diff --git a/src/Lean/Meta/Tactic/Grind/AC.lean b/src/Lean/Meta/Tactic/Grind/AC.lean
index b232708d43..13a160eed3 100644
--- a/src/Lean/Meta/Tactic/Grind/AC.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC.lean
@@ -16,8 +16,9 @@ public import Lean.Meta.Tactic.Grind.AC.DenoteExpr
 public import Lean.Meta.Tactic.Grind.AC.ToExpr
 public import Lean.Meta.Tactic.Grind.AC.VarRename
 public import Lean.Meta.Tactic.Grind.AC.PP
+public import Lean.Meta.Tactic.Grind.AC.Inv
 public section
-namespace Lean
+namespace Lean.Meta.Grind.AC
 builtin_initialize registerTraceClass `grind.ac
 builtin_initialize registerTraceClass `grind.ac.assert
 builtin_initialize registerTraceClass `grind.ac.internalize
@@ -29,4 +30,13 @@ builtin_initialize registerTraceClass `grind.debug.ac.check
 builtin_initialize registerTraceClass `grind.debug.ac.queue
 builtin_initialize registerTraceClass `grind.debug.ac.superpose
 builtin_initialize registerTraceClass `grind.debug.ac.eq
-end Lean
+
+builtin_initialize
+  acExt.setMethods
+    (internalize := AC.internalize)
+    (newEq := AC.processNewEq)
+    (newDiseq := AC.processNewDiseq)
+    (check := AC.check)
+    (checkInv := AC.checkInvariants)
+
+end Lean.Meta.Grind.AC
diff --git a/src/Lean/Meta/Tactic/Grind/AC/Eq.lean b/src/Lean/Meta/Tactic/Grind/AC/Eq.lean
index 5febc23087..4eb6058c1f 100644
--- a/src/Lean/Meta/Tactic/Grind/AC/Eq.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC/Eq.lean
@@ -360,8 +360,7 @@ private def EqCnstr.assert (c : EqCnstr) : ACM Unit := do
   else
     c.addToQueue
 
-@[export lean_process_ac_eq]
-def processNewEqImpl (a b : Expr) : GoalM Unit := withExprs a b do
+def processNewEq(a b : Expr) : GoalM Unit := withExprs a b do
   let ea ← asACExpr a
   let lhs ← norm ea
   let eb ← asACExpr b
@@ -369,8 +368,7 @@ def processNewEqImpl (a b : Expr) : GoalM Unit := withExprs a b do
   let c ← mkEqCnstr lhs rhs (.core a b ea eb)
   c.assert
 
-@[export lean_process_ac_diseq]
-def processNewDiseqImpl (a b : Expr) : GoalM Unit := withExprs a b do
+def processNewDiseq (a b : Expr) : GoalM Unit := withExprs a b do
   let ea ← asACExpr a
   let lhs ← norm ea
   let eb ← asACExpr b
diff --git a/src/Lean/Meta/Tactic/Grind/AC/Internalize.lean b/src/Lean/Meta/Tactic/Grind/AC/Internalize.lean
index 2b69690a76..d6fca3b7f8 100644
--- a/src/Lean/Meta/Tactic/Grind/AC/Internalize.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC/Internalize.lean
@@ -22,8 +22,7 @@ partial def reify (e : Expr) : ACM Grind.AC.Expr := do
   else
     return .var (← mkVar e)
 
-@[export lean_grind_ac_internalize]
-def internalizeImpl (e : Expr) (parent? : Option Expr) : GoalM Unit := do
+def internalize (e : Expr) (parent? : Option Expr) : GoalM Unit := do
   unless (← getConfig).ac do return ()
   unless e.isApp && e.appFn!.isApp do return ()
   let op := e.appFn!.appFn!
@@ -39,6 +38,6 @@ def internalizeImpl (e : Expr) (parent? : Option Expr) : GoalM Unit := do
     }
     trace[grind.ac.internalize] "[{id}] {← e'.denoteExpr}"
     addTermOpId e
-    markAsACTerm e
+    acExt.markTerm e
 
 end Lean.Meta.Grind.AC
diff --git a/src/Lean/Meta/Tactic/Grind/AC/PP.lean b/src/Lean/Meta/Tactic/Grind/AC/PP.lean
index 2ff59c8cb6..41ad7b0b1e 100644
--- a/src/Lean/Meta/Tactic/Grind/AC/PP.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC/PP.lean
@@ -53,7 +53,7 @@ private def ppStruct? : M (Option MessageData) := do
 
 def pp? (goal : Goal) : MetaM (Option MessageData) := do
   let mut msgs := #[]
-  for struct in goal.ac.structs do
+  for struct in (← acExt.getStateCore goal).structs do
     let some msg ← ppStruct? |>.run' struct | pure ()
     msgs := msgs.push msg
   if msgs.isEmpty then
diff --git a/src/Lean/Meta/Tactic/Grind/AC/Types.lean b/src/Lean/Meta/Tactic/Grind/AC/Types.lean
index 781abb308e..994c21652d 100644
--- a/src/Lean/Meta/Tactic/Grind/AC/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC/Types.lean
@@ -8,9 +8,7 @@ prelude
 public import Init.Core
 public import Init.Grind.AC
 public import Std.Data.HashMap
-public import Lean.Expr
-public import Lean.Data.PersistentArray
-public import Lean.Meta.Tactic.Grind.ExprPtr
+public import Lean.Meta.Tactic.Grind.Types
 import Lean.Meta.Tactic.Grind.AC.Seq
 public section
 namespace Lean.Meta.Grind.AC
@@ -98,7 +96,7 @@ structure Struct where
   /-- Mapping from Lean expressions to their representations as `AC.Expr` -/
   denote           : PHashMap ExprPtr AC.Expr := {}
   /-- `denoteEntries` is `denote` as a `PArray` for deterministic traversal. -/
-  denoteEntries  : PArray (Expr × AC.Expr) := {}
+  denoteEntries    : PArray (Expr × AC.Expr) := {}
   /-- Equations to process. -/
   queue            : Queue := {}
   /-- Processed equations. -/
@@ -130,4 +128,6 @@ structure State where
   steps := 0
   deriving Inhabited
 
+builtin_initialize acExt : SolverExtension State ← registerSolverExtension (return {})
+
 end Lean.Meta.Grind.AC
diff --git a/src/Lean/Meta/Tactic/Grind/AC/Util.lean b/src/Lean/Meta/Tactic/Grind/AC/Util.lean
index 224b62823c..8b26259c4e 100644
--- a/src/Lean/Meta/Tactic/Grind/AC/Util.lean
+++ b/src/Lean/Meta/Tactic/Grind/AC/Util.lean
@@ -5,7 +5,7 @@ Authors: Leonardo de Moura
 -/
 module
 prelude
-public import Lean.Meta.Tactic.Grind.Types
+public import Lean.Meta.Tactic.Grind.AC.Types
 public import Lean.Meta.Tactic.Grind.ProveEq
 public import Lean.Meta.Tactic.Grind.SynthInstance
 public import Lean.Meta.Tactic.Grind.Arith.CommRing.RingId
@@ -15,10 +15,10 @@ namespace Lean.Meta.Grind.AC
 open Lean.Grind
 
 def get' : GoalM State := do
-  return (← get).ac
+  acExt.getState
 
 @[inline] def modify' (f : State → State) : GoalM Unit := do
-  modify fun s => { s with ac := f s.ac }
+  acExt.modifyState f
 
 def checkMaxSteps : GoalM Bool := do
   return (← get').steps >= (← getConfig).acSteps
@@ -115,7 +115,7 @@ def mkVar (e : Expr) : ACM AC.Var := do
     varMap     := s.varMap.insert { expr := e } var
   }
   addTermOpId e
-  markAsACTerm e
+  acExt.markTerm e
   return var
 
 def getOpId? (op : Expr) : GoalM (Option Nat) := do
diff --git a/src/Lean/Meta/Tactic/Grind/Check.lean b/src/Lean/Meta/Tactic/Grind/Check.lean
index 7fa00485b5..618744c669 100644
--- a/src/Lean/Meta/Tactic/Grind/Check.lean
+++ b/src/Lean/Meta/Tactic/Grind/Check.lean
@@ -7,12 +7,11 @@ module
 prelude
 public import Lean.Meta.Tactic.Grind.Types
 import Lean.Meta.Tactic.Grind.Arith.Main
-import Lean.Meta.Tactic.Grind.AC.Eq
 namespace Lean.Meta.Grind
 /--
 Checks whether satellite solvers can make progress (e.g., detect unsatisfiability, propagate equations, etc)
 -/
 public def check : GoalM Bool := do
-  Arith.check <||> AC.check
+  Arith.check
 
 namespace Lean.Meta.Grind
diff --git a/src/Lean/Meta/Tactic/Grind/Core.lean b/src/Lean/Meta/Tactic/Grind/Core.lean
index 284aaefb08..170a6184a3 100644
--- a/src/Lean/Meta/Tactic/Grind/Core.lean
+++ b/src/Lean/Meta/Tactic/Grind/Core.lean
@@ -215,31 +215,6 @@ def propagateLinarith : PendingTheoryPropagation → GoalM Unit
   | .diseqs ps => propagateLinarithDiseqs ps
   | _ => return ()
 
-/--
-Helper function for combining `ENode.ac?` fields and detecting what needs to be
-propagated to the ac module.
--/
-private def checkACEq (rhsRoot lhsRoot : ENode) : GoalM PendingTheoryPropagation := do
-  match lhsRoot.ac? with
-  | some lhs =>
-    if let some rhs := rhsRoot.ac? then
-      return .eq lhs rhs
-    else
-      -- We have to retrieve the node because other fields have been updated
-      let rhsRoot ← getENode rhsRoot.self
-      setENode rhsRoot.self { rhsRoot with ac? := lhs }
-      return .diseqs (← getParents rhsRoot.self)
-  | none =>
-    if rhsRoot.ac?.isSome then
-      return .diseqs (← getParents lhsRoot.self)
-    else
-      return .none
-
-def propagateAC : PendingTheoryPropagation → GoalM Unit
-  | .eq lhs rhs => AC.processNewEq lhs rhs
-  | .diseqs ps => propagateACDiseqs ps
-  | _ => return ()
-
 /--
 Tries to apply beta-reduction using the parent applications of the functions in `fns` with
 the lambda expressions in `lams`.
@@ -374,7 +349,7 @@ where
     let cutsatTodo ← checkCutsatEq rhsRoot lhsRoot
     let ringTodo ← checkCommRingEq rhsRoot lhsRoot
     let linarithTodo ← checkLinarithEq rhsRoot lhsRoot
-    let ACTodo ← checkACEq rhsRoot lhsRoot
+    let todo ← Solvers.mergeTerms rhsRoot lhsRoot
     resetParentsOf lhsRoot.self
     copyParentsTo parents rhsNode.root
     unless (← isInconsistent) do
@@ -389,7 +364,7 @@ where
       propagateCutsat cutsatTodo
       propagateCommRing ringTodo
       propagateLinarith linarithTodo
-      propagateAC ACTodo
+      todo.propagate
   updateRoots (lhs : Expr) (rootNew : Expr) : GoalM Unit := do
     let isFalseRoot ← isFalseExpr rootNew
     traverseEqc lhs fun n => do
diff --git a/src/Lean/Meta/Tactic/Grind/Internalize.lean b/src/Lean/Meta/Tactic/Grind/Internalize.lean
index 6b32061f97..d76c53ede2 100644
--- a/src/Lean/Meta/Tactic/Grind/Internalize.lean
+++ b/src/Lean/Meta/Tactic/Grind/Internalize.lean
@@ -21,8 +21,6 @@ import Lean.Meta.Tactic.Grind.MarkNestedSubsingletons
 public section
 namespace Lean.Meta.Grind
 
-@[extern "lean_grind_ac_internalize"] -- forward definition
-opaque AC.internalize (e : Expr) (parent? : Option Expr) : GoalM Unit
 @[extern "lean_grind_arith_internalize"] -- forward definition
 opaque Arith.internalize (e : Expr) (parent? : Option Expr) : GoalM Unit
 
@@ -366,7 +364,6 @@ private def tryEta (e : Expr) (generation : Nat) : GoalM Unit := do
 
 private def internalizeTheories (e : Expr) (parent? : Option Expr := none) : GoalM Unit := do
   Arith.internalize e parent?
-  AC.internalize e parent?
 
 @[export lean_grind_internalize]
 private partial def internalizeImpl (e : Expr) (generation : Nat) (parent? : Option Expr := none) : GoalM Unit := withIncRecDepth do
@@ -381,6 +378,7 @@ private partial def internalizeImpl (e : Expr) (generation : Nat) (parent? : Opt
     Otherwise, it will not be able to propagate that `a + 1 = 1` when `a = 0`
     -/
     internalizeTheories e parent?
+    Solvers.internalize e parent?
   else
     go
     propagateEtaStruct e generation
@@ -467,6 +465,7 @@ where
             registerParent e arg
         addCongrTable e
         internalizeTheories e parent?
+        Solvers.internalize e parent?
         propagateUp e
         propagateBetaForNewApp e
 
diff --git a/src/Lean/Meta/Tactic/Grind/Inv.lean b/src/Lean/Meta/Tactic/Grind/Inv.lean
index e599865af3..15cfa8f071 100644
--- a/src/Lean/Meta/Tactic/Grind/Inv.lean
+++ b/src/Lean/Meta/Tactic/Grind/Inv.lean
@@ -9,7 +9,6 @@ public import Lean.Meta.Tactic.Grind.Types
 import Lean.Meta.Tactic.Grind.Proof
 import Lean.Meta.Tactic.Grind.MatchCond
 import Lean.Meta.Tactic.Grind.Arith.Inv
-import Lean.Meta.Tactic.Grind.AC.Inv
 namespace Lean.Meta.Grind
 /-!
 Debugging support code for checking basic invariants.
@@ -126,7 +125,7 @@ public def checkInvariants (expensive := false) : GoalM Unit := do
     if expensive then
       checkPtrEqImpliesStructEq
     Arith.checkInvariants
-    AC.checkInvariants
+    Solvers.checkInvariants
   if expensive && grind.debug.proofs.get (← getOptions) then
     checkProofs
 
diff --git a/src/Lean/Meta/Tactic/Grind/Lookahead.lean b/src/Lean/Meta/Tactic/Grind/Lookahead.lean
index e9b757b57f..6d659e9a77 100644
--- a/src/Lean/Meta/Tactic/Grind/Lookahead.lean
+++ b/src/Lean/Meta/Tactic/Grind/Lookahead.lean
@@ -19,7 +19,7 @@ private partial def solve (generation : Nat) : SearchM Bool := withIncRecDepth d
     return false -- `splitNext` should have been configured to not create choice points
   if (← getGoal).inconsistent then
     return true
-  if (← intros' generation <||> assertAll <||> check <||> splitNext <||> ematch) then
+  if (← intros' generation <||> assertAll <||> check <||> Solvers.check <||> splitNext <||> ematch) then
     solve generation
   else
     return false
diff --git a/src/Lean/Meta/Tactic/Grind/Main.lean b/src/Lean/Meta/Tactic/Grind/Main.lean
index 054a6e7c1d..46cf108ddf 100644
--- a/src/Lean/Meta/Tactic/Grind/Main.lean
+++ b/src/Lean/Meta/Tactic/Grind/Main.lean
@@ -109,7 +109,8 @@ private def mkGoal (mvarId : MVarId) (params : Params) : GrindM Goal := do
   let thmMap := params.ematch
   let casesTypes := params.casesTypes
   let clean ← mkCleanState mvarId params
-  GoalM.run' { mvarId, ematch.thmMap := thmMap, split.casesTypes := casesTypes, clean } do
+  let sstates ← Solvers.mkInitialStates
+  GoalM.run' { mvarId, ematch.thmMap := thmMap, split.casesTypes := casesTypes, clean, sstates } do
     mkENodeCore falseExpr (interpreted := true) (ctor := false) (generation := 0)
     mkENodeCore trueExpr (interpreted := true) (ctor := false) (generation := 0)
     mkENodeCore btrueExpr (interpreted := false) (ctor := true) (generation := 0)
diff --git a/src/Lean/Meta/Tactic/Grind/PP.lean b/src/Lean/Meta/Tactic/Grind/PP.lean
index 6fb4490b07..e3d6b4fc61 100644
--- a/src/Lean/Meta/Tactic/Grind/PP.lean
+++ b/src/Lean/Meta/Tactic/Grind/PP.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 module
-
 prelude
 public import Init.Grind.Util
 public import Init.Grind.PP
@@ -14,7 +13,6 @@ public import Lean.Meta.Tactic.Grind.Arith.CommRing.PP
 public import Lean.Meta.Tactic.Grind.Arith.Linear.PP
 public import Lean.Meta.Tactic.Grind.AC.PP
 import Lean.PrettyPrinter
-
 public section
 
 namespace Lean.Meta.Grind
diff --git a/src/Lean/Meta/Tactic/Grind/Propagate.lean b/src/Lean/Meta/Tactic/Grind/Propagate.lean
index 35d3a0b7fc..56dabcfcae 100644
--- a/src/Lean/Meta/Tactic/Grind/Propagate.lean
+++ b/src/Lean/Meta/Tactic/Grind/Propagate.lean
@@ -186,7 +186,7 @@ builtin_grind_propagator propagateEqDown ↓Eq := fun e => do
     propagateCutsatDiseq lhs rhs
     propagateCommRingDiseq lhs rhs
     propagateLinarithDiseq lhs rhs
-    propagateACDiseq lhs rhs
+    Solvers.propagateDiseqs lhs rhs
     let thms ← getExtTheorems α
     if !thms.isEmpty then
       /-
diff --git a/src/Lean/Meta/Tactic/Grind/Solve.lean b/src/Lean/Meta/Tactic/Grind/Solve.lean
index 7427143fec..c59f78a476 100644
--- a/src/Lean/Meta/Tactic/Grind/Solve.lean
+++ b/src/Lean/Meta/Tactic/Grind/Solve.lean
@@ -51,8 +51,8 @@ where
           intros gen
         else
           break
-      if (← assertAll <||> check <||> ematch <||> lookahead <||> splitNext <||> Arith.Cutsat.mbtc
-          <||> Arith.Linear.mbtc <||> tryFallback) then
+      if (← assertAll <||> check <||> Solvers.check <||> ematch <||> lookahead <||> splitNext <||> Arith.Cutsat.mbtc
+          <||> Arith.Linear.mbtc <||> Solvers.mbtc <||> tryFallback) then
         continue
       return some (← getGoal) -- failed
     return none -- solved
diff --git a/src/Lean/Meta/Tactic/Grind/Types.lean b/src/Lean/Meta/Tactic/Grind/Types.lean
index a3b28b30f1..0ac9bef6b8 100644
--- a/src/Lean/Meta/Tactic/Grind/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/Types.lean
@@ -16,7 +16,6 @@ public import Lean.Meta.Tactic.Grind.AlphaShareCommon
 public import Lean.Meta.Tactic.Grind.Attr
 public import Lean.Meta.Tactic.Grind.ExtAttr
 public import Lean.Meta.Tactic.Grind.Arith.Types
-public import Lean.Meta.Tactic.Grind.AC.Types
 public import Lean.Meta.Tactic.Grind.EMatchTheorem
 meta import Lean.Parser.Do
 import Lean.Meta.Match.MatchEqsExt
@@ -379,6 +378,23 @@ private def expandReportIssueMacro (s : Syntax) : MacroM (TSyntax `doElem) := do
 macro "reportIssue!" s:(interpolatedStr(term) <|> term) : doElem => do
   expandReportIssueMacro s.raw
 
+/--
+Each E-node may have "solver terms" attached to them.
+Each term is an element of the equivalence class that the
+solver cares about. Each solver is responsible for marking the terms they care about.
+The `grind` core propagates equalities and disequalities to the theory solvers
+using these "marked" terms. The root of the equivalence class
+contains a list of representatives sorted by solver id. Note that many E-nodes
+do not have any solver terms attached to them.
+
+"Solver terms" are referenced as "theory variables" in the SMT literature.
+The SMT solver Z3 uses a similar representation.
+-/
+inductive SolverTerms where
+  | nil
+  | next (solverId : Nat) (e : Expr) (rest : SolverTerms)
+  deriving Inhabited, Repr
+
 /--
 Stores information for a node in the E-graph.
 Each internalized expression `e` has an `ENode` associated with it.
@@ -445,13 +461,8 @@ structure ENode where
   to the linarith module. Its implementation is similar to the `offset?` field.
   -/
   linarith? : Option Expr := none
-  /--
-  The `ac?` field is used to propagate equalities from the `grind` congruence closure module
-  to the ac module. Its implementation is similar to the `offset?` field.
-  -/
-  ac? : Option Expr := none
-  -- Remark: we expect to have builtin support for offset constraints, cutsat, comm ring, linarith, and ac.
-  -- If the number of satellite solvers increases, we may add support for an arbitrary solvers like done in Z3.
+  /-- Solver terms attached to this E-node. -/
+  sTerms : SolverTerms := .nil
   deriving Inhabited, Repr
 
 def ENode.isRoot (n : ENode) :=
@@ -773,8 +784,6 @@ structure Goal where
   split        : Split.State := {}
   /-- State of arithmetic procedures. -/
   arith        : Arith.State := {}
-  /-- State of the ac solver. -/
-  ac           : AC.State := {}
   /-- State of the clean name generator. -/
   clean        : Clean.State := {}
   /-- Solver states. -/
@@ -1171,14 +1180,10 @@ and invokes process `Arith.Cutsat.processNewDiseq e₁ e₂`
 def propagateCutsatDiseq (lhs rhs : Expr) : GoalM Unit := do
   let some lhs ← get? lhs | return ()
   let some rhs ← get? rhs | return ()
-  -- Recall that core can take care of disequalities of the form `1≠2`.
-  unless isNum lhs && isNum rhs do
-    Arith.Cutsat.processNewDiseq lhs rhs
+  Arith.Cutsat.processNewDiseq lhs rhs
 where
   get? (a : Expr) : GoalM (Option Expr) := do
     let root ← getRootENode a
-    if isNum root.self then
-      return some root.self
     return root.cutsat?
 
 /--
@@ -1295,53 +1300,6 @@ def markAsLinarithTerm (e : Expr) : GoalM Unit := do
     setENode root.self { root with linarith? := some e }
     propagateLinarithDiseqs (← getParents root.self)
 
-/--
-Notifies the ac module that `a = b` where
-`a` and `b` are terms that have been internalized by this module.
--/
-@[extern "lean_process_ac_eq"] -- forward definition
-opaque AC.processNewEq (a b : Expr) : GoalM Unit
-
-/--
-Notifies the ac module that `a ≠ b` where
-`a` and `b` are terms that have been internalized by this module.
--/
-@[extern "lean_process_ac_diseq"] -- forward definition
-opaque AC.processNewDiseq (a b : Expr) : GoalM Unit
-
-/--
-Given `lhs` and `rhs` that are known to be disequal, checks whether
-`lhs` and `rhs` have ac terms `e₁` and `e₂` attached to them,
-and invokes process `AC.processNewDiseq e₁ e₂`
--/
-def propagateACDiseq (lhs rhs : Expr) : GoalM Unit := do
-  let some lhs ← get? lhs | return ()
-  let some rhs ← get? rhs | return ()
-  AC.processNewDiseq lhs rhs
-where
-  get? (a : Expr) : GoalM (Option Expr) := do
-    return (← getRootENode a).ac?
-
-/--
-Traverses disequalities in `parents`, and propagate the ones relevant to the
-ac module.
--/
-def propagateACDiseqs (parents : ParentSet) : GoalM Unit := do
-  forEachDiseq parents propagateACDiseq
-
-/--
-Marks `e` as a term of interest to the ac module.
-If the root of `e`s equivalence class has already a term of interest,
-a new equality is propagated to the ac module.
--/
-def markAsACTerm (e : Expr) : GoalM Unit := do
-  let root ← getRootENode e
-  if let some e' := root.ac? then
-    AC.processNewEq e e'
-  else
-    setENode root.self { root with ac? := some e }
-    propagateACDiseqs (← getParents root.self)
-
 /-- Returns `true` is `e` is the root of its congruence class. -/
 def isCongrRoot (e : Expr) : GoalM Bool := do
   return (← getENode e).isCongrRoot
@@ -1659,29 +1617,52 @@ structure SolverExtension (σ : Type) where private mk ::
   checkInv    : GoalM Unit
   deriving Inhabited
 
-private builtin_initialize solverExtensionsRef : IO.Ref (Array (SolverExtension EnvExtensionState)) ← IO.mkRef #[]
+private builtin_initialize solverExtensionsRef : IO.Ref (Array (SolverExtension SolverExtensionState)) ← IO.mkRef #[]
 
 /--
+Registers a new solver extension for `grind`.
 Solver extensions can only be registered during initialization.
 Reason: We do not use any synchronization primitive to access `solverExtensionsRef`.
 -/
-def registerSolverExtension {σ : Type}
-    (mkInitial   : IO σ)
-    (internalize : Expr → (parent? : Option Expr) → GoalM Unit)
-    (newEq       : Expr → Expr → GoalM Unit)
-    (newDiseq    : Expr → Expr → GoalM Unit := fun _ _ => return ())
-    (mbtc        : GoalM Bool := return false)
-    (check       : GoalM Bool := return false)
-    (checkInv    : GoalM Unit := return ()) : IO (SolverExtension σ) := do
+def registerSolverExtension {σ : Type} (mkInitial   : IO σ) : IO (SolverExtension σ) := do
   unless (← initializing) do
     throw (IO.userError "failed to register `grind` solver, extensions can only be registered during initialization")
   let exts ← solverExtensionsRef.get
   let id := exts.size
-  let ext : SolverExtension σ := { id, mkInitial, internalize, newEq, newDiseq, check, checkInv, mbtc }
+  let ext : SolverExtension σ := {
+    id, mkInitial
+    internalize := fun _ _ => return ()
+    newEq := fun _ _ => return ()
+    newDiseq := fun _ _ => return ()
+    check := fun _ _ => return false
+    checkInv := fun _ _ => return ()
+    mbtc := fun _ _ => return false
+  }
   solverExtensionsRef.modify fun exts => exts.push (unsafe unsafeCast ext)
   return ext
 
-def mkInitialExtStates : IO (Array EnvExtensionState) := do
+/--
+Sets methods/handlers for solver extension `ext`.
+Solver extension methods can only be registered during initialization.
+Reason: We do not use any synchronization primitive to access `solverExtensionsRef`.
+-/
+def SolverExtension.setMethods (ext : SolverExtension σ)
+    (internalize : Expr → (parent? : Option Expr) → GoalM Unit := fun _ _  => return ())
+    (newEq       : Expr → Expr → GoalM Unit := fun _ _ => return ())
+    (newDiseq    : Expr → Expr → GoalM Unit := fun _ _ => return ())
+    (mbtc        : GoalM Bool := return false)
+    (check       : GoalM Bool := return false)
+    (checkInv    : GoalM Unit := return ()) : IO Unit := do
+  unless (← initializing) do
+    throw (IO.userError "failed to register `grind` solver, extensions can only be registered during initialization")
+  unless ext.id < (← solverExtensionsRef.get).size do
+    throw (IO.userError "failed to register `grind` solver methods, invalid solver id")
+  solverExtensionsRef.modify fun exts => exts.modify ext.id fun s => { s with
+    internalize, newEq, newDiseq, mbtc, check, checkInv
+  }
+
+/-- Returns initial state for registered solvers. -/
+def Solvers.mkInitialStates : IO (Array SolverExtensionState) := do
   let exts ← solverExtensionsRef.get
   exts.mapM fun ext => ext.mkInitial
 
@@ -1696,11 +1677,14 @@ private unsafe def SolverExtension.modifyStateImpl (ext : SolverExtension σ) (f
 @[implemented_by SolverExtension.modifyStateImpl]
 opaque SolverExtension.modifyState (ext : SolverExtension σ) (f : σ → σ) : GoalM Unit
 
-private unsafe def SolverExtension.getStateImpl (ext : SolverExtension σ) : GoalM σ := do
-  return unsafeCast (← get).sstates[ext.id]!
+private unsafe def SolverExtension.getStateCoreImpl (ext : SolverExtension σ) (goal : Goal) : IO σ :=
+  return unsafeCast goal.sstates[ext.id]!
 
-@[implemented_by SolverExtension.getStateImpl]
-opaque SolverExtension.getState (ext : SolverExtension σ) : GoalM σ
+@[implemented_by SolverExtension.getStateCoreImpl]
+opaque SolverExtension.getStateCore (ext : SolverExtension σ) (goal : Goal) : IO σ
+
+def SolverExtension.getState (ext : SolverExtension σ) : GoalM σ := do
+  ext.getStateCore (← get)
 
 /-- Internalizes given expression in all registered solvers. -/
 def Solvers.internalize (e : Expr) (parent? : Option Expr) : GoalM Unit := do
@@ -1726,4 +1710,125 @@ def Solvers.mbtc : GoalM Bool := do
       result := true
   return result
 
+/--
+Given a new disequality `lhs ≠ rhs`, propagates it to relevant theories.
+-/
+def Solvers.propagateDiseqs (lhs rhs : Expr) : GoalM Unit := do
+  go (← getRootENode lhs).sTerms (← getRootENode rhs).sTerms
+where
+  go (lhsTerms rhsTerms : SolverTerms) : GoalM Unit := do
+    match lhsTerms, rhsTerms with
+    | .nil, _ => return ()
+    | _, .nil => return ()
+    | .next id₁ lhs lhsTerms, .next id₂ rhs rhsTerms =>
+      if id₁ == id₂ then
+        (← solverExtensionsRef.get)[id₁]!.newDiseq lhs rhs
+        go lhsTerms rhsTerms
+      else if id₁ < id₂ then
+        go lhsTerms (.next id₂ rhs rhsTerms)
+      else
+        go (.next id₁ lhs lhsTerms) rhsTerms
+
+private def propagateDiseqOf (id : Nat) (lhs rhs : Expr) : GoalM Unit := do
+  visitLhs (← getRootENode lhs).sTerms
+where
+  visitLhs (sTerms : SolverTerms) : GoalM Unit := do
+    match sTerms with
+    | .nil => return ()
+    | .next id' e sTerms =>
+      if id == id' then
+        visitRhs e (← getRootENode rhs).sTerms
+      else if id < id' then
+        return ()
+      else
+        visitLhs sTerms
+
+  visitRhs (lhsTerm : Expr) (sTerms : SolverTerms) : GoalM Unit := do
+    match sTerms with
+    | .nil => return ()
+    | .next id' e sTerms =>
+      if id == id' then
+        let rhsTerm := e
+        (← solverExtensionsRef.get)[id]!.newDiseq lhsTerm rhsTerm
+      else if id < id' then
+        return ()
+      else
+        visitRhs lhsTerm sTerms
+
+def isSameSolverTerms (a b : SolverTerms) : Bool :=
+  unsafe ptrEq a b
+
+def SolverExtension.markTerm (ext : SolverExtension σ) (e : Expr) : GoalM Unit := do
+  let root ← getRootENode e
+  let id := ext.id
+  let rec go (sTerms : SolverTerms) : GoalM SolverTerms := do
+    match sTerms with
+    | .nil => return .next id e .nil
+    | .next id' e' sTerms' =>
+      if id == id' then
+        (← solverExtensionsRef.get)[id]!.newEq e e'
+        return sTerms
+      else if id < id' then
+        return .next id e sTerms
+      else
+        let sTermsNew ← go sTerms'
+        if isSameSolverTerms sTermsNew sTerms' then
+          return sTerms
+        else
+          return .next id' e' sTermsNew
+  let sTermsNew ← go root.sTerms
+  unless isSameSolverTerms sTermsNew root.sTerms do
+    setENode root.self { root with sTerms := sTermsNew }
+    forEachDiseq (← getParents root.self) (propagateDiseqOf id)
+
+private inductive PendingSolverPropagationsData where
+  | nil
+  | eq (solverId : Nat) (lhs rhs : Expr) (rest : PendingSolverPropagationsData)
+  | diseqs (solverId : Nat) (ps : ParentSet) (rest : PendingSolverPropagationsData)
+
+structure PendingSolverPropagations where private mk ::
+  private data : PendingSolverPropagationsData
+
+def Solvers.mergeTerms (rhsRoot lhsRoot : ENode) : GoalM PendingSolverPropagations := do
+  let (sTerms, data) ← go rhsRoot.sTerms lhsRoot.sTerms
+  unless sTerms matches .nil do
+    -- We have to retrieve the node because other fields have been updated
+    let rhsRoot ← getENode rhsRoot.self
+    setENode rhsRoot.self { rhsRoot with sTerms }
+  return { data }
+where
+  toPendingDiseqs (sTerms : SolverTerms) (ps : ParentSet) : PendingSolverPropagationsData :=
+    match sTerms with
+    | .nil => .nil
+    | .next id _ sTerms => .diseqs id ps (toPendingDiseqs sTerms ps)
+
+  go (rhsTerms : SolverTerms) (lhsTerms : SolverTerms) : GoalM (SolverTerms × PendingSolverPropagationsData) := do
+    match rhsTerms, lhsTerms with
+    | .nil,     .nil     => return (.nil, .nil)
+    | .nil,     .next .. => return (lhsTerms, toPendingDiseqs lhsTerms (← getParents rhsRoot.self))
+    | .next .., .nil     => return (rhsTerms, toPendingDiseqs rhsTerms (← getParents lhsRoot.self))
+    | .next id₁ rhs rhsTerms, .next id₂ lhs lhsTerms =>
+      if id₁ == id₂ then
+        let (s, p) ← go rhsTerms lhsTerms
+        return (.next id₁ rhs s, .eq id₁ lhs rhs p)
+      else if id₁ < id₂ then
+        let (s, p) ← go rhsTerms (.next id₂ lhs lhsTerms)
+        return (.next id₁ rhs s, .diseqs id₁ (← getParents lhsRoot.self) p)
+      else
+        let (s, p) ← go (.next id₁ rhs rhsTerms) lhsTerms
+        return (.next id₂ lhs s, .diseqs id₂ (← getParents rhsRoot.self) p)
+
+def PendingSolverPropagations.propagate (p : PendingSolverPropagations) : GoalM Unit := do
+  go p.data
+where
+  go (p : PendingSolverPropagationsData) : GoalM Unit := do
+    match p with
+    | .nil => return ()
+    | .eq solverId lhs rhs rest =>
+      (← solverExtensionsRef.get)[solverId]!.newEq lhs rhs
+      go rest
+    | .diseqs solverId parentSet rest =>
+      forEachDiseq parentSet (propagateDiseqOf solverId)
+      go rest
+
 end Lean.Meta.Grind
diff --git a/stage0/src/stdlib_flags.h b/stage0/src/stdlib_flags.h
index 79a0e58edd..ad491b0de1 100644
--- a/stage0/src/stdlib_flags.h
+++ b/stage0/src/stdlib_flags.h
@@ -1,3 +1,4 @@
+// update me!
 #include "util/options.h"
 
 namespace lean {