feat: beta reduction in grind (#6700)

This PR adds support for beta reduction in the `grind` tactic. `grind`
can now solve goals such as
```lean
example (f : Nat → Nat) : f = (fun x : Nat => x + 5) → f 2 > 5 := by
  grind
```

src/Lean/Meta/Tactic/Grind.lean

@@ -53,6 +53,7 @@ builtin_initialize registerTraceClass `grind.offset.propagate
 builtin_initialize registerTraceClass `grind.offset.eq
 builtin_initialize registerTraceClass `grind.offset.eq.to (inherited := true)
 builtin_initialize registerTraceClass `grind.offset.eq.from (inherited := true)
+builtin_initialize registerTraceClass `grind.beta
 
 /-! Trace options for `grind` developers -/
 builtin_initialize registerTraceClass `grind.debug
@@ -68,5 +69,6 @@ builtin_initialize registerTraceClass `grind.debug.canon
 builtin_initialize registerTraceClass `grind.debug.offset
 builtin_initialize registerTraceClass `grind.debug.offset.proof
 builtin_initialize registerTraceClass `grind.debug.ematch.pattern
+builtin_initialize registerTraceClass `grind.debug.beta
 
 end Lean

src/Lean/Meta/Tactic/Grind/Beta.lean

@@ -0,0 +1,77 @@
+/-
+Copyright (c) 2025 Amazon.com, Inc. or its affiliates. All Rights Reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import Lean.Meta.Tactic.Grind.Types
+
+namespace Lean.Meta.Grind
+
+/-- Returns all lambda expressions in the equivalence class with root `root`. -/
+def getEqcLambdas (root : ENode) : GoalM (Array Expr) := do
+  unless root.hasLambdas do return #[]
+  foldEqc root.self (init := #[]) fun n lams =>
+    if n.self.isLambda then return lams.push n.self else return lams
+
+/--
+Returns the root of the functions in the equivalence class containing `e`.
+That is, if `f a` is in `root`s equivalence class, results contains the root of `f`.
+-/
+def getFnRoots (e : Expr) : GoalM (Array Expr) := do
+  foldEqc e (init := #[]) fun n fns => do
+    let fn := n.self.getAppFn
+    let fnRoot := (← getRoot? fn).getD fn
+    if Option.isNone <| fns.find? (isSameExpr · fnRoot) then
+      return fns.push fnRoot
+    else
+      return fns
+
+/--
+For each `lam` in `lams` s.t. `lam` and `f` are in the same equivalence class,
+propagate `f args = lam args`.
+-/
+def propagateBetaEqs (lams : Array Expr) (f : Expr) (args : Array Expr) : GoalM Unit := do
+  if args.isEmpty then return ()
+  for lam in lams do
+    let rhs := lam.beta args
+    unless rhs.isLambda do
+      let mut gen := Nat.max (← getGeneration lam) (← getGeneration f)
+      let lhs := mkAppN f args
+      if (← hasSameType f lam) then
+        let mut h ← mkEqProof f lam
+        for arg in args do
+          gen := Nat.max gen (← getGeneration arg)
+          h ← mkCongrFun h arg
+        let eq ← mkEq lhs rhs
+        trace[grind.beta] "{eq}, using {lam}"
+        addNewFact h eq (gen+1)
+
+private def isPropagateBetaTarget (e : Expr) : GoalM Bool := do
+  let .app f _ := e | return false
+  go f
+where
+  go (f : Expr) : GoalM Bool := do
+    if let some root ← getRootENode? f then
+      return root.hasLambdas
+    let .app f _ := f | return false
+    go f
+
+/--
+Applies beta-reduction for lambdas in `f`s equivalence class.
+We use this function while internalizing new applications.
+-/
+def propagateBetaForNewApp (e : Expr) : GoalM Unit := do
+  unless (← isPropagateBetaTarget e) do return ()
+  let mut e := e
+  let mut args := #[]
+  repeat
+    unless args.isEmpty do
+      if let some root ← getRootENode? e then
+        if root.hasLambdas then
+          propagateBetaEqs (← getEqcLambdas root) e args.reverse
+    let .app f arg := e | return ()
+    e := f
+    args := args.push arg
+
+end Lean.Meta.Grind

src/Lean/Meta/Tactic/Grind/Core.lean

@@ -11,6 +11,7 @@ import Lean.Meta.Tactic.Grind.Inv
 import Lean.Meta.Tactic.Grind.PP
 import Lean.Meta.Tactic.Grind.Ctor
 import Lean.Meta.Tactic.Grind.Util
+import Lean.Meta.Tactic.Grind.Beta
 import Lean.Meta.Tactic.Grind.Internalize
 
 namespace Lean.Meta.Grind
@@ -40,7 +41,7 @@ Remove `root` parents from the congruence table.
 This is an auxiliary function performed while merging equivalence classes.
 -/
 private def removeParents (root : Expr) : GoalM ParentSet := do
-  let parents ← getParentsAndReset root
+  let parents ← getParents root
   for parent in parents do
     -- Recall that we may have `Expr.forallE` in `parents` because of `ForallProp.lean`
     if (← pure parent.isApp <&&> isCongrRoot parent) then
@@ -107,6 +108,31 @@ private def propagateOffsetEq (rhsRoot lhsRoot : ENode) : GoalM Unit := do
     if let some rhsOffset := rhsRoot.offset? then
       Arith.processNewOffsetEqLit rhsOffset lhsRoot.self
 
+/--
+Tries to apply beta-reductiong using the parent applications of the functions in `fns` with
+the lambda expressions in `lams`.
+-/
+def propagateBeta (lams : Array Expr) (fns : Array Expr) : GoalM Unit := do
+  if lams.isEmpty then return ()
+  let lamRoot ← getRoot lams.back!
+  trace[grind.debug.beta] "fns: {fns}, lams: {lams}"
+  for fn in fns do
+    trace[grind.debug.beta] "fn: {fn}, parents: {(← getParents fn).toArray}"
+    for parent in (← getParents fn) do
+      let mut args := #[]
+      let mut curr := parent
+      trace[grind.debug.beta] "parent: {parent}"
+      repeat
+        trace[grind.debug.beta] "curr: {curr}"
+        if (← isEqv curr lamRoot) then
+          propagateBetaEqs lams curr args.reverse
+        let .app f arg := curr
+          | break
+        -- Remark: recall that we do not eagerly internalize partial applications.
+        internalize curr (← getGeneration parent)
+        args := args.push arg
+        curr := f
+
 private partial def addEqStep (lhs rhs proof : Expr) (isHEq : Bool) : GoalM Unit := do
   let lhsNode ← getENode lhs
   let rhsNode ← getENode rhs
@@ -158,6 +184,10 @@ where
       proof?  := proof
       flipped
     }
+    let lams₁ ← getEqcLambdas lhsRoot
+    let lams₂ ← getEqcLambdas rhsRoot
+    let fns₁  ← if lams₁.isEmpty then pure #[] else getFnRoots rhsRoot.self
+    let fns₂  ← if lams₂.isEmpty then pure #[] else getFnRoots lhsRoot.self
     let parents ← removeParents lhsRoot.self
     updateRoots lhs rhsNode.root
     trace_goal[grind.debug] "{← ppENodeRef lhs} new root {← ppENodeRef rhsNode.root}, {← ppENodeRef (← getRoot lhs)}"
@@ -172,6 +202,9 @@ where
       hasLambdas := rhsRoot.hasLambdas || lhsRoot.hasLambdas
       heqProofs  := isHEq || rhsRoot.heqProofs || lhsRoot.heqProofs
     }
+    propagateBeta lams₁ fns₁
+    propagateBeta lams₂ fns₂
+    resetParentsOf lhsRoot.self
     copyParentsTo parents rhsNode.root
     unless (← isInconsistent) do
       updateMT rhsRoot.self

src/Lean/Meta/Tactic/Grind/Internalize.lean

@@ -12,6 +12,7 @@ import Lean.Meta.Match.MatchEqsExt
 import Lean.Meta.Tactic.Grind.Types
 import Lean.Meta.Tactic.Grind.Util
 import Lean.Meta.Tactic.Grind.Canon
+import Lean.Meta.Tactic.Grind.Beta
 import Lean.Meta.Tactic.Grind.Arith.Internalize
 
 namespace Lean.Meta.Grind
@@ -194,7 +195,7 @@ partial def internalize (e : Expr) (generation : Nat) (parent? : Option Expr :=
           activateTheoremPatterns fName generation
         else
           internalize f generation e
-          registerParent e f
+        registerParent e f
         for h : i in [: args.size] do
           let arg := args[i]
           internalize arg generation e
@@ -204,6 +205,8 @@ partial def internalize (e : Expr) (generation : Nat) (parent? : Option Expr :=
       updateAppMap e
       Arith.internalize e parent?
       propagateUp e
+      propagateBetaForNewApp e
+
 end
 
 end Lean.Meta.Grind

src/Lean/Meta/Tactic/Grind/Types.lean

@@ -501,8 +501,9 @@ def getENode (e : Expr) : GoalM ENode := do
   (← get).getENode e
 
 /-- Returns the generation of the given term. Is assumes it has been internalized -/
-def getGeneration (e : Expr) : GoalM Nat :=
-  return (← getENode e).generation
+def getGeneration (e : Expr) : GoalM Nat := do
+  let some n ← getENode? e | return 0
+  return n.generation
 
 /-- Returns `true` if `e` is in the equivalence class of `True`. -/
 def isEqTrue (e : Expr) : GoalM Bool := do
@@ -519,8 +520,8 @@ def isEqv (a b : Expr) : GoalM Bool := do
   if isSameExpr a b then
     return true
   else
-    let na ← getENode a
-    let nb ← getENode b
+    let some na ← getENode? a | return false
+    let some nb ← getENode? b | return false
     return isSameExpr na.root nb.root
 
 /-- Returns `true` if the root of its equivalence class. -/
@@ -549,6 +550,11 @@ def getRoot (e : Expr) : GoalM Expr := do
 def getRootENode (e : Expr) : GoalM ENode := do
   getENode (← getRoot e)
 
+/-- Returns the root enode in the equivalence class of `e` if it is in an equivalence class. -/
+def getRootENode? (e : Expr) : GoalM (Option ENode) := do
+  let some n ← getENode? e | return none
+  getENode? n.root
+
 /--
 Returns the next element in the equivalence class of `e`
 if `e` has been internalized in the given goal.
@@ -614,7 +620,7 @@ Records that `parent` is a parent of `child`. This function actually stores the
 information in the root (aka canonical representative) of `child`.
 -/
 def registerParent (parent : Expr) (child : Expr) : GoalM Unit := do
-  let some childRoot ← getRoot? child | return ()
+  let childRoot := (← getRoot? child).getD child
   let parents := if let some parents := (← get).parents.find? { expr := childRoot } then parents else {}
   modify fun s => { s with parents := s.parents.insert { expr := childRoot } (parents.insert parent) }
 
@@ -628,12 +634,10 @@ def getParents (e : Expr) : GoalM ParentSet := do
   return parents
 
 /--
-Similar to `getParents`, but also removes the entry `e ↦ parents` from the parent map.
+Removes the entry `e ↦ parents` from the parent map.
 -/
-def getParentsAndReset (e : Expr) : GoalM ParentSet := do
-  let parents ← getParents e
+def resetParentsOf (e : Expr) : GoalM Unit := do
   modify fun s => { s with parents := s.parents.erase { expr := e } }
-  return parents
 
 /--
 Copy `parents` to the parents of `root`.
@@ -800,6 +804,18 @@ def getENodes : GoalM (Array ENode) := do
     if isSameExpr n.next e then return ()
     curr := n.next
 
+/-- Folds using `f` and `init` over the equivalence class containing `e` -/
+@[inline] def foldEqc (e : Expr) (init : α) (f : ENode → α → GoalM α) : GoalM α := do
+  let mut curr := e
+  let mut r := init
+  repeat
+    let n ← getENode curr
+    r ← f n r
+    if isSameExpr n.next e then return r
+    curr := n.next
+  unreachable!
+  return r
+
 def forEachENode (f : ENode → GoalM Unit) : GoalM Unit := do
   let nodes ← getENodes
   for n in nodes do

tests/lean/run/grind_beta.lean

@@ -0,0 +1,72 @@
+def f (x : Nat) : Nat → Nat → Nat :=
+  match x with
+  | 0 => fun _ _ => 0
+  | _+1 => fun a b => a + b
+
+example : f 0 b c = 0 := by
+  grind [f]
+
+example : f (a+1) b c = b + c := by
+  grind [f]
+
+example : f x b c ≠ b + c → x = a + 1 → False := by
+  grind [f]
+
+example : x = a + 1 → f x b c ≠ b + c → False := by
+  grind [f]
+
+example : x = a + 1 → f x b c ≠ b + c → False := by
+  grind [f]
+
+example : f x b c > 0 → x = 0 → False := by
+  grind [f]
+
+example : f x b c > 0 → x ≠ 0 := by
+  grind [f]
+
+example (f : Nat → Nat → Nat) : f 2 3 ≠ 5 → f = (fun x y : Nat => x + y) → False := by
+  grind
+
+opaque bla : Nat → Nat → Nat → Nat
+
+/--
+info: [grind.beta] f 2 3 = bla 2 3 2, using fun x y => bla x y x
+[grind.beta] f 2 3 = 2 + 3, using fun x y => x + y
+-/
+#guard_msgs (info) in
+set_option trace.grind.beta true in
+example (g h f : Nat → Nat → Nat) :
+        f 2 3 ≠ 5 →
+          g = (fun x y : Nat => x + y) →
+          h = (fun x y => bla x y x) →
+          g = h →
+          f = h →
+          False := by
+  grind
+
+example (g h f : Nat → Nat → Nat) :
+        f 2 3 ≠ 5 →
+          h = (fun x y => bla x y x) →
+          g = (fun x y : Nat => x + y) →
+          g = h →
+          h = f →
+          False := by
+  grind
+
+
+example (f : Nat → Nat → Nat) : f = (fun x y : Nat => x + y) → f 2 3 = 5 := by
+  grind
+
+example (f g h : Nat → Nat → Nat) :
+          h = (fun x y => bla x y x) →
+          g = (fun x y : Nat => x + y) →
+          g = h →
+          h = f →
+          f 2 3 = 5 := by
+  grind
+
+example (f : Nat → Nat) : f = (fun x : Nat => x + 5) → f 2 > 5 := by
+  grind
+
+example (f : Nat → Nat → Nat) : f a = (fun x : Nat => x + 5) → f a 2 > 5 := by
+  grind

tests/lean/run/grind_canon_insts.lean

@@ -52,15 +52,13 @@ theorem left_comm [CommMonoid α] (a b c : α) : a * (b * c) = b * (a * c) := by
 
 open Lean Meta Elab Tactic Grind in
 def fallback : Fallback := do
-  let nodes ← filterENodes fun e => return e.self.isAppOf ``HMul.hMul
+  let nodes ← filterENodes fun e => return e.self.isApp && e.self.isAppOf ``HMul.hMul
   trace[Meta.debug] "{nodes.toList.map (·.self)}"
   (← get).mvarId.admit
 
 set_option trace.Meta.debug true
 
-/--
-info: [Meta.debug] [b * c, a * (b * c), d * (b * c)]
--/
+/-- info: [Meta.debug] [b * c, a * (b * c), d * (b * c)] -/
 #guard_msgs (info) in
 example (a b c d : Nat) : b * (a * c) = d * (b * c) → False := by
   rw [left_comm] -- Introduces a new (non-canonical) instance for `Mul Nat`

tests/lean/run/grind_canon_types.lean

@@ -5,7 +5,7 @@ def f (a : α) := a
 
 open Lean Meta Grind in
 def fallback : Fallback := do
-  let nodes ← filterENodes fun e => return e.self.isAppOf ``f
+  let nodes ← filterENodes fun e => return e.self.isApp && e.self.isAppOf ``f
   trace[Meta.debug] "{nodes.toList.map (·.self)}"
   (← get).mvarId.admit
 

tests/lean/run/grind_congr.lean

@@ -6,7 +6,7 @@ def g (a : Nat) := a + a
 -- Prints the equivalence class containing a `f` application
 open Lean Meta Grind in
 def fallback : Fallback := do
-  let #[n, _] ← filterENodes fun e => return e.self.isAppOf ``f | unreachable!
+  let #[n, _] ← filterENodes fun e => return e.self.isApp && e.self.isAppOf ``f | unreachable!
   let eqc ← getEqc n.self
   trace[Meta.debug] eqc
   (← get).mvarId.admit

tests/lean/run/grind_many_eqs.lean

@@ -14,7 +14,7 @@ def fallback (n : Nat) : Fallback := do
   -- The `f 0` equivalence class contains `n+1` elements
   assert! (← getEqc f0).length == n + 1
   forEachENode fun node => do
-    if node.self.isAppOf ``g then
+    if node.self.isApp && node.self.isAppOf ``g then
       -- Any equivalence class containing a `g`-application contains 2 elements
       assert! (← getEqc (← getRoot node.self)).length == 2
   (← get).mvarId.admit

tests/lean/run/grind_nested_proofs.lean

@@ -6,7 +6,7 @@ open Lean Meta Grind in
 def fallback : Fallback := do
   let nodes ← filterENodes fun e => return e.self.isAppOf ``Lean.Grind.nestedProof
   trace[Meta.debug] "{nodes.toList.map (·.self)}"
-  let nodes ← filterENodes fun e => return e.self.isAppOf ``GetElem.getElem
+  let nodes ← filterENodes fun e => return e.self.isApp && e.self.isAppOf ``GetElem.getElem
   let [_, n, _] := nodes.toList | unreachable!
   trace[Meta.debug] "{← getEqc n.self}"
   (← get).mvarId.admit