diff --git a/paddle/cinn/common/integer_set.cc b/paddle/cinn/common/integer_set.cc
index 762c273caef7c5..1887238c2eb4a8 100644
--- a/paddle/cinn/common/integer_set.cc
+++ b/paddle/cinn/common/integer_set.cc
@@ -13,6 +13,8 @@
 // limitations under the License.
 
 #include "paddle/cinn/common/integer_set.h"
+
+#include "paddle/cinn/common/arithmatic.h"
 #include "paddle/cinn/ir/ir_mutator.h"
 #include "paddle/cinn/ir/op/ir_operators.h"
 #include "paddle/cinn/ir/utils/ir_copy.h"
@@ -164,11 +166,115 @@ std::optional<bool> SymbolicExprAnalyzer::ProveLT(const ir::Expr& lhs,
   return ProveGT(rhs, lhs);
 }
 
+// Tell whether lhs can be divisible by rhs, lhs must be a pure math expression
+// and rhs must be a var
+std::optional<bool> SymbolicExprAnalyzer::ProveDivisible(
+    const ir::Expr& lhs, const ir::Expr& rhs) const {
+  CHECK(rhs.is_var()) << "Rhs in ProveDivisible must be a var temporarily!\n";
+  CHECK(lhs.defined());
+  CHECK(rhs.defined());
+  CHECK(cinn::common::IsPureMath(lhs));
+
+  ir::Expr lhs_copy = ir::ir_utils::IRCopy(lhs);
+  if (cinn::common::is_zero(lhs_copy)) return true;
+
+  auto OptionalAnd = [](const std::optional<bool>& lhs,
+                        const std::optional<bool>& rhs) -> std::optional<bool> {
+    if (lhs.has_value() && rhs.has_value()) {
+      return lhs.value() && rhs.value();
+    } else {
+      return std::nullopt;
+    }
+  };
+  auto OptionalOr = [](const std::optional<bool>& lhs,
+                       const std::optional<bool>& rhs) -> std::optional<bool> {
+    if (lhs.has_value() && rhs.has_value()) {
+      return lhs.value() || rhs.value();
+    } else if ((!lhs.has_value()) && (!rhs.has_value())) {
+      return std::nullopt;
+    } else if (lhs.has_value() && (!rhs.has_value())) {
+      return lhs.value() ? std::optional<bool>(lhs.value())
+                         : std::optional<bool>(std::nullopt);
+    } else {
+      return rhs.value() ? std::optional<bool>(rhs.value())
+                         : std::optional<bool>(std::nullopt);
+    }
+  };
+
+  std::vector<ir::Expr> ops{};
+  std::optional<bool> res = std::nullopt;
+  ir::Expr zero(0);
+  ir::Expr tmp_expr;
+
+  auto is_ge = ProveGE(lhs, rhs);
+
+  switch (lhs.node_type()) {
+    case cinn::ir::IrNodeTy::_Var_:
+      return ProveEQ(lhs, rhs);
+    case cinn::ir::IrNodeTy::IntImm:
+      return false;
+    case cinn::ir::IrNodeTy::Sum:
+      res = true;
+      ops = lhs.As<ir::Sum>()->operands();
+      CHECK(!ops.empty());
+      std::for_each(ops.begin(), ops.end(), [&](const ir::Expr& expr) {
+        res = OptionalAnd(res, this->ProveDivisible(expr, rhs));
+      });
+      res = OptionalAnd(res, is_ge);
+      return res;
+    case cinn::ir::IrNodeTy::Product:
+      res = false;
+      ops = lhs.As<ir::Product>()->operands();
+      CHECK(!ops.empty());
+      std::for_each(ops.begin(), ops.end(), [&](const ir::Expr& expr) {
+        res = OptionalOr(res, this->ProveDivisible(expr, rhs));
+        if (res.has_value() && res.value()) return;
+      });
+      res = OptionalAnd(res, is_ge);
+      return res;
+    case cinn::ir::IrNodeTy::FracOp:
+      tmp_expr = cinn::common::AutoSimplify(lhs);
+      if (tmp_expr.node_type() == cinn::ir::IrNodeTy::FracOp)
+        return std::nullopt;
+      return OptionalAnd(ProveDivisible(tmp_expr, rhs), is_ge);
+    case cinn::ir::IrNodeTy::FloatImm:
+      return false;
+    case cinn::ir::IrNodeTy::Add:
+      return OptionalAnd(
+          OptionalAnd(ProveDivisible(lhs.As<ir::Add>()->a(), rhs),
+                      ProveDivisible(lhs.As<ir::Add>()->b(), rhs)),
+          is_ge);
+    case cinn::ir::IrNodeTy::Sub:
+      return OptionalAnd(
+          OptionalAnd(ProveDivisible(lhs.As<ir::Sub>()->a(), rhs),
+                      ProveDivisible(lhs.As<ir::Sub>()->b(), rhs)),
+          is_ge);
+    case cinn::ir::IrNodeTy::Div:
+      tmp_expr = cinn::common::AutoSimplify(lhs);
+      if (tmp_expr.node_type() == cinn::ir::IrNodeTy::Div) return std::nullopt;
+      return OptionalAnd(ProveDivisible(tmp_expr, rhs), is_ge);
+    case cinn::ir::IrNodeTy::Mul:
+      return OptionalAnd(
+          OptionalOr(ProveDivisible(lhs.As<ir::Mul>()->a(), rhs),
+                     ProveDivisible(lhs.As<ir::Mul>()->b(), rhs)),
+          is_ge);
+    case cinn::ir::IrNodeTy::Mod:
+      return false;
+    case cinn::ir::IrNodeTy::Minus:
+      return ProveDivisible(lhs.As<ir::Minus>()->v(), rhs);
+    default:
+      LOG(FATAL) << "Not supported yet!";
+      break;
+  }
+}
+
 class BoundReplacer : public ir::IRMutator<> {
  public:
   explicit BoundReplacer(const cas_intervals_t& var_intervals,
                          bool is_lower_bound)
-      : var_intervals_(var_intervals), sign_(is_lower_bound) {}
+      : var_intervals_(var_intervals),
+        sign_(is_lower_bound),
+        var_visited_({}) {}
 
   void operator()(ir::Expr* expr) { IRMutator::Visit(expr, expr); }
 
@@ -183,10 +289,16 @@ class BoundReplacer : public ir::IRMutator<> {
       upper_bound =
           interval.e_r.defined() ? interval.e_r : ir::Expr(interval.r);
     }
-    if (sign_) {
-      *op = ir::ir_utils::IRCopy(lower_bound);
+    if (!var_visited_.count(var->name)) {
+      if (sign_) {
+        *op = ir::ir_utils::IRCopy(lower_bound);
+        var_visited_.insert({var->name, lower_bound});
+      } else {
+        *op = ir::ir_utils::IRCopy(upper_bound);
+        var_visited_.insert({var->name, upper_bound});
+      }
     } else {
-      *op = ir::ir_utils::IRCopy(upper_bound);
+      *op = ir::ir_utils::IRCopy(var_visited_.at(var->name));
     }
   }
 
@@ -248,6 +360,7 @@ class BoundReplacer : public ir::IRMutator<> {
 
  private:
   const cas_intervals_t& var_intervals_;
+  std::unordered_map<std::string, ir::Expr> var_visited_;
   // Determine replacing with upper or lower bound,
   // True means lower bound and False means upper bound.
   bool sign_;
diff --git a/paddle/cinn/common/integer_set.h b/paddle/cinn/common/integer_set.h
index e0f23da2e744f8..6d095b12083f11 100644
--- a/paddle/cinn/common/integer_set.h
+++ b/paddle/cinn/common/integer_set.h
@@ -41,6 +41,8 @@ class SymbolicExprAnalyzer {
   std::optional<bool> ProveLE(const ir::Expr& lhs, const ir::Expr& rhs) const;
   std::optional<bool> ProveGT(const ir::Expr& lhs, const ir::Expr& rhs) const;
   std::optional<bool> ProveLT(const ir::Expr& lhs, const ir::Expr& rhs) const;
+  std::optional<bool> ProveDivisible(const ir::Expr& lhs,
+                                     const ir::Expr& rhs) const;
 
   ir::Expr LowerBound(const ir::Expr& expr) const;
   ir::Expr UpperBound(const ir::Expr& expr) const;
diff --git a/paddle/cinn/common/integer_set_test.cc b/paddle/cinn/common/integer_set_test.cc
index 23406ec2f770ea..6d57f2dd0ed257 100644
--- a/paddle/cinn/common/integer_set_test.cc
+++ b/paddle/cinn/common/integer_set_test.cc
@@ -136,6 +136,50 @@ TEST_F(TestSymbolicExprAnalyzer, compare) {
                analyzer.Prove(e3 < e4).value());
 }
 
+TEST_F(TestSymbolicExprAnalyzer, Divisible) {
+  auto x = ir::Var(ir::Expr(1), ir::Expr(7), "x");
+  auto y = ir::Var(ir::Expr(1), ir::Expr(15), "y");
+  auto S = ir::Var(ir::Expr(16), ir::Expr(256), "S");
+
+  cas_intervals_t divisible_var_intervals = {
+      {"x", CasInterval(x->lower_bound, x->upper_bound)},
+      {"y", CasInterval(y->lower_bound, y->upper_bound)},
+      {"S", CasInterval(S->lower_bound, S->upper_bound)},
+  };
+  SymbolicExprAnalyzer divisible_analyzer{divisible_var_intervals};
+
+  // case 1
+  ir::Expr e1 = 4 * x + 2 * y * x;
+  ir::Expr e2 = x;
+  ir::Expr e3 = y;
+
+  EXPECT_TRUE(divisible_analyzer.ProveDivisible(e1, e2).value_or(false));
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e1, e3).value_or(false));
+
+  // case 2
+  ir::Expr e4 = y + y * x + 4 * y - x * y;
+
+  EXPECT_TRUE(divisible_analyzer.ProveDivisible(e4, e3).value_or(false));
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e4, e2).value_or(false));
+
+  // case 3
+  ir::Expr e5 = x / y + x + y;
+
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e5, e3).value_or(false));
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e5, e2).value_or(false));
+
+  // case 4
+  ir::Expr e6 = S * x / 4 + x * y;
+
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e6, e2).value_or(false));
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e6, e3).value_or(false));
+
+  ir::Expr e7 = 16 * x / 4 + x * y;
+
+  EXPECT_TRUE(divisible_analyzer.ProveDivisible(e7, e2).value_or(false));
+  EXPECT_FALSE(divisible_analyzer.ProveDivisible(e7, e3).value_or(false));
+}
+
 TEST(SingleIntervalIntSet, constant) {
   SingleIntervalIntSet empty_set(ir::Expr(0), ir::Expr(-1));
   SingleIntervalIntSet all_set(SymbolicExprLimit::negative_inf,