(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.arity : Key → Nat

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.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.

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)