File size: 2,806 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Sampling.TreeCategorical
namespace OAI
namespace MatchingFPRAS.TreeFraction
lemma cumulativeFraction_val_ofFn {r : ℕ} (q : Fin r → Fraction) (hq : ∀ i,0<(q i).2) (n : ℕ) :
val (cumulativeFraction (List.ofFn q,n))=FiniteChoice.cumulative (fun i=>val (q i)) n := by
rw [cumulativeFraction,sum_val]
· simp only [List.map_take,List.map_ofFn,Function.comp_apply,List.sum_take_ofFn,
FiniteChoice.cumulative,Finset.sum_filter]
· intro b hb
obtain ⟨i,hi⟩ := List.mem_ofFn.mp (List.mem_of_mem_take hb)
subst b
exact hq i
end MatchingFPRAS.TreeFraction
namespace MatchingFPRAS.ExecutableChoice
open TreeTyped TreeFraction
lemma choose_values {r K : ℕ} (q : Fin r → Fraction) (hq : ∀ i,0<(q i).2)
(hs : ∑ i,val (q i)=1) (hK : 0<K) (J : Fin K) :
choose (List.ofFn q,(J.val,K))=(FiniteChoice.choose (fun i=>val (q i)) hs hK J).val := by
let j := FiniteChoice.choose (fun i=>val (q i)) hs hK J
have hpred (i : ℕ) : predicate ((List.ofFn q,(J.val,K)),i)=
decide ((J.val:ℚ)/K<FiniteChoice.cumulative (fun i=>val (q i)) (i+1)) := by
rw [predicate,less_val (J.val,K) (cumulativeFraction (List.ofFn q,i+1)) hK (sum_den _),cumulativeFraction_val_ofFn q hq]
rfl
unfold choose
simp only [List.length_ofFn]
apply le_antisymm
· apply listMin_le
simp only [List.mem_filter,List.mem_range,hpred,decide_eq_true_eq]
exact ⟨j.isLt,FiniteChoice.choose_upper _ hs hK J⟩
· apply le_listMin j.isLt.le
intro i hi
simp only [List.mem_filter,List.mem_range,hpred,decide_eq_true_eq] at hi
exact FiniteChoice.choose_le _ hs hK J ⟨i,hi.1⟩ hi.2
lemma bitValue_ofFn (k : ℕ) (bits : Fin k → Bool) :
StackAddition.value (List.ofFn bits)=(FiniteChoice.tapeIndex k bits).val := by
have hv' (k : ℕ) (bits : Fin k → Bool) :
StackAddition.value (List.ofFn bits)=∑ i, (if bits i then 1 else 0)*2^i.val := by
induction k with
| zero => simp [StackAddition.value]
| succ k ih =>
rw [List.ofFn_succ,StackAddition.value,ih,Fin.sum_univ_succ]
simp only [Fin.val_zero,pow_zero,Nat.mul_one,Fin.val_succ,pow_succ]
rw [Finset.mul_sum]
rw [Bool.toNat,Bool.cond_eq_ite]
congr 1
apply Finset.sum_congr rfl
intro i _
ring
rw [FiniteChoice.tapeIndex_value,hv']
lemma chooseBits_values {r k : ℕ} (q : Fin r → Fraction) (hq : ∀ i,0<(q i).2)
(hs : ∑ i,val (q i)=1) (bits : Fin k → Bool) :
chooseBits (List.ofFn q,List.ofFn bits)=
(FiniteChoice.chooseBits (fun i=>val (q i)) hs k bits).val := by
rw [chooseBits,bitDenominator_eq,List.length_ofFn,bitValue_ofFn]
exact choose_values q hq hs (by positivity) (FiniteChoice.tapeIndex k bits)
end MatchingFPRAS.ExecutableChoice
end OAI
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