f45c19b428
This PR extends the notion of “fixed parameter” of a recursive function
also to parameters that come after varying function. The main benefit is
that we get nicer induction principles.
Before the definition
```lean
def app (as : List α) (bs : List α) : List α :=
match as with
| [] => bs
| a::as => a :: app as bs
```
produced
```lean
app.induct.{u_1} {α : Type u_1} (motive : List α → List α → Prop) (case1 : ∀ (bs : List α), motive [] bs)
(case2 : ∀ (bs : List α) (a : α) (as : List α), motive as bs → motive (a :: as) bs) (as bs : List α) : motive as bs
```
and now you get
```lean
app.induct.{u_1} {α : Type u_1} (motive : List α → Prop) (case1 : motive [])
(case2 : ∀ (a : α) (as : List α), motive as → motive (a :: as)) (as : List α) : motive as
```
because `bs` is fixed throughout the recursion (and can completely be
dropped from the principle).
This is a breaking change when such an induction principle is used
explicitly. Using `fun_induction` makes proof tactics robust against
this change.
The rules for when a parameter is fixed are now:
1. A parameter is fixed if it is reducibly defq to the the corresponding
argument in each recursive call, so we have to look at each such call.
2. With mutual recursion, it is not clear a-priori which arguments of
another function correspond to the parameter. This requires an analysis
with some graph algorithms to determine.
3. A parameter can only be fixed if all parameters occurring in its type
are fixed as well.
This dependency graph on parameters can be different for the different
functions in a recursive group, even leading to cycles.
4. For structural recursion, we kinda want to know the fixed parameters
before investigating which argument to actually recurs on. But once we
have that we may find that we fixed an index of the recursive
parameter’s type, and these cannot be fixed. So we have to un-fix them
5. … and all other fixed parameters that have dependencies on them.
Lean tries to identify the largest set of parameters that satisfies
these criteria.
Note that in a definition like
```lean
def app : List α → List α → List α
| [], bs => bs
| a::as, bs => a :: app as bs
```
the `bs` is not considered fixes, as it goes through the matcher
machinery.
Fixes #7027
Fixes #2113
76 lines
2.2 KiB
Text
76 lines
2.2 KiB
Text
/-
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Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joachim Breitner
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-/
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prelude
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import Lean.Meta.Basic
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import Lean.ScopedEnvExtension
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import Lean.ReservedNameAction
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/-!
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This module defines the data structure and environment extension to remember how to map the
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function's arguments to the functional induction principle's arguments.
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Also used for functional cases.
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-/
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namespace Lean.Meta
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inductive FunIndParamKind where
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| dropped
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| param
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| target
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deriving BEq, Repr, Inhabited
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/--
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A `FunIndInfo` indicates how a function's arguments map to the arguments of the functional induction
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(resp. cases) theorem.
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The size of `params` also indicates the arity of the function.
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-/
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structure FunIndInfo where
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funIndName : Name
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/--
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`true` means that the corresponding level parameter of the function is also a level param
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of the induction principle.
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-/
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levelMask : Array Bool
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params : Array FunIndParamKind
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deriving Inhabited, Repr
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builtin_initialize funIndInfoExt : MapDeclarationExtension FunIndInfo ← mkMapDeclarationExtension
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def getFunInductName (declName : Name) : Name :=
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declName ++ `induct
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def getFunCasesName (declName : Name) : Name :=
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declName ++ `fun_cases
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def getMutualInductName (declName : Name) : Name :=
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declName ++ `mutual_induct
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def getFunInduct? (cases : Bool) (declName : Name) : CoreM (Option Name) := do
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let .defnInfo _ ← getConstInfo declName | return none
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try
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let thmName := if cases then
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getFunCasesName declName
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else
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getFunInductName declName
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let result ← realizeGlobalConstNoOverloadCore thmName
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return some result
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catch _ =>
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return none
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def setFunIndInfo (funIndInfo : FunIndInfo) : CoreM Unit := do
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assert! !(funIndInfoExt.contains (← getEnv) funIndInfo.funIndName)
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modifyEnv fun env => funIndInfoExt.insert env funIndInfo.funIndName funIndInfo
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def getFunIndInfoForInduct? (inductName : Name) : CoreM (Option FunIndInfo) := do
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return funIndInfoExt.find? (← getEnv) inductName
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def getFunIndInfo? (cases : Bool) (funName : Name) : CoreM (Option FunIndInfo) := do
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let some inductName ← getFunInduct? cases funName | return none
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getFunIndInfoForInduct? inductName
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end Lean.Meta
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