diff --git a/mlir/include/mlir/Analysis/AffineStructures.h b/mlir/include/mlir/Analysis/AffineStructures.h
--- a/mlir/include/mlir/Analysis/AffineStructures.h
+++ b/mlir/include/mlir/Analysis/AffineStructures.h
@@ -475,6 +475,14 @@
     return success(gaussianEliminateIds(position, position + 1) == 1);
   }
 
+  /// Removes local variables using equalities. Each equality is checked if it
+  /// can be reduced to the form: `e = affine-expr`, where `e` is a local
+  /// variable and `affine-expr` is an affine expression not containing `e`.
+  /// If an equality satisfies this form, the local variable is replaced in
+  /// each constraint and then removed. The equality used to replace this local
+  /// variable is also removed.
+  void removeRedundantLocalVars();
+
   /// Eliminates identifiers from equality and inequality constraints
   /// in column range [posStart, posLimit).
   /// Returns the number of variables eliminated.
diff --git a/mlir/lib/Analysis/AffineStructures.cpp b/mlir/lib/Analysis/AffineStructures.cpp
--- a/mlir/lib/Analysis/AffineStructures.cpp
+++ b/mlir/lib/Analysis/AffineStructures.cpp
@@ -940,9 +940,14 @@
   if (isEmptyByGCDTest() || hasInvalidConstraint())
     return true;
 
-  // First, eliminate as many identifiers as possible using Gaussian
-  // elimination.
   FlatAffineConstraints tmpCst(*this);
+
+  // First, eliminate as many local variables as possible using equalities.
+  tmpCst.removeRedundantLocalVars();
+  if (tmpCst.isEmptyByGCDTest() || tmpCst.hasInvalidConstraint())
+    return true;
+
+  // Eliminate as many identifiers as possible using Gaussian elimination.
   unsigned currentPos = 0;
   while (currentPos < tmpCst.getNumIds()) {
     tmpCst.gaussianEliminateIds(currentPos, tmpCst.getNumIds());
@@ -1780,6 +1785,54 @@
   equalities.resizeVertically(pos);
 }
 
+/// Removes local variables using equalities. Each equality is checked if it
+/// can be reduced to the form: `e = affine-expr`, where `e` is a local
+/// variable and `affine-expr` is an affine expression not containing `e`.
+/// If an equality satisfies this form, the local variable is replaced in
+/// each constraint and then removed. The equality used to replace this local
+/// variable is also removed.
+void FlatAffineConstraints::removeRedundantLocalVars() {
+  // Normalize the equality constraints to reduce coefficients of local
+  // variables to 1 wherever possible.
+  for (unsigned i = 0, e = getNumEqualities(); i < e; ++i)
+    normalizeConstraintByGCD</*isEq=*/true>(this, i);
+
+  while (true) {
+    unsigned i, e, j, f;
+    for (i = 0, e = getNumEqualities(); i < e; ++i) {
+      // Find a local variable to eliminate using ith equality.
+      for (j = getNumDimAndSymbolIds(), f = getNumIds(); j < f; ++j)
+        if (std::abs(atEq(i, j)) == 1)
+          break;
+
+      // Local variable can be eliminated using ith equality.
+      if (j < f)
+        break;
+    }
+
+    // No equality can be used to eliminate a local variable.
+    if (i == e)
+      break;
+
+    // Use the ith equality to simplify other equalities. If any changes
+    // are made to an equality constraint, it is normalized by GCD.
+    for (unsigned k = 0, t = getNumEqualities(); k < t; ++k) {
+      if (atEq(k, j) != 0) {
+        eliminateFromConstraint(this, k, i, j, j, /*isEq=*/true);
+        normalizeConstraintByGCD</*isEq=*/true>(this, k);
+      }
+    }
+
+    // Use the ith equality to simplify inequalities.
+    for (unsigned k = 0, t = getNumInequalities(); k < t; ++k)
+      eliminateFromConstraint(this, k, i, j, j, /*isEq=*/false);
+
+    // Remove the ith equality and the found local variable.
+    removeId(j);
+    removeEquality(i);
+  }
+}
+
 std::pair<AffineMap, AffineMap> FlatAffineConstraints::getLowerAndUpperBound(
     unsigned pos, unsigned offset, unsigned num, unsigned symStartPos,
     ArrayRef<AffineExpr> localExprs, MLIRContext *context) const {
diff --git a/mlir/unittests/Analysis/AffineStructuresTest.cpp b/mlir/unittests/Analysis/AffineStructuresTest.cpp
--- a/mlir/unittests/Analysis/AffineStructuresTest.cpp
+++ b/mlir/unittests/Analysis/AffineStructuresTest.cpp
@@ -769,4 +769,30 @@
   EXPECT_THAT(fac.getInequality(0), testing::ElementsAre(12, 20, 40));
 }
 
+TEST(FlatAffineConstraintsTest, simplifyLocalsTest) {
+  // (x) : (exists y: 2x + y = 1 and y = 2).
+  FlatAffineConstraints fac(1, 0, 1);
+  fac.addEquality({2, 1, -1});
+  fac.addEquality({0, 1, -2});
+
+  EXPECT_TRUE(fac.isEmpty());
+
+  // (x) : (exists y, z, w: 3x + y = 1 and 2y = z and 3y = w and z = w).
+  FlatAffineConstraints fac2(1, 0, 3);
+  fac2.addEquality({3, 1, 0, 0, -1});
+  fac2.addEquality({0, 2, -1, 0, 0});
+  fac2.addEquality({0, 3, 0, -1, 0});
+  fac2.addEquality({0, 0, 1, -1, 0});
+
+  EXPECT_TRUE(fac2.isEmpty());
+
+  // (x) : (exists y: x >= y + 1 and 2x + y = 0 and y >= -1).
+  FlatAffineConstraints fac3(1, 0, 1);
+  fac3.addInequality({1, -1, -1});
+  fac3.addInequality({0, 1, 1});
+  fac3.addEquality({2, 1, 0});
+
+  EXPECT_TRUE(fac3.isEmpty());
+}
+
 } // namespace mlir