| 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 |
|
|
