(Imperfect) discrimination trees. We use a hybrid representation.

• A PersistentHashMap for the root node which usually contains many children.
• A sorted array of key/node pairs for inner nodes.

The edges are labeled by keys:

• Constant names (and arity). Universe levels are ignored.
• Free variables (and arity). Thus, an entry in the discrimination tree may reference hypotheses from the local context.
• Literals
• Star/Wildcard. We use them to represent metavariables and terms we want to ignore. We ignore implicit arguments and proofs.
• Other. We use to represent other kinds of terms (e.g., nested lambda, forall, sort, etc).

We reduce terms using TransparencyMode.reducible. Thus, all reducible definitions in an expression e are unfolded before we insert it into the discrimination tree.

Recall that projections from classes are NOT reducible. For example, the expressions Add.add α (ringAdd ?α ?s) ?x ?x and Add.add Nat Nat.hasAdd a b generates paths with the following keys respectively

⟨Add.add, 4⟩, α, *, *, *
⟨Add.add, 4⟩, Nat, *, ⟨a,0⟩, ⟨b,0⟩

That is, we don't reduce Add.add Nat inst a b into Nat.add a b. We say the Add.add applications are the de-facto canonical forms in the metaprogramming framework. Moreover, it is the metaprogrammer's responsibility to re-pack applications such as Nat.add a b into Add.add Nat inst a b.

Remark: we store the arity in the keys 1- To be able to implement the "skip" operation when retrieving "candidate" unifiers. 2- Distinguish partial applications f a, f a b, and f a b c.

def Lean.Meta.DiscrTree.Key.ctorIdx : Key → Nat

def Lean.Meta.DiscrTree.Key.lt : Key → Key → Bool

instance Lean.Meta.DiscrTree.instLTKey : LT Key

instance Lean.Meta.DiscrTree.instDecidableLtKey (a b : Key) : Decidable (a < b)

def Lean.Meta.DiscrTree.Key.format : Key → Format

instance Lean.Meta.DiscrTree.instToFormatKey : ToFormat Key

def Lean.Meta.DiscrTree.keysAsPattern (keys : Array Key) : CoreM MessageData

Helper function for converting an entry (i.e., Array Key) to the discrimination tree into MessageData that is more user-friendly. We use this function to implement diagnostic information.

def Lean.Meta.DiscrTree.keysAsPattern.next? : StateRefT' IO.RealWorld (List Key) CoreM (Option Key)

def Lean.Meta.DiscrTree.keysAsPattern.mkApp (f : MessageData) (args : Array MessageData) (parenIfNonAtomic : Bool) : CoreM MessageData

partial def Lean.Meta.DiscrTree.keysAsPattern.go (parenIfNonAtomic : Bool := true) : StateRefT' IO.RealWorld (List Key) CoreM MessageData

partial def Lean.Meta.DiscrTree.keysAsPattern.goN (num : Nat) : StateRefT' IO.RealWorld (List Key) CoreM (Array MessageData)

def Lean.Meta.DiscrTree.Key.arity : Key → Nat

instance Lean.Meta.DiscrTree.instInhabitedTrie {α : Type} : Inhabited (Trie α)

def Lean.Meta.DiscrTree.empty {α : Type} : DiscrTree α

partial def Lean.Meta.DiscrTree.Trie.format {α : Type} [ToFormat α] : Trie α → Format

instance Lean.Meta.DiscrTree.instToFormatTrie {α : Type} [ToFormat α] : ToFormat (Trie α)

def Lean.Meta.DiscrTree.format {α : Type} [ToFormat α] (d : DiscrTree α) : Format

instance Lean.Meta.DiscrTree.instToFormat {α : Type} [ToFormat α] : ToFormat (DiscrTree α)

instance Lean.Meta.DiscrTree.instInhabited {α : Type} : Inhabited (DiscrTree α)

def Lean.Meta.DiscrTree.mkNoindexAnnotation (e : Expr) : Expr

def Lean.Meta.DiscrTree.hasNoindexAnnotation (e : Expr) : Bool

partial def Lean.Meta.DiscrTree.reduce (e : Expr) : MetaM Expr

Reduction procedure for the discrimination tree indexing.

def Lean.Meta.DiscrTree.reduceDT (e : Expr) (root : Bool) : MetaM Expr

whnf for the discrimination tree module

partial def Lean.Meta.DiscrTree.mkPathAux (root : Bool) (todo : Array Expr) (keys : Array Key) (noIndexAtArgs : Bool) : MetaM (Array Key)

When noIndexAtArgs := true, pushArgs assumes function application arguments have a no_index annotation. That is, f a b is indexed as it was f (no_index a) (no_index b). This feature is used when indexing local proofs in the simplifier. This is useful in examples like the one described on issue #2670. In this issue, we have a local hypotheses (h : ∀ p : α × β, f p p.2 = p.2), and users expect it to be applicable to f (a, b) b = b. This worked in Lean 3 since no indexing was used. We can retrieve Lean 3 behavior by writing (h : ∀ p : α × β, f p (no_index p.2) = p.2), but this is very inconvenient when the hypotheses was not written by the user in first place. For example, it was introduced by another tactic. Thus, when populating the discrimination tree explicit arguments provided to simp (e.g., simp [h]), we use noIndexAtArgs := true. See comment: https://github.com/leanprover/lean4/issues/2670#issuecomment-1758889365

def Lean.Meta.DiscrTree.mkPath (e : Expr) (noIndexAtArgs : Bool := false) : MetaM (Array Key)

When noIndexAtArgs := true, pushArgs assumes function application arguments have a no_index annotation. That is, f a b is indexed as it was f (no_index a) (no_index b). This feature is used when indexing local proofs in the simplifier. This is useful in examples like the one described on issue #2670. In this issue, we have a local hypotheses (h : ∀ p : α × β, f p p.2 = p.2), and users expect it to be applicable to f (a, b) b = b. This worked in Lean 3 since no indexing was used. We can retrieve Lean 3 behavior by writing (h : ∀ p : α × β, f p (no_index p.2) = p.2), but this is very inconvenient when the hypotheses was not written by the user in first place. For example, it was introduced by another tactic. Thus, when populating the discrimination tree explicit arguments provided to simp (e.g., simp [h]), we use noIndexAtArgs := true. See comment: https://github.com/leanprover/lean4/issues/2670#issuecomment-1758889365

def Lean.Meta.DiscrTree.insertCore {α : Type} [BEq α] (d : DiscrTree α) (keys : Array Key) (v : α) : DiscrTree α

def Lean.Meta.DiscrTree.insert {α : Type} [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (noIndexAtArgs : Bool := false) : MetaM (DiscrTree α)

def Lean.Meta.DiscrTree.insertIfSpecific {α : Type} [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (noIndexAtArgs : Bool := false) : MetaM (DiscrTree α)

Inserts a value into a discrimination tree, but only if its key is not of the form #[*] or #[=, *, *, *].

def Lean.Meta.DiscrTree.getMatch {α : Type} (d : DiscrTree α) (e : Expr) : MetaM (Array α)

Find values that match e in d.

def Lean.Meta.DiscrTree.getMatchWithExtra {α : Type} (d : DiscrTree α) (e : Expr) : MetaM (Array (α × Nat))

Similar to getMatch, but returns solutions that are prefixes of e. We store the number of ignored arguments in the result.

partial def Lean.Meta.DiscrTree.getMatchWithExtra.mayMatchPrefix {α : Type} (d : DiscrTree α) (k : Key) : MetaM Bool

partial def Lean.Meta.DiscrTree.getMatchWithExtra.go {α : Type} (d : DiscrTree α) (e : Expr) (numExtra : Nat) (result : Array (α × Nat)) : MetaM (Array (α × Nat))

def Lean.Meta.DiscrTree.getMatchKeyRootFor (e : Expr) : MetaM (Key × Nat)

Return the root symbol for e, and the number of arguments after reduceDT.

def Lean.Meta.DiscrTree.getMatchLiberal {α : Type} (d : DiscrTree α) (e : Expr) : MetaM (Array α × Nat)

A liberal version of getMatch which only takes the root symbol of e into account. We use this method to simulate Lean 3's indexing.

The natural number in the result is the number of arguments in e after reduceDT.

def Lean.Meta.DiscrTree.getUnify {α : Type} (d : DiscrTree α) (e : Expr) : MetaM (Array α)

partial def Lean.Meta.DiscrTree.getUnify.process {α : Type} (skip : Nat) (todo : Array Expr) (c : Trie α) (result : Array α) : MetaM (Array α)

partial def Lean.Meta.DiscrTree.Trie.foldM {m : Type u_1 → Type u_2} {σ : Type u_1} {α : Type} [Monad m] (initialKeys : Array Key) (f : σ → Array Key → α → m σ) (init : σ) : Trie α → m σ

Monadically fold the keys and values stored in a Trie.

@[inline]

def Lean.Meta.DiscrTree.Trie.fold {σ : Type u_1} {α : Type} (initialKeys : Array Key) (f : σ → Array Key → α → σ) (init : σ) (t : Trie α) : σ

Fold the keys and values stored in a Trie.

partial def Lean.Meta.DiscrTree.Trie.foldValuesM {m : Type u_1 → Type u_2} {σ : Type u_1} {α : Type} [Monad m] (f : σ → α → m σ) (init : σ) : Trie α → m σ

Monadically fold the values stored in a Trie.

@[inline]

def Lean.Meta.DiscrTree.Trie.foldValues {σ : Type u_1} {α : Type} (f : σ → α → σ) (init : σ) (t : Trie α) : σ

Fold the values stored in a Trie.

partial def Lean.Meta.DiscrTree.Trie.size {α : Type} : Trie α → Nat

The number of values stored in a Trie.

@[inline]

def Lean.Meta.DiscrTree.foldM {m : Type u_1 → Type u_2} {σ : Type u_1} {α : Type} [Monad m] (f : σ → Array Key → α → m σ) (init : σ) (t : DiscrTree α) : m σ

Monadically fold over the keys and values stored in a DiscrTree.

@[inline]

def Lean.Meta.DiscrTree.fold {σ : Type u_1} {α : Type} (f : σ → Array Key → α → σ) (init : σ) (t : DiscrTree α) : σ

Fold over the keys and values stored in a DiscrTree

@[inline]

def Lean.Meta.DiscrTree.foldValuesM {m : Type u_1 → Type u_2} {σ : Type u_1} {α : Type} [Monad m] (f : σ → α → m σ) (init : σ) (t : DiscrTree α) : m σ

Monadically fold over the values stored in a DiscrTree.

@[inline]

def Lean.Meta.DiscrTree.foldValues {σ : Type u_1} {α : Type} (f : σ → α → σ) (init : σ) (t : DiscrTree α) : σ

Fold over the values stored in a DiscrTree.

@[inline]

def Lean.Meta.DiscrTree.containsValueP {α : Type} (t : DiscrTree α) (f : α → Bool) : Bool

Check for the presence of a value satisfying a predicate.

@[inline]

def Lean.Meta.DiscrTree.values {α : Type} (t : DiscrTree α) : Array α

Extract the values stored in a DiscrTree.

@[inline]

def Lean.Meta.DiscrTree.toArray {α : Type} (t : DiscrTree α) : Array (Array Key × α)

Extract the keys and values stored in a DiscrTree.

@[inline]

def Lean.Meta.DiscrTree.size {α : Type} (t : DiscrTree α) : Nat

Get the number of values stored in a DiscrTree. O(n) in the size of the tree.

partial def Lean.Meta.DiscrTree.Trie.mapArraysM {m : Type → Type} [Monad m] {α β : Type} (t : Trie α) (f : Array α → m (Array β)) : m (Trie β)

Apply a monadic function to the array of values at each node in a DiscrTree.

def Lean.Meta.DiscrTree.mapArraysM {m : Type → Type} [Monad m] {α β : Type} (d : DiscrTree α) (f : Array α → m (Array β)) : m (DiscrTree β)

Apply a monadic function to the array of values at each node in a DiscrTree.

def Lean.Meta.DiscrTree.mapArrays {α β : Type} (d : DiscrTree α) (f : Array α → Array β) : DiscrTree β

Apply a function to the array of values at each node in a DiscrTree.