diff --git a/tests/filecheck/dialects/arith/canonicalize.mlir b/tests/filecheck/dialects/arith/canonicalize.mlir
index 01a2ea0c86..b87efd39d8 100644
--- a/tests/filecheck/dialects/arith/canonicalize.mlir
+++ b/tests/filecheck/dialects/arith/canonicalize.mlir
@@ -109,6 +109,22 @@ func.func @test_const_var_const() {
 %foldable_times = arith.muli %c2, %c2 : i32
 "test.op"(%foldable_times) : (i32) -> ()
 
+%c0 = arith.constant 0 : i32
+
+%zero_plus = arith.addi %c0, %a : i32
+%plus_zero = arith.addi %a, %c0 : i32
+
+// CHECK: "test.op"(%a, %a) {"identity addition check"} : (i32, i32) -> ()
+"test.op"(%zero_plus, %plus_zero) {"identity addition check"} : (i32, i32) -> ()
+
+// CHECK: %plus_const = arith.addi %a, %c2 : i32
+%plus_const = arith.addi %c2, %a : i32
+"test.op"(%plus_const) : (i32) -> ()
+
+// CHECK: %foldable_plus = arith.constant 4 : i32
+%foldable_plus = arith.addi %c2, %c2 : i32
+"test.op"(%foldable_plus) : (i32) -> ()
+
 // CHECK: %int = "test.op"() : () -> i32
 %int = "test.op"() : () -> i32
 // CHECK-NEXT: %{{.*}} = arith.constant true
diff --git a/tests/filecheck/dialects/csl/csl-canonicalize.mlir b/tests/filecheck/dialects/csl/csl-canonicalize.mlir
index 5075afe5c9..713595bf39 100644
--- a/tests/filecheck/dialects/csl/csl-canonicalize.mlir
+++ b/tests/filecheck/dialects/csl/csl-canonicalize.mlir
@@ -41,14 +41,13 @@ builtin.module {
 "test.op"(%19) : (!csl<dsd mem1d_dsd>) -> ()
 
 // CHECK-NEXT:  %3 = "test.op"() : () -> !csl<dsd mem1d_dsd>
-// CHECK-NEXT:  %4 = arith.constant 2 : si16
-// CHECK-NEXT:  %5 = arith.addi %4, %4 : si16
-// CHECK-NEXT:  %6 = "csl.increment_dsd_offset"(%3, %5) <{"elem_type" = f32}> : (!csl<dsd mem1d_dsd>, si16) -> !csl<dsd mem1d_dsd>
-// CHECK-NEXT:  %7 = arith.constant 511 : ui16
-// CHECK-NEXT:  %8 = "csl.set_dsd_length"(%6, %7) : (!csl<dsd mem1d_dsd>, ui16) -> !csl<dsd mem1d_dsd>
-// CHECK-NEXT:  %9 = arith.constant 3 : si8
-// CHECK-NEXT:  %10 = "csl.set_dsd_stride"(%8, %9) : (!csl<dsd mem1d_dsd>, si8) -> !csl<dsd mem1d_dsd>
-// CHECK-NEXT:  "test.op"(%10) : (!csl<dsd mem1d_dsd>) -> ()
+// CHECK-NEXT:  %4 = arith.constant 4 : si16
+// CHECK-NEXT:  %5 = "csl.increment_dsd_offset"(%3, %4) <{"elem_type" = f32}> : (!csl<dsd mem1d_dsd>, si16) -> !csl<dsd mem1d_dsd>
+// CHECK-NEXT:  %6 = arith.constant 511 : ui16
+// CHECK-NEXT:  %7 = "csl.set_dsd_length"(%5, %6) : (!csl<dsd mem1d_dsd>, ui16) -> !csl<dsd mem1d_dsd>
+// CHECK-NEXT:  %8 = arith.constant 3 : si8
+// CHECK-NEXT:  %9 = "csl.set_dsd_stride"(%7, %8) : (!csl<dsd mem1d_dsd>, si8) -> !csl<dsd mem1d_dsd>
+// CHECK-NEXT:  "test.op"(%9) : (!csl<dsd mem1d_dsd>) -> ()
 
 }
 // CHECK-NEXT: }
diff --git a/tests/interactive/test_app.py b/tests/interactive/test_app.py
index c8401e03b6..5f2d24290e 100644
--- a/tests/interactive/test_app.py
+++ b/tests/interactive/test_app.py
@@ -323,8 +323,8 @@ async def test_rewrites():
         app.input_text_area.insert(
             """
         func.func @hello(%n : i32) -> i32 {
-  %two = arith.constant 0 : i32
-  %res = arith.addi %two, %n : i32
+  %c0 = arith.constant 0 : i32
+  %res = arith.addi %n, %c0 : i32
   func.return %res : i32
 }
         """
@@ -334,11 +334,11 @@ async def test_rewrites():
         await pilot.click("#condense_button")
 
         addi_pass = AvailablePass(
-            display_name="AddiOp(%res = arith.addi %two, %n : i32):arith.addi:AddImmediateZero",
+            display_name="AddiOp(%res = arith.addi %n, %c0 : i32):arith.addi:AddiIdentityRight",
             module_pass=individual_rewrite.ApplyIndividualRewritePass,
             pass_spec=list(
                 parse_pipeline(
-                    'apply-individual-rewrite{matched_operation_index=3 operation_name="arith.addi" pattern_name="AddImmediateZero"}'
+                    'apply-individual-rewrite{matched_operation_index=3 operation_name="arith.addi" pattern_name="AddiIdentityRight"}'
                 )
             )[0],
         )
@@ -359,7 +359,7 @@ async def test_rewrites():
                 individual_rewrite.ApplyIndividualRewritePass,
                 list(
                     parse_pipeline(
-                        'apply-individual-rewrite{matched_operation_index=3 operation_name="arith.addi" pattern_name="AddImmediateZero"}'
+                        'apply-individual-rewrite{matched_operation_index=3 operation_name="arith.addi" pattern_name="AddiIdentityRight"}'
                     )
                 )[0],
             ),
@@ -371,7 +371,7 @@ async def test_rewrites():
             app.output_text_area.text
             == """builtin.module {
   func.func @hello(%n : i32) -> i32 {
-    %two = arith.constant 0 : i32
+    %c0 = arith.constant 0 : i32
     func.return %n : i32
   }
 }
diff --git a/xdsl/dialects/arith.py b/xdsl/dialects/arith.py
index dc0fe4cf49..5f609abcc0 100644
--- a/xdsl/dialects/arith.py
+++ b/xdsl/dialects/arith.py
@@ -316,9 +316,12 @@ def print(self, printer: Printer):
 class AddiOpHasCanonicalizationPatternsTrait(HasCanonicalizationPatternsTrait):
     @classmethod
     def get_canonicalization_patterns(cls) -> tuple[RewritePattern, ...]:
-        from xdsl.transforms.canonicalization_patterns.arith import AddImmediateZero
+        from xdsl.transforms.canonicalization_patterns.arith import (
+            AddiConstantProp,
+            AddiIdentityRight,
+        )
 
-        return (AddImmediateZero(),)
+        return (AddiIdentityRight(), AddiConstantProp())
 
 
 @irdl_op_definition
diff --git a/xdsl/transforms/canonicalization_patterns/arith.py b/xdsl/transforms/canonicalization_patterns/arith.py
index cf12cdefc9..85fb9d7093 100644
--- a/xdsl/transforms/canonicalization_patterns/arith.py
+++ b/xdsl/transforms/canonicalization_patterns/arith.py
@@ -9,15 +9,30 @@
 from xdsl.utils.hints import isa
 
 
-class AddImmediateZero(RewritePattern):
+class AddiIdentityRight(RewritePattern):
     @op_type_rewrite_pattern
     def match_and_rewrite(self, op: arith.AddiOp, rewriter: PatternRewriter) -> None:
-        if (
-            isinstance(op.lhs.owner, arith.ConstantOp)
-            and isinstance(value := op.lhs.owner.value, IntegerAttr)
-            and value.value.data == 0
-        ):
-            rewriter.replace_matched_op([], [op.rhs])
+        if (rhs := const_evaluate_operand(op.rhs)) is None:
+            return
+        if rhs != 0:
+            return
+        rewriter.replace_matched_op((), (op.lhs,))
+
+
+class AddiConstantProp(RewritePattern):
+    @op_type_rewrite_pattern
+    def match_and_rewrite(self, op: arith.AddiOp, rewriter: PatternRewriter):
+        if (lhs := const_evaluate_operand(op.lhs)) is None:
+            return
+        if (rhs := const_evaluate_operand(op.rhs)) is None:
+            # Swap inputs if lhs is constant and rhs is not
+            rewriter.replace_matched_op(arith.AddiOp(op.rhs, op.lhs))
+            return
+
+        assert isinstance(op.result.type, IntegerType | IndexType)
+        rewriter.replace_matched_op(
+            arith.ConstantOp.from_int_and_width(lhs + rhs, op.result.type)
+        )
 
 
 def _fold_const_operation(