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 +475,13 @@
     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.
+  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 +940,15 @@
   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 +1786,46 @@
   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
+/// variables 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.
+void FlatAffineConstraints::removeRedundantLocalVars() {
+  bool change = true;
+  while (change) {
+    change = false;
+    for (int64_t i = 0; i < getNumEqualities(); ++i) {
+      bool foundOne = false;
+      unsigned eliminateVar;
+      for (unsigned j = getNumDimAndSymbolIds(), f = getNumIds(); j < f; ++j) {
+        if (std::abs(atEq(i, j)) == 1) {
+          foundOne = true;
+          eliminateVar = j;
+          break;
+        }
+      }
+
+      if (!foundOne)
+        continue;
+
+      change = true;
+      // Use equality to simplify other constraints
+      for (unsigned j = 0, f = getNumEqualities(); j < f; ++j)
+        eliminateFromConstraint(this, j, i, eliminateVar, eliminateVar,
+                                /*isEq=*/true);
+      for (unsigned j = 0, f = getNumInequalities(); j < f; ++j)
+        eliminateFromConstraint(this, j, i, eliminateVar, eliminateVar,
+                                /*isEq=*/false);
+      removeId(eliminateVar);
+      removeEquality(i);
+      --i;
+      normalizeConstraintsByGCD();
+      break;
+    }
+  }
+}
+
 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 +769,13 @@
   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());
+}
+
 } // namespace mlir