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
@@ -374,7 +374,6 @@
 
   /// Returns the constant bound for the pos^th identifier if there is one;
   /// None otherwise.
-  // TODO: Support EQ bounds.
   Optional<int64_t> getConstantBound(BoundType type, unsigned pos) const;
 
   /// Gets the lower and upper bound of the `offset` + `pos`th identifier
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
@@ -2828,11 +2828,22 @@
 
 Optional<int64_t> FlatAffineConstraints::getConstantBound(BoundType type,
                                                           unsigned pos) const {
-  assert(type != BoundType::EQ && "EQ not implemented");
   FlatAffineConstraints tmpCst(*this);
   if (type == BoundType::LB)
-    return tmpCst.computeConstantLowerOrUpperBound</*isLower=*/true>(pos);
-  return tmpCst.computeConstantLowerOrUpperBound</*isLower=*/false>(pos);
+    return FlatAffineConstraints(*this)
+        .computeConstantLowerOrUpperBound</*isLower=*/true>(pos);
+  if (type == BoundType::UB)
+    return FlatAffineConstraints(*this)
+        .computeConstantLowerOrUpperBound</*isLower=*/false>(pos);
+
+  assert(type == BoundType::EQ && "expected EQ");
+  Optional<int64_t> lb =
+      FlatAffineConstraints(*this)
+          .computeConstantLowerOrUpperBound</*isLower=*/true>(pos);
+  Optional<int64_t> ub =
+      FlatAffineConstraints(*this)
+          .computeConstantLowerOrUpperBound</*isLower=*/false>(pos);
+  return (lb && ub && *lb == *ub) ? Optional<int64_t>(*ub) : None;
 }
 
 // A simple (naive and conservative) check for hyper-rectangularity.
diff --git a/mlir/lib/Dialect/SCF/Transforms/LoopSpecialization.cpp b/mlir/lib/Dialect/SCF/Transforms/LoopSpecialization.cpp
--- a/mlir/lib/Dialect/SCF/Transforms/LoopSpecialization.cpp
+++ b/mlir/lib/Dialect/SCF/Transforms/LoopSpecialization.cpp
@@ -305,6 +305,16 @@
   AffineMap newMap = alignedBoundMap;
   SmallVector<Value> newOperands;
   unpackOptionalValues(constraints.getMaybeDimAndSymbolValues(), newOperands);
+  // If dims/symbols have known constant values, use those in order to simplify
+  // the affine map further.
+  for (int64_t i = 0; i < constraints.getNumDimAndSymbolIds(); ++i) {
+    // Skip unused operands and operands that are already constants.
+    if (!newOperands[i] || getConstantIntValue(newOperands[i]))
+      continue;
+    if (auto bound = constraints.getConstantBound(FlatAffineConstraints::EQ, i))
+      newOperands[i] =
+          rewriter.create<arith::ConstantIndexOp>(op->getLoc(), *bound);
+  }
   mlir::canonicalizeMapAndOperands(&newMap, &newOperands);
   rewriter.setInsertionPoint(op);
   rewriter.replaceOpWithNewOp<AffineApplyOp>(op, newMap, newOperands);
@@ -457,19 +467,30 @@
     if (ubInt)
       constraints.addBound(FlatAffineConstraints::EQ, dimUb, *ubInt);
 
-    // iv >= lb (equiv.: iv - lb >= 0)
+    // Lower bound: iv >= lb (equiv.: iv - lb >= 0)
     SmallVector<int64_t> ineqLb(constraints.getNumCols(), 0);
     ineqLb[dimIv] = 1;
     ineqLb[dimLb] = -1;
     constraints.addInequality(ineqLb);
 
-    // iv < lb + step * ((ub - lb - 1) floorDiv step) + 1
-    AffineExpr exprLb = lbInt ? rewriter.getAffineConstantExpr(*lbInt)
-                              : rewriter.getAffineDimExpr(dimLb);
-    AffineExpr exprUb = ubInt ? rewriter.getAffineConstantExpr(*ubInt)
-                              : rewriter.getAffineDimExpr(dimUb);
-    AffineExpr ivUb =
-        exprLb + 1 + (*stepInt * ((exprUb - exprLb - 1).floorDiv(*stepInt)));
+    // Upper bound
+    AffineExpr ivUb;
+    if (lbInt && ubInt && (*lbInt + *stepInt >= *ubInt)) {
+      // The loop has at most one iteration.
+      // iv < lb + 1
+      // TODO: Try to derive this constraint by simplifying the expression in
+      // the else-branch.
+      ivUb = rewriter.getAffineDimExpr(dimLb) + 1;
+    } else {
+      // The loop may have more than one iteration.
+      // iv < lb + step * ((ub - lb - 1) floorDiv step) + 1
+      AffineExpr exprLb = lbInt ? rewriter.getAffineConstantExpr(*lbInt)
+                                : rewriter.getAffineDimExpr(dimLb);
+      AffineExpr exprUb = ubInt ? rewriter.getAffineConstantExpr(*ubInt)
+                                : rewriter.getAffineDimExpr(dimUb);
+      ivUb =
+          exprLb + 1 + (*stepInt * ((exprUb - exprLb - 1).floorDiv(*stepInt)));
+    }
     auto map = AffineMap::get(
         /*dimCount=*/constraints.getNumDimIds(),
         /*symbolCount=*/constraints.getNumSymbolIds(), /*result=*/ivUb);
diff --git a/mlir/test/Dialect/SCF/for-loop-canonicalization.mlir b/mlir/test/Dialect/SCF/for-loop-canonicalization.mlir
--- a/mlir/test/Dialect/SCF/for-loop-canonicalization.mlir
+++ b/mlir/test/Dialect/SCF/for-loop-canonicalization.mlir
@@ -348,3 +348,22 @@
   %dim = tensor.dim %1, %c0 : tensor<?x?xf32>
   return %dim : index
 }
+
+// -----
+
+// CHECK-LABEL: func @one_trip_scf_for_canonicalize_min
+//       CHECK:   %[[C4:.*]] = arith.constant 4 : i64
+//       CHECK:   scf.for
+//       CHECK:     memref.store %[[C4]], %{{.*}}[] : memref<i64>
+func @one_trip_scf_for_canonicalize_min(%A : memref<i64>) {
+  %c0 = arith.constant 0 : index
+  %c2 = arith.constant 2 : index
+  %c4 = arith.constant 4 : index
+
+  scf.for %i = %c0 to %c4 step %c4 {
+    %1 = affine.min affine_map<(d0, d1)[] -> (4, d1 - d0)> (%i, %c4)
+    %2 = arith.index_cast %1: index to i64
+    memref.store %2, %A[]: memref<i64>
+  }
+  return
+}