(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 : Lean.Meta.DiscrTree.Key → Nat

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

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instance Lean.Meta.DiscrTree.instLTKey : LT Lean.Meta.DiscrTree.Key

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• Lean.Meta.DiscrTree.instLTKey = { lt := fun (a b : Lean.Meta.DiscrTree.Key) => Lean.Meta.DiscrTree.Key.lt a b = true }

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

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• Lean.Meta.DiscrTree.instDecidableLtKeyInstLTKey a b = inferInstanceAs (Decidable (Lean.Meta.DiscrTree.Key.lt a b = true))

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

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instance Lean.Meta.DiscrTree.instToFormatKey : Lean.ToFormat Lean.Meta.DiscrTree.Key

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• Lean.Meta.DiscrTree.instToFormatKey = { format := Lean.Meta.DiscrTree.Key.format }

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

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

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• Lean.Meta.DiscrTree.instInhabitedTrie = { default := Lean.Meta.DiscrTree.Trie.node #[] #[] }

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

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• Lean.Meta.DiscrTree.empty = { root := { root := Lean.PersistentHashMap.Node.entries Lean.PersistentHashMap.mkEmptyEntriesArray, size := 0 } }

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

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

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• Lean.Meta.DiscrTree.instToFormatTrie = { format := Lean.Meta.DiscrTree.Trie.format }

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

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

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• Lean.Meta.DiscrTree.instToFormatDiscrTree = { format := Lean.Meta.DiscrTree.format }

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

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

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• Lean.Meta.DiscrTree.mkNoindexAnnotation e = Lean.mkAnnotation `noindex e

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

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• Lean.Meta.DiscrTree.hasNoindexAnnotation e = Option.isSome (Lean.annotation? `noindex e)

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

Reduction procedure for the discrimination tree indexing. The parameter config controls how aggressively the term is reduced. The parameter at type DiscrTree controls this value. See comment at DiscrTree.

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

whnf for the discrimination tree module

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• Lean.Meta.DiscrTree.reduceDT e root config = if root = true then Lean.Meta.DiscrTree.reduceUntilBadKey e config else Lean.Meta.DiscrTree.reduce e config

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

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

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

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• Lean.Meta.DiscrTree.insert d e v config noIndexAtArgs = do let keys ← Lean.Meta.DiscrTree.mkPath e config noIndexAtArgs pure (Lean.Meta.DiscrTree.insertCore d keys v)

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

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

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

Find values that match e in d.

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• Lean.Meta.DiscrTree.getMatch d e config = do let __do_lift ← Lean.Meta.DiscrTree.getMatchCore d e config pure __do_lift.snd

def Lean.Meta.DiscrTree.getMatchWithExtra {α : Type} (d : Lean.Meta.DiscrTree α) (e : Lean.Expr) (config : Lean.Meta.WhnfCoreConfig) : Lean.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.

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

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

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

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partial def Lean.Meta.DiscrTree.getUnify.process {α : Type} (config : Lean.Meta.WhnfCoreConfig) (skip : Nat) (todo : Array Lean.Expr) (c : Lean.Meta.DiscrTree.Trie α) (result : Array α) : Lean.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 Lean.Meta.DiscrTree.Key) (f : σ → Array Lean.Meta.DiscrTree.Key → α → m σ) (init : σ) : Lean.Meta.DiscrTree.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 Lean.Meta.DiscrTree.Key) (f : σ → Array Lean.Meta.DiscrTree.Key → α → σ) (init : σ) (t : Lean.Meta.DiscrTree.Trie α) : σ

Fold the keys and values stored in a Trie.

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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) : Lean.Meta.DiscrTree.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 : Lean.Meta.DiscrTree.Trie α) : σ

Fold the values stored in a Trie.

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• Lean.Meta.DiscrTree.Trie.foldValues f init t = Id.run (Lean.Meta.DiscrTree.Trie.foldValuesM f init t)

partial def Lean.Meta.DiscrTree.Trie.size {α : Type} : Lean.Meta.DiscrTree.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 Lean.Meta.DiscrTree.Key → α → m σ) (init : σ) (t : Lean.Meta.DiscrTree α) : m σ

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

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@[inline]

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

Fold over the keys and values stored in a DiscrTree

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• Lean.Meta.DiscrTree.fold f init t = Id.run (Lean.Meta.DiscrTree.foldM (fun (s : σ) (keys : Array Lean.Meta.DiscrTree.Key) (a : α) => pure (f s keys a)) init t)

@[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 : Lean.Meta.DiscrTree α) : m σ

Monadically fold over the values stored in a DiscrTree.

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@[inline]

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

Fold over the values stored in a DiscrTree.

Equations

• Lean.Meta.DiscrTree.foldValues f init t = Id.run (Lean.Meta.DiscrTree.foldValuesM f init t)

@[inline]

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

Check for the presence of a value satisfying a predicate.

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• Lean.Meta.DiscrTree.containsValueP t f = Lean.Meta.DiscrTree.foldValues (fun (r : Bool) (a : α) => r || f a) false t

@[inline]

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

Extract the values stored in a DiscrTree.

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• Lean.Meta.DiscrTree.values t = Lean.Meta.DiscrTree.foldValues (fun (as : Array α) (a : α) => Array.push as a) #[] t

@[inline]

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

Extract the keys and values stored in a DiscrTree.

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@[inline]

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

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

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

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

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

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

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• Lean.Meta.DiscrTree.mapArrays d f = Id.run (Lean.Meta.DiscrTree.mapArraysM d fun (A : Array α) => pure (f A))