diff --git a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Proof.lean b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Proof.lean
index e3a4b2605e..689e0d58c8 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Proof.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Proof.lean
@@ -45,25 +45,30 @@ structure PreNullCert where
   Thus, we need to track a denominator to justify the proof step `div`.
   -/
   d  : Int := 1
+  deriving Inhabited
 
 def PreNullCert.unit (i : Nat) (n : Nat) : PreNullCert :=
   let qs := Array.replicate n (.num 0)
   let qs := qs.set! i (.num 1)
   { qs }
 
-def PreNullCert.mul (c : PreNullCert) (k : Int) (char? : Option Nat) : PreNullCert :=
-  if k == 1 then c
+def PreNullCert.div (c : PreNullCert) (k : Int) : RingM PreNullCert := do
+  return { c with d := c.d * k }
+
+def PreNullCert.mul (c : PreNullCert) (k : Int) : RingM PreNullCert := do
+  if k == 1 then
+    return c
   else
     let g := Int.gcd k c.d
     let k := k / g
     let d := c.d / g
     if k == 1 then
-      { c with d }
+      return { c with d }
     else
-      let qs := c.qs.map fun q => if q.isZero then q else q.mulConst' k char?
-      { qs, d }
+      let qs ← c.qs.mapM fun q => q.mulConstM k
+      return { qs, d }
 
-def PreNullCert.combine (k₁ : Int) (m₁ : Mon) (c₁ : PreNullCert) (k₂ : Int) (m₂ : Mon) (c₂ : PreNullCert) (char? : Option Nat) : PreNullCert := Id.run do
+def PreNullCert.combine (k₁ : Int) (m₁ : Mon) (c₁ : PreNullCert) (k₂ : Int) (m₂ : Mon) (c₂ : PreNullCert) : RingM PreNullCert := do
   let d₁    := c₁.d
   let d₂    := c₂.d
   let k₁_d₂ := k₁*d₂
@@ -79,17 +84,17 @@ def PreNullCert.combine (k₁ : Int) (m₁ : Mon) (c₁ : PreNullCert) (k₂ : I
   let mut qs : Vector Poly n := Vector.replicate n (.num 0)
   for h : i in [:n] do
     if h₁ : i < qs₁.size then
-      let q₁ := qs₁[i].mulMon' k₁ m₁ char?
+      let q₁ ← qs₁[i].mulMonM k₁ m₁
       if h₂ : i < qs₂.size then
-        let q₂ := qs₂[i].mulMon' k₂ m₂ char?
-        qs := qs.set i (q₁.combine' q₂ char?)
+        let q₂ ← qs₂[i].mulMonM k₂ m₂
+        qs := qs.set i (← q₁.combineM q₂)
       else
         qs := qs.set i q₁
     else
       have : i < n := h.upper
       have : qs₁.size = n ∨ qs₂.size = n := by simp +zetaDelta [Nat.max_def]; split <;> simp [*]
       have : i < qs₂.size := by omega
-      let q₂ := qs₂[i].mulMon' k₂ m₂ char?
+      let q₂ ← qs₂[i].mulMonM k₂ m₂
       qs := qs.set i q₂
   return { qs := qs.toArray, d }
 
@@ -101,9 +106,57 @@ structure NullCertHypothesis where
 structure ProofM.State where
   /-- Mapping from `EqCnstr` to `PreNullCert` -/
   cache       : Std.HashMap UInt64 PreNullCert := {}
-  hypToId     : Std.HashMap UInt64 Nat := {}
   hyps        : Array NullCertHypothesis := #[]
 
+abbrev ProofM := StateRefT ProofM.State RingM
+
+private abbrev caching (c : α) (k : ProofM PreNullCert) : ProofM PreNullCert := do
+  let addr := unsafe (ptrAddrUnsafe c).toUInt64 >>> 2
+  if let some h := (← get).cache[addr]? then
+    return h
+  else
+    let h ← k
+    modify fun s => { s with cache := s.cache.insert addr h }
+    return h
+
+partial def EqCnstr.toPreNullCert (c : EqCnstr) : ProofM PreNullCert := caching c do
+  match c.h with
+  | .core a b lhs rhs =>
+    let i := (← get).hyps.size
+    let h ← mkEqProof a b
+    modify fun s => { s with hyps := s.hyps.push { h, lhs, rhs } }
+    return PreNullCert.unit i (i+1)
+  | .superpose c₁ c₂ k₁ k₂ m₁ m₂ => (← c₁.toPreNullCert).combine k₁ m₁ k₂ m₂ (← c₂.toPreNullCert)
+  | .simp c₁ c₂ k₁ k₂ m => (← c₁.toPreNullCert).combine k₁ m k₂ .unit (← c₂.toPreNullCert)
+  | .mul k c => (← c.toPreNullCert).mul k
+  | .div k c => (← c.toPreNullCert).div k
+
+structure NullCertExt where
+  d   : Int
+  qhs : Array (Poly × NullCertHypothesis)
+
+def EqCnstr.mkNullCertExt (c : EqCnstr) : RingM NullCertExt := do
+  let (nc, s) ← c.toPreNullCert.run {}
+  return { d := nc.d, qhs := nc.qs.zip s.hyps }
+
+def NullCertExt.toPoly (nc : NullCertExt) : RingM Poly := do
+  let mut p : Poly := .num 0
+  for (q, h) in nc.qhs do
+    p ← p.combineM (← q.mulM (← (h.lhs.sub h.rhs).toPolyM))
+  return p
+
+def NullCertExt.check (c : EqCnstr) (nc : NullCertExt) : RingM Bool := do
+  let p₁ := c.p.mulConst' nc.d (← nonzeroChar?)
+  let p₂ ← nc.toPoly
+  return p₁ == p₂
+
+def setInconsistent (c : EqCnstr) : RingM Unit := do
+  trace_goal[grind.ring.assert.unsat] "{← c.denoteExpr}"
+  let nc ← c.mkNullCertExt
+  trace_goal[grind.ring.assert.unsat] "{nc.d}*({← c.p.denoteExpr}), {← (← nc.toPoly).denoteExpr}"
+  trace_goal[grind.ring.assert.unsat] "{nc.d}*({← c.p.denoteExpr}), {← nc.qhs.mapM fun (p, h) => return (← p.denoteExpr, ← h.lhs.denoteExpr, ← h.rhs.denoteExpr) }"
+  -- TODO
+
 private def mkLemmaPrefix (declName declNameC : Name) : RingM Expr := do
   let ring ← getRing
   let ctx ← toContextExpr
@@ -123,8 +176,5 @@ def setEqUnsat (k : Int) (a b : Expr) (ra rb : RingExpr) : RingM Unit := do
     h := mkApp h charInst
   closeGoal <| mkApp5 h (toExpr ra) (toExpr rb) (toExpr k) reflBoolTrue (← mkEqProof a b)
 
-def setInconsistent (c : EqCnstr) : RingM Unit := do
-  trace_goal[grind.ring.assert.unsat] "{← c.denoteExpr}"
-  -- TODO
 
 end Lean.Meta.Grind.Arith.CommRing
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Types.lean b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Types.lean
index 1ec4d0b046..d5bc0afbcc 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Types.lean
@@ -26,7 +26,7 @@ structure EqCnstr where
 
 inductive EqCnstrProof where
   | core (a b : Expr) (ra rb : RingExpr)
-  | superpose (c₁ c₂ : EqCnstr)
+  | superpose (c₁ c₂ : EqCnstr) (k₁ k₂ : Int) (m₁ m₂ : Mon)
   | simp (c₁ c₂ : EqCnstr) (k₁ k₂ : Int) (m : Mon)
   | mul (k : Int) (e : EqCnstr)
   | div (k : Int) (e : EqCnstr)
@@ -104,7 +104,7 @@ inductive SimpChain where
     ```
     If we have a commutative ring where
     ```
-    ∀ (k : Int) (a b : α), k ≠ 0 → (intCast k) * a = 0 → a = 0
+    ∀ (k : Int) (a : α), k ≠ 0 → (intCast k) * a = 0 → a = 0
     ```
     grind can deduce that `x+y+z = 0`
     -/
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Util.lean b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Util.lean
index a1295cda6c..4da0951f90 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Util.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/CommRing/Util.lean
@@ -5,6 +5,7 @@ Authors: Leonardo de Moura
 -/
 prelude
 import Lean.Meta.Tactic.Grind.Types
+import Lean.Meta.Tactic.Grind.Arith.CommRing.Poly
 
 namespace Lean.Meta.Grind.Arith.CommRing
 
@@ -77,9 +78,18 @@ Converts the given ring expression into a multivariate polynomial.
 If the ring has a nonzero characteristic, it is used during normalization.
 -/
 def _root_.Lean.Grind.CommRing.Expr.toPolyM (e : RingExpr) : RingM Poly := do
-  if let some c ← nonzeroChar? then
-    return e.toPolyC c
-  else
-    return e.toPoly
+  if let some c ← nonzeroChar? then return e.toPolyC c else return e.toPoly
+
+def _root_.Lean.Grind.CommRing.Poly.mulConstM (p : Poly) (k : Int) : RingM Poly :=
+  return p.mulConst' k (← nonzeroChar?)
+
+def _root_.Lean.Grind.CommRing.Poly.mulMonM (p : Poly) (k : Int) (m : Mon) : RingM Poly :=
+  return p.mulMon' k m (← nonzeroChar?)
+
+def _root_.Lean.Grind.CommRing.Poly.mulM (p₁ p₂ : Poly) : RingM Poly := do
+  if let some c ← nonzeroChar? then return p₁.mulC p₂ c else return p₁.mul p₂
+
+def _root_.Lean.Grind.CommRing.Poly.combineM (p₁ p₂ : Poly) : RingM Poly :=
+  return p₁.combine' p₂ (← nonzeroChar?)
 
 end Lean.Meta.Grind.Arith.CommRing