49551036ed
Old `Nat.repeat` => `Nat.for`
Old `Nat.mrepeat` => `Nat.mfor`
New `Nat.repeat` has type
```
def repeat {α : Type u} (f : α → α) (n : Nat) (a : α) : α :=
``
`List.repeat` => `List.replicate` (like in Haskell)
Avoid weird `ℕ` in List library
210 lines
7.3 KiB
Text
210 lines
7.3 KiB
Text
/-
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Copyright (c) 2018 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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-/
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prelude
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import init.data.nat.basic init.data.fin.basic init.data.uint
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import init.data.repr init.data.tostring init.control.id
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universes u v w
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/-
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The Compiler has special support for arrays.
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They are implemented using dynamic arrays: https://en.wikipedia.org/wiki/Dynamic_array
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-/
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-- TODO(Leo): mark as opaque
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structure Array (α : Type u) :=
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(sz : Nat)
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(data : Fin sz → α)
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attribute [extern cpp inline "lean::array_sz(#2)"] Array.sz
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@[extern cpp inline "lean::array_get_size(#2)"]
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def Array.size {α : Type u} (a : @& Array α) : Nat :=
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a.sz
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namespace Array
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variables {α : Type u} {β : Type v}
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/- The parameter `c` is the initial capacity -/
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@[extern cpp inline "lean::mk_empty_array(#1)"]
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def mkEmpty (c : @& Nat) : Array α :=
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{ sz := 0,
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data := λ ⟨x, h⟩, absurd h (Nat.notLtZero x) }
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@[extern cpp inline "lean::array_push(#2, #3)"]
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def push (a : Array α) (v : α) : Array α :=
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{ sz := Nat.succ a.sz,
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data := λ ⟨j, h₁⟩,
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if h₂ : j = a.sz then v
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else a.data ⟨j, Nat.ltOfLeOfNe (Nat.leOfLtSucc h₁) h₂⟩ }
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def mkArray {α : Type u} (n : Nat) (v : α) : Array α :=
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let a : Array α := mkEmpty n in
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n.repeat (λ a, a.push v) a
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def empty : Array α :=
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mkEmpty 0
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instance : HasEmptyc (Array α) :=
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⟨Array.empty⟩
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instance : Inhabited (Array α) :=
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⟨Array.empty⟩
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def isEmpty (a : Array α) : Bool :=
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a.size = 0
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def singleton (v : α) : Array α :=
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mkArray 1 v
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@[extern cpp inline "lean::array_index(#2, #3)"]
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def index (a : @& Array α) (i : @& Fin a.sz) : α :=
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a.data i
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/- Low-level version of `index` which is as fast as a C array read.
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`Fin` values are represented as tag pointers in the Lean runtime. Thus,
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`index` may be slightly slower than `idx`. -/
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@[extern cpp inline "lean::array_idx(#2, #3)"]
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def idx (a : @& Array α) (i : USize) (h : i.toNat < a.sz) : α :=
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a.index ⟨i.toNat, h⟩
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/- "Comfortable" version of `index`. It performs a bound check at runtime. -/
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@[extern cpp inline "lean::array_get(#2, #3, #4)"]
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def get [Inhabited α] (a : @& Array α) (i : @& Nat) : α :=
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if h : i < a.sz then a.index ⟨i, h⟩ else default α
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def getOpt (a : Array α) (i : Nat) : Option α :=
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if h : i < a.size then some (a.index ⟨i, h⟩) else none
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@[extern cpp inline "lean::array_update(#2, #3, #4)"]
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def update (a : Array α) (i : @& Fin a.sz) (v : α) : Array α :=
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{ sz := a.sz,
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data := λ j, if h : i = j then v else a.data j }
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/- Low-level version of `update` which is as fast as a C array update.
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`Fin` values are represented as tag pointers in the Lean runtime. Thus,
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`update` may be slightly slower than `updt`. -/
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@[extern cpp inline "lean::array_updt(#2, #3, #4)"]
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def updt (a : @& Array α) (i : USize) (v : @& α) (h : i.toNat < a.sz) : Array α :=
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a.update ⟨i.toNat, h⟩ v
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/- "Comfortable" version of `update`. It performs a bound check at runtime. -/
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@[extern cpp inline "lean::array_set(#2, #3, #4)"]
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def set (a : Array α) (i : @& Nat) (v : α) : Array α :=
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if h : i < a.sz then a.update ⟨i, h⟩ v else a
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theorem szUpdateEq (a : Array α) (i : Fin a.sz) (v : α) : (update a i v).sz = a.sz :=
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rfl
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@[extern cpp inline "lean::array_pop(#2)"]
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def pop (a : Array α) : Array α :=
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{ sz := Nat.pred a.sz,
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data := λ ⟨j, h⟩, a.index ⟨j, Nat.ltOfLtOfLe h (Nat.predLe _)⟩ }
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section
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variables {m : Type v → Type v} [Monad m]
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local attribute [instance] monadInhabited'
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-- TODO(Leo): justify termination using wf-rec
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@[specialize] partial def miterateAux (a : Array α) (f : Π i : Fin a.sz, α → β → m β) : Nat → β → m β
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| i b :=
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if h : i < a.sz then
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let idx : Fin a.sz := ⟨i, h⟩ in
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f idx (a.index idx) b >>= miterateAux (i+1)
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else pure b
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@[inline] def miterate (a : Array α) (b : β) (f : Π i : Fin a.sz, α → β → m β) : m β :=
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miterateAux a f 0 b
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@[inline] def mfoldl (a : Array α) (b : β) (f : α → β → m β) : m β :=
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miterate a b (λ _, f)
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end
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@[inline] def iterate (a : Array α) (b : β) (f : Π i : Fin a.sz, α → β → β) : β :=
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id.run $ miterateAux a f 0 b
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@[inline] def foldl (a : Array α) (f : α → β → β) (b : β) : β :=
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iterate a b (λ _, f)
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@[specialize] private def revIterateAux (a : Array α) (f : Π i : Fin a.sz, α → β → β) : Π (i : Nat), i ≤ a.sz → β → β
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| 0 h b := b
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| (j+1) h b :=
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let i : Fin a.sz := ⟨j, h⟩ in
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revIterateAux j (Nat.leOfLt h) (f i (a.index i) b)
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@[inline] def revIterate (a : Array α) (b : β) (f : Π i : Fin a.sz, α → β → β) : β :=
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revIterateAux a f a.size (Nat.leRefl _) b
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@[inline] def revFoldl (a : Array α) (b : β) (f : α → β → β) : β :=
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revIterate a b (λ _, f)
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def toList (a : Array α) : List α :=
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a.revFoldl [] (::)
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instance [HasRepr α] : HasRepr (Array α) :=
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⟨repr ∘ toList⟩
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instance [HasToString α] : HasToString (Array α) :=
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⟨toString ∘ toList⟩
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section
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variables {m : Type u → Type u} [Monad m]
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@[inline] private def mforeachAux (a : Array α) (f : Π i : Fin a.sz, α → m α) : m { a' : Array α // a'.sz = a.sz } :=
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miterate a ⟨a, rfl⟩ $ λ i v ⟨a', h⟩, do
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let i' : Fin a'.sz := Eq.recOn h.symm i,
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x ← f i v,
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pure $ ⟨a'.update i' x, (szUpdateEq a' i' x) ▸ h⟩
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@[inline] def mforeach (a : Array α) (f : Π i : Fin a.sz, α → m α) : m (Array α) :=
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Subtype.val <$> mforeachAux a f
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@[inline] def mmap (f : α → m α) (a : Array α) : m (Array α) :=
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mforeach a (λ _, f)
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end
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@[inline] def foreach (a : Array α) (f : Π i : Fin a.sz, α → α) : Array α :=
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id.run $ mforeach a f
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theorem szForeachEq (a : Array α) (f : Π i : Fin a.sz, α → α) : (foreach a f).sz = a.sz :=
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(id.run $ mforeachAux a f).property
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@[inline] def map (f : α → α) (a : Array α) : Array α :=
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foreach a (λ _, f)
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@[inline] def map₂ (f : α → α → α) (a b : Array α) : Array α :=
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if h : a.size ≤ b.size
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then foreach a (λ ⟨i, h'⟩, f (b.index ⟨i, Nat.ltOfLtOfLe h' h⟩))
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else foreach b (λ ⟨i, h'⟩, f (a.index ⟨i, Nat.ltTrans h' (Nat.gtOfNotLe h)⟩))
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end Array
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export Array (mkArray)
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private theorem repeatCorePushSz {α : Type u} : ∀ (n : Nat) (v : α) (a : Array α),
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(Nat.repeatCore (λ (a : Array α), a.push v) n (a.push v)).sz =
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(Nat.repeatCore (λ (a : Array α), a.push v) n a).sz.succ
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| 0 _ _ := rfl
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| (Nat.succ n) v a :=
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show (Nat.repeatCore (λ (a : Array α), a.push v) n ((a.push v).push v)).sz =
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(Nat.repeatCore (λ (a : Array α), a.push v) n (a.push v)).sz.succ, from
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repeatCorePushSz n v (a.push v)
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theorem szMkArrayEq {α : Type u} (n : Nat) (v : α) : (mkArray n v).sz = n :=
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Nat.recOn n rfl $ λ n ih,
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have aux₁ : (Nat.repeatCore (λ (a : Array α), a.push v) n (Array.mkEmpty n)).sz = n, from ih,
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have aux₂ : (Nat.repeatCore (λ (a : Array α), a.push v) n ((Array.mkEmpty n).push v)).sz =
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(Nat.repeatCore (λ (a : Array α), a.push v) n (Array.mkEmpty n)).sz.succ, from
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repeatCorePushSz _ _ _,
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have aux₃ : (Nat.repeatCore (λ (a : Array α), a.push v) n (Array.mkEmpty n)).sz.succ = n.succ, from
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congrArg _ aux₁,
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Eq.trans aux₂ aux₃
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@[inlineIfReduce] def List.toArrayAux {α : Type u} : List α → Array α → Array α
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| [] r := r
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| (a::as) r := List.toArrayAux as (r.push a)
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@[inline] def List.toArray {α : Type u} (as : List α) : Array α :=
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as.toArrayAux (Array.mkEmpty as.length)
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