opaque Lean.Meta.IndPredBelow.maxBackwardChainingDepth : Lean.Option Nat

structure Lean.Meta.IndPredBelow.Context : Type

The context used in the creation of the below scheme for inductive predicates.

• motives : Array (Lean.Name × Lean.Expr)
• typeInfos : Array Lean.InductiveVal
• belowNames : Array Lean.Name
• headers : Array Lean.Expr
• numParams : Nat

structure Lean.Meta.IndPredBelow.Variables : Type

Collection of variables used to keep track of the positions of binders in the construction of below motives and constructors.

• target : Array Lean.Expr
• indVal : Array Lean.Expr
• params : Array Lean.Expr
• args : Array Lean.Expr
• motives : Array Lean.Expr
• innerType : Lean.Expr

instance Lean.Meta.IndPredBelow.instInhabitedVariables : Inhabited Lean.Meta.IndPredBelow.Variables

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structure Lean.Meta.IndPredBelow.BrecOnVariables : Type

Collection of variables used to keep track of the local context used in the brecOn proof.

• params : Array Lean.FVarId
• motives : Array Lean.FVarId
• indices : Array Lean.FVarId
• witness : Lean.FVarId
• indHyps : Array Lean.FVarId

def Lean.Meta.IndPredBelow.mkContext (declName : Lean.Name) : Lean.MetaM Lean.Meta.IndPredBelow.Context

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def Lean.Meta.IndPredBelow.mkContext.motiveName (motiveTypes : Array Lean.Expr) (i : Nat) : Lean.MetaM Lean.Name

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def Lean.Meta.IndPredBelow.mkContext.mkHeader (motives : Array (Lean.Name × Lean.Expr)) (numParams : Nat) (indVal : Lean.InductiveVal) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkContext.addMotives (motives : Array (Lean.Name × Lean.Expr)) (numParams : Nat) : Lean.Expr → Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkContext.motiveType (indVal : Lean.InductiveVal) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkContext.mkIndValConst (indVal : Lean.InductiveVal) : Lean.Expr

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• Lean.Meta.IndPredBelow.mkContext.mkIndValConst indVal = Lean.mkConst indVal.name (List.map Lean.mkLevelParam indVal.levelParams)

def Lean.Meta.IndPredBelow.mkCtorType (ctx : Lean.Meta.IndPredBelow.Context) (belowIdx : Nat) (originalCtor : Lean.ConstructorVal) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkCtorType.addHeaderVars (ctx : Lean.Meta.IndPredBelow.Context) (belowIdx : Nat) (originalCtor : Lean.ConstructorVal) (vars : Lean.Meta.IndPredBelow.Variables) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkCtorType.addMotives (ctx : Lean.Meta.IndPredBelow.Context) (belowIdx : Nat) (originalCtor : Lean.ConstructorVal) (vars : Lean.Meta.IndPredBelow.Variables) : Lean.MetaM Lean.Expr

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partial def Lean.Meta.IndPredBelow.mkCtorType.modifyBinders (ctx : Lean.Meta.IndPredBelow.Context) (belowIdx : Nat) (originalCtor : Lean.ConstructorVal) (vars : Lean.Meta.IndPredBelow.Variables) (i : Nat) : Lean.MetaM Lean.Expr

def Lean.Meta.IndPredBelow.mkCtorType.rebuild (ctx : Lean.Meta.IndPredBelow.Context) (belowIdx : Nat) (originalCtor : Lean.ConstructorVal) (vars : Lean.Meta.IndPredBelow.Variables) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkCtorType.replaceTempVars (ctx : Lean.Meta.IndPredBelow.Context) (vars : Lean.Meta.IndPredBelow.Variables) (ctor : Lean.Expr) : Lean.Expr

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def Lean.Meta.IndPredBelow.mkCtorType.checkCount (ctx : Lean.Meta.IndPredBelow.Context) (domain : Lean.Expr) : Lean.MetaM Bool

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def Lean.Meta.IndPredBelow.mkCtorType.mkBelowBinder (ctx : Lean.Meta.IndPredBelow.Context) (vars : Lean.Meta.IndPredBelow.Variables) (binder domain : Lean.Expr) {α : Type} (k : Nat → Lean.Expr → Lean.MetaM α) : Lean.MetaM α

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def Lean.Meta.IndPredBelow.mkCtorType.mkMotiveBinder (ctx : Lean.Meta.IndPredBelow.Context) (vars : Lean.Meta.IndPredBelow.Variables) (indValIdx : Nat) (binder domain : Lean.Expr) {α : Type} (k : Lean.Expr → Lean.MetaM α) : Lean.MetaM α

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def Lean.Meta.IndPredBelow.mkCtorType.copyVarName (oldName : Lean.FVarId) : Lean.MetaM Lean.Name

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• Lean.Meta.IndPredBelow.mkCtorType.copyVarName oldName = do let binderDecl ← oldName.getDecl liftM (Lean.mkFreshUserName binderDecl.userName)

def Lean.Meta.IndPredBelow.mkConstructor (ctx : Lean.Meta.IndPredBelow.Context) (i : Nat) (ctor : Lean.Name) : Lean.MetaM Lean.Constructor

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def Lean.Meta.IndPredBelow.mkInductiveType (ctx : Lean.Meta.IndPredBelow.Context) (i : Nat) (indVal : Lean.InductiveVal) : Lean.MetaM Lean.InductiveType

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def Lean.Meta.IndPredBelow.mkBelowDecl (ctx : Lean.Meta.IndPredBelow.Context) : Lean.MetaM Lean.Declaration

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partial def Lean.Meta.IndPredBelow.backwardsChaining (m : Lean.MVarId) (depth : Nat) : Lean.MetaM Bool

def Lean.Meta.IndPredBelow.proveBrecOn (ctx : Lean.Meta.IndPredBelow.Context) (indVal : Lean.InductiveVal) (type : Lean.Expr) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.proveBrecOn.intros (ctx : Lean.Meta.IndPredBelow.Context) (indVal : Lean.InductiveVal) (m : Lean.MVarId) : Lean.MetaM (Lean.MVarId × Lean.Meta.IndPredBelow.BrecOnVariables)

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def Lean.Meta.IndPredBelow.proveBrecOn.applyIH (m : Lean.MVarId) (vars : Lean.Meta.IndPredBelow.BrecOnVariables) : Lean.MetaM (List Lean.MVarId)

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def Lean.Meta.IndPredBelow.proveBrecOn.induction (ctx : Lean.Meta.IndPredBelow.Context) (indVal : Lean.InductiveVal) (m : Lean.MVarId) (vars : Lean.Meta.IndPredBelow.BrecOnVariables) : Lean.MetaM (List Lean.MVarId)

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def Lean.Meta.IndPredBelow.proveBrecOn.applyCtors (ms : List Lean.MVarId) : Lean.MetaM (List Lean.MVarId)

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partial def Lean.Meta.IndPredBelow.proveBrecOn.introNPRec (m : Lean.MVarId) : Lean.MetaM Lean.MVarId

def Lean.Meta.IndPredBelow.proveBrecOn.closeGoal (maxDepth : Nat) (m : Lean.MVarId) : Lean.MetaM Unit

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def Lean.Meta.IndPredBelow.mkBrecOnDecl (ctx : Lean.Meta.IndPredBelow.Context) (idx : Nat) : Lean.MetaM Lean.Declaration

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def Lean.Meta.IndPredBelow.mkBrecOnDecl.mkType (ctx : Lean.Meta.IndPredBelow.Context) (idx : Nat) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.mkBrecOnDecl.mkIH (ctx : Lean.Meta.IndPredBelow.Context) (params motives : Array Lean.Expr) (idx : Nat) (motive : Lean.Name × Lean.Expr) : Lean.MetaM (Lean.Name × (Array Lean.Expr → Lean.MetaM Lean.Expr))

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def Lean.Meta.IndPredBelow.getBelowIndices (ctorName : Lean.Name) : Lean.MetaM (Array Nat)

Given a constructor name, find the indices of the corresponding below version thereof.

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partial def Lean.Meta.IndPredBelow.getBelowIndices.loop (xs : Array Lean.Expr) (rest : Lean.Expr) (belowIndices : Array Nat) (xIdx yIdx : Nat) : Lean.MetaM (Array Nat)

def Lean.Meta.IndPredBelow.mkBelowMatcher (matcherApp : Lean.Meta.MatcherApp) (belowMotive below : Lean.Expr) (idx : Nat) : Lean.MetaM (Lean.Expr × Lean.MetaM Unit)

This function adds an additional below discriminant to a matcher application. It is used for modifying the patterns, such that the structural recursion can use the new below predicate instead of the original one and thus be used prove structural recursion.

It takes as parameters:

• matcherApp: a matcher application
• belowMotive: the motive, that the below type should carry
• below: an expression from the local context that is the below version of a predicate and can be used for structural recursion
• idx: the index of the original predicate discriminant.

It returns:

• A matcher application as an expression
• A side-effect for adding the matcher to the environment

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def Lean.Meta.IndPredBelow.mkBelowMatcher.addBelowPattern (belowMotive : Lean.Expr) (idx : Nat) (indName : Lean.Name) (lhs : Lean.Meta.Match.AltLHS) : Lean.MetaM Lean.Meta.Match.AltLHS

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partial def Lean.Meta.IndPredBelow.mkBelowMatcher.convertToBelow (belowMotive : Lean.Expr) (indName : Lean.Name) (originalPattern : Lean.Meta.Match.Pattern) : Lean.MetaM (Array Lean.LocalDecl × Lean.Meta.Match.Pattern)

this function changes the type of the pattern from the original type to the below version thereof. Most of the cases don't apply. In order to change the type and the pattern to be type correct, we don't simply recursively change all occurrences, but rather, we recursively change occurrences of the constructor. As such there are only a few cases:

• the pattern is a constructor from the original type. Here we need to:
  • replace the constructor
  • copy original recursive patterns and convert them to below and reintroduce them in the correct position
  • turn original recursive patterns inaccessible
  • introduce free variables as needed.
• it is an as pattern. Here the constructor could be hidden inside of it.
• it is a variable. Here you we need to introduce a fresh variable of a different type.

partial def Lean.Meta.IndPredBelow.mkBelowMatcher.transformFields (belowMotive belowCtor : Lean.Expr) (indName : Lean.Name) (belowFieldOpts : Array (Option Lean.Meta.Match.Pattern)) : Lean.MetaM (Array Lean.LocalDecl × Array Lean.Meta.Match.Pattern)

partial def Lean.Meta.IndPredBelow.mkBelowMatcher.transformFields.loop (belowMotive : Lean.Expr) (indName : Lean.Name) (belowCtor : Lean.Expr) (belowFieldOpts : Array (Option Lean.Meta.Match.Pattern)) (belowFields : Array Lean.Meta.Match.Pattern) (additionalFVars : Array Lean.LocalDecl) : Lean.MetaM (Array Lean.LocalDecl × Array Lean.Meta.Match.Pattern)

def Lean.Meta.IndPredBelow.mkBelowMatcher.toInaccessible : Lean.Meta.Match.Pattern → Lean.MetaM Lean.Meta.Match.Pattern

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• Lean.Meta.IndPredBelow.mkBelowMatcher.toInaccessible (Lean.Meta.Match.Pattern.inaccessible p) = pure (Lean.Meta.Match.Pattern.inaccessible p)
• Lean.Meta.IndPredBelow.mkBelowMatcher.toInaccessible (Lean.Meta.Match.Pattern.var v) = pure (Lean.Meta.Match.Pattern.var v)
• Lean.Meta.IndPredBelow.mkBelowMatcher.toInaccessible x✝ = do let __do_lift ← x✝.toExpr pure (Lean.Meta.Match.Pattern.inaccessible __do_lift)

def Lean.Meta.IndPredBelow.mkBelowMatcher.newMotive (matcherApp : Lean.Meta.MatcherApp) (belowType : Lean.Expr) (ys : Array Lean.Expr) : Lean.MetaM Lean.Expr

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def Lean.Meta.IndPredBelow.findBelowIdx (xs : Array Lean.Expr) (motive : Lean.Expr) : Lean.MetaM (Option (Lean.Expr × Nat))

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def Lean.Meta.IndPredBelow.mkBelow (declName : Lean.Name) : Lean.MetaM Unit

Generates the auxiliary lemmas below and brecOn for a recursive inductive predicate. The current generator doesn't support nested predicates, but pattern-matching on them still works thanks to well-founded recursion.

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