diff --git a/lib/Dialect/Poly/BUILD b/lib/Dialect/Poly/BUILD
index c1c36f7..9dfb6d4 100644
--- a/lib/Dialect/Poly/BUILD
+++ b/lib/Dialect/Poly/BUILD
@@ -13,9 +13,9 @@ td_library(
     ],
     includes = ["@heir//include"],
     deps = [
-        # the base mlir target for defining operations and dialects in tablegen
-        "@llvm-project//mlir:OpBaseTdFiles",
         "@llvm-project//mlir:BuiltinDialectTdFiles",
+        "@llvm-project//mlir:InferTypeOpInterfaceTdFiles",
+        "@llvm-project//mlir:OpBaseTdFiles",
         "@llvm-project//mlir:SideEffectInterfacesTdFiles",
     ],
 )
@@ -89,6 +89,7 @@ cc_library(
     hdrs = [
         "PolyDialect.h",
         "PolyOps.h",
+        "PolyTraits.h",
         "PolyTypes.h",
     ],
     deps = [
@@ -97,6 +98,7 @@ cc_library(
         ":types_inc_gen",
         "@llvm-project//mlir:Dialect",
         "@llvm-project//mlir:IR",
+        "@llvm-project//mlir:InferTypeOpInterface",
         "@llvm-project//mlir:Support",
     ],
 )
diff --git a/lib/Dialect/Poly/PolyOps.cpp b/lib/Dialect/Poly/PolyOps.cpp
index b54b117..b92c58a 100644
--- a/lib/Dialect/Poly/PolyOps.cpp
+++ b/lib/Dialect/Poly/PolyOps.cpp
@@ -61,6 +61,12 @@ OpFoldResult FromTensorOp::fold(FromTensorOp::FoldAdaptor adaptor) {
   return dyn_cast<DenseIntElementsAttr>(adaptor.getInput());
 }
 
+LogicalResult EvalOp::verify() {
+  return getPoint().getType().isSignlessInteger(32)
+             ? success()
+             : emitOpError("argument point must be a 32-bit integer");
+}
+
 } // namespace poly
 } // namespace tutorial
 } // namespace mlir
diff --git a/lib/Dialect/Poly/PolyOps.h b/lib/Dialect/Poly/PolyOps.h
index 5ebeebd..c562244 100644
--- a/lib/Dialect/Poly/PolyOps.h
+++ b/lib/Dialect/Poly/PolyOps.h
@@ -2,12 +2,14 @@
 #define LIB_DIALECT_POLY_POLYOPS_H_
 
 #include "lib/Dialect/Poly/PolyDialect.h"
+#include "lib/Dialect/Poly/PolyTraits.h"
 #include "lib/Dialect/Poly/PolyTypes.h"
-#include "mlir/include/mlir/IR/BuiltinOps.h"    // from @llvm-project
-#include "mlir/include/mlir/IR/BuiltinTypes.h"  // from @llvm-project
-#include "mlir/include/mlir/IR/Dialect.h"       // from @llvm-project
+#include "mlir/Interfaces/InferTypeOpInterface.h" // from @llvm-project
+#include "mlir/include/mlir/IR/BuiltinOps.h"      // from @llvm-project
+#include "mlir/include/mlir/IR/BuiltinTypes.h"    // from @llvm-project
+#include "mlir/include/mlir/IR/Dialect.h"         // from @llvm-project
 
 #define GET_OP_CLASSES
 #include "lib/Dialect/Poly/PolyOps.h.inc"
 
-#endif  // LIB_DIALECT_POLY_POLYOPS_H_
+#endif // LIB_DIALECT_POLY_POLYOPS_H_
diff --git a/lib/Dialect/Poly/PolyOps.td b/lib/Dialect/Poly/PolyOps.td
index f5e1dcd..520ab09 100644
--- a/lib/Dialect/Poly/PolyOps.td
+++ b/lib/Dialect/Poly/PolyOps.td
@@ -5,16 +5,22 @@ include "PolyDialect.td"
 include "PolyTypes.td"
 include "mlir/IR/BuiltinAttributes.td"
 include "mlir/IR/OpBase.td"
+include "mlir/Interfaces/InferTypeOpInterface.td"
 include "mlir/Interfaces/SideEffectInterfaces.td"
 
 // Type constraint for poly binop arguments: polys, vectors of polys, or
 // tensors of polys.
 def PolyOrContainer : TypeOrContainer<Polynomial, "poly-or-container">;
 
-class Poly_BinOp<string mnemonic> : Op<Poly_Dialect, mnemonic, [Pure, ElementwiseMappable, SameOperandsAndResultElementType]> {
+// Inject verification that all integer-like arguments are 32-bits
+def Has32BitArguments : NativeOpTrait<"Has32BitArguments"> {
+  let cppNamespace = "::mlir::tutorial::poly";
+}
+
+class Poly_BinOp<string mnemonic> : Op<Poly_Dialect, mnemonic, [Pure, ElementwiseMappable, SameOperandsAndResultType]> {
   let arguments = (ins PolyOrContainer:$lhs, PolyOrContainer:$rhs);
   let results = (outs PolyOrContainer:$output);
-  let assemblyFormat = "$lhs `,` $rhs attr-dict `:` `(` type($lhs) `,` type($rhs) `)` `->` type($output)";
+  let assemblyFormat = "$lhs `,` $rhs attr-dict `:` qualified(type($output))";
   let hasFolder = 1;
 }
 
@@ -34,22 +40,23 @@ def Poly_FromTensorOp : Op<Poly_Dialect, "from_tensor", [Pure]> {
   let summary = "Creates a Polynomial from integer coefficients stored in a tensor.";
   let arguments = (ins TensorOf<[AnyInteger]>:$input);
   let results = (outs Polynomial:$output);
-  let assemblyFormat = "$input attr-dict `:` type($input) `->` type($output)";
+  let assemblyFormat = "$input attr-dict `:` type($input) `->` qualified(type($output))";
   let hasFolder = 1;
 }
 
-def Poly_EvalOp : Op<Poly_Dialect, "eval"> {
+def Poly_EvalOp : Op<Poly_Dialect, "eval", [AllTypesMatch<["point", "output"]>, Has32BitArguments]> {
   let summary = "Evaluates a Polynomial at a given input value.";
   let arguments = (ins Polynomial:$input, AnyInteger:$point);
   let results = (outs AnyInteger:$output);
-  let assemblyFormat = "$input `,` $point attr-dict `:` `(` type($input) `,` type($point) `)` `->` type($output)";
+  let assemblyFormat = "$input `,` $point attr-dict `:` `(` qualified(type($input)) `,` type($point) `)` `->` type($output)";
+  let hasVerifier = 1;
 }
 
 def Poly_ConstantOp : Op<Poly_Dialect, "constant", [Pure, ConstantLike]> {
   let summary = "Define a constant polynomial via an attribute.";
   let arguments = (ins AnyIntElementsAttr:$coefficients);
   let results = (outs Polynomial:$output);
-  let assemblyFormat = "$coefficients attr-dict `:` type($output)";
+  let assemblyFormat = "$coefficients attr-dict `:` qualified(type($output))";
   let hasFolder = 1;
 }
 
diff --git a/lib/Dialect/Poly/PolyTraits.h b/lib/Dialect/Poly/PolyTraits.h
new file mode 100644
index 0000000..d220f1c
--- /dev/null
+++ b/lib/Dialect/Poly/PolyTraits.h
@@ -0,0 +1,28 @@
+#ifndef LIB_DIALECT_POLY_POLYTRAITS_H_
+#define LIB_DIALECT_POLY_POLYTRAITS_H_
+
+#include "mlir/include/mlir/IR/OpDefinition.h"
+
+namespace mlir::tutorial::poly {
+
+template <typename ConcreteType>
+class Has32BitArguments : public OpTrait::TraitBase<ConcreteType, Has32BitArguments> {
+ public:
+  static LogicalResult verifyTrait(Operation *op) {
+    for (auto type : op->getOperandTypes()) {
+      // OK to skip non-integer operand types
+      if (!type.isIntOrIndex()) continue;
+
+      if (!type.isInteger(32)) {
+        return op->emitOpError()
+               << "requires each numeric operand to be a 32-bit integer";
+      }
+    }
+
+    return success();
+  }
+};
+
+}
+
+#endif  // LIB_DIALECT_POLY_POLYTRAITS_H_
diff --git a/tests/code_motion.mlir b/tests/code_motion.mlir
index 3fc8b1a..51004a3 100644
--- a/tests/code_motion.mlir
+++ b/tests/code_motion.mlir
@@ -15,8 +15,8 @@ module {
     %ret_val = affine.for %i = 0 to 100 iter_args(%sum_iter = %p0) -> !poly.poly<10> {
       // The poly.mul should be hoisted out of the loop.
       // CHECK-NOT: poly.mul
-      %2 = poly.mul %p0, %p1 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-      %sum_next = poly.add %sum_iter, %2 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+      %2 = poly.mul %p0, %p1 : !poly.poly<10>
+      %sum_next = poly.add %sum_iter, %2 : !poly.poly<10>
       affine.yield %sum_next : !poly.poly<10>
     }
 
diff --git a/tests/control_flow_sink.mlir b/tests/control_flow_sink.mlir
index 4354b48..d90fb68 100644
--- a/tests/control_flow_sink.mlir
+++ b/tests/control_flow_sink.mlir
@@ -12,12 +12,12 @@ func.func @test_simple_sink(%arg0: i1) -> !poly.poly<10> {
   // CHECK: scf.if
   %4 = scf.if %arg0 -> (!poly.poly<10>) {
     // CHECK: poly.from_tensor
-    %2 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+    %2 = poly.mul %p0, %p0 : !poly.poly<10>
     scf.yield %2 : !poly.poly<10>
   // CHECK: else
   } else {
     // CHECK: poly.from_tensor
-    %3 = poly.mul %p1, %p1 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+    %3 = poly.mul %p1, %p1 : !poly.poly<10>
     scf.yield %3 : !poly.poly<10>
   }
   return %4 : !poly.poly<10>
diff --git a/tests/cse.mlir b/tests/cse.mlir
index 80d23f2..899e402 100644
--- a/tests/cse.mlir
+++ b/tests/cse.mlir
@@ -8,8 +8,8 @@ func.func @test_simple_cse() -> !poly.poly<10> {
   // exactly one mul op
   // CHECK-NEXT: poly.mul
   // CHECK-NEXT: poly.add
-  %2 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %3 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %4 = poly.add %2, %3 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+  %2 = poly.mul %p0, %p0 : !poly.poly<10>
+  %3 = poly.mul %p0, %p0 : !poly.poly<10>
+  %4 = poly.add %2, %3 : !poly.poly<10>
   return %4 : !poly.poly<10>
 }
diff --git a/tests/poly_canonicalize.mlir b/tests/poly_canonicalize.mlir
index 8d8636a..0b219ab 100644
--- a/tests/poly_canonicalize.mlir
+++ b/tests/poly_canonicalize.mlir
@@ -6,8 +6,8 @@ func.func @test_simple() -> !poly.poly<10> {
   // CHECK-NEXT: return
   %0 = arith.constant dense<[1, 2, 3]> : tensor<3xi32>
   %p0 = poly.from_tensor %0 : tensor<3xi32> -> !poly.poly<10>
-  %2 = poly.add %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %3 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %4 = poly.add %2, %3 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+  %2 = poly.add %p0, %p0 : !poly.poly<10>
+  %3 = poly.mul %p0, %p0 : !poly.poly<10>
+  %4 = poly.add %2, %3 : !poly.poly<10>
   return %2 : !poly.poly<10>
 }
diff --git a/tests/poly_syntax.mlir b/tests/poly_syntax.mlir
index 0eb66b3..8ab6db8 100644
--- a/tests/poly_syntax.mlir
+++ b/tests/poly_syntax.mlir
@@ -11,11 +11,11 @@ module {
   // CHECK-LABEL: test_op_syntax
   func.func @test_op_syntax(%arg0: !poly.poly<10>, %arg1: !poly.poly<10>) -> !poly.poly<10> {
     // CHECK: poly.add
-    %0 = poly.add %arg0, %arg1 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+    %0 = poly.add %arg0, %arg1 : !poly.poly<10>
     // CHECK: poly.sub
-    %1 = poly.sub %arg0, %arg1 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+    %1 = poly.sub %arg0, %arg1 : !poly.poly<10>
     // CHECK: poly.mul
-    %2 = poly.mul %arg0, %arg1 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+    %2 = poly.mul %arg0, %arg1 : !poly.poly<10>
 
     %3 = arith.constant dense<[1, 2, 3]> : tensor<3xi32>
     // CHECK: poly.from_tensor
@@ -27,9 +27,7 @@ module {
 
     %7 = tensor.from_elements %arg0, %arg1 : tensor<2x!poly.poly<10>>
     // CHECK: poly.add
-    %8 = poly.add %7, %7 : (tensor<2x!poly.poly<10>>, tensor<2x!poly.poly<10>>) -> tensor<2x!poly.poly<10>>
-    // CHECK: poly.add
-    %9 = poly.add %7, %4 : (tensor<2x!poly.poly<10>>, !poly.poly<10>) -> tensor<2x!poly.poly<10>>
+    %8 = poly.add %7, %7 : tensor<2x!poly.poly<10>>
 
     // CHECK: poly.constant
     %10 = poly.constant dense<[2, 3, 4]> : tensor<3xi32> : !poly.poly<10>
diff --git a/tests/poly_verifier.mlir b/tests/poly_verifier.mlir
new file mode 100644
index 0000000..fadca35
--- /dev/null
+++ b/tests/poly_verifier.mlir
@@ -0,0 +1,10 @@
+// RUN: tutorial-opt %s 2>%t; FileCheck %s < %t
+
+func.func @test_invalid_evalop(%arg0: !poly.poly<10>, %cst: i64) -> i64 {
+  // This is a little brittle, since it matches both the error message
+  // emitted by Has32BitArguments as well as that of EvalOp::verify.
+  // I manually tested that they both fire when the input is as below.
+  // CHECK: to be a 32-bit integer
+  %0 = poly.eval %arg0, %cst : (!poly.poly<10>, i64) -> i64
+  return %0 : i64
+}
diff --git a/tests/sccp.mlir b/tests/sccp.mlir
index 39b3a24..2243119 100644
--- a/tests/sccp.mlir
+++ b/tests/sccp.mlir
@@ -28,8 +28,8 @@ func.func @test_poly_sccp() -> !poly.poly<10> {
   // CHECK: poly.constant dense<[1, 2, 3]>
   // CHECK-NOT: poly.mul
   // CHECK-NOT: poly.add
-  %2 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %3 = poly.mul %p0, %p0 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
-  %4 = poly.add %2, %3 : (!poly.poly<10>, !poly.poly<10>) -> !poly.poly<10>
+  %2 = poly.mul %p0, %p0 : !poly.poly<10>
+  %3 = poly.mul %p0, %p0 : !poly.poly<10>
+  %4 = poly.add %2, %3 : !poly.poly<10>
   return %2 : !poly.poly<10>
 }