openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionFractionSize.lean
| 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 | |