Library search

This file defines tactics exact? and apply?, (formerly known as library_search) and a term elaborator exact?% that tries to find a lemma solving the current goal (subgoals are solved using solveByElim).

example : x < x + 1 := exact?%
example : Nat := by exact?

def Lean.Meta.LibrarySearch.solveByElim (required : List Expr) (exfalso : Bool) (goals : List MVarId) (maxDepth : Nat) : MetaM (List MVarId)

Wrapper for calling `Lean.Meta.SolveByElim.solveByElim with appropriate arguments for library search.

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inductive Lean.Meta.LibrarySearch.DeclMod : Type

A "modifier" for a declaration.

• none indicates the original declaration,
• mp indicates that (possibly after binders) the declaration is an ↔, and we want to consider the forward direction,
• mpr similarly, but for the backward direction.

• none : DeclMod

  the original declaration

• mp : DeclMod

  the forward direction of an iff

• mpr : DeclMod

  the backward direction of an iff

instance Lean.Meta.LibrarySearch.instDecidableEqDeclMod : DecidableEq DeclMod

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• Lean.Meta.LibrarySearch.instDecidableEqDeclMod x✝ y✝ = if h : x✝.toCtorIdx = y✝.toCtorIdx then isTrue ⋯ else isFalse ⋯

instance Lean.Meta.LibrarySearch.instInhabitedDeclMod : Inhabited DeclMod

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• Lean.Meta.LibrarySearch.instInhabitedDeclMod = { default := Lean.Meta.LibrarySearch.DeclMod.none }

instance Lean.Meta.LibrarySearch.instOrdDeclMod : Ord DeclMod

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• Lean.Meta.LibrarySearch.instOrdDeclMod = { compare := Lean.Meta.LibrarySearch.ordDeclMod✝ }

instance Lean.Meta.LibrarySearch.instHashableDeclMod : Hashable DeclMod

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• Lean.Meta.LibrarySearch.instHashableDeclMod = { hash := Lean.Meta.LibrarySearch.hashDeclMod✝ }

@[reducible]

def Lean.Meta.LibrarySearch.CandidateFinder : Type

LibrarySearch has an extension mechanism for replacing the function used to find candidate lemmas.

Equations

• Lean.Meta.LibrarySearch.CandidateFinder = (Lean.Expr → Lean.MetaM (Array (Lean.Name × Lean.Meta.LibrarySearch.DeclMod)))

def Lean.Meta.LibrarySearch.droppedKeys : List (List LazyDiscrTree.Key)

We drop .star and Eq * * * from the discriminator trees because they match too much.

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def Lean.Meta.LibrarySearch.libSearchFindDecls : Expr → MetaM (Array (Name × DeclMod))

Create function for finding relevant declarations.

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def Lean.Meta.LibrarySearch.mkHeartbeatCheck (leavePercent : Nat) : MetaM (MetaM Bool)

Return an action that returns true when the remaining heartbeats is less than the currently remaining heartbeats * leavePercent / 100.

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def Lean.Meta.LibrarySearch.interleaveWith {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → γ) (x : Array α) (g : β → γ) (y : Array β) : Array γ

Interleave x y interleaves the elements of x and y until one is empty and then returns final elements in other list.

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def Lean.Meta.LibrarySearch.abortSpeculation {m : Type u_1 → Type u_2} {α : Type u_1} [MonadExcept Exception m] : m α

Called to abort speculative execution in library search.

Equations

• Lean.Meta.LibrarySearch.abortSpeculation = throw (Lean.Exception.internal Lean.Meta.LibrarySearch.abortSpeculationId✝)

def Lean.Meta.LibrarySearch.isAbortSpeculation : Exception → Bool

Returns true if this is an abort speculation exception.

Equations

• Lean.Meta.LibrarySearch.isAbortSpeculation (Lean.Exception.internal id extra) = (id == Lean.Meta.LibrarySearch.abortSpeculationId✝)
• Lean.Meta.LibrarySearch.isAbortSpeculation x✝ = false

@[reducible]

def Lean.Meta.LibrarySearch.Candidate : Type

A library search candidate using symmetry includes the goal to solve, the metavar context for that goal, and the name and orientation of a rule to try using with goal.

Equations

• Lean.Meta.LibrarySearch.Candidate = ((Lean.MVarId × Lean.MetavarContext) × Lean.Name × Lean.Meta.LibrarySearch.DeclMod)

def Lean.Meta.LibrarySearch.librarySearchSymm (searchFn : CandidateFinder) (goal : MVarId) : MetaM (Array Candidate)

Run searchFn on both the goal and symm applied to the goal.

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def Lean.Meta.LibrarySearch.mkLibrarySearchLemma (lem : Name) (mod : DeclMod) : MetaM Expr

Create lemma from name and mod.

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def Lean.Meta.LibrarySearch.tryOnEach (act : Candidate → MetaM (List MVarId)) (candidates : Array Candidate) : MetaM (Option (Array (List MVarId × MetavarContext)))

Sequentially invokes a tactic act on each value in candidates on the current state.

The tactic act should return a list of meta-variables that still need to be resolved. If this list is empty, then no variables remain to be solved, and tryOnEach returns none with the environment set so each goal is resolved.

If the action throws an internal exception with the abortSpeculationId id then further computation is stopped and intermediate results returned. If any other exception is thrown, then it is silently discarded.

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def Lean.Meta.LibrarySearch.librarySearch (goal : MVarId) (tactic : Bool → List MVarId → MetaM (List MVarId) := fun (initial : Bool) (g : List MVarId) => solveByElim [] initial g 6) (allowFailure : MVarId → MetaM Bool := fun (x : MVarId) => pure true) (leavePercentHeartbeats : Nat := 10) : MetaM (Option (Array (List MVarId × MetavarContext)))

Tries to solve the goal either by:

• calling tactic true
• or applying a library lemma then calling tactic false on the resulting goals.

Typically here tactic is solveByElim, with the Bool flag indicating whether it may retry with exfalso.

If it successfully closes the goal, returns none. Otherwise, it returns some a, where a : Array (List MVarId × MetavarContext), with an entry for each library lemma which was successfully applied, containing a list of the subsidiary goals, and the metavariable context after the application.

(Always succeeds, and the metavariable context stored in the monad is reverted, unless the goal was completely solved.)

(Note that if solveByElim solves some but not all subsidiary goals, this is not currently tracked.)

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