Monadic operations #
mapM #
Alternate (non-tail-recursive) form of mapM for proofs.
Equations
- List.mapM' f [] = pure []
- List.mapM' f (a :: l) = do let __do_lift ← f a let __do_lift_1 ← List.mapM' f l pure (__do_lift :: __do_lift_1)
Instances For
@[simp]
theorem
List.mapM'_nil
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
{f : α → m β}
:
List.mapM' f [] = pure []
@[simp]
theorem
List.mapM'_cons
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
{a : α}
{l : List α}
[Monad m]
{f : α → m β}
:
List.mapM' f (a :: l) = do
let __do_lift ← f a
let __do_lift_1 ← List.mapM' f l
pure (__do_lift :: __do_lift_1)
theorem
List.mapM'_eq_mapM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(f : α → m β)
(l : List α)
:
List.mapM' f l = List.mapM f l
theorem
List.mapM'_eq_mapM.go
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(f : α → m β)
(l : List α)
(acc : List β)
:
List.mapM.loop f l acc = do
let __do_lift ← List.mapM' f l
pure (acc.reverse ++ __do_lift)
theorem
List.foldlM_cons_eq_append
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(f : α → m β)
(as : List α)
(b : β)
(bs : List β)
:
Auxiliary lemma for mapM_eq_reverse_foldlM_cons.
foldlM and foldrM #
theorem
List.foldlM_map
{m : Type u_1 → Type u_2}
{β₁ : Type u_3}
{β₂ : Type u_4}
{α : Type u_1}
[Monad m]
(f : β₁ → β₂)
(g : α → β₂ → m α)
(l : List β₁)
(init : α)
:
List.foldlM g init (List.map f l) = List.foldlM (fun (x : α) (y : β₁) => g x (f y)) init l
theorem
List.foldrM_map
{m : Type u_1 → Type u_2}
{β₁ : Type u_3}
{β₂ : Type u_4}
{α : Type u_1}
[Monad m]
[LawfulMonad m]
(f : β₁ → β₂)
(g : β₂ → α → m α)
(l : List β₁)
(init : α)
:
List.foldrM g init (List.map f l) = List.foldrM (fun (x : β₁) (y : α) => g (f x) y) init l
forM #
@[simp]
theorem
List.forM_nil'
{m : Type u_1 → Type u_2}
{α : Type u_3}
{f : α → m PUnit}
[Monad m]
:
[].forM f = pure PUnit.unit
forIn' #
theorem
List.forIn'_loop_congr
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
{xs : List α}
[Monad m]
{as : List α}
{bs : List α}
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
{b : β}
(ha : ∃ (ys : List α), ys ++ xs = as)
(hb : ∃ (ys : List α), ys ++ xs = bs)
(h : ∀ (a : α) (m_1 : a ∈ as) (m' : a ∈ bs) (b : β), f a m_1 b = g a m' b)
:
List.forIn'.loop as f xs b ha = List.forIn'.loop bs g xs b hb
@[simp]
theorem
List.forIn'_cons
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
{a : α}
{as : List α}
(f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β))
(b : β)
:
forIn' (a :: as) b f = do
let x ← f a ⋯ b
match x with
| ForInStep.done b => pure b
| ForInStep.yield b => forIn' as b fun (a' : α) (m : a' ∈ as) (b : β) => f a' ⋯ b
theorem
List.forIn'_congr
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
{as : List α}
{bs : List α}
(w : as = bs)
{b : β}
{b' : β}
(hb : b = b')
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
(h : ∀ (a : α) (m_1 : a ∈ bs) (b : β), f a ⋯ b = g a m_1 b)
:
theorem
List.forIn'_eq_foldlM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(l : List α)
(f : (a : α) → a ∈ l → β → m (ForInStep β))
(init : β)
:
forIn' l init f = ForInStep.value <$> List.foldlM
(fun (b : ForInStep β) (a : { x : α // x ∈ l }) =>
match b with
| ForInStep.yield b => f a.val ⋯ b
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield init) l.attach
We can express a for loop over a list as a fold,
in which whenever we reach .done b we keep that value through the rest of the fold.
theorem
List.forIn_eq_foldlM
{m : Type u_1 → Type u_2}
{α : Type u_3}
{β : Type u_1}
[Monad m]
[LawfulMonad m]
(f : α → β → m (ForInStep β))
(init : β)
(l : List α)
:
forIn l init f = ForInStep.value <$> List.foldlM
(fun (b : ForInStep β) (a : α) =>
match b with
| ForInStep.yield b => f a b
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield init) l
We can express a for loop over a list as a fold,
in which whenever we reach .done b we keep that value through the rest of the fold.