diff --git a/src/Lean/Meta/Tactic/Grind/EMatch.lean b/src/Lean/Meta/Tactic/Grind/EMatch.lean
index 5ca3b7aaa9..37a3ecc63d 100644
--- a/src/Lean/Meta/Tactic/Grind/EMatch.lean
+++ b/src/Lean/Meta/Tactic/Grind/EMatch.lean
@@ -8,14 +8,14 @@ import Lean.Meta.Tactic.Grind.Types
 import Lean.Meta.Tactic.Grind.Internalize
 
 namespace Lean.Meta.Grind
-
-/-- Returns maximum term generation that is considered during ematching -/
-private def getMaxGeneration : GoalM Nat := do
-  return 10000 -- TODO
-
-/-- Returns `true` if the maximum number of instances has been reached. -/
-private def checkMaxInstancesExceeded : GoalM Bool := do
-  return false -- TODO
+/--
+Theorem instance found using E-matching.
+Recall that we only internalize new instances after we complete a full round of E-matching. -/
+structure EMatchTheoremInstance where
+  proof      : Expr
+  prop       : Expr
+  generation : Nat
+  deriving Inhabited
 
 namespace EMatch
 /-! This module implements a simple E-matching procedure as a backtracking search. -/
@@ -51,13 +51,6 @@ structure Choice where
   assignment : Array Expr
   deriving Inhabited
 
-/-- Theorem instances found so far. We only internalize them after we complete a full round of E-matching. -/
-structure TheoremInstance where
-  proof      : Expr
-  prop       : Expr
-  generation : Nat
-  deriving Inhabited
-
 /-- Context for the E-matching monad. -/
 structure Context where
   /-- `useMT` is `true` if we are using the mod-time optimization. It is always set to false for new `EMatchTheorem`s. -/
@@ -70,7 +63,7 @@ structure Context where
 structure State where
   /-- Choices that still have to be processed. -/
   choiceStack  : List Choice := []
-  newInstances : PArray TheoremInstance := {}
+  newInstances : Array EMatchTheoremInstance := #[]
   deriving Inhabited
 
 abbrev M := ReaderT Context $ StateRefT State GoalM
@@ -181,6 +174,8 @@ Missing parameters are synthesized using type inference and type class synthesis
 -/
 private partial def instantiateTheorem (c : Choice) : M Unit := withDefault do
   let thm := (← read).thm
+  unless (← markTheorenInstance thm.proof c.assignment) do
+    return ()
   trace[grind.ematch.instance.assignment] "{← thm.origin.pp}: {assignmentToMessageData c.assignment}"
   let proof ← thm.getProofWithFreshMVarLevels
   let numParams := thm.numParams
@@ -285,22 +280,26 @@ where
 def ematchTheorems (thms : PArray EMatchTheorem) : M Unit := do
   thms.forM ematchTheorem
 
-def internalizeNewInstances : M Unit := do
-  -- TODO
-  return ()
-
 end EMatch
 
 open EMatch
 
-/-- Performs one round of E-matching, and internalizes new instances. -/
-def ematch : GoalM Unit := do
-  let go (thms newThms : PArray EMatchTheorem) : EMatch.M Unit := do
+/-- Performs one round of E-matching, and returns new instances. -/
+def ematch : GoalM (Array EMatchTheoremInstance) := do
+  let go (thms newThms : PArray EMatchTheorem) : EMatch.M (Array EMatchTheoremInstance) := do
     withReader (fun ctx => { ctx with useMT := true }) <| ematchTheorems thms
     withReader (fun ctx => { ctx with useMT := false }) <| ematchTheorems newThms
-    internalizeNewInstances
-  unless (← checkMaxInstancesExceeded) do
-    go (← get).thms (← get).newThms |>.run'
-    modify fun s => { s with thms := s.thms ++ s.newThms, newThms := {}, gmt := s.gmt + 1 }
+    return (← get).newInstances
+  if (← checkMaxInstancesExceeded) then
+    return #[]
+  else
+    let insts ← go (← get).thms (← get).newThms |>.run'
+    modify fun s => { s with
+      thms         := s.thms ++ s.newThms
+      newThms      := {}
+      gmt          := s.gmt + 1
+      numInstances := s.numInstances + insts.size
+    }
+    return insts
 
 end Lean.Meta.Grind
diff --git a/src/Lean/Meta/Tactic/Grind/Internalize.lean b/src/Lean/Meta/Tactic/Grind/Internalize.lean
index 341e2b7e50..75d650ea2a 100644
--- a/src/Lean/Meta/Tactic/Grind/Internalize.lean
+++ b/src/Lean/Meta/Tactic/Grind/Internalize.lean
@@ -59,7 +59,10 @@ private partial def activateTheoremPatterns (fName : Name) (generation : Nat) :
       let thm := { thm with symbols }
       match symbols with
       | [] =>
-        let thm := { thm with patterns := (← thm.patterns.mapM (internalizePattern · generation)) }
+        -- Recall that we use the proof as part of the key for a set of instances found so far.
+        -- We don't want to use structural equality when comparing keys.
+        let proof ← shareCommon thm.proof
+        let thm := { thm with proof, patterns := (← thm.patterns.mapM (internalizePattern · generation)) }
         trace[grind.ematch] "activated `{thm.origin.key}`, {thm.patterns.map ppPattern}"
         modify fun s => { s with newThms := s.newThms.push thm }
       | _ =>
diff --git a/src/Lean/Meta/Tactic/Grind/Preprocessor.lean b/src/Lean/Meta/Tactic/Grind/Preprocessor.lean
index c8ebf593a1..bb638daffe 100644
--- a/src/Lean/Meta/Tactic/Grind/Preprocessor.lean
+++ b/src/Lean/Meta/Tactic/Grind/Preprocessor.lean
@@ -144,7 +144,7 @@ def preprocess (mvarId : MVarId) : PreM State := do
   loop (← mkGoal mvarId)
   let goals := (← get).goals
   -- Testing `ematch` module here. We will rewrite this part later.
-  let goals ← goals.mapM fun goal => GoalM.run' goal ematch
+  let goals ← goals.mapM fun goal => GoalM.run' goal (discard <| ematch)
   if (← isTracingEnabledFor `grind.pre) then
     trace[grind.debug.pre] (← ppGoals goals)
   for goal in goals do
diff --git a/src/Lean/Meta/Tactic/Grind/Types.lean b/src/Lean/Meta/Tactic/Grind/Types.lean
index 438f665bd4..754f2c1e57 100644
--- a/src/Lean/Meta/Tactic/Grind/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/Types.lean
@@ -101,6 +101,11 @@ def getMainDeclName : GrindM Name :=
 @[inline] def getMethodsRef : GrindM MethodsRef :=
   read
 
+/--
+Returns maximum term generation that is considered during ematching. -/
+def getMaxGeneration : GrindM Nat := do
+  return 10000 -- TODO
+
 /--
 Abtracts nested proofs in `e`. This is a preprocessing step performed before internalization.
 -/
@@ -193,31 +198,44 @@ structure NewEq where
   proof : Expr
   isHEq : Bool
 
-abbrev ENodes := PHashMap USize ENode
+/--
+Key for the `ENodeMap` and `ParentMap` map.
+We use pointer addresses and rely on the fact all internalized expressions
+have been hash-consed, i.e., we have applied `shareCommon`.
+-/
+private structure ENodeKey where
+  expr : Expr
 
-structure CongrKey (enodes : ENodes) where
+instance : Hashable ENodeKey where
+  hash k := unsafe (ptrAddrUnsafe k.expr).toUInt64
+
+instance : BEq ENodeKey where
+  beq k₁ k₂ := isSameExpr k₁.expr k₂.expr
+
+abbrev ENodeMap := PHashMap ENodeKey ENode
+
+/--
+Key for the congruence table.
+We need access to the `enodes` to be able to retrieve the equivalence class roots.
+-/
+structure CongrKey (enodes : ENodeMap) where
   e : Expr
 
-private abbrev toENodeKey (e : Expr) : USize :=
-  unsafe ptrAddrUnsafe e
-
-private def hashRoot (enodes : ENodes) (e : Expr) : UInt64 :=
-  if let some node := enodes.find? (toENodeKey e) then
-    toENodeKey node.root |>.toUInt64
+private def hashRoot (enodes : ENodeMap) (e : Expr) : UInt64 :=
+  if let some node := enodes.find? { expr := e } then
+    unsafe (ptrAddrUnsafe node.root).toUInt64
   else
     13
 
-private def hasSameRoot (enodes : ENodes) (a b : Expr) : Bool := Id.run do
-  let ka := toENodeKey a
-  let kb := toENodeKey b
-  if ka == kb then
+private def hasSameRoot (enodes : ENodeMap) (a b : Expr) : Bool := Id.run do
+  if isSameExpr a b then
     return true
   else
-    let some n1 := enodes.find? ka | return false
-    let some n2 := enodes.find? kb | return false
-    toENodeKey n1.root == toENodeKey n2.root
+    let some n1 := enodes.find? { expr := a } | return false
+    let some n2 := enodes.find? { expr := b } | return false
+    isSameExpr n1.root n2.root
 
-def congrHash (enodes : ENodes) (e : Expr) : UInt64 :=
+def congrHash (enodes : ENodeMap) (e : Expr) : UInt64 :=
   if e.isAppOfArity ``Lean.Grind.nestedProof 2 then
     -- We only hash the proposition
     hashRoot enodes (e.getArg! 0)
@@ -229,7 +247,7 @@ where
     | .app f a => go f (mixHash r (hashRoot enodes a))
     | _ => mixHash r (hashRoot enodes e)
 
-partial def isCongruent (enodes : ENodes) (a b : Expr) : Bool :=
+partial def isCongruent (enodes : ENodeMap) (a b : Expr) : Bool :=
   if a.isAppOfArity ``Lean.Grind.nestedProof 2 && b.isAppOfArity ``Lean.Grind.nestedProof 2 then
     hasSameRoot enodes (a.getArg! 0) (b.getArg! 0)
   else
@@ -249,15 +267,43 @@ instance : Hashable (CongrKey enodes) where
 instance : BEq (CongrKey enodes) where
   beq k1 k2 := isCongruent enodes k1.e k2.e
 
-abbrev CongrTable (enodes : ENodes) := PHashSet (CongrKey enodes)
+abbrev CongrTable (enodes : ENodeMap) := PHashSet (CongrKey enodes)
 
 -- Remark: we cannot use pointer addresses here because we have to traverse the tree.
 abbrev ParentSet := RBTree Expr Expr.quickComp
-abbrev ParentMap := PHashMap USize ParentSet
+abbrev ParentMap := PHashMap ENodeKey ParentSet
+
+/--
+The E-matching module instantiates theorems using the `EMatchTheorem proof` and a (partial) assignment.
+We want to avoid instantiating the same theorem with the same assignment more than once.
+Therefore, we store the (pre-)instance information in set.
+Recall that the proofs of activated theorems have been hash-consed.
+The assignment contains internalized expressions, which have also been hash-consed.
+-/
+structure PreInstance where
+  proof      : Expr
+  assignment : Array Expr
+
+instance : Hashable PreInstance where
+  hash i := Id.run do
+    let mut r := unsafe (ptrAddrUnsafe i.proof >>> 3).toUInt64
+    for v in i.assignment do
+      r := mixHash r (unsafe (ptrAddrUnsafe v >>> 3).toUInt64)
+    return r
+
+instance : BEq PreInstance where
+  beq i₁ i₂ := Id.run do
+    unless isSameExpr i₁.proof i₂.proof do return false
+    unless i₁.assignment.size == i₂.assignment.size do return false
+    for v₁ in i₁.assignment, v₂ in i₂.assignment do
+      unless isSameExpr v₁ v₂ do return false
+    return true
+
+abbrev PreInstanceSet := PHashSet PreInstance
 
 structure Goal where
   mvarId       : MVarId
-  enodes       : ENodes := {}
+  enodes       : ENodeMap := {}
   parents      : ParentMap := {}
   congrTable   : CongrTable enodes := {}
   /--
@@ -285,6 +331,8 @@ structure Goal where
   thmMap       : EMatchTheorems
   /-- Number of theorem instances generated so far -/
   numInstances : Nat := 0
+  /-- (pre-)instances found so far -/
+  instances    : PreInstanceSet := {}
   deriving Inhabited
 
 def Goal.admit (goal : Goal) : MetaM Unit :=
@@ -294,6 +342,21 @@ abbrev GoalM := StateRefT Goal GrindM
 
 abbrev Propagator := Expr → GoalM Unit
 
+/--
+A helper function used to mark a theorem instance found by the E-matching module.
+It returns `true` if it is a new instance and `false` otherwise.
+-/
+def markTheorenInstance (proof : Expr) (assignment : Array Expr) : GoalM Bool := do
+  let k := { proof, assignment }
+  if (← get).instances.contains k then
+    return false
+  modify fun s => { s with instances := s.instances.insert k }
+  return true
+
+/-- Returns `true` if the maximum number of instances has been reached. -/
+def checkMaxInstancesExceeded : GoalM Bool := do
+  return false -- TODO
+
 /-- Returns `true` if `e` is the internalized `True` expression.  -/
 def isTrueExpr (e : Expr) : GrindM Bool :=
   return isSameExpr e (← getTrueExpr)
@@ -307,11 +370,11 @@ Returns `some n` if `e` has already been "internalized" into the
 Otherwise, returns `none`s.
 -/
 def getENode? (e : Expr) : GoalM (Option ENode) :=
-  return (← get).enodes.find? (unsafe ptrAddrUnsafe e)
+  return (← get).enodes.find? { expr := e }
 
 /-- Returns node associated with `e`. It assumes `e` has already been internalized. -/
 def getENode (e : Expr) : GoalM ENode := do
-  let some n := (← get).enodes.find? (unsafe ptrAddrUnsafe e)
+  let some n := (← get).enodes.find? { expr := e }
     | throwError "internal `grind` error, term has not been internalized{indentExpr e}"
   return n
 
@@ -362,7 +425,7 @@ def getNext (e : Expr) : GoalM Expr :=
 
 /-- Returns `true` if `e` has already been internalized. -/
 def alreadyInternalized (e : Expr) : GoalM Bool :=
-  return (← get).enodes.contains (unsafe ptrAddrUnsafe e)
+  return (← get).enodes.contains { expr := e }
 
 def getTarget? (e : Expr) : GoalM (Option Expr) := do
   let some n ← getENode? e | return none
@@ -407,9 +470,8 @@ 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 key := toENodeKey childRoot
-  let parents := if let some parents := (← get).parents.find? key then parents else {}
-  modify fun s => { s with parents := s.parents.insert key (parents.insert parent) }
+  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) }
 
 /--
 Returns the set of expressions `e` is a child of, or an expression in
@@ -417,7 +479,7 @@ Returns the set of expressions `e` is a child of, or an expression in
 The information is only up to date if `e` is the root (aka canonical representative) of the equivalence class.
 -/
 def getParents (e : Expr) : GoalM ParentSet := do
-  let some parents := (← get).parents.find? (toENodeKey e) | return {}
+  let some parents := (← get).parents.find? { expr := e } | return {}
   return parents
 
 /--
@@ -425,7 +487,7 @@ Similar to `getParents`, but also removes the entry `e ↦ parents` from the par
 -/
 def getParentsAndReset (e : Expr) : GoalM ParentSet := do
   let parents ← getParents e
-  modify fun s => { s with parents := s.parents.erase (toENodeKey e) }
+  modify fun s => { s with parents := s.parents.erase { expr := e } }
   return parents
 
 /--
@@ -433,15 +495,14 @@ Copy `parents` to the parents of `root`.
 `root` must be the root of its equivalence class.
 -/
 def copyParentsTo (parents : ParentSet) (root : Expr) : GoalM Unit := do
-  let key := toENodeKey root
-  let mut curr := if let some parents := (← get).parents.find? key then parents else {}
+  let mut curr := if let some parents := (← get).parents.find? { expr := root } then parents else {}
   for parent in parents do
     curr := curr.insert parent
-  modify fun s => { s with parents := s.parents.insert key curr }
+  modify fun s => { s with parents := s.parents.insert { expr := root } curr }
 
 def setENode (e : Expr) (n : ENode) : GoalM Unit :=
   modify fun s => { s with
-    enodes := s.enodes.insert (unsafe ptrAddrUnsafe e) n
+    enodes := s.enodes.insert { expr := e } n
     congrTable := unsafe unsafeCast s.congrTable
   }