File size: 3,166 Bytes
97c3405 | 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 | import OAI.Combinatorics.ContingencyTables.Counting.CompanionPhaseFinishProgram
namespace OAI
/-! # Additive binary-size bounds for stored phase arithmetic -/
namespace ContingencyTables.CompanionFractionSize
open MatchingFPRAS MatchingFPRAS.TreeTyped CompanionMultiplierProgram CompanionPhaseCounts
lemma height_mul (a b : TreeFraction.Fraction) :
TreeFraction.height (TreeFraction.mul (a,b)) ≤ TreeFraction.height a+TreeFraction.height b+1 := by
have hn := TreeFraction.normalize_size_num (a.1*b.1,a.2*b.2)
have hd := TreeFraction.normalize_size_den (a.1*b.1,a.2*b.2)
have hnum := nat_size_mul a.1 b.1
have hden := nat_size_mul a.2 b.2
have ha := le_max_left a.1.size a.2.size
have ha' := le_max_right a.1.size a.2.size
have hb := le_max_left b.1.size b.2.size
have hb' := le_max_right b.1.size b.2.size
dsimp only at hn hd
change (TreeFraction.mul (a,b)).1.size ≤ _ at hn
change (TreeFraction.mul (a,b)).2.size ≤ _ at hd
unfold TreeFraction.height
apply max_le <;> omega
lemma height_div (a b : TreeFraction.Fraction) :
TreeFraction.height (TreeFraction.div (a,b)) ≤ TreeFraction.height a+TreeFraction.height b+1 := by
have hn := TreeFraction.normalize_size_num (a.1*b.2,a.2*b.1)
have hd := TreeFraction.normalize_size_den (a.1*b.2,a.2*b.1)
have hnum := nat_size_mul a.1 b.2
have hden := nat_size_mul a.2 b.1
have ha := le_max_left a.1.size a.2.size
have ha' := le_max_right a.1.size a.2.size
have hb := le_max_left b.1.size b.2.size
have hb' := le_max_right b.1.size b.2.size
dsimp only at hn hd
change (TreeFraction.div (a,b)).1.size ≤ _ at hn
change (TreeFraction.div (a,b)).2.size ≤ _ at hd
unfold TreeFraction.height
apply max_le <;> omega
lemma countRatio_height (z : DefectCountInput) : TreeFraction.height (countRatio z) ≤ z.1.2.length := by
have hp : plainCountCode z.1 ≤ z.1.2.length := List.length_filter_le _ _
have hd : defectCountCode z ≤ z.1.2.length := List.length_filter_le _ _
have hs (n : ℕ) : n.size ≤ n := Nat.size_le.mpr Nat.lt_two_pow_self
exact max_le ((hs _).trans hp) ((hs _).trans hd)
lemma updateCode_height (z : UpdateInput) :
TreeFraction.height (updateCode z) ≤ TreeFraction.height z.2+z.1.1.2.length+1 := by
unfold updateCode
split_ifs
· omega
· exact (height_mul _ _).trans (Nat.add_le_add_right
(Nat.add_le_add_left (countRatio_height _) _) 1)
lemma sum_height (xs : List TreeFraction.Fraction) (M : ℕ)
(hm : ∀ x∈xs,TreeFraction.height x ≤ M) :
TreeFraction.height (TreeFraction.sum xs) ≤ 1+xs.length*(M+2) := by
simpa only [TreeFraction.sum,TreeFraction.height,Nat.size_zero,Nat.size_one,Nat.zero_max] using
TreeFraction.fractionFold_height xs (0,1) M hm
lemma term_height (z : CompanionPhaseFinishProgram.TermInput) (M : ℕ)
(h₁ : TreeFraction.height z.2.1 ≤ M) (h₂ : TreeFraction.height z.2.2 ≤ M) :
TreeFraction.height (CompanionPhaseFinishProgram.term z) ≤ 2*M+1 := by
unfold CompanionPhaseFinishProgram.term
split_ifs
· have h := height_div z.2.2 z.2.1
omega
· norm_num [TreeFraction.height]
end ContingencyTables.CompanionFractionSize
end OAI
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