k := by
let mark : (E → Fin k → Bool) → E → I := fun bits e=>(c e).draw k (bits e)
have hs (e : E) : ∑ b,portion c e b=1 := WeightedOptions.real_law_sum _
have hn (e : E) (b : I) : 0≤portion c e b := (Rat.cast_nonneg (K:=ℝ)).mpr ((c e).law_nonneg b)
have hm : tv (imageLaw mark) (product (portion c))≤(∑ e : E,(2:ℝ)*(c e).size)/(2:ℝ)^k := by
rw [show mark=(fun bits : E → Fin k → Bool=>fun e=>(c e).draw k (bits e)) from rfl,
imageLaw_componentwise]
have hh:=product_tv (fun e=>imageLaw ((c e).draw k)) (portion c)
(fun e=>imageLaw_nonneg _) (fun e=>imageLaw_sum _) hn hs
refine hh.trans ?_
rw [sum_div]
apply sum_le_sum
intro e _
exact (c e).draw_tv k
have htv:=tensor_tv (imageLaw f) (gibbs w) (imageLaw mark) (product (portion c))
(imageLaw_nonneg f) (imageLaw_sum f) (product_nonneg _ hn) (product_sum _ hs)
have hc:=push_contraction (tensor (imageLaw f) (imageLaw mark))
(tensor (gibbs w) (product (portion c))) (fun x : PM inc×(E → I)=>reveal inc x.1 x.2)
rw [reveal_law inc _ c,←imageLaw_tensor f mark,←imageLaw_comp] at hc
change tv (imageLaw (draw inc f c k)) (law inc w c)≤_ at hc
rw [imageLaw_tensor f mark] at hc
exact hc.trans (htv.trans (add_le_add_right hm _))
lemma draw_tv_size (f : A → PM inc) (w : E → ℝ) (c : E → WeightedOptions I) (k s : ℕ)
(h : ∀ e,(c e).size=s) :
tv (imageLaw (draw inc f c k)) (law inc w c)≤
tv (imageLaw f) (gibbs w)+2*s*Fintype.card E/(2:ℝ)^k := by
have hh:=draw_tv inc f w c k
simpa [h,mul_comm,mul_left_comm,mul_assoc] using hh
end MatchingFPRAS.TableMarking
end OAI
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