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 (Name × Expr)
• typeInfos : Array InductiveVal
• belowNames : Array Name
• headers : Array 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 Expr
• indVal : Array Expr
• params : Array Expr
• args : Array Expr
• motives : Array Expr
• innerType : Expr

instance Lean.Meta.IndPredBelow.instInhabitedVariables : Inhabited 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 FVarId
• motives : Array FVarId
• indices : Array FVarId
• witness : FVarId
• indHyps : Array FVarId

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

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

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

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

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

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

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

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

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

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

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def Lean.Meta.IndPredBelow.mkBelowMatcher (matcherApp : MatcherApp) (belowMotive below : Expr) (idx : Nat) : MetaM (Expr × 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.findBelowIdx (xs : Array Expr) (motive : Expr) : MetaM (Option (Expr × Nat))

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