| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Tapes.ProductBits |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TableMarking |
|
|
| open Finset Transport Variation |
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| variable {V E I : Type*} [Fintype V] [Fintype E] [DecidableEq E] [Fintype I] [DecidableEq I] |
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| variable (inc : E → V → Prop) |
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| abbrev Marked := Σ M : PM inc, (↥M.val → I) |
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| def reveal (M : PM inc) (marks : E → I) : Marked (I:=I) inc := ⟨M,fun e=>marks e.val⟩ |
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| variable {A : Type*} [Fintype A] [Nonempty A] |
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| 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)) |
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| noncomputable def portion (c : E → WeightedOptions I) (e : E) (i : I) : ℝ := (c e).law i |
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| 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 _)) |
|
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| 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 |
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| end MatchingFPRAS.TableMarking |
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| end OAI |
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