Loop-based consumers

This module provides consumers that iterate over a given iterator, applying a certain user-supplied function in every iteration. Concretely, the following operations are provided:

• ForIn instances
• IterM.fold, the analogue of List.foldl
• IterM.foldM, the analogue of List.foldlM
• IterM.drain, which iterates over the whole iterator and discards all emitted values. It can be used to apply the monadic effects of the iterator.

Some producers and combinators provide specialized implementations. These are captured by the IteratorLoop and IteratorLoopPartial typeclasses. They should be implemented by all types of iterators. A default implementation is provided. The typeclass LawfulIteratorLoop asserts that an IteratorLoop instance equals the default implementation.

def Std.Iterators.IteratorLoop.rel (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] {γ : Type x} (plausible_forInStep : β → γ → ForInStep γ → Prop) (p' p : IterM m β × γ) : Prop

Relation that needs to be well-formed in order for a loop over an iterator to terminate. It is assumed that the plausible_forInStep predicate relates the input and output of the stepper function.

def Std.Iterators.IteratorLoop.WellFounded (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] {γ : Type x} (plausible_forInStep : β → γ → ForInStep γ → Prop) : Prop

Asserts that IteratorLoop.rel is well-founded.

class Std.Iterators.IteratorLoop (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] (n : Type w → Type w'') : Type (max (max (w + 1) w') w'')

IteratorLoop α m provides efficient implementations of loop-based consumers for α-based iterators. The basis is a ForIn-style loop construct with the complication that it can be used for infinite iterators, too -- given a proof that the given loop will nevertheless terminate.

This class is experimental and users of the iterator API should not explicitly depend on it. They can, however, assume that consumers that require an instance will work for all iterators provided by the standard library.

• forIn (_lift : (γ : Type w) → m γ → n γ) (γ : Type w) (plausible_forInStep : β → γ → ForInStep γ → Prop) : WellFounded α m plausible_forInStep → IterM m β → γ → ((b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) → n γ

class Std.Iterators.IteratorLoopPartial (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β] (n : Type w → Type w'') : Type (max (max (w + 1) w') w'')

IteratorLoopPartial α m provides efficient implementations of loop-based consumers for α-based iterators. The basis is a partial, i.e. potentially nonterminating, ForIn instance.

This class is experimental and users of the iterator API should not explicitly depend on it. They can, however, assume that consumers that require an instance will work for all iterators provided by the standard library.

• forInPartial (_lift : (γ : Type w) → m γ → n γ) {γ : Type w} : IterM m β → γ → (β → γ → n (ForInStep γ)) → n γ

instance Std.Iterators.instWellFoundedRelationWFRel {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] {γ : Type x} {plausible_forInStep : β → γ → ForInStep γ → Prop} (wf : IteratorLoop.WellFounded α m plausible_forInStep) : WellFoundedRelation (Std.Iterators.IteratorLoop.WFRel✝ wf)

@[irreducible, specialize #[]]

def Std.Iterators.IterM.DefaultConsumers.forIn {m : Type w → Type w'} {α β : Type w} [Iterator α m β] {n : Type w → Type w''} [Monad n] (lift : (γ : Type w) → m γ → n γ) (γ : Type w) (plausible_forInStep : β → γ → ForInStep γ → Prop) (wf : IteratorLoop.WellFounded α m plausible_forInStep) (it : IterM m β) (init : γ) (f : (b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) : n γ

This is the loop implementation of the default instance IteratorLoop.defaultImplementation.

@[inline]

def Std.Iterators.IteratorLoop.defaultImplementation {β α : Type w} {m : Type w → Type w'} {n : Type w → Type w''} [Monad n] [Iterator α m β] : IteratorLoop α m n

This is the default implementation of the IteratorLoop class. It simply iterates through the iterator using IterM.step. For certain iterators, more efficient implementations are possible and should be used instead.

class Std.Iterators.LawfulIteratorLoop {β : Type w} (α : Type w) (m : Type w → Type w') (n : Type w → Type w'') [Monad n] [Iterator α m β] [Finite α m] [i : IteratorLoop α m n] : Prop

Asserts that a given IteratorLoop instance is equal to IteratorLoop.defaultImplementation. (Even though equal, the given instance might be vastly more efficient.)

• lawful : i = IteratorLoop.defaultImplementation

@[specialize #[]]

partial def Std.Iterators.IterM.DefaultConsumers.forInPartial {m : Type w → Type w'} {α β : Type w} [Iterator α m β] {n : Type w → Type w''} [Monad n] (lift : (γ : Type w) → m γ → n γ) (γ : Type w) (it : IterM m β) (init : γ) (f : β → γ → n (ForInStep γ)) : n γ

This is the loop implementation of the default instance IteratorLoopPartial.defaultImplementation.

@[inline]

def Std.Iterators.IteratorLoopPartial.defaultImplementation {β α : Type w} {m : Type w → Type w'} {n : Type w → Type w''} [Monad m] [Monad n] [Iterator α m β] : IteratorLoopPartial α m n

This is the default implementation of the IteratorLoopPartial class. It simply iterates through the iterator using IterM.step. For certain iterators, more efficient implementations are possible and should be used instead.

instance Std.Iterators.instLawfulIteratorLoopOfMonad {β : Type w} (α : Type w) (m n : Type w → Type w') [Monad m] [Monad n] [Iterator α m β] [Finite α m] : LawfulIteratorLoop α m n

theorem Std.Iterators.IteratorLoop.wellFounded_of_finite {m : Type w → Type w'} {α β γ : Type w} [Iterator α m β] [Finite α m] : WellFounded α m fun (x : β) (x : γ) (x : ForInStep γ) => True

@[inline]

def Std.Iterators.IteratorLoop.finiteForIn {m : Type w → Type w'} {n : Type w → Type w''} {α β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n] (lift : (γ : Type w) → m γ → n γ) : ForIn n (IterM m β) β

This ForIn-style loop construct traverses a finite iterator using an IteratorLoop instance.

instance Std.Iterators.instForInIterMOfFiniteOfIteratorLoopOfMonadLiftT {m : Type w → Type w'} {n : Type w → Type w''} {α β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n] [MonadLiftT m n] : ForIn n (IterM m β) β

instance Std.Iterators.instForInPartialOfIteratorLoopPartialOfMonadLiftT {m : Type w → Type w'} {n : Type w → Type w''} {α β : Type w} [Iterator α m β] [IteratorLoopPartial α m n] [MonadLiftT m n] : ForIn n (IterM.Partial m β) β

instance Std.Iterators.instForMIterMOfFiniteOfIteratorLoopOfMonadLiftT {m : Type w → Type w'} {n : Type w → Type w''} {α β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n] [MonadLiftT m n] : ForM n (IterM m β) β

instance Std.Iterators.instForMPartialOfFiniteOfIteratorLoopPartialOfMonadLiftT {m : Type w → Type w'} {n : Type w → Type w''} {α β : Type w} [Iterator α m β] [Finite α m] [IteratorLoopPartial α m n] [MonadLiftT m n] : ForM n (IterM.Partial m β) β

@[inline]

def Std.Iterators.IterM.foldM {m : Type w → Type w'} {n : Type w → Type w''} [Monad n] {α β γ : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n] [MonadLiftT m n] (f : γ → β → n γ) (init : γ) (it : IterM m β) : n γ

Folds a monadic function over an iterator from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

The monadic effects of f are interleaved with potential effects caused by the iterator's step function. Therefore, it may not be equivalent to (← it.toList).foldlM.

This function requires a Finite instance proving that the iterator will finish after a finite number of steps. If the iterator is not finite or such an instance is not available, consider using it.allowNontermination.foldM instead of it.foldM. However, it is not possible to formally verify the behavior of the partial variant.

@[inline]

def Std.Iterators.IterM.Partial.foldM {m n : Type w → Type w'} [Monad n] {α β γ : Type w} [Iterator α m β] [IteratorLoopPartial α m n] [MonadLiftT m n] (f : γ → β → n γ) (init : γ) (it : Partial m β) : n γ

Folds a monadic function over an iterator from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

The monadic effects of f are interleaved with potential effects caused by the iterator's step function. Therefore, it may not be equivalent to it.toList.foldlM.

This is a partial, potentially nonterminating, function. It is not possible to formally verify its behavior. If the iterator has a Finite instance, consider using IterM.foldM instead.

@[inline]

def Std.Iterators.IterM.fold {m : Type w → Type w'} {α β γ : Type w} [Monad m] [Iterator α m β] [Finite α m] [IteratorLoop α m m] (f : γ → β → γ) (init : γ) (it : IterM m β) : m γ

Folds a function over an iterator from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

It is equivalent to it.toList.foldl.

This function requires a Finite instance proving that the iterator will finish after a finite number of steps. If the iterator is not finite or such an instance is not available, consider using it.allowNontermination.fold instead of it.fold. However, it is not possible to formally verify the behavior of the partial variant.

@[inline]

def Std.Iterators.IterM.Partial.fold {m : Type w → Type w'} {α β γ : Type w} [Monad m] [Iterator α m β] [IteratorLoopPartial α m m] (f : γ → β → γ) (init : γ) (it : Partial m β) : m γ

Folds a function over an iterator from the left, accumulating a value starting with init. The accumulated value is combined with the each element of the list in order, using f.

It is equivalent to it.toList.foldl.

This is a partial, potentially nonterminating, function. It is not possible to formally verify its behavior. If the iterator has a Finite instance, consider using IterM.fold instead.

@[inline]

def Std.Iterators.IterM.drain {α : Type w} {m : Type w → Type w'} [Monad m] {β : Type w} [Iterator α m β] [Finite α m] (it : IterM m β) [IteratorLoop α m m] : m PUnit

Iterates over the whole iterator, applying the monadic effects of each step, discarding all emitted values.

This function requires a Finite instance proving that the iterator will finish after a finite number of steps. If the iterator is not finite or such an instance is not available, consider using it.allowNontermination.drain instead of it.drain. However, it is not possible to formally verify the behavior of the partial variant.

@[inline]

def Std.Iterators.IterM.Partial.drain {α : Type w} {m : Type w → Type w'} [Monad m] {β : Type w} [Iterator α m β] (it : Partial m β) [IteratorLoopPartial α m m] : m PUnit

Iterates over the whole iterator, applying the monadic effects of each step, discarding all emitted values.

This is a partial, potentially nonterminating, function. It is not possible to formally verify its behavior. If the iterator has a Finite instance, consider using IterM.drain instead.