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e63eb7f | 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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
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