diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Internalize.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Internalize.lean
index 9243b40532..e02c73d77b 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Internalize.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Internalize.lean
@@ -91,6 +91,7 @@ def internalize (e : Expr) (parent? : Option Expr) : GoalM Unit := do
   let some type := getType? e | return ()
   if isForbiddenParent parent? then return ()
   if let some structId ← getStructId? type then LinearM.run structId do
+    trace[grind.linarith.internalize] "{e}"
     setTermStructId e
     linearExt.markTerm e
     markVars e
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Model.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Model.lean
index 6ae8d4f287..ed00fb7cdb 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Model.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Model.lean
@@ -6,6 +6,7 @@ Authors: Leonardo de Moura
 module
 prelude
 public import Lean.Meta.Tactic.Grind.Arith.Linear.Types
+import Lean.Meta.Tactic.Grind.Arith.Linear.Reify
 import Lean.Meta.Tactic.Grind.Arith.ModelUtil
 import Init.Grind.Module.Envelope
 public section
@@ -28,6 +29,58 @@ private def toQ? (e : Expr) : Option Expr :=
   | Grind.IntModule.OfNatModule.toQ _ _ a => some a
   | _ => none
 
+/--
+Helper function for evaluating terms that have been processed by `internalize`, but
+we did not added them to constraints. See comment at `assignTerms`.
+-/
+private partial def evalTermAt? (e : Expr) (s : Struct) (model : Std.HashMap Expr Rat) : MetaM (Option Rat) := do
+  go e
+where
+  go (e : Expr) : OptionT MetaM Rat := do
+    if let some val := model.get? e then
+      return val
+    match_expr e with
+    | Neg.neg _ i a => if isNegInst s i then return - (← go a) else failure
+    | HAdd.hAdd _ _ _ i a b => if isAddInst s i then return (← go a) + (← go b) else failure
+    | HSub.hSub _ _ _ i a b => if isSubInst s i then return (← go a) - (← go b) else failure
+    | HMul.hMul _ _ _ i a b => if isHomoMulInst s i then return (← go a) * (← go b) else failure
+    | HSMul.hSMul _ _ _ i a b =>
+      if isSMulIntInst s i then
+        let k ← getIntValue? a
+        return k * (← go b)
+      else if isSMulNatInst s i then
+        let k ← getNatValue? a
+        return k * (← go b)
+      else
+        failure
+    | Zero.zero _ i => if isZeroInst s i then return 0 else failure
+    | OfNat.ofNat _ n _ => let k ← getNatValue? n; return k
+    | _ => failure
+
+/--
+Assigns terms that do not have an assignment associated with them because they were used only as markers
+for communicating with the `grind` core.
+For example, suppose we have `a + b = c`. The `internalize` function marks `a + b`, `a`, and `b`
+with theory variables. Let's assume we also have `c = 2*b`. In this case, `internalize` marks `2*b`
+with a theory variable, and `b` is already marked. Then, when both equalities are asserted
+in the `grind` core, the `linarith` module is notified that `a + b = 2*b` is true, and it is then
+normalized as `a + -1*b = 0`. The terms `a` and `b` are assigned rational values by the search
+procedure, but `a + b` and `2*b` are not. This procedure assigns them using the model computed by the
+search procedure.
+
+**Note**: We should reconsider whether to add the equalities `「a+b」 = a + b` and `「2*b」 = 2*b`
+to force the search procedure to assign interpretations to these terms.
+-/
+private def assignTerms (goal : Goal) (structId : Nat) (model : Std.HashMap Expr Rat) : MetaM (Std.HashMap Expr Rat) := do
+  let mut model := model
+  let s ← linearExt.getStateCore goal
+  let struct := s.structs[structId]!
+  for (e, structId') in s.exprToStructIdEntries do
+    if structId == structId' && !model.contains e then
+      if let some v ← evalTermAt? e struct model then
+        model := assignEqc goal e v model
+  return model
+
 /--
 Construct a model that satisfies all constraints in the linarith model for the structure with id `structId`.
 It also assigns values to (integer) terms that have not been internalized by the linarith model.
@@ -42,6 +95,7 @@ def mkModel (goal : Goal) (structId : Nat) : MetaM (Array (Expr × Rat)) := do
     if (← hasType s.type node) then
       if let some v := getAssignment? s node.self then
         model := assignEqc goal node.self v model
+  model ← assignTerms goal structId model
   -- Assign `toQ a` terms
   for e in goal.exprs do
     let node ← goal.getENode e
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Search.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Search.lean
index aeaff83698..aaf7182f03 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Search.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Search.lean
@@ -18,6 +18,9 @@ def IneqCnstr.throwUnexpected (c : IneqCnstr) : LinearM α := do
 def DiseqCnstr.throwUnexpected (c : DiseqCnstr) : LinearM α := do
   throwError "`grind linarith` internal error, unexpected{indentD (← c.denoteExpr)}"
 
+def EqCnstr.throwUnexpected (c :EqCnstr) : LinearM α := do
+  throwError "`grind linarith` internal error, unexpected{indentD (← c.denoteExpr)}"
+
 private def checkIsNextVar (x : Var) : LinearM Unit := do
   if x != (← getStruct).assignment.size then
     throwError "`grind linarith` internal error, assigning variable out of order"
@@ -246,6 +249,28 @@ private def resetDecisionStack : SearchM Unit := do
   let first := (← get).cases[0]!
   modifyStruct fun s => { first.saved with assignment := s.assignment }
 
+/-- Assign eliminated variables using `elimEqs` field. -/
+private def assignElimVars : SearchM Unit := do
+  if (← inconsistent) then return ()
+  go (← getStruct).elimStack
+where
+  go (xs : List Var) : SearchM Unit := do
+    match xs with
+    | [] => return ()
+    | x :: xs =>
+      let some c := (← getStruct).elimEqs[x]!
+        | throwError "`grind` internal error, eliminated variable must have equation associated with it"
+      -- `x` may not be the max variable
+      let a := c.p.coeff x
+      if a == 0 then c.throwUnexpected
+      -- ensure `x` is 0 when evaluating `c.p`
+      modifyStruct fun s => { s with assignment := s.assignment.set x 0 }
+      let some v ← c.p.eval? | c.throwUnexpected
+      let v := (-v) / a
+      traceAssignment x v
+      modifyStruct fun s => { s with assignment := s.assignment.set x v }
+      go xs
+
 /-- Search for an assignment/model for the linear constraints. -/
 private def searchAssignmentMain : SearchM Unit := do
   repeat
@@ -253,6 +278,7 @@ private def searchAssignmentMain : SearchM Unit := do
     checkSystem "linarith"
     if (← hasAssignment) then
       trace[grind.debug.linarith.search] "found assignment"
+      assignElimVars
       return ()
     if (← isInconsistent) then
       -- `grind` state is inconsistent
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
index c901bab209..1240e82e62 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
@@ -245,6 +245,8 @@ structure State where
   typeIdOf : PHashMap ExprPtr (Option Nat) := {}
   /- Mapping from expressions/terms to their structure ids. -/
   exprToStructId : PHashMap ExprPtr Nat := {}
+  /-- `exprToStructId` content as an array for traversal. -/
+  exprToStructIdEntries : PArray (Expr × Nat) := {}
   /--
   Some types are unordered rings, so we do not process them in `linarith`.
   When such types are detected in `getStructId?`, we add them to the set
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Util.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Util.lean
index 87c470928c..a9237e6814 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Util.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Util.lean
@@ -40,7 +40,10 @@ def setTermStructId (e : Expr) : LinearM Unit := do
     unless structId' == structId do
       reportIssue! "expression in two different structure in linarith module{indentExpr e}"
     return ()
-  modify' fun s => { s with exprToStructId := s.exprToStructId.insert { expr := e } structId }
+  modify' fun s => { s with
+    exprToStructId := s.exprToStructId.insert { expr := e } structId
+    exprToStructIdEntries := s.exprToStructIdEntries.push (e, structId)
+  }
 
 def getNoNatDivInst : LinearM Expr := do
   let some inst := (← getStruct).noNatDivInst?
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/ModelUtil.lean b/src/Lean/Meta/Tactic/Grind/Arith/ModelUtil.lean
index da9f939c58..a7d3a6a3ea 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/ModelUtil.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/ModelUtil.lean
@@ -59,7 +59,7 @@ Returns `true` if `e` should be treated as an interpreted value by the arithmeti
 def isInterpretedTerm (e : Expr) : Bool :=
   isNatNum e || isIntNum e || e.isAppOf ``HAdd.hAdd || e.isAppOf ``HMul.hMul || e.isAppOf ``HSub.hSub || e.isAppOf ``HSMul.hSMul
   || e.isAppOf ``Neg.neg || e.isAppOf ``HDiv.hDiv || e.isAppOf ``HMod.hMod || e.isAppOf ``One.one || e.isAppOf ``Zero.zero
-  || e.isAppOf ``NatCast.natCast || e.isIte || e.isDIte || e.isAppOf ``OfNat.ofNat || e.isAppOf ``Grind.ToInt.toInt
+  || e.isAppOf ``Inv.inv || e.isAppOf ``NatCast.natCast || e.isIte || e.isDIte || e.isAppOf ``OfNat.ofNat || e.isAppOf ``Grind.ToInt.toInt
   || e.isAppOf ``Grind.IntModule.OfNatModule.toQ || e matches .lit (.natVal _)
 
 /--
diff --git a/tests/lean/run/grind_10500.lean b/tests/lean/run/grind_10500.lean
new file mode 100644
index 0000000000..091b26aadc
--- /dev/null
+++ b/tests/lean/run/grind_10500.lean
@@ -0,0 +1,80 @@
+open Lean Grind Std
+
+set_option warn.sorry false
+
+namespace Ex1
+variable [Field Q] [LT Q] [LE Q] [LawfulOrderLT Q] [IsLinearOrder Q] [OrderedRing Q]
+
+variable (a₁ a₂ a₃ a₄ a₅ : Q)
+variable (α L ν : Q)
+
+/--
+trace: [grind.linarith.model] a₁ := 0
+[grind.linarith.model] a₂ := 0
+[grind.linarith.model] a₃ := 0
+[grind.linarith.model] a₄ := 0
+[grind.linarith.model] a₅ := 0
+[grind.linarith.model] α := 0
+[grind.linarith.model] L := 0
+[grind.linarith.model] ν := 2
+-/
+#guard_msgs in
+set_option trace.grind.linarith.model true in
+theorem upper_bound
+    (hL : L = α) (hL1 : L ≤ 1)
+    (ha₁ : 0 ≤ a₁) (ha₂ : 0 ≤ a₂) (ha₃ : 0 ≤ a₃) (ha₄ : 0 ≤ a₄) (ha₅ : 0 ≤ a₅)
+    (hα : α = a₁ + a₂ + a₃ + a₄ + a₅) :
+    ν ≤ 9/10 := by
+  fail_if_success grind
+  sorry
+
+end Ex1
+
+namespace Ex2
+
+variable (a₁ a₂ a₃ a₄ a₅ : Rat)
+variable (α L ν : Rat)
+
+/--
+trace: [grind.linarith.model] a₁ := 0
+[grind.linarith.model] a₂ := 0
+[grind.linarith.model] a₃ := 0
+[grind.linarith.model] a₄ := 0
+[grind.linarith.model] a₅ := 0
+[grind.linarith.model] α := 0
+[grind.linarith.model] L := 0
+[grind.linarith.model] ν := 2
+-/
+#guard_msgs in
+set_option trace.grind.linarith.model true in
+theorem upper_bound
+    (hL : L = α) (hL1 : L ≤ 1)
+    (ha₁ : 0 ≤ a₁) (ha₂ : 0 ≤ a₂) (ha₃ : 0 ≤ a₃) (ha₄ : 0 ≤ a₄) (ha₅ : 0 ≤ a₅)
+    (hα : α = a₁ + a₂ + a₃ + a₄ + a₅) :
+    ν ≤ 9/10 := by
+  fail_if_success grind
+  sorry
+
+end Ex2
+
+/--
+trace: [grind.linarith.model] a := 0
+[grind.linarith.model] b := 0
+[grind.linarith.model] c := 0
+-/
+#guard_msgs in
+set_option trace.grind.linarith.model true in
+example [Field α] [LE α] [LT α] [Std.IsPreorder α] [OrderedRing α] (a b c : α) (h : a = b + c) : False := by
+  fail_if_success grind
+  sorry
+
+/--
+trace: [grind.linarith.model] a := 0
+[grind.linarith.model] b := 0
+[grind.linarith.model] c := 0
+-/
+#guard_msgs in
+set_option trace.grind.linarith.model true in
+example (a b c : Rat) (h : a = b + c) : False := by
+  fail_if_success grind
+  sorry