Topological / Suspending Recursive Builder

This module defines a recursive build function that topologically (ι.e., via a depth-first search with memoization) builds the elements of a build store.

This is called a suspending scheduler in Build systems à la carte.

Recursive Fetching

In this section, we define the primitives that make up a builder.

@[inline, reducible]

abbrev Lake.DFetchFn (α : Type u) (β : α → Type v) (m : Type v → Type w) : Type (max u w)

A dependently typed monadic fetch function.

That is, a function within the monad m and takes an input a : α describing what to fetch and and produces some output b : β a (dependently typed) or b : B (not) describing what was fetched. All build functions are fetch functions, but not all fetch functions need build something.

Equations

• Lake.DFetchFn α β m = ((a : α) → m (β a))

In order to nest builds / fetches within one another, we equip the monad m with a fetch function of its own.

@[inline, reducible]

abbrev Lake.DFetchT (α : Type u) (β : α✝ → Type v) (m : Type v → Type w) (α : Type v) : Type (max (max w u) w)

A transformer that equips a monad with a DFetchFn.

Equations

• Lake.DFetchT α β m = Lake.EquipT (Lake.DFetchFn α β m) m

@[inline, reducible]

abbrev Lake.FetchT (α : Type u) (β : Type v) (m : Type v → Type w) (α : Type v) : Type (max (max w u) w)

A DFetchT that is not dependently typed.

Equations

• Lake.FetchT α β m = Lake.DFetchT α (fun (x : α) => β) m

We can then use the such a monad as the basis for a fetch function itself.

@[inline, reducible]

abbrev Lake.DRecFetchFn (α : Type u) (β : α → Type v) (m : Type v → Type w) : Type (max u w u)

Equations

• Lake.DRecFetchFn α β m = Lake.DFetchFn α β (Lake.DFetchT α β m)

@[inline, reducible]

abbrev Lake.RecFetchFn (α : Type u) (β : Type v) (m : Type v → Type w) : Type (max u w)

A DRecFetchFn that is not dependently typed.

Equations

• Lake.RecFetchFn α β m = (α → Lake.FetchT α β m β)

@[specialize #[]]

partial def Lake.recFetch {m : Type u → Type u_1} {α : Type u_2} {β : α → Type u} [∀ (α : Type u), Nonempty (m α)] (fetch : Lake.DRecFetchFn α β m) : Lake.DFetchFn α β m

A DFetchFn that provides its base DRecFetchFn with itself.

The basic recFetch can fail to terminate in a variety of ways, it can even cycle (i.e., a fetches b which fetches a). Thus, we define the acyclicRecFetch below to guard against such cases.

@[inline]

def Lake.recFetchAcyclic {κ : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_3} {β : α → Type u_1} [BEq κ] [Monad m] (keyOf : α → κ) (fetch : Lake.DRecFetchFn α β (Lake.CycleT κ m)) : Lake.DFetchFn α β (Lake.CycleT κ m)

A recFetch augmented by a CycleT to guard against recursive cycles. If the set of visited keys is finite, this function should provably terminate.

We use keyOf to the derive the unique key of a fetch from its descriptor a : α. We do this because descriptors may not be comparable and/or contain more information than necessary to determine uniqueness.

Equations

• One or more equations did not get rendered due to their size.

When building, we usually do not want to build the same thing twice during a single build pass. At the same time, separate builds may both wish to fetch the same thing. Thus, we need to store past build results to return them upon future fetches. This is what recFetchMemoize below does.

@[inline]

def Lake.recFetchMemoize {κ : Type u_1} {m : Type u_1 → Type u_2} {β : κ → Type u_1} {α : Type u_3} [BEq κ] [Monad m] [Lake.MonadDStore κ β m] (keyOf : α → κ) (fetch : Lake.DRecFetchFn α (fun (a : α) => β (keyOf a)) (Lake.CycleT κ m)) : Lake.DFetchFn α (fun (a : α) => β (keyOf a)) (Lake.CycleT κ m)

recFetchAcyclic augmented with a MonadDStore to memoize fetch results and thus avoid computing the same result twice.

Equations

• One or more equations did not get rendered due to their size.

Building

In this section, we use the abstractions we have just created to define the desired topological recursive build function (a.k.a. a suspending scheduler).

@[inline]

def Lake.buildAcyclic {κ : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [BEq κ] [Monad m] (keyOf : α → κ) (a : α) (build : Lake.RecFetchFn α β (Lake.CycleT κ m)) : ExceptT (Lake.Cycle κ) m β

Recursively builds objects for the keys κ, avoiding cycles.

Equations

• Lake.buildAcyclic keyOf a build = Lake.recFetchAcyclic keyOf build a []

@[inline]

def Lake.buildDTop {κ : Type u_1} {m : Type u_1 → Type u_2} {α : Type u_3} (β : κ → Type u_1) [BEq κ] [Monad m] [Lake.MonadDStore κ β m] (keyOf : α → κ) (a : α) (build : Lake.DRecFetchFn α (fun (a : α) => β (keyOf a)) (Lake.CycleT κ m)) : ExceptT (Lake.Cycle κ) m (β (keyOf a))

Dependently typed version of buildTop.

Equations

• Lake.buildDTop β keyOf a build = Lake.recFetchMemoize keyOf build a []

@[inline]

def Lake.buildTop {κ : Type u_1} {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [BEq κ] [Monad m] [Lake.MonadStore κ β m] (keyOf : α → κ) (a : α) (build : Lake.RecFetchFn α β (Lake.CycleT κ m)) : ExceptT (Lake.Cycle κ) m β

Recursively fills a MonadStore of key-object pairs by building objects topologically (ι.e., depth-first with memoization). If a cycle is detected, the list of keys traversed is thrown.

Equations

• Lake.buildTop keyOf a build = Lake.recFetchMemoize keyOf build a []