diff --git a/src/Lean/Elab/Tactic/Induction.lean b/src/Lean/Elab/Tactic/Induction.lean
index 960485c90d..9448b93e8f 100644
--- a/src/Lean/Elab/Tactic/Induction.lean
+++ b/src/Lean/Elab/Tactic/Induction.lean
@@ -10,6 +10,7 @@ import Lean.Meta.CollectMVars
 import Lean.Meta.Tactic.ElimInfo
 import Lean.Meta.Tactic.Induction
 import Lean.Meta.Tactic.Cases
+import Lean.Meta.GeneralizeVars
 import Lean.Elab.App
 import Lean.Elab.Tactic.ElabTerm
 import Lean.Elab.Tactic.Generalize
@@ -243,56 +244,6 @@ where
 
 end ElimApp
 
-/--
-  Return a set of `FVarId`s containing `targets` and all variables they depend on.
-
-  Remark: this method assumes `targets` are free variables.
--/
-private partial def mkForbiddenSet (targets : Array Expr) : MetaM NameSet := do
-  loop (targets.toList.map Expr.fvarId!) {}
-where
-  visit (fvarId : FVarId) (todo : List FVarId) (s : NameSet) : MetaM (List FVarId × NameSet) := do
-    let localDecl ← getLocalDecl fvarId
-    let mut s' := collectFVars {} (← instantiateMVars localDecl.type)
-    if let some val := localDecl.value? then
-      s' := collectFVars s' (← instantiateMVars val)
-    let mut todo := todo
-    let mut s := s
-    for fvarId in s'.fvarSet do
-      unless s.contains fvarId do
-        todo := fvarId :: todo
-        s := s.insert fvarId
-    return (todo, s)
-
-  loop (todo : List FVarId) (s : NameSet) : MetaM NameSet := do
-    match todo with
-    | [] => return s
-    | fvarId::todo =>
-      if s.contains fvarId then
-        loop todo s
-      else
-        let (todo, s) ← visit fvarId todo <| s.insert fvarId
-        loop todo s
-
-/--
-  Collect forward dependencies that are not in the forbidden set, and depend on some variable in `targets`.
-
-  Remark: this method assumes `targets` are free variables.
-
-  Remark: we *not* collect instance implicit arguments nor auxiliary declarations for compiling
-  recursive declarations.
--/
-private def collectForwardDeps (targets : Array Expr) (forbidden : NameSet) : MetaM NameSet := do
-  let mut s : NameSet := targets.foldl (init := {}) fun s target => s.insert target.fvarId!
-  let mut r : NameSet := {}
-  for localDecl in (← getLCtx) do
-    unless forbidden.contains localDecl.fvarId do
-      unless localDecl.isAuxDecl || localDecl.binderInfo.isInstImplicit do
-      if (← getMCtx).findLocalDeclDependsOn localDecl fun fvarId => s.contains fvarId then
-        r := r.insert localDecl.fvarId
-        s := s.insert localDecl.fvarId
-  return r
-
 /-
   Recall that
   ```
@@ -300,7 +251,7 @@ private def collectForwardDeps (targets : Array Expr) (forbidden : NameSet) : Me
   «induction»  := leading_parser nonReservedSymbol "induction " >> majorPremise >> usingRec >> generalizingVars >> optional inductionAlts
   ```
   `stx` is syntax for `induction`. -/
-private def getGeneralizingFVarIds (stx : Syntax) : TacticM (Array FVarId) :=
+private def getUserGeneralizingFVarIds (stx : Syntax) : TacticM (Array FVarId) :=
   withRef stx do
     let generalizingStx := stx[3]
     if generalizingStx.isNone then
@@ -313,18 +264,16 @@ private def getGeneralizingFVarIds (stx : Syntax) : TacticM (Array FVarId) :=
 -- process `generalizingVars` subterm of induction Syntax `stx`.
 private def generalizeVars (mvarId : MVarId) (stx : Syntax) (targets : Array Expr) : TacticM (Nat × MVarId) :=
   withMVarContext mvarId do
-    let userFVarIds ← getGeneralizingFVarIds stx
-    let forbidden ← mkForbiddenSet targets
-    let mut s ← collectForwardDeps targets forbidden
+    let userFVarIds ← getUserGeneralizingFVarIds stx
+    let forbidden ← mkGeneralizationForbiddenSet targets
+    let mut s ← getFVarSetToGeneralize targets forbidden
     for userFVarId in userFVarIds do
       if forbidden.contains userFVarId then
         throwError "variable cannot be generalized because target depends on it{indentExpr (mkFVar userFVarId)}"
       if s.contains userFVarId then
         throwError "unnecessary 'generalizing' argument, variable '{mkFVar userFVarId}' is generalized automatically"
       s := s.insert userFVarId
-    let fvarIds := s.fold (init := #[]) fun s fvarId => s.push fvarId
-    let lctx ← getLCtx
-    let fvarIds ← fvarIds.qsort fun x y => (lctx.get! x).index < (lctx.get! y).index
+    let fvarIds ← sortFVars s
     let (fvarIds, mvarId') ← Meta.revert mvarId fvarIds
     return (fvarIds.size, mvarId')
 
diff --git a/src/Lean/Meta.lean b/src/Lean/Meta.lean
index 1045c27416..5d453b3a10 100644
--- a/src/Lean/Meta.lean
+++ b/src/Lean/Meta.lean
@@ -32,3 +32,4 @@ import Lean.Meta.SizeOf
 import Lean.Meta.Coe
 import Lean.Meta.SortLocalDecls
 import Lean.Meta.CollectFVars
+import Lean.Meta.GeneralizeVars
diff --git a/src/Lean/Meta/GeneralizeVars.lean b/src/Lean/Meta/GeneralizeVars.lean
new file mode 100644
index 0000000000..aa21f93482
--- /dev/null
+++ b/src/Lean/Meta/GeneralizeVars.lean
@@ -0,0 +1,75 @@
+/-
+Copyright (c) 2021 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+import Lean.Meta.Basic
+import Lean.Util.CollectFVars
+
+namespace Lean.Meta
+
+/--
+  Return a set of `FVarId`s containing `targets` and all variables they depend on.
+
+  Remark: this method assumes `targets` are free variables.
+-/
+partial def mkGeneralizationForbiddenSet (targets : Array Expr) : MetaM NameSet := do
+  loop (targets.toList.map Expr.fvarId!) {}
+where
+  visit (fvarId : FVarId) (todo : List FVarId) (s : NameSet) : MetaM (List FVarId × NameSet) := do
+    let localDecl ← getLocalDecl fvarId
+    let mut s' := collectFVars {} (← instantiateMVars localDecl.type)
+    if let some val := localDecl.value? then
+      s' := collectFVars s' (← instantiateMVars val)
+    let mut todo := todo
+    let mut s := s
+    for fvarId in s'.fvarSet do
+      unless s.contains fvarId do
+        todo := fvarId :: todo
+        s := s.insert fvarId
+    return (todo, s)
+
+  loop (todo : List FVarId) (s : NameSet) : MetaM NameSet := do
+    match todo with
+    | [] => return s
+    | fvarId::todo =>
+      if s.contains fvarId then
+        loop todo s
+      else
+        let (todo, s) ← visit fvarId todo <| s.insert fvarId
+        loop todo s
+
+/--
+  Collect variables to be generalized.
+  It uses the following heuristic
+  - Collect forward dependencies that are not in the forbidden set, and depend on some variable in `targets`.
+
+  - We use `mkForbiddenSet` to compute `forbidden`.
+
+  Remark: this method assumes `targets` are free variables.
+
+  Remark: we *not* collect instance implicit arguments nor auxiliary declarations for compiling
+  recursive declarations.
+-/
+def getFVarSetToGeneralize (targets : Array Expr) (forbidden : NameSet) : MetaM NameSet := do
+  let mut s : NameSet := targets.foldl (init := {}) fun s target => s.insert target.fvarId!
+  let mut r : NameSet := {}
+  for localDecl in (← getLCtx) do
+    unless forbidden.contains localDecl.fvarId do
+      unless localDecl.isAuxDecl || localDecl.binderInfo.isInstImplicit do
+      if (← getMCtx).findLocalDeclDependsOn localDecl fun fvarId => s.contains fvarId then
+        r := r.insert localDecl.fvarId
+        s := s.insert localDecl.fvarId
+  return r
+
+def sortFVars (fvars : NameSet) : MetaM (Array FVarId) := do
+  let fvarIds := fvars.fold (init := #[]) fun s fvarId => s.push fvarId
+  let lctx ← getLCtx
+  return fvarIds.qsort fun x y => (lctx.get! x).index < (lctx.get! y).index
+
+def getFVarsToGeneralize (targets : Array Expr) : MetaM (Array FVarId) := do
+  let forbidden ← mkGeneralizationForbiddenSet targets
+  let s ← getFVarSetToGeneralize targets forbidden
+  sortFVars s
+
+end Lean.Meta
\ No newline at end of file