openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Numerics /ScalarControlSpace.lean
| import OAI.Combinatorics.EditApproximation.Binary.FractionControlSpace | |
| namespace OAI | |
| namespace EditApproximation | |
| def bitCeilDivInvocations (divisor bits : List Bool) : ℕ := | |
| let divided := bitDivModWithWork divisor bits | |
| bitDivModInvocations divisor bits + bitCompareInvocations divided.2.1 [] + 1 + | |
| if (bitCompareWithWork divided.2.1 []).1 = .eq then 0 | |
| else bitAddInvocations divided.1 [true] false | |
| theorem bitCeilDivInvocations_le_work (divisor bits : List Bool) : | |
| bitCeilDivInvocations divisor bits ≤ (bitCeilDivWithWork divisor bits).2 := by | |
| have hd := bitDivModInvocations_le_work divisor bits | |
| have hc := bitCompareInvocations_le_work (bitDivModWithWork divisor bits).2.1 [] | |
| have ha := bitAddInvocations_le_work (bitDivModWithWork divisor bits).1 [true] false | |
| unfold bitCeilDivInvocations bitCeilDivWithWork | |
| dsimp only | |
| split_ifs <;> dsimp only <;> omega | |
| def coarseDenominatorInvocations (allowanceDen q L branching : List Bool) (D : ℕ) : ℕ := | |
| let first := bitMulWithWork L branching | |
| let base := bitMulWithWork first.1 q | |
| let trimmed := trimBitWordWithWork base.1 | |
| let power := bitPowerWithWork trimmed.1 D | |
| let initial := bitMulWithWork allowanceDen q | |
| let product := bitMulWithWork initial.1 power.1 | |
| bitMulInvocations L branching + bitMulInvocations first.1 q + trimBitWordInvocations base.1 + | |
| bitPowerInvocations trimmed.1 D + bitMulInvocations allowanceDen q + | |
| bitMulInvocations initial.1 power.1 + trimBitWordInvocations product.1 + 1 | |
| theorem coarseDenominatorInvocations_le_work (allowanceDen q L branching : List Bool) (D : ℕ) : | |
| coarseDenominatorInvocations allowanceDen q L branching D ≤ | |
| (coarseDenominatorWithWork allowanceDen q L branching D).2 := by | |
| have h1 := bitMulInvocations_le_work L branching | |
| have h2 := bitMulInvocations_le_work (bitMulWithWork L branching).1 q | |
| have h3 := trimBitWordInvocations_le_work (bitMulWithWork (bitMulWithWork L branching).1 q).1 | |
| have h4 := bitPowerInvocations_le_work | |
| (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D | |
| have h5 := bitMulInvocations_le_work allowanceDen q | |
| have h6 := bitMulInvocations_le_work (bitMulWithWork allowanceDen q).1 | |
| (bitPowerWithWork (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D).1 | |
| have h7 := trimBitWordInvocations_le_work | |
| (bitMulWithWork (bitMulWithWork allowanceDen q).1 | |
| (bitPowerWithWork (trimBitWordWithWork (bitMulWithWork (bitMulWithWork L branching).1 q).1).1 D).1).1 | |
| unfold coarseDenominatorInvocations coarseDenominatorWithWork | |
| dsimp only | |
| omega | |
| /-- Includes the map's base frame and every recursive traversal frame. -/ | |
| def arithmeticMapInvocations {α : Type*} (invocations : α → ℕ) : List α → ℕ | |
| | [] => 1 | |
| | a :: rest => invocations a + arithmeticMapInvocations invocations rest + 1 | |
| theorem arithmeticMapInvocations_le_work {α β : Type*} (f : α → β × ℕ) | |
| (invocations : α → ℕ) (values : List α) | |
| (h : ∀ a ∈ values, invocations a ≤ (f a).2) : | |
| arithmeticMapInvocations invocations values ≤ (arithmeticMapWithWork f values).2 + 1 := by | |
| induction values with | |
| | nil => rfl | |
| | cons a rest ih => | |
| have hh := h a List.mem_cons_self | |
| have ht := ih (fun b hb => h b (List.mem_cons_of_mem _ hb)) | |
| simp only [arithmeticMapInvocations, arithmeticMapWithWork] | |
| omega | |
| namespace BinaryFraction | |
| def invInvocations (a : BinaryFraction) : ℕ := bitCompareInvocations a.numerator.bits [] + 1 | |
| theorem invInvocations_le_work (a : BinaryFraction) : | |
| invInvocations a ≤ a.invWithWork.2 := by | |
| have h := bitCompareInvocations_le_work a.numerator.bits [] | |
| unfold invInvocations invWithWork | |
| dsimp only | |
| split_ifs <;> dsimp only <;> omega | |
| def divInvocations (a b : BinaryFraction) : ℕ := | |
| invInvocations b + canonicalMulInvocations a b.invWithWork.1 + 1 | |
| theorem divInvocations_le_work (a b : BinaryFraction) : | |
| divInvocations a b ≤ (divWithWork a b).2 := by | |
| have hi := invInvocations_le_work b | |
| have hm := canonicalMulInvocations_le_work a b.invWithWork.1 | |
| unfold divInvocations divWithWork | |
| dsimp only | |
| omega | |
| def fixedDenominatorInvocations (a : BinaryFraction) (denominator : List Bool) | |
| (hden : 0 < bitWordValue denominator) : ℕ := | |
| let product := bitMulWithWork a.numerator.bits denominator | |
| let quotient := bitDivModWithWork a.denominator product.1 | |
| let raw : BinaryFraction := ⟨⟨a.numerator.negative, quotient.1⟩, denominator, hden⟩ | |
| bitMulInvocations a.numerator.bits denominator + bitDivModInvocations a.denominator product.1 + | |
| canonicalizeInvocations raw + 1 | |
| theorem fixedDenominatorInvocations_le_work (a : BinaryFraction) (denominator : List Bool) | |
| (hden : 0 < bitWordValue denominator) : | |
| fixedDenominatorInvocations a denominator hden ≤ (fixedDenominatorWithWork a denominator hden).2 := by | |
| have hm := bitMulInvocations_le_work a.numerator.bits denominator | |
| have hd := bitDivModInvocations_le_work a.denominator | |
| (bitMulWithWork a.numerator.bits denominator).1 | |
| have hc := canonicalizeInvocations_le_work | |
| (⟨⟨a.numerator.negative, | |
| (bitDivModWithWork a.denominator (bitMulWithWork a.numerator.bits denominator).1).1⟩, | |
| denominator, hden⟩ : BinaryFraction) | |
| unfold fixedDenominatorInvocations fixedDenominatorWithWork | |
| dsimp only | |
| omega | |
| def naturalCeilingInvocations (a : BinaryFraction) : ℕ := | |
| if a.numerator.negative then 1 else bitCeilDivInvocations a.denominator a.numerator.bits + 1 | |
| theorem naturalCeilingInvocations_le_work (a : BinaryFraction) : | |
| naturalCeilingInvocations a ≤ a.naturalCeilingWithWork.2 := by | |
| have h := bitCeilDivInvocations_le_work a.denominator a.numerator.bits | |
| unfold naturalCeilingInvocations naturalCeilingWithWork | |
| split_ifs <;> dsimp only <;> omega | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |