This module finds lexicographic termination arguments for well-founded recursion.

Starting with basic measures (sizeOf xᵢ for all parameters xᵢ), and complex measures (e.g. e₂ - e₁ if e₁ < e₂ is found in the context of a recursive call) it tries all combinations until it finds one where all proof obligations go through with the given tactic (decerasing_by), if given, or the default decreasing_tactic.

For mutual recursion, a single measure is not just one parameter, but one from each recursive function. Enumerating these can lead to a combinatoric explosion, so we bound the number of measures tried.

In addition to measures derived from sizeOf xᵢ, it also considers measures that assign an order to the functions themselves. This way we can support mutual function definitions where no arguments decrease from one function to another.

The result of this module is a TerminationWF, which is then passed on to wfRecursion; this design is crucial so that whatever we infer in this module could also be written manually by the user. It would be bad if there are function definitions that can only be processed with the guessed lexicographic order.

The following optimizations are applied to make this feasible:

1. The crucial optimization is to look at each argument of each recursive call once, try to prove < and (if that fails ≤), and then look at that table to pick a suitable measure.

2. The next-crucial optimization is to fill that table lazily. This way, we run the (likely expensive) tactics as few times as possible, while still being able to consider a possibly large number of combinations.

3. Before we even try to prove <, we check if the arguments are equal (=). No well-founded measure will relate equal terms, likely this check is faster than firing up the tactic engine, and it adds more signal to the output.

4. Instead of traversing the whole function body over and over, we traverse it once and store the arguments (in unpacked form) and the local MetaM state at each recursive call (see collectRecCalls), which we then re-use for the possibly many proof attempts.

The logic here is based on “Finding Lexicographic Orders for Termination Proofs in Isabelle/HOL” by Lukas Bulwahn, Alexander Krauss, and Tobias Nipkow, 10.1007/978-3-540-74591-4_5 https://www21.in.tum.de/~nipkow/pubs/tphols07.pdf.

We got the idea of considering the measure e₂ - e₁ if we see e₁ < e₂ from “Termination Analysis with Calling Context Graphs” by Panagiotis Manolios & Daron Vroon, https://doi.org/10.1007/11817963_36.

opaque Lean.Elab.WF.GuessLex.showInferredTerminationBy : Lean.Option Bool

def Lean.Elab.WF.GuessLex.originalVarNames (preDef : PreDefinition) : MetaM (Array Name)

Given a predefinition, return the variable names in the outermost lambdas. Includes the “fixed prefix”.

The length of the returned array is also used to determine the arity of the function, so it should match what packDomain does.

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• Lean.Elab.WF.GuessLex.originalVarNames preDef = Lean.Meta.lambdaTelescope preDef.value fun (xs : Array Lean.Expr) (x : Lean.Expr) => Array.mapM (fun (x : Lean.Expr) => x.fvarId!.getUserName) xs

def Lean.Elab.WF.GuessLex.naryVarNames (xs : Array Name) : MetaM (Array Name)

Given the original parameter names from originalVarNames, find good variable names to be used when talking about termination arguments: Use user-given parameter names if present; use x1...xn otherwise.

The names ought to accessible (no macro scopes) and fresh wrt to the current environment, so that with showInferredTerminationBy we can print them to the user reliably. We do that by appending ' as needed.

It is possible (but unlikely without malice) that some of the user-given names shadow each other, and the guessed relation refers to the wrong one. In that case, the user gets to keep both pieces (and may have to rename variables).

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partial def Lean.Elab.WF.GuessLex.naryVarNames.freshen (ns : Array Name) (n : Name) : MetaM Name

structure Lean.Elab.WF.GuessLex.Measureextends Lean.Elab.TerminationArgument : Type

A termination measure with extra fields for use within GuessLex

• ref : Syntax
• structural : Bool
• fn : Expr
• natFn : Expr

  Like .fn, but unconditionally with sizeOf at the right type. We use this one when in evalRecCall

instance Lean.Elab.WF.GuessLex.instInhabitedMeasure : Inhabited Measure

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• Lean.Elab.WF.GuessLex.instInhabitedMeasure = { default := { toTerminationArgument := default, natFn := default } }

def Lean.Elab.WF.GuessLex.Measure.toString (measure : Measure) : MetaM String

String description of this measure

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• measure.toString = Lean.Meta.lambdaTelescope measure.fn fun (_xs : Array Lean.Expr) (e : Lean.Expr) => do let __do_lift ← Lean.Meta.ppExpr e pure __do_lift.pretty

def Lean.Elab.WF.GuessLex.mayOmitSizeOf (is_mutual : Bool) (args : Array Expr) (x : Expr) : MetaM Bool

Determine if the measure for parameter x should be sizeOf x or just x.

For non-mutual definitions, we omit sizeOf when the argument does not depend on the other varying parameters, and its WellFoundedRelation instance goes via SizeOf.

For mutual definitions, we omit sizeOf only when the argument is (at reducible transparency!) of type Nat (else we'd have to worry about differently-typed measures from different functions to line up).

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def Lean.Elab.WF.GuessLex.withUserNames {α : Type} (xs : Array Expr) (ns : Array Name) (k : MetaM α) : MetaM α

Sets the user names for the given freevars in xs.

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def Lean.Elab.WF.GuessLex.simpleMeasures (preDefs : Array PreDefinition) (fixedPrefixSize : Nat) (userVarNamess : Array (Array Name)) : MetaM (Array (Array Measure))

Create one measure for each (eligible) parameter of the given predefintion.

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@[reducible, inline]

abbrev Lean.Elab.WF.GuessLex.M (recFnName : Name) (α β : Type) : Type

Internal monad used by withRecApps

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• Lean.Elab.WF.GuessLex.M recFnName α β = StateRefT' IO.RealWorld (Array α) (StateRefT' IO.RealWorld (Lean.HasConstCache #[recFnName]) Lean.MetaM) β

def Lean.Elab.WF.GuessLex.withRecApps {α : Type} (recFnName : Name) (fixedPrefixSize : Nat) (param e : Expr) (k : Expr → Array Expr → MetaM α) : MetaM (Array α)

Traverses the given expression e, and invokes the continuation k at every saturated call to recFnName.

The expression param is passed along, and refined when going under a matcher or casesOn application.

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partial def Lean.Elab.WF.GuessLex.withRecApps.processRec {α : Type} (recFnName : Name) (fixedPrefixSize : Nat) (k : Expr → Array Expr → MetaM α) (param e : Expr) : M recFnName α Unit

partial def Lean.Elab.WF.GuessLex.withRecApps.processApp {α : Type} (recFnName : Name) (fixedPrefixSize : Nat) (k : Expr → Array Expr → MetaM α) (param e : Expr) : M recFnName α Unit

def Lean.Elab.WF.GuessLex.withRecApps.containsRecFn {α : Type} (recFnName : Name) (e : Expr) : M recFnName α Bool

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• Lean.Elab.WF.GuessLex.withRecApps.containsRecFn recFnName e = modifyGetThe (Lean.HasConstCache #[recFnName]) fun (x : Lean.HasConstCache #[recFnName]) => Lean.HasConstCache.contains e x

partial def Lean.Elab.WF.GuessLex.withRecApps.loop {α : Type} (recFnName : Name) (fixedPrefixSize : Nat) (k : Expr → Array Expr → MetaM α) (param e : Expr) : M recFnName α Unit

structure Lean.Elab.WF.GuessLex.SavedLocalContext : Type

A SavedLocalContext captures the state and local context of a MetaM, to be continued later.

• savedLocalContext : LocalContext
• savedLocalInstances : LocalInstances
• savedState : Meta.SavedState

def Lean.Elab.WF.GuessLex.SavedLocalContext.create : MetaM SavedLocalContext

Capture the MetaM state including local context.

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def Lean.Elab.WF.GuessLex.SavedLocalContext.run {α : Type} (slc : SavedLocalContext) (k : MetaM α) : MetaM α

Run a MetaM action in the saved state.

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• slc.run k = Lean.withoutModifyingState (Lean.Meta.withLCtx slc.savedLocalContext slc.savedLocalInstances do slc.savedState.restore k)

structure Lean.Elab.WF.GuessLex.RecCallWithContext : Type

A RecCallWithContext focuses on a single recursive call in a unary predefinition, and runs the given action in the context of that call.

• ref : Syntax

  Syntax location of recursive call

• caller : Nat

  Function index of caller

• params : Array Expr

  Parameters of caller (including fixed prefix)

• callee : Nat

  Function index of callee

• args : Array Expr

  Arguments to callee (including fixed prefix)

• ctxt : SavedLocalContext

def Lean.Elab.WF.GuessLex.RecCallWithContext.create (ref : Syntax) (caller : Nat) (params : Array Expr) (callee : Nat) (args : Array Expr) : MetaM RecCallWithContext

Store the current recursive call and its context.

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def Lean.Elab.WF.GuessLex.filterSubsumed (rcs : Array RecCallWithContext) : Array RecCallWithContext

The elaborator is prone to duplicate terms, including recursive calls, even if the user only wrote a single one. This duplication is wasteful if we run the tactics on duplicated calls, and confusing in the output of GuessLex. So prune the list of recursive calls, and remove those where another call exists that has the same goal and context that is no more specific.

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def Lean.Elab.WF.GuessLex.collectRecCalls (unaryPreDef : PreDefinition) (fixedPrefixSize : Nat) (argsPacker : Meta.ArgsPacker) : MetaM (Array RecCallWithContext)

Traverse a unary PreDefinition, and returns a WithRecCall closure for each recursive call site.

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def Lean.Elab.WF.GuessLex.isNatCmp (e : Expr) : Option (Expr × Expr)

Is the expression a <-like comparison of Nat expressions

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def Lean.Elab.WF.GuessLex.complexMeasures (preDefs : Array PreDefinition) (fixedPrefixSize : Nat) (userVarNamess : Array (Array Name)) (recCalls : Array RecCallWithContext) : MetaM (Array (Array Measure))

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inductive Lean.Elab.WF.GuessLex.GuessLexRel : Type

A GuessLexRel described how a recursive call affects a measure; whether it decreases strictly, non-strictly, is equal, or else.

• lt : GuessLexRel
• eq : GuessLexRel
• le : GuessLexRel
• no_idea : GuessLexRel

instance Lean.Elab.WF.GuessLex.instReprGuessLexRel : Repr GuessLexRel

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• Lean.Elab.WF.GuessLex.instReprGuessLexRel = { reprPrec := Lean.Elab.WF.GuessLex.reprGuessLexRel✝ }

instance Lean.Elab.WF.GuessLex.instDecidableEqGuessLexRel : DecidableEq GuessLexRel

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• Lean.Elab.WF.GuessLex.instDecidableEqGuessLexRel x✝ y✝ = if h : x✝.toCtorIdx = y✝.toCtorIdx then isTrue ⋯ else isFalse ⋯

instance Lean.Elab.WF.GuessLex.instToStringGuessLexRel : ToString GuessLexRel

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instance Lean.Elab.WF.GuessLex.instToFormatGuessLexRel : ToFormat GuessLexRel

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• Lean.Elab.WF.GuessLex.instToFormatGuessLexRel = { format := fun (r : Lean.Elab.WF.GuessLex.GuessLexRel) => Std.Format.text (toString r) }

def Lean.Elab.WF.GuessLex.GuessLexRel.toNatRel : GuessLexRel → Expr

Given a GuessLexRel, produce a binary Expr that relates two Nat values accordingly.

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• Lean.Elab.WF.GuessLex.GuessLexRel.lt.toNatRel = Lean.mkAppN (Lean.mkConst `LT.lt [Lean.levelZero]) #[Lean.mkConst `Nat, Lean.mkConst `instLTNat]
• Lean.Elab.WF.GuessLex.GuessLexRel.eq.toNatRel = Lean.mkAppN (Lean.mkConst `Eq [Lean.levelOne]) #[Lean.mkConst `Nat]
• Lean.Elab.WF.GuessLex.GuessLexRel.le.toNatRel = Lean.mkAppN (Lean.mkConst `LE.le [Lean.levelZero]) #[Lean.mkConst `Nat, Lean.mkConst `instLENat]

def Lean.Elab.WF.GuessLex.evalRecCall (decrTactic? : Option DecreasingBy) (callerMeasures calleeMeasures : Array Measure) (rcc : RecCallWithContext) (callerMeasureIdx calleeMeasureIdx : Nat) : MetaM GuessLexRel

For a given recursive call, and a choice of parameter and argument index, try to prove equality, < or ≤.

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structure Lean.Elab.WF.GuessLex.RecCallCache : Type

• mk'' :: (
• decrTactic? : Option DecreasingBy
• callerMeasures : Array Measure
• calleeMeasures : Array Measure
• rcc : RecCallWithContext
• cache : IO.Ref (Array (Array (Option GuessLexRel)))
• )

def Lean.Elab.WF.GuessLex.RecCallCache.mk (decrTactics : Array (Option DecreasingBy)) (measuress : Array (Array Measure)) (rcc : RecCallWithContext) : BaseIO RecCallCache

Create a cache to memoize calls to evalRecCall descTactic? rcc

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def Lean.Elab.WF.GuessLex.RecCallCache.eval (rc : RecCallCache) (callerMeasureIdx calleeMeasureIdx : Nat) : MetaM GuessLexRel

Run evalRecCall and cache there result

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def Lean.Elab.WF.GuessLex.RecCallCache.prettyEntry (rcc : RecCallCache) (callerMeasureIdx calleeMeasureIdx : Nat) : MetaM String

Print a single cache entry as a string, without forcing it

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inductive Lean.Elab.WF.GuessLex.MutualMeasure : Type

The measures that we order lexicographically can be comparing arguments, or numbering the functions

• args : Array Nat → MutualMeasure

  For every function, the given argument index

• func : Nat → MutualMeasure

  The given function index is assigned 1, the rest 0

def Lean.Elab.WF.GuessLex.inspectCall (rc : RecCallCache) : MutualMeasure → MetaM GuessLexRel

Evaluate a recursive call at a given MutualMeasure

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• Lean.Elab.WF.GuessLex.inspectCall rc (Lean.Elab.WF.GuessLex.MutualMeasure.args taIdxs) = rc.eval taIdxs[rc.rcc.caller]! taIdxs[rc.rcc.callee]!

def Lean.Elab.WF.GuessLex.generateCombinations? (numMeasures : Array Nat) (threshold : Nat := 32) : Option (Array (Array Nat))

Generate all combination of measures. Assumes we have numbered the measures of each function, and their counts is in numMeasures.

This puts the uniform combinations ([0,0,0], [1,1,1]) to the front; they are commonly most useful to try first, when the mutually recursive functions have similar argument structures

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partial def Lean.Elab.WF.GuessLex.generateCombinations?.goUniform (numMeasures : Array Nat) (idx : Nat) : OptionT (StateM (Array (Array Nat))) Unit

partial def Lean.Elab.WF.GuessLex.generateCombinations?.go (numMeasures : Array Nat) (threshold : Nat := 32) (fidx : Nat) : OptionT (ReaderT (Array Nat) (StateM (Array (Array Nat)))) Unit

def Lean.Elab.WF.GuessLex.generateMeasures (numTermArgs : Array Nat) : MetaM (Array MutualMeasure)

Enumerate all measures we want to try.

All arguments (resp. combinations thereof) and possible orderings of functions (if more than one)

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def Lean.Elab.WF.GuessLex.solve {m : Type → Type} {α : Type} [Monad m] (measures : Array α) (calls : Array (α → m GuessLexRel)) : m (Option (Array α))

The core logic of guessing the lexicographic order Given a matrix that for each call and measure indicates whether that measure is decreasing, equal, less-or-equal or unknown, It finds a sequence of measures that is lexicographically decreasing.

The matrix is implemented here as an array of monadic query methods only so that we can fill is lazily. Morally, this is a pure function

Equations

• Lean.Elab.WF.GuessLex.solve measures calls = Lean.Elab.WF.GuessLex.solve.go measures calls #[]

partial def Lean.Elab.WF.GuessLex.solve.go {m : Type → Type} {α : Type} [Monad m] (measures : Array α) (calls : Array (α → m GuessLexRel)) (acc : Array α) : m (Option (Array α))

def Lean.Elab.WF.GuessLex.formatTable : Array (Array String) → String

Given a matrix (row-major) of strings, arranges them in tabular form. First column is left-aligned, others right-aligned. Single space as column separator.

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def Lean.Elab.WF.GuessLex.RecCallWithContext.posString (rcc : RecCallWithContext) : MetaM String

Concise textual representation of the source location of a recursive call

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def Lean.Elab.WF.GuessLex.measureHeader (measure : Measure) : StateT (Nat × String) MetaM String

How to present the measure in the table header, possibly abbreviated.

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def Lean.Elab.WF.GuessLex.collectHeaders {α : Type} (a : StateT (Nat × String) MetaM α) : MetaM (α × String)

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• Lean.Elab.WF.GuessLex.collectHeaders a = do let __discr ← a.run (0, "") match __discr with | (x, fst, footer) => pure (x, footer)

def Lean.Elab.WF.GuessLex.explainNonMutualFailure (measures : Array Measure) (rcs : Array RecCallCache) : MetaM Format

Explain what we found out about the recursive calls (non-mutual case)

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def Lean.Elab.WF.GuessLex.explainMutualFailure (declNames : Array Name) (measuress : Array (Array Measure)) (rcs : Array RecCallCache) : MetaM Format

Explain what we found out about the recursive calls (mutual case)

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def Lean.Elab.WF.GuessLex.explainFailure (declNames : Array Name) (measuress : Array (Array Measure)) (rcs : Array RecCallCache) : MetaM Format

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def Lean.Elab.WF.GuessLex.mkProdElem (xs : Array Expr) : MetaM Expr

For #[x₁, .., xₙ] create (x₁, .., xₙ).

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def Lean.Elab.WF.GuessLex.toTerminationArguments (preDefs : Array PreDefinition) (fixedPrefixSize : Nat) (userVarNamess : Array (Array Name)) (measuress : Array (Array Measure)) (solution : Array MutualMeasure) : MetaM TerminationArguments

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def Lean.Elab.WF.GuessLex.reportTermArgs (preDefs : Array PreDefinition) (termArgs : TerminationArguments) : MetaM Unit

Shows the inferred termination argument to the user, and implements termination_by?

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def Lean.Elab.WF.guessLex (preDefs : Array PreDefinition) (unaryPreDef : PreDefinition) (fixedPrefixSize : Nat) (argsPacker : Meta.ArgsPacker) : MetaM TerminationArguments

Main entry point of this module:

Try to find a lexicographic ordering of the arguments for which the recursive definition terminates. See the module doc string for a high-level overview.

The preDefs are used to determine arity and types of arguments; the bodies are ignored.

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