diff --git a/src/arith/rewrite_simplify.cc b/src/arith/rewrite_simplify.cc
index f1838f5a9099..6418f28cf87c 100644
--- a/src/arith/rewrite_simplify.cc
+++ b/src/arith/rewrite_simplify.cc
@@ -1096,24 +1096,26 @@ PrimExpr RewriteSimplifier::Impl::VisitExpr_(const MinNode* op) {
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
     TVM_TRY_REWRITE_IF(min(truncdiv(x + c1, c2) * c2, max(x, c2)), max(x, c2),
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value &&
-                           CanProveGreaterEqual(x.Eval(), 0));
+                           CanProveGreaterEqual(x.Eval(), 1));
 
     TVM_TRY_REWRITE_IF(min(x, truncdiv(x + c1, c2) * c2), x,
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
     TVM_TRY_REWRITE_IF(min(max(x, c2), truncdiv(x + c1, c2) * c2), max(x, c2),
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value &&
-                           CanProveGreaterEqual(x.Eval(), 0));
+                           CanProveGreaterEqual(x.Eval(), 1));
 
     // Divide up rounding: floor div
     TVM_TRY_REWRITE_IF(min(floordiv(x + c1, c2) * c2, x), x,
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
     TVM_TRY_REWRITE_IF(min(floordiv(x + c1, c2) * c2, max(x, c2)), max(x, c2),
-                       c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
+                       c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value &&
+                           CanProveGreaterEqual(x.Eval(), 1));
 
     TVM_TRY_REWRITE_IF(min(x, floordiv(x + c1, c2) * c2), x,
                        c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
     TVM_TRY_REWRITE_IF(min(max(x, c2), floordiv(x + c1, c2) * c2), max(x, c2),
-                       c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value);
+                       c2.Eval()->value > 0 && c1.Eval()->value + 1 == c2.Eval()->value &&
+                           CanProveGreaterEqual(x.Eval(), 1));
 
     TVM_TRY_REWRITE_IF(min(x, floordiv(x, c2) * c2), floordiv(x, c2) * c2, c2.Eval()->value > 0);
     TVM_TRY_REWRITE_IF(min(floordiv(x, c2) * c2, x), floordiv(x, c2) * c2, c2.Eval()->value > 0);
diff --git a/tests/python/unittest/test_arith_rewrite_simplify.py b/tests/python/unittest/test_arith_rewrite_simplify.py
index 975af097c030..84340ec031c7 100644
--- a/tests/python/unittest/test_arith_rewrite_simplify.py
+++ b/tests/python/unittest/test_arith_rewrite_simplify.py
@@ -649,20 +649,23 @@ def test_min_index_simplify():
     # truc div
     ck.analyzer.update(x, tvm.arith.ConstIntBound(0, 1000))
     ck.verify(tvm.te.min(tdiv(x + 3, 4) * 4, x), x)
-    ck.verify(tvm.te.min(tdiv(x + 3, 4) * 4, tvm.te.max(x, 4)), tvm.te.max(x, 4))
     ck.verify(tvm.te.min(x, tdiv(x + 3, 4) * 4), x)
-    ck.verify(tvm.te.min(tvm.te.max(x, 4), tdiv(x + 3, 4) * 4), tvm.te.max(x, 4))
     ck.analyzer.update(x, tvm.arith.ConstIntBound(-1000, 1000), True)
     ck.verify(tvm.te.min(tdiv(x, 10), tdiv(y, 10)), tdiv(tvm.te.min(x, y), 10))
     ck.verify(tvm.te.min(tdiv(x, (-10)), tdiv(y, (-10))), tdiv(tvm.te.max(x, y), (-10)))
+    ck.analyzer.update(x, tvm.arith.ConstIntBound(1, 1000), True)
+    ck.verify(tvm.te.min(tdiv(x + 3, 4) * 4, tvm.te.max(x, 4)), tvm.te.max(x, 4))
+    ck.verify(tvm.te.min(tvm.te.max(x, 4), tdiv(x + 3, 4) * 4), tvm.te.max(x, 4))
 
     # floor div
     ck.analyzer.update(x, tvm.arith.ConstIntBound(-1000, 1000), True)
     ck.verify(tvm.te.min(fld(x + 3, 4) * 4, x), x)
-    ck.verify(tvm.te.min(fld(x + 3, 4) * 4, tvm.te.max(x, 4)), tvm.te.max(x, 4))
     ck.verify(tvm.te.min(x, fld(x + 3, 4) * 4), x)
     ck.verify(tvm.te.min(x, fld(x, 4) * 4), fld(x, 4) * 4)
+    ck.analyzer.update(x, tvm.arith.ConstIntBound(1, 1000), True)
+    ck.verify(tvm.te.min(fld(x + 3, 4) * 4, tvm.te.max(x, 4)), tvm.te.max(x, 4))
     ck.verify(tvm.te.min(tvm.te.max(x, 4), fld(x + 3, 4) * 4), tvm.te.max(x, 4))
+    ck.analyzer.update(x, tvm.arith.ConstIntBound(-1000, 1000), True)
     ck.verify(tvm.te.min(fld(x, 10), fld(y, 10)), fld(tvm.te.min(x, y), 10))
     ck.verify(tvm.te.min(fld(x, (-10)), fld(y, (-10))), fld(tvm.te.max(x, y), (-10)))