[Mlir-commits] [mlir] [MLIR] Add `IntegerDivisibilityAnalysis` and `InferIntDivisibilityOpInterface` (PR #197728)

Alan Li llvmlistbot at llvm.org
Thu May 14 08:59:14 PDT 2026


https://github.com/lialan updated https://github.com/llvm/llvm-project/pull/197728

>From 813e2e40a7131bfee2fef40eaa6b37ee19356b87 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 09:21:50 -0700
Subject: [PATCH 1/7] [MLIR] Add InferIntDivisibilityOpInterface and
 divisibility lattice types

Ports the op interface and lattice value types (ConstantIntDivisibility,
IntegerDivisibility, SetIntDivisibilityFn) from IREE's
IREE::Util::IntegerDivisibilityAnalysis into upstream MLIR. No analysis or
op implementations yet; subsequent commits will add those.

Mirrors the structure of InferIntRangeInterface.
---
 mlir/include/mlir/Interfaces/CMakeLists.txt   |   1 +
 .../InferIntDivisibilityOpInterface.h         | 120 ++++++++++++++++++
 .../InferIntDivisibilityOpInterface.td        |  41 ++++++
 mlir/lib/Interfaces/CMakeLists.txt            |   2 +
 .../InferIntDivisibilityOpInterface.cpp       |  11 ++
 5 files changed, 175 insertions(+)
 create mode 100644 mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
 create mode 100644 mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.td
 create mode 100644 mlir/lib/Interfaces/InferIntDivisibilityOpInterface.cpp

diff --git a/mlir/include/mlir/Interfaces/CMakeLists.txt b/mlir/include/mlir/Interfaces/CMakeLists.txt
index 3cbc9df05f3d7..6461c68423c73 100644
--- a/mlir/include/mlir/Interfaces/CMakeLists.txt
+++ b/mlir/include/mlir/Interfaces/CMakeLists.txt
@@ -6,6 +6,7 @@ add_mlir_interface(DerivedAttributeOpInterface)
 add_mlir_interface(DestinationStyleOpInterface)
 add_mlir_interface(FunctionInterfaces)
 add_mlir_interface(IndexingMapOpInterface)
+add_mlir_interface(InferIntDivisibilityOpInterface)
 add_mlir_interface(InferIntRangeInterface)
 add_mlir_interface(InferStridedMetadataInterface)
 add_mlir_interface(InferTypeOpInterface)
diff --git a/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
new file mode 100644
index 0000000000000..6050ccdfbf99c
--- /dev/null
+++ b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
@@ -0,0 +1,120 @@
+//===- InferIntDivisibilityOpInterface.h - Integer Divisibility -*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// This file contains definitions of the integer divisibility inference
+// interface defined in `InferIntDivisibilityOpInterface.td`.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE_H
+#define MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE_H
+
+#include "mlir/IR/OpDefinition.h"
+#include <numeric>
+#include <optional>
+
+namespace mlir {
+
+/// Statically known divisibility information for an integer SSA value.
+/// Tracks separate divisors for the unsigned and signed interpretations of
+/// the value so that subsequent analyses can use whichever is more precise.
+class ConstantIntDivisibility {
+public:
+  ConstantIntDivisibility() = default;
+  ConstantIntDivisibility(uint64_t udiv, uint64_t sdiv)
+      : udivVal(udiv), sdivVal(sdiv) {}
+
+  bool operator==(const ConstantIntDivisibility &other) const {
+    return udivVal == other.udivVal && sdivVal == other.sdivVal;
+  }
+
+  uint64_t udiv() const { return this->udivVal; }
+  uint64_t sdiv() const { return this->sdivVal; }
+
+  // Returns the union (computed separately for signed and unsigned bounds)
+  // for this range and `other`.
+  ConstantIntDivisibility getUnion(const ConstantIntDivisibility &other) const {
+    return ConstantIntDivisibility(
+        /*udiv=*/std::gcd(udiv(), other.udiv()),
+        /*sdiv=*/std::gcd(sdiv(), other.sdiv()));
+  }
+
+private:
+  uint64_t udivVal;
+  uint64_t sdivVal;
+
+  friend raw_ostream &operator<<(raw_ostream &os,
+                                 const ConstantIntDivisibility &div);
+};
+
+inline raw_ostream &operator<<(raw_ostream &os,
+                               const ConstantIntDivisibility &div) {
+  os << "ConstantIntDivisibility(udiv = " << div.udivVal
+     << ", sdiv = " << div.sdivVal << ")";
+  return os;
+}
+
+/// This lattice value represents the integer divisibility of an SSA value.
+class IntegerDivisibility {
+public:
+  IntegerDivisibility(ConstantIntDivisibility value)
+      : value(std::move(value)) {}
+  IntegerDivisibility(
+      std::optional<ConstantIntDivisibility> value = std::nullopt)
+      : value(std::move(value)) {}
+  // Gets the minimum divisibility of 1 that is used to indicate that the value
+  // cannot be analyzed further.
+  static IntegerDivisibility getMinDivisibility() {
+    return IntegerDivisibility(ConstantIntDivisibility(1, 1));
+  }
+
+  bool isUninitialized() const { return !value.has_value(); }
+  const ConstantIntDivisibility &getValue() const {
+    assert(!isUninitialized());
+    return *value;
+  }
+
+  bool operator==(const IntegerDivisibility &rhs) const {
+    return value == rhs.value;
+  }
+
+  static IntegerDivisibility join(const IntegerDivisibility &lhs,
+                                  const IntegerDivisibility &rhs) {
+    if (lhs.isUninitialized()) {
+      return rhs;
+    }
+    if (rhs.isUninitialized()) {
+      return lhs;
+    }
+    return IntegerDivisibility(lhs.getValue().getUnion(rhs.getValue()));
+  }
+
+  void print(raw_ostream &os) const { os << value; }
+
+private:
+  std::optional<ConstantIntDivisibility> value;
+};
+
+inline raw_ostream &operator<<(raw_ostream &os,
+                               const IntegerDivisibility &div) {
+  div.print(os);
+  return os;
+}
+
+/// The type of the `setResultDivs` callback provided to ops implementing
+/// InferIntDivisibilityOpInterface. It should be called once for each integer
+/// result value and be passed the ConstantIntDivisibility corresponding to
+/// that value.
+using SetIntDivisibilityFn =
+    llvm::function_ref<void(Value, const ConstantIntDivisibility &)>;
+
+} // end namespace mlir
+
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.h.inc"
+
+#endif // MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE_H
diff --git a/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.td b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.td
new file mode 100644
index 0000000000000..c665475e0fd7f
--- /dev/null
+++ b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.td
@@ -0,0 +1,41 @@
+//===- InferIntDivisibilityOpInterface.td - Integer Divisibility -*- tablegen -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// Defines the interface for divisibility analysis on scalar integers.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE
+#define MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE
+
+include "mlir/IR/OpBase.td"
+
+def InferIntDivisibilityOpInterface :
+    OpInterface<"InferIntDivisibilityOpInterface"> {
+  let description = [{
+    Allows operations to participate in integer divisibility analysis.
+  }];
+  let cppNamespace = "::mlir";
+
+  let methods = [
+    InterfaceMethod<
+      /*desc=*/[{
+        Infer the divisibility of the results of this op given the
+        divisibility of its arguments. For each result value, the method
+        should call `setResultDivs` with that `Value` as an argument.
+      }],
+      /*retTy=*/"void",
+      /*methodName=*/"inferResultDivisibility",
+      /*args=*/(ins
+        "::llvm::ArrayRef<::mlir::IntegerDivisibility>":$argDivs,
+        "::mlir::SetIntDivisibilityFn":$setResultDivs)
+    >
+  ];
+}
+
+#endif // MLIR_INTERFACES_INFERINTDIVISIBILITYOPINTERFACE
diff --git a/mlir/lib/Interfaces/CMakeLists.txt b/mlir/lib/Interfaces/CMakeLists.txt
index 41e890cb408ba..267c5b945da62 100644
--- a/mlir/lib/Interfaces/CMakeLists.txt
+++ b/mlir/lib/Interfaces/CMakeLists.txt
@@ -9,6 +9,7 @@ set(LLVM_OPTIONAL_SOURCES
   FunctionImplementation.cpp
   FunctionInterfaces.cpp
   IndexingMapOpInterface.cpp
+  InferIntDivisibilityOpInterface.cpp
   InferIntRangeInterface.cpp
   InferStridedMetadataInterface.cpp
   InferTypeOpInterface.cpp
@@ -67,6 +68,7 @@ add_mlir_library(MLIRFunctionInterfaces
 
 add_mlir_interface_library(IndexingMapOpInterface)
 add_mlir_interface_library(InferIntRangeInterface)
+add_mlir_interface_library(InferIntDivisibilityOpInterface)
 
 add_mlir_library(MLIRInferStridedMetadataInterface
   InferStridedMetadataInterface.cpp
diff --git a/mlir/lib/Interfaces/InferIntDivisibilityOpInterface.cpp b/mlir/lib/Interfaces/InferIntDivisibilityOpInterface.cpp
new file mode 100644
index 0000000000000..acd7cd9530b5c
--- /dev/null
+++ b/mlir/lib/Interfaces/InferIntDivisibilityOpInterface.cpp
@@ -0,0 +1,11 @@
+//===- InferIntDivisibilityOpInterface.cpp - Integer divisibility inference ==//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.h"
+
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.cpp.inc"

>From 15ae61944fe90f2e66c8d295e8d897b1bb123692 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 09:34:07 -0700
Subject: [PATCH 2/7] [MLIR] Address review feedback on
 InferIntDivisibilityOpInterface

- Alphabetize lib/Interfaces/CMakeLists.txt entry (Div < Range).
- Fix stale "range" wording in ConstantIntDivisibility::getUnion comment.
- Add explicit to IntegerDivisibility's optional-bearing constructor to
  match IntegerValueRange.
---
 .../include/mlir/Interfaces/InferIntDivisibilityOpInterface.h | 4 ++--
 mlir/lib/Interfaces/CMakeLists.txt                            | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
index 6050ccdfbf99c..374acee05cb10 100644
--- a/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
+++ b/mlir/include/mlir/Interfaces/InferIntDivisibilityOpInterface.h
@@ -37,7 +37,7 @@ class ConstantIntDivisibility {
   uint64_t sdiv() const { return this->sdivVal; }
 
   // Returns the union (computed separately for signed and unsigned bounds)
-  // for this range and `other`.
+  // for this divisibility and `other`.
   ConstantIntDivisibility getUnion(const ConstantIntDivisibility &other) const {
     return ConstantIntDivisibility(
         /*udiv=*/std::gcd(udiv(), other.udiv()),
@@ -64,7 +64,7 @@ class IntegerDivisibility {
 public:
   IntegerDivisibility(ConstantIntDivisibility value)
       : value(std::move(value)) {}
-  IntegerDivisibility(
+  explicit IntegerDivisibility(
       std::optional<ConstantIntDivisibility> value = std::nullopt)
       : value(std::move(value)) {}
   // Gets the minimum divisibility of 1 that is used to indicate that the value
diff --git a/mlir/lib/Interfaces/CMakeLists.txt b/mlir/lib/Interfaces/CMakeLists.txt
index 267c5b945da62..d20d290c45c01 100644
--- a/mlir/lib/Interfaces/CMakeLists.txt
+++ b/mlir/lib/Interfaces/CMakeLists.txt
@@ -67,8 +67,8 @@ add_mlir_library(MLIRFunctionInterfaces
 )
 
 add_mlir_interface_library(IndexingMapOpInterface)
-add_mlir_interface_library(InferIntRangeInterface)
 add_mlir_interface_library(InferIntDivisibilityOpInterface)
+add_mlir_interface_library(InferIntRangeInterface)
 
 add_mlir_library(MLIRInferStridedMetadataInterface
   InferStridedMetadataInterface.cpp

>From 6614fed94f584845bf61909f5e9dbaa81e2afaea Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 09:38:59 -0700
Subject: [PATCH 3/7] [MLIR] Add IntegerDivisibilityAnalysis

Ports IREE's IntegerDivisibilityAnalysis (sparse forward dataflow analysis
over the InferIntDivisibilityOpInterface) into upstream MLIR. Mirrors the
shape of IntegerRangeAnalysis. The analysis is a silent no-op until per-op
external models are added in a follow-up commit.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply at anthropic.com>
---
 .../DataFlow/IntegerDivisibilityAnalysis.h    |  57 ++++++++
 mlir/lib/Analysis/CMakeLists.txt              |   3 +
 .../DataFlow/IntegerDivisibilityAnalysis.cpp  | 135 ++++++++++++++++++
 3 files changed, 195 insertions(+)
 create mode 100644 mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
 create mode 100644 mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp

diff --git a/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
new file mode 100644
index 0000000000000..d4f4f6ae5df0b
--- /dev/null
+++ b/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
@@ -0,0 +1,57 @@
+//===- IntegerDivisibilityAnalysis.h - Integer divisibility -----*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// This file declares the dataflow analysis class for integer divisibility
+// inference. Operations participate in the analysis by implementing
+// `InferIntDivisibilityOpInterface`.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_ANALYSIS_DATAFLOW_INTEGERDIVISIBILITYANALYSIS_H
+#define MLIR_ANALYSIS_DATAFLOW_INTEGERDIVISIBILITYANALYSIS_H
+
+#include "mlir/Analysis/DataFlow/SparseAnalysis.h"
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.h"
+
+#include <optional>
+
+namespace mlir::dataflow {
+
+class IntegerDivisibilityLattice : public Lattice<IntegerDivisibility> {
+public:
+  using Lattice::Lattice;
+};
+
+class IntegerDivisibilityAnalysis
+    : public SparseForwardDataFlowAnalysis<IntegerDivisibilityLattice> {
+public:
+  using SparseForwardDataFlowAnalysis::SparseForwardDataFlowAnalysis;
+
+  // At an entry point, set the lattice to the most pessimistic state,
+  // indicating that no further reasoning can be done.
+  void setToEntryState(IntegerDivisibilityLattice *lattice) override;
+
+  // Visit an operation, invoking the transfer function.
+  LogicalResult
+  visitOperation(Operation *op,
+                 ArrayRef<const IntegerDivisibilityLattice *> operands,
+                 ArrayRef<IntegerDivisibilityLattice *> results) override;
+
+  /// Visit block arguments or operation results of an operation with region
+  /// control-flow for which values are not defined by region control-flow. This
+  /// function tries to infer the divisibility of loop induction variables based
+  /// on known loop bounds and steps.
+  void visitNonControlFlowArguments(
+      Operation *op, const RegionSuccessor &successor,
+      ValueRange successorInputs,
+      ArrayRef<IntegerDivisibilityLattice *> argLattices) override;
+};
+
+} // namespace mlir::dataflow
+
+#endif // MLIR_ANALYSIS_DATAFLOW_INTEGERDIVISIBILITYANALYSIS_H
diff --git a/mlir/lib/Analysis/CMakeLists.txt b/mlir/lib/Analysis/CMakeLists.txt
index db10ebcf2c311..596ffaff428b5 100644
--- a/mlir/lib/Analysis/CMakeLists.txt
+++ b/mlir/lib/Analysis/CMakeLists.txt
@@ -13,6 +13,7 @@ set(LLVM_OPTIONAL_SOURCES
   DataFlow/ConstantPropagationAnalysis.cpp
   DataFlow/DeadCodeAnalysis.cpp
   DataFlow/DenseAnalysis.cpp
+  DataFlow/IntegerDivisibilityAnalysis.cpp
   DataFlow/IntegerRangeAnalysis.cpp
   DataFlow/LivenessAnalysis.cpp
   DataFlow/SparseAnalysis.cpp
@@ -37,6 +38,7 @@ add_mlir_library(MLIRAnalysis
   DataFlow/ConstantPropagationAnalysis.cpp
   DataFlow/DeadCodeAnalysis.cpp
   DataFlow/DenseAnalysis.cpp
+  DataFlow/IntegerDivisibilityAnalysis.cpp
   DataFlow/IntegerRangeAnalysis.cpp
   DataFlow/LivenessAnalysis.cpp
   DataFlow/SparseAnalysis.cpp
@@ -53,6 +55,7 @@ add_mlir_library(MLIRAnalysis
   MLIRControlFlowInterfaces
   MLIRDataLayoutInterfaces
   MLIRFunctionInterfaces
+  MLIRInferIntDivisibilityOpInterface
   MLIRInferIntRangeInterface
   MLIRInferStridedMetadataInterface
   MLIRInferTypeOpInterface
diff --git a/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp b/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp
new file mode 100644
index 0000000000000..4a45f7f3aed22
--- /dev/null
+++ b/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp
@@ -0,0 +1,135 @@
+//===- IntegerDivisibilityAnalysis.cpp - Integer divisibility ---*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// This file defines the dataflow analysis class for integer divisibility
+// inference. Operations participate in the analysis by implementing
+// `InferIntDivisibilityOpInterface`.
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h"
+
+#include "mlir/Dialect/Utils/StaticValueUtils.h"
+#include "mlir/Interfaces/LoopLikeInterface.h"
+#include "llvm/Support/Debug.h"
+
+#define DEBUG_TYPE "int-divisibility-analysis"
+
+using llvm::dbgs;
+
+namespace mlir::dataflow {
+
+void IntegerDivisibilityAnalysis::setToEntryState(
+    IntegerDivisibilityLattice *lattice) {
+  propagateIfChanged(lattice,
+                     lattice->join(IntegerDivisibility::getMinDivisibility()));
+}
+
+LogicalResult IntegerDivisibilityAnalysis::visitOperation(
+    Operation *op, ArrayRef<const IntegerDivisibilityLattice *> operands,
+    ArrayRef<IntegerDivisibilityLattice *> results) {
+  auto inferrable = dyn_cast<InferIntDivisibilityOpInterface>(op);
+  if (!inferrable) {
+    setAllToEntryStates(results);
+    return success();
+  }
+
+  LLVM_DEBUG(dbgs() << "Inferring divisibility for " << *op << "\n");
+  auto argDivs = llvm::map_to_vector(
+      operands, [](const IntegerDivisibilityLattice *lattice) {
+        return lattice->getValue();
+      });
+  auto joinCallback = [&](Value v, const IntegerDivisibility &newDiv) {
+    auto result = dyn_cast<OpResult>(v);
+    if (!result) {
+      return;
+    }
+    assert(llvm::is_contained(op->getResults(), result));
+
+    LLVM_DEBUG(dbgs() << "Inferred divisibility " << newDiv << "\n");
+    IntegerDivisibilityLattice *lattice = results[result.getResultNumber()];
+    IntegerDivisibility oldDiv = lattice->getValue();
+
+    ChangeResult changed = lattice->join(newDiv);
+
+    // Catch loop results with loop variant bounds and conservatively make
+    // them [-inf, inf] so we don't circle around infinitely often (because
+    // the dataflow analysis in MLIR doesn't attempt to work out trip counts
+    // and often can't).
+    bool isYieldedResult = llvm::any_of(v.getUsers(), [](Operation *op) {
+      return op->hasTrait<OpTrait::IsTerminator>();
+    });
+    if (isYieldedResult && !oldDiv.isUninitialized() &&
+        !(lattice->getValue() == oldDiv)) {
+      LLVM_DEBUG(llvm::dbgs() << "Loop variant loop result detected\n");
+      changed |= lattice->join(IntegerDivisibility::getMinDivisibility());
+    }
+    propagateIfChanged(lattice, changed);
+  };
+
+  inferrable.inferResultDivisibility(argDivs, joinCallback);
+  return success();
+}
+
+void IntegerDivisibilityAnalysis::visitNonControlFlowArguments(
+    Operation *op, const RegionSuccessor &successor, ValueRange successorInputs,
+    ArrayRef<IntegerDivisibilityLattice *> argLattices) {
+  // Get the constant divisibility, or query the lattice for Values.
+  auto getDivFromOfr = [&](std::optional<OpFoldResult> ofr, Block *block,
+                           bool isUnsigned) -> uint64_t {
+    if (ofr.has_value()) {
+      if (auto constBound = getConstantIntValue(*ofr)) {
+        return constBound.value();
+      }
+      auto value = cast<Value>(ofr.value());
+      const IntegerDivisibilityLattice *lattice =
+          getLatticeElementFor(getProgramPointBefore(block), value);
+      if (lattice != nullptr && !lattice->getValue().isUninitialized()) {
+        return isUnsigned ? lattice->getValue().getValue().udiv()
+                          : lattice->getValue().getValue().sdiv();
+      }
+    }
+    return isUnsigned
+               ? IntegerDivisibility::getMinDivisibility().getValue().udiv()
+               : IntegerDivisibility::getMinDivisibility().getValue().sdiv();
+  };
+
+  // Infer bounds for loop arguments that have static bounds
+  if (auto loop = dyn_cast<LoopLikeOpInterface>(op)) {
+    std::optional<SmallVector<Value>> ivs = loop.getLoopInductionVars();
+    std::optional<SmallVector<OpFoldResult>> lbs = loop.getLoopLowerBounds();
+    std::optional<SmallVector<OpFoldResult>> steps = loop.getLoopSteps();
+    if (!ivs || !lbs || !steps) {
+      return SparseForwardDataFlowAnalysis::visitNonControlFlowArguments(
+          op, successor, successorInputs, argLattices);
+    }
+    for (auto [iv, lb, step] : llvm::zip_equal(*ivs, *lbs, *steps)) {
+      IntegerDivisibilityLattice *ivEntry = getLatticeElement(iv);
+      Block *block = iv.getParentBlock();
+      uint64_t stepUDiv = getDivFromOfr(step, block, /*unsigned=*/true);
+      uint64_t stepSDiv = getDivFromOfr(step, block, /*unsigned=*/false);
+      uint64_t lbUDiv = getDivFromOfr(lb, block, /*unsigned=*/true);
+      uint64_t lbSDiv = getDivFromOfr(lb, block, /*unsigned=*/false);
+      ConstantIntDivisibility lbDiv(lbUDiv, lbSDiv);
+      ConstantIntDivisibility stepDiv(stepUDiv, stepSDiv);
+
+      // Loop induction variables are computed as `lb + i * step`. The
+      // divisibility for `i * step` is just the divisibility of `step`, so
+      // the total divisibility is obtained by unioning the step divisibility
+      // with the lower bound divisibility, which takes the GCD of the two.
+      ConstantIntDivisibility ivDiv = stepDiv.getUnion(lbDiv);
+      propagateIfChanged(ivEntry, ivEntry->join(ivDiv));
+    }
+    return;
+  }
+
+  return SparseForwardDataFlowAnalysis::visitNonControlFlowArguments(
+      op, successor, successorInputs, argLattices);
+}
+
+} // namespace mlir::dataflow

>From 312d469bc16ddd985053cd4762e0bd1de83a36f2 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 09:44:43 -0700
Subject: [PATCH 4/7] [MLIR] Address review feedback on
 IntegerDivisibilityAnalysis

- Replace stale "[-inf, inf]" widening comment with divisibility-accurate
  wording.
- Promote two method comments to /// Doxygen style.
- Add class-level /// doc comments for IntegerDivisibilityLattice and
  IntegerDivisibilityAnalysis, mirroring IntegerRangeAnalysis.
- Drop redundant <optional> include from the header.
---
 .../DataFlow/IntegerDivisibilityAnalysis.h      | 17 ++++++++++++-----
 .../DataFlow/IntegerDivisibilityAnalysis.cpp    |  8 ++++----
 2 files changed, 16 insertions(+), 9 deletions(-)

diff --git a/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h b/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
index d4f4f6ae5df0b..3a877647490a3 100644
--- a/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
+++ b/mlir/include/mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h
@@ -18,25 +18,32 @@
 #include "mlir/Analysis/DataFlow/SparseAnalysis.h"
 #include "mlir/Interfaces/InferIntDivisibilityOpInterface.h"
 
-#include <optional>
-
 namespace mlir::dataflow {
 
+/// This lattice element represents the integer divisibility of an SSA value.
 class IntegerDivisibilityLattice : public Lattice<IntegerDivisibility> {
 public:
   using Lattice::Lattice;
 };
 
+/// Integer divisibility analysis determines, for each integer-typed SSA
+/// value, a divisor that the value is guaranteed to be a multiple of. It
+/// uses operations that implement `InferIntDivisibilityOpInterface` and
+/// also sets the divisibility of induction variables of loops with known
+/// lower bounds and steps.
+///
+/// This analysis depends on DeadCodeAnalysis, and will be a silent no-op
+/// if DeadCodeAnalysis is not loaded in the same solver context.
 class IntegerDivisibilityAnalysis
     : public SparseForwardDataFlowAnalysis<IntegerDivisibilityLattice> {
 public:
   using SparseForwardDataFlowAnalysis::SparseForwardDataFlowAnalysis;
 
-  // At an entry point, set the lattice to the most pessimistic state,
-  // indicating that no further reasoning can be done.
+  /// At an entry point, set the lattice to the most pessimistic state,
+  /// indicating that no further reasoning can be done.
   void setToEntryState(IntegerDivisibilityLattice *lattice) override;
 
-  // Visit an operation, invoking the transfer function.
+  /// Visit an operation, invoking the transfer function.
   LogicalResult
   visitOperation(Operation *op,
                  ArrayRef<const IntegerDivisibilityLattice *> operands,
diff --git a/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp b/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp
index 4a45f7f3aed22..ba10a8b5a0060 100644
--- a/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp
+++ b/mlir/lib/Analysis/DataFlow/IntegerDivisibilityAnalysis.cpp
@@ -57,10 +57,10 @@ LogicalResult IntegerDivisibilityAnalysis::visitOperation(
 
     ChangeResult changed = lattice->join(newDiv);
 
-    // Catch loop results with loop variant bounds and conservatively make
-    // them [-inf, inf] so we don't circle around infinitely often (because
-    // the dataflow analysis in MLIR doesn't attempt to work out trip counts
-    // and often can't).
+    // Catch loop results with loop-variant divisibility and conservatively
+    // set them to divisibility 1 (no information) so we don't ratchet
+    // indefinitely (the dataflow analysis in MLIR doesn't attempt to work
+    // out trip counts and often can't).
     bool isYieldedResult = llvm::any_of(v.getUsers(), [](Operation *op) {
       return op->hasTrait<OpTrait::IsTerminator>();
     });

>From cd6a541b0b3064f472b30c77f8e1d1b8c84f6df7 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 09:50:44 -0700
Subject: [PATCH 5/7] [MLIR] Add InferIntDivisibilityOpInterface external
 models for arith and affine

Ports the per-op divisibility models from IREE's UtilExternalModels.cpp.
Each dialect now provides registerInferIntDivisibilityOpInterfaceExternalModels.
Registration into the global DialectRegistry is wired up in a follow-up commit.

Arith ops covered: ConstantOp, AddIOp, SubIOp, MulIOp, DivUIOp,
  MinUIOp/MaxUIOp/MinSIOp/MaxSIOp, SelectOp.
Affine ops covered: AffineApplyOp, AffineMinOp, AffineMaxOp,
  AffineDelinearizeIndexOp (the last using AffineExprDivisibilityFinder).
---
 .../IR/InferIntDivisibilityOpInterfaceImpl.h  |  21 +
 .../IR/InferIntDivisibilityOpInterfaceImpl.h  |  21 +
 mlir/lib/Dialect/Affine/IR/CMakeLists.txt     |   2 +
 .../InferIntDivisibilityOpInterfaceImpl.cpp   | 368 ++++++++++++++++++
 mlir/lib/Dialect/Arith/IR/CMakeLists.txt      |   3 +
 .../InferIntDivisibilityOpInterfaceImpl.cpp   | 162 ++++++++
 6 files changed, 577 insertions(+)
 create mode 100644 mlir/include/mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h
 create mode 100644 mlir/include/mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h
 create mode 100644 mlir/lib/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.cpp
 create mode 100644 mlir/lib/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.cpp

diff --git a/mlir/include/mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h b/mlir/include/mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h
new file mode 100644
index 0000000000000..560bef63b56a8
--- /dev/null
+++ b/mlir/include/mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h
@@ -0,0 +1,21 @@
+//===- InferIntDivisibilityOpInterfaceImpl.h --------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_DIALECT_AFFINE_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
+#define MLIR_DIALECT_AFFINE_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
+
+namespace mlir {
+class DialectRegistry;
+
+namespace affine {
+void registerInferIntDivisibilityOpInterfaceExternalModels(
+    DialectRegistry &registry);
+} // namespace affine
+} // namespace mlir
+
+#endif // MLIR_DIALECT_AFFINE_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
diff --git a/mlir/include/mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h b/mlir/include/mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h
new file mode 100644
index 0000000000000..0909790b8b7d0
--- /dev/null
+++ b/mlir/include/mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h
@@ -0,0 +1,21 @@
+//===- InferIntDivisibilityOpInterfaceImpl.h --------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_DIALECT_ARITH_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
+#define MLIR_DIALECT_ARITH_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
+
+namespace mlir {
+class DialectRegistry;
+
+namespace arith {
+void registerInferIntDivisibilityOpInterfaceExternalModels(
+    DialectRegistry &registry);
+} // namespace arith
+} // namespace mlir
+
+#endif // MLIR_DIALECT_ARITH_IR_INFERINTDIVISIBILITYOPINTERFACEIMPL_H
diff --git a/mlir/lib/Dialect/Affine/IR/CMakeLists.txt b/mlir/lib/Dialect/Affine/IR/CMakeLists.txt
index 566bc060e5d38..1caf2fa396797 100644
--- a/mlir/lib/Dialect/Affine/IR/CMakeLists.txt
+++ b/mlir/lib/Dialect/Affine/IR/CMakeLists.txt
@@ -2,6 +2,7 @@ add_mlir_dialect_library(MLIRAffineDialect
   AffineMemoryOpInterfaces.cpp
   AffineOps.cpp
   AffineValueMap.cpp
+  InferIntDivisibilityOpInterfaceImpl.cpp
   InferIntRangeInterfaceImpls.cpp
   ValueBoundsOpInterfaceImpl.cpp
 
@@ -16,6 +17,7 @@ add_mlir_dialect_library(MLIRAffineDialect
   MLIRArithDialect
   MLIRDialectUtils
   MLIRIR
+  MLIRInferIntDivisibilityOpInterface
   MLIRInferIntRangeInterface
   MLIRInferTypeOpInterface
   MLIRLoopLikeInterface
diff --git a/mlir/lib/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.cpp b/mlir/lib/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.cpp
new file mode 100644
index 0000000000000..30850cf0a4df0
--- /dev/null
+++ b/mlir/lib/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.cpp
@@ -0,0 +1,368 @@
+//===- InferIntDivisibilityOpInterfaceImpl.cpp ----------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h"
+#include "mlir/Dialect/Affine/IR/AffineOps.h"
+#include "mlir/IR/AffineExprVisitor.h"
+#include "mlir/IR/DialectRegistry.h"
+#include "mlir/IR/Matchers.h"
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.h"
+
+#include <cstdlib>
+#include <numeric>
+
+using namespace mlir;
+
+namespace {
+
+static ConstantIntDivisibility
+getDivisibilityOfOperand(Value v, IntegerDivisibility divisibility) {
+  if (!divisibility.isUninitialized()) {
+    return divisibility.getValue();
+  }
+  APInt intVal;
+  if (matchPattern(v, m_ConstantInt(&intVal))) {
+    uint64_t udiv = intVal.getZExtValue();
+    uint64_t sdiv = std::abs(intVal.getSExtValue());
+    return ConstantIntDivisibility(udiv, sdiv);
+  }
+  return ConstantIntDivisibility(1, 1);
+}
+
+/// Visits affine expressions and recursively calculates the divisibilities of
+/// each subexpression. The final divisibilities of the expression and its
+/// subexpressions will be stored in the map for which a reference is provided
+/// to the AffineExprDivisibilityFinder (i.e., `divisibilityMap`).
+class AffineExprDivisibilityFinder
+    : public AffineExprVisitor<AffineExprDivisibilityFinder,
+                               ConstantIntDivisibility> {
+public:
+  using ExprDivisibilityMap =
+      llvm::DenseMap<AffineExpr, ConstantIntDivisibility>;
+  AffineExprDivisibilityFinder(ExprDivisibilityMap &divisibilityMap)
+      : divisibilityMap(divisibilityMap) {}
+
+  ConstantIntDivisibility visitConstantExpr(AffineConstantExpr expr) {
+    // Constant expressions are trivial, since they are always static.
+    uint64_t constValue = std::abs(expr.getValue());
+    return ConstantIntDivisibility(constValue, constValue);
+  }
+
+  ConstantIntDivisibility visitDimExpr(AffineDimExpr expr) {
+    // Dim expressions cannot be analyzed further, so return the divisibility
+    // in `divisibilityMap` if it has been populated by the caller, or fallback
+    // to the minimum divisibility.
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    return IntegerDivisibility::getMinDivisibility().getValue();
+  }
+
+  ConstantIntDivisibility visitSymbolExpr(AffineSymbolExpr expr) {
+    // Symbol expressions cannot be analyzed further, so return the divisibility
+    // in `divisibilityMap` if it has been populated by the caller, or fallback
+    // to the minimum divisibility.
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    return IntegerDivisibility::getMinDivisibility().getValue();
+  }
+
+  /// Infer the divisibility of an addition or subtraction expression by
+  /// recursively visiting the LHS and RHS, and then unioning the results.
+  ConstantIntDivisibility visitAddExpr(AffineBinaryOpExpr expr) {
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    // The divisibility of an addition is the GCD of its constituents'
+    // divisibilities.
+    ConstantIntDivisibility lhsDiv = visit(expr.getLHS());
+    ConstantIntDivisibility rhsDiv = visit(expr.getRHS());
+    return lhsDiv.getUnion(rhsDiv);
+  }
+
+  /// Infer the divisibility of a multiplication expression by recursively
+  /// visiting the LHS and RHS, and then multiplying the results.
+  ConstantIntDivisibility visitMulExpr(AffineBinaryOpExpr expr) {
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    // The divisibility of a multiplication is the product of its constituents'
+    // divisibilities.
+    ConstantIntDivisibility lhsDiv = visit(expr.getLHS());
+    ConstantIntDivisibility rhsDiv = visit(expr.getRHS());
+    return ConstantIntDivisibility(lhsDiv.udiv() * rhsDiv.udiv(),
+                                   lhsDiv.sdiv() * rhsDiv.sdiv());
+  }
+
+  ConstantIntDivisibility visitFloorDivExpr(AffineBinaryOpExpr expr) {
+    return visitDivExpr(expr);
+  }
+
+  ConstantIntDivisibility visitCeilDivExpr(AffineBinaryOpExpr expr) {
+    return visitDivExpr(expr);
+  }
+
+  /// Infer the divisibility of a mod expression. If the RHS is a constant,
+  /// the result divisibility is gcd(lhs_divisibility, rhs_constant), since
+  /// (d * k) mod c is always divisible by gcd(d, c). Furthermore, if the
+  /// LHS divisibility is itself divisible by the constant (i.e., d % c == 0),
+  /// then (d * k) mod c is always zero, represented as divisibility 0.
+  ConstantIntDivisibility visitModExpr(AffineBinaryOpExpr expr) {
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    auto constRhs = dyn_cast<AffineConstantExpr>(expr.getRHS());
+    if (!constRhs || constRhs.getValue() == 0) {
+      return ConstantIntDivisibility(1, 1);
+    }
+    auto constValue = static_cast<uint64_t>(std::abs(constRhs.getValue()));
+    ConstantIntDivisibility lhsDiv = visit(expr.getLHS());
+    // If the LHS is always a multiple of constValue, x mod constValue is
+    // always zero. Divisibility 0 is the lattice top ("divides everything").
+    uint64_t modUDiv = (lhsDiv.udiv() % constValue == 0)
+                           ? 0
+                           : std::gcd(lhsDiv.udiv(), constValue);
+    uint64_t modSDiv = (lhsDiv.sdiv() % constValue == 0)
+                           ? 0
+                           : std::gcd(lhsDiv.sdiv(), constValue);
+    return ConstantIntDivisibility(modUDiv, modSDiv);
+  }
+
+private:
+  ConstantIntDivisibility visitInvalidExpr(AffineBinaryOpExpr expr) {
+    return IntegerDivisibility::getMinDivisibility().getValue();
+  }
+
+  /// Helper shared by ceildiv and floordiv implementations. Returns the minimum
+  /// divisibility as a fallback if the divisor is not a constant, because the
+  /// divisibility cannot be inferred in this case. If the divisor is a
+  /// constant, then this function recursively visits the dividend, and returns
+  /// the quotient of the dividend's divisibility with the divisor.
+  ConstantIntDivisibility visitDivExpr(AffineBinaryOpExpr expr) {
+    if (divisibilityMap.contains(expr)) {
+      return divisibilityMap[expr];
+    }
+    auto constRhs = dyn_cast<AffineConstantExpr>(expr.getRHS());
+    // Division by zero is undefined, so return the minimum divisibility.
+    if (!constRhs || constRhs.getValue() == 0) {
+      return ConstantIntDivisibility(1, 1);
+    }
+    auto constValue = static_cast<uint64_t>(std::abs(constRhs.getValue()));
+    ConstantIntDivisibility lhsDiv = visit(expr.getLHS());
+    uint64_t divUDiv =
+        lhsDiv.udiv() % constValue == 0 ? lhsDiv.udiv() / constValue : 1;
+    uint64_t divSDiv =
+        lhsDiv.sdiv() % constValue == 0 ? lhsDiv.sdiv() / constValue : 1;
+    return ConstantIntDivisibility(divUDiv, divSDiv);
+  }
+
+  ExprDivisibilityMap &divisibilityMap;
+};
+
+/// Returns the divisibilities of each AffineMap result based on the
+/// divisibilities of its dims and symbols. The `dimAndSymbolDivisibilities`
+/// should contain the divisibilities of the dims, followed by the
+/// divisibilities of the symbols in ascending order by their positions.
+static SmallVector<ConstantIntDivisibility> getResultDivisibilities(
+    AffineMap map,
+    ArrayRef<ConstantIntDivisibility> dimAndSymbolDivisibilities) {
+  // Seed the AffineExprDivisibilityFinder with the dimAndSymbolDivisibilities.
+  llvm::DenseMap<AffineExpr, ConstantIntDivisibility> exprDivisibilityMap;
+  SmallVector<AffineExpr> inputExprs;
+  inputExprs.append(llvm::map_to_vector(
+      llvm::seq<int64_t>(map.getNumDims()),
+      [&](int64_t dim) { return getAffineDimExpr(dim, map.getContext()); }));
+  inputExprs.append(llvm::map_to_vector(
+      llvm::seq<int64_t>(map.getNumSymbols()),
+      [&](int64_t sym) { return getAffineSymbolExpr(sym, map.getContext()); }));
+  for (auto [expr, divisibility] :
+       llvm::zip_equal(inputExprs, dimAndSymbolDivisibilities)) {
+    exprDivisibilityMap[expr] = divisibility;
+  }
+  AffineExprDivisibilityFinder divisibilityFinder(exprDivisibilityMap);
+
+  // Walk each result expression and compute their divisibilities.
+  SmallVector<ConstantIntDivisibility> resultDivisibilities;
+  for (AffineExpr resultExpr : map.getResults()) {
+    resultDivisibilities.push_back(divisibilityFinder.visit(resultExpr));
+  }
+  return resultDivisibilities;
+}
+
+struct AffineApplyInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          AffineApplyInferIntDivisibilityOpInterface, affine::AffineApplyOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto affineApplyOp = cast<affine::AffineApplyOp>(op);
+    SmallVector<ConstantIntDivisibility> operandDivisibilities;
+    for (auto [operand, divisibility] :
+         llvm::zip(affineApplyOp.getOperands(), argDivs)) {
+      operandDivisibilities.push_back(
+          getDivisibilityOfOperand(operand, divisibility));
+    }
+
+    SmallVector<ConstantIntDivisibility> resultDivisibilities =
+        getResultDivisibilities(affineApplyOp.getMap(), operandDivisibilities);
+    for (auto [result, divisibility] :
+         llvm::zip_equal(affineApplyOp->getResults(), resultDivisibilities)) {
+      setResultDivs(result, divisibility);
+    }
+  }
+};
+
+/// Infer the result divisibility of an affine.min or affine.max operation
+/// based on its operand divisibilities. The result divisibility is the GCD
+/// of the divisibilities of each of the affine map results, because the result
+/// of the affine.min/max op could be any of these results.
+template <typename MinOrMaxTy>
+static void
+inferAffineMinOrMaxResultDivisibility(MinOrMaxTy minOrMaxOp,
+                                      ArrayRef<IntegerDivisibility> argDivs,
+                                      SetIntDivisibilityFn setResultDivs) {
+  static_assert(
+      llvm::is_one_of<MinOrMaxTy, affine::AffineMinOp,
+                      affine::AffineMaxOp>::value,
+      "MinOrMaxTy must be affine::AffineMinOp or affine::AffineMaxOp");
+  SmallVector<ConstantIntDivisibility> operandDivisibilities;
+  for (auto [operand, divisibility] :
+       llvm::zip(minOrMaxOp.getOperands(), argDivs)) {
+    operandDivisibilities.push_back(
+        getDivisibilityOfOperand(operand, divisibility));
+  }
+
+  SmallVector<ConstantIntDivisibility> resultDivisibilities =
+      getResultDivisibilities(minOrMaxOp.getMap(), operandDivisibilities);
+
+  ConstantIntDivisibility resultDivisibility =
+      resultDivisibilities.pop_back_val();
+  for (auto divisibility : resultDivisibilities) {
+    resultDivisibility = resultDivisibility.getUnion(divisibility);
+  }
+  setResultDivs(minOrMaxOp.getResult(), resultDivisibility);
+}
+
+struct AffineMinInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          AffineMinInferIntDivisibilityOpInterface, affine::AffineMinOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto affineMinOp = cast<affine::AffineMinOp>(op);
+    inferAffineMinOrMaxResultDivisibility(affineMinOp, argDivs, setResultDivs);
+  }
+};
+
+struct AffineMaxInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          AffineMaxInferIntDivisibilityOpInterface, affine::AffineMaxOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto affineMaxOp = cast<affine::AffineMaxOp>(op);
+    inferAffineMinOrMaxResultDivisibility(affineMaxOp, argDivs, setResultDivs);
+  }
+};
+
+struct AffineDelinearizeIndexInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          AffineDelinearizeIndexInferIntDivisibilityOpInterface,
+          affine::AffineDelinearizeIndexOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto delinearizeOp = cast<affine::AffineDelinearizeIndexOp>(op);
+    MLIRContext *ctx = op->getContext();
+
+    // Operands are: [linear_index, dynamic_basis_values...]
+    ConstantIntDivisibility linearDiv =
+        getDivisibilityOfOperand(delinearizeOp.getLinearIndex(), argDivs[0]);
+
+    ArrayRef<int64_t> staticBasis = delinearizeOp.getStaticBasis();
+    int64_t numResults = delinearizeOp.getNumResults();
+
+    // Build affine expressions for each result.
+    // Dim 0 = linear index, symbols = dynamic basis values.
+    AffineExpr linearExpr = getAffineDimExpr(0, ctx);
+
+    // Collect operand divisibilities: [linear_index_div, dynamic_basis_divs...]
+    SmallVector<ConstantIntDivisibility> operandDivs;
+    operandDivs.push_back(linearDiv);
+
+    // Map static/dynamic basis values to affine expressions.
+    int64_t dynIdx = 0;
+    SmallVector<AffineExpr> basisExprs;
+    for (int64_t i = 0, e = static_cast<int64_t>(staticBasis.size()); i < e;
+         ++i) {
+      if (ShapedType::isDynamic(staticBasis[i])) {
+        basisExprs.push_back(getAffineSymbolExpr(dynIdx, ctx));
+        operandDivs.push_back(getDivisibilityOfOperand(
+            delinearizeOp.getDynamicBasis()[dynIdx], argDivs[1 + dynIdx]));
+        dynIdx++;
+      } else {
+        basisExprs.push_back(getAffineConstantExpr(staticBasis[i], ctx));
+      }
+    }
+
+    // The computation basis skips the outer bound if present.
+    bool hasOuter = delinearizeOp.hasOuterBound();
+    int64_t basisStart = hasOuter ? 1 : 0;
+
+    // Each result[i] can be expressed as an affine expression of the linear
+    // index using the effective basis (after dropping outer bound if present).
+    // Effective basis B[k] = basisExprs[basisStart + k], for k = 0..N-2.
+    // Stride s[i] = product of B[i..N-2] = product of
+    //               basisExprs[basisStart+i .. end].
+    //
+    // result[0]   = x floordiv s[0]
+    // result[i>0] = (x floordiv s[i]) mod B[i-1]
+    // For i=N-1, s[N-1]=1, so result[N-1] = x mod B[N-2].
+
+    AffineExpr stride = getAffineConstantExpr(1, ctx);
+    for (int64_t i = numResults - 1; i >= 0; --i) {
+      AffineExpr resultExpr;
+      if (i == 0) {
+        resultExpr = linearExpr.floorDiv(stride);
+      } else {
+        resultExpr =
+            (linearExpr.floorDiv(stride)) % basisExprs[basisStart + i - 1];
+      }
+
+      AffineMap resultMap = AffineMap::get(1, dynIdx, resultExpr, ctx);
+      SmallVector<ConstantIntDivisibility> divs =
+          getResultDivisibilities(resultMap, operandDivs);
+      setResultDivs(delinearizeOp.getResult(i), divs[0]);
+
+      if (i > 0) {
+        stride = basisExprs[basisStart + i - 1] * stride;
+      }
+    }
+  }
+};
+
+} // namespace
+
+void mlir::affine::registerInferIntDivisibilityOpInterfaceExternalModels(
+    DialectRegistry &registry) {
+  registry.addExtension(+[](MLIRContext *context, AffineDialect *dialect) {
+    AffineApplyOp::attachInterface<AffineApplyInferIntDivisibilityOpInterface>(
+        *context);
+    AffineMinOp::attachInterface<AffineMinInferIntDivisibilityOpInterface>(
+        *context);
+    AffineMaxOp::attachInterface<AffineMaxInferIntDivisibilityOpInterface>(
+        *context);
+    AffineDelinearizeIndexOp::attachInterface<
+        AffineDelinearizeIndexInferIntDivisibilityOpInterface>(*context);
+  });
+}
diff --git a/mlir/lib/Dialect/Arith/IR/CMakeLists.txt b/mlir/lib/Dialect/Arith/IR/CMakeLists.txt
index 4beb99ccfdfba..3423e11a7d0f0 100644
--- a/mlir/lib/Dialect/Arith/IR/CMakeLists.txt
+++ b/mlir/lib/Dialect/Arith/IR/CMakeLists.txt
@@ -1,6 +1,7 @@
 set(LLVM_OPTIONAL_SOURCES
   ArithOps.cpp
   ArithDialect.cpp
+  InferIntDivisibilityOpInterfaceImpl.cpp
   InferIntRangeInterfaceImpls.cpp
   ValueBoundsOpInterfaceImpl.cpp
   )
@@ -12,6 +13,7 @@ add_public_tablegen_target(MLIRArithCanonicalizationIncGen)
 add_mlir_dialect_library(MLIRArithDialect
   ArithOps.cpp
   ArithDialect.cpp
+  InferIntDivisibilityOpInterfaceImpl.cpp
   InferIntRangeInterfaceImpls.cpp
 
   ADDITIONAL_HEADER_DIRS
@@ -24,6 +26,7 @@ add_mlir_dialect_library(MLIRArithDialect
   LINK_LIBS PUBLIC
   MLIRCastInterfaces
   MLIRDialect
+  MLIRInferIntDivisibilityOpInterface
   MLIRInferIntRangeCommon
   MLIRInferIntRangeInterface
   MLIRInferTypeOpInterface
diff --git a/mlir/lib/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.cpp b/mlir/lib/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.cpp
new file mode 100644
index 0000000000000..f06971ae88004
--- /dev/null
+++ b/mlir/lib/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.cpp
@@ -0,0 +1,162 @@
+//===- InferIntDivisibilityOpInterfaceImpl.cpp ----------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h"
+#include "mlir/Dialect/Arith/IR/Arith.h"
+#include "mlir/IR/DialectRegistry.h"
+#include "mlir/IR/Matchers.h"
+#include "mlir/Interfaces/InferIntDivisibilityOpInterface.h"
+
+#include <cstdlib>
+
+using namespace mlir;
+
+namespace {
+
+static ConstantIntDivisibility
+getDivisibilityOfOperand(Value v, IntegerDivisibility divisibility) {
+  if (!divisibility.isUninitialized()) {
+    return divisibility.getValue();
+  }
+  APInt intVal;
+  if (matchPattern(v, m_ConstantInt(&intVal))) {
+    uint64_t udiv = intVal.getZExtValue();
+    uint64_t sdiv = std::abs(intVal.getSExtValue());
+    return ConstantIntDivisibility(udiv, sdiv);
+  }
+  return ConstantIntDivisibility(1, 1);
+}
+
+/// Helper for binary arith ops whose result divisibility is the GCD (union) of
+/// their operands' divisibilities. This covers add, sub, min, and max.
+template <typename OpTy>
+struct ArithBinaryGCDInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          ArithBinaryGCDInferIntDivisibilityOpInterface<OpTy>, OpTy> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto binOp = cast<OpTy>(op);
+    auto lhsDiv = getDivisibilityOfOperand(binOp.getLhs(), argDivs[0]);
+    auto rhsDiv = getDivisibilityOfOperand(binOp.getRhs(), argDivs[1]);
+    setResultDivs(binOp.getResult(), lhsDiv.getUnion(rhsDiv));
+  }
+};
+
+/// For arith.select, the result divisibility is the GCD of the true and false
+/// operands' divisibilities. The condition (operand 0) is i1 and irrelevant.
+struct ArithSelectInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          ArithSelectInferIntDivisibilityOpInterface, arith::SelectOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto selectOp = cast<arith::SelectOp>(op);
+    // argDivs[0] is the condition (i1), argDivs[1] is true, argDivs[2] is
+    // false.
+    auto trueDiv =
+        getDivisibilityOfOperand(selectOp.getTrueValue(), argDivs[1]);
+    auto falseDiv =
+        getDivisibilityOfOperand(selectOp.getFalseValue(), argDivs[2]);
+    setResultDivs(selectOp.getResult(), trueDiv.getUnion(falseDiv));
+  }
+};
+
+struct ArithConstantInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          ArithConstantInferIntDivisibilityOpInterface, arith::ConstantOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto constOp = cast<arith::ConstantOp>(op);
+    auto constAttr = dyn_cast_if_present<IntegerAttr>(constOp.getValue());
+    if (constAttr) {
+      const APInt &value = constAttr.getValue();
+      uint64_t udiv = value.getZExtValue();
+      uint64_t sdiv = std::abs(value.getSExtValue());
+      setResultDivs(constOp.getResult(), ConstantIntDivisibility(udiv, sdiv));
+    }
+  }
+};
+
+struct ArithMulIInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          ArithMulIInferIntDivisibilityOpInterface, arith::MulIOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto mulOp = cast<arith::MulIOp>(op);
+
+    auto lhsDivisibility = getDivisibilityOfOperand(mulOp.getLhs(), argDivs[0]);
+    auto rhsDivisibility = getDivisibilityOfOperand(mulOp.getRhs(), argDivs[1]);
+
+    uint64_t mulUDiv = lhsDivisibility.udiv() * rhsDivisibility.udiv();
+    uint64_t mulSDiv = lhsDivisibility.sdiv() * rhsDivisibility.sdiv();
+
+    setResultDivs(mulOp.getResult(), ConstantIntDivisibility(mulUDiv, mulSDiv));
+  }
+};
+
+struct ArithDivUIInferIntDivisibilityOpInterface
+    : InferIntDivisibilityOpInterface::ExternalModel<
+          ArithDivUIInferIntDivisibilityOpInterface, arith::DivUIOp> {
+
+  void inferResultDivisibility(Operation *op,
+                               ArrayRef<IntegerDivisibility> argDivs,
+                               SetIntDivisibilityFn setResultDivs) const {
+    auto divOp = cast<arith::DivUIOp>(op);
+
+    APInt intVal;
+    if (!matchPattern(divOp.getRhs(), m_ConstantInt(&intVal))) {
+      return;
+    }
+
+    auto lhsDivisibility = getDivisibilityOfOperand(divOp.getLhs(), argDivs[0]);
+
+    uint64_t divUDiv = lhsDivisibility.udiv() % intVal.getZExtValue() == 0
+                           ? lhsDivisibility.udiv() / intVal.getZExtValue()
+                           : 1;
+    uint64_t divSDiv =
+        lhsDivisibility.sdiv() % std::abs(intVal.getSExtValue()) == 0
+            ? lhsDivisibility.sdiv() / std::abs(intVal.getSExtValue())
+            : 1;
+
+    setResultDivs(divOp, ConstantIntDivisibility(divUDiv, divSDiv));
+  }
+};
+
+} // namespace
+
+void mlir::arith::registerInferIntDivisibilityOpInterfaceExternalModels(
+    DialectRegistry &registry) {
+  registry.addExtension(+[](MLIRContext *context, ArithDialect *dialect) {
+    ConstantOp::attachInterface<ArithConstantInferIntDivisibilityOpInterface>(
+        *context);
+    MulIOp::attachInterface<ArithMulIInferIntDivisibilityOpInterface>(*context);
+    DivUIOp::attachInterface<ArithDivUIInferIntDivisibilityOpInterface>(
+        *context);
+    AddIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<AddIOp>>(*context);
+    SubIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<SubIOp>>(*context);
+    MinUIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<MinUIOp>>(*context);
+    MaxUIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<MaxUIOp>>(*context);
+    MinSIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<MinSIOp>>(*context);
+    MaxSIOp::attachInterface<
+        ArithBinaryGCDInferIntDivisibilityOpInterface<MaxSIOp>>(*context);
+    SelectOp::attachInterface<ArithSelectInferIntDivisibilityOpInterface>(
+        *context);
+  });
+}

>From 664c3bc9ef0a22b9a4ecb0b3c7d04315ebbdcb26 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Thu, 14 May 2026 05:33:29 -0700
Subject: [PATCH 6/7] [MLIR] Add divisibility-analysis test pass and lit tests

Adds the TestIntegerDivisibilityAnalysis pass, registers it in
mlir-opt, and provides lit tests under
mlir/test/Analysis/DataFlow/integer-divisibility.mlir.
---
 .../DataFlow/integer-divisibility.mlir        | 152 ++++++++++++++++++
 mlir/test/lib/Analysis/CMakeLists.txt         |   2 +
 .../TestIntegerDivisibilityAnalysis.cpp       |  97 +++++++++++
 mlir/tools/mlir-opt/mlir-opt.cpp              |   2 +
 4 files changed, 253 insertions(+)
 create mode 100644 mlir/test/Analysis/DataFlow/integer-divisibility.mlir
 create mode 100644 mlir/test/lib/Analysis/DataFlow/TestIntegerDivisibilityAnalysis.cpp

diff --git a/mlir/test/Analysis/DataFlow/integer-divisibility.mlir b/mlir/test/Analysis/DataFlow/integer-divisibility.mlir
new file mode 100644
index 0000000000000..7f9466e949d4c
--- /dev/null
+++ b/mlir/test/Analysis/DataFlow/integer-divisibility.mlir
@@ -0,0 +1,152 @@
+// RUN: mlir-opt --split-input-file --test-int-divisibility-analysis --allow-unregistered-dialect %s | FileCheck %s
+
+// CHECK-LABEL: @constant
+func.func @constant() -> index {
+  %0 = arith.constant 8 : index
+  // CHECK: divisibility = "udiv = 8, sdiv = 8"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @muli_constant
+func.func @muli_constant(%arg0 : index) -> index {
+  %c4 = arith.constant 4 : index
+  %0 = arith.muli %arg0, %c4 : index
+  // CHECK: divisibility = "udiv = 4, sdiv = 4"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @addi_gcd_of_muli_operands
+func.func @addi_gcd_of_muli_operands(%arg0 : index, %arg1 : index) -> index {
+  %c8 = arith.constant 8 : index
+  %c12 = arith.constant 12 : index
+  %a = arith.muli %arg0, %c8 : index
+  %b = arith.muli %arg1, %c12 : index
+  %0 = arith.addi %a, %b : index
+  // gcd(8, 12) = 4.
+  // CHECK: divisibility = "udiv = 4, sdiv = 4"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @addi_same_divisibility
+func.func @addi_same_divisibility(%arg0 : index, %arg1 : index) -> index {
+  %c16 = arith.constant 16 : index
+  %a = arith.muli %arg0, %c16 : index
+  %b = arith.muli %arg1, %c16 : index
+  %0 = arith.addi %a, %b : index
+  // CHECK: divisibility = "udiv = 16, sdiv = 16"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @affine_apply_mul
+func.func @affine_apply_mul(%arg0 : index) -> index {
+  %c2 = arith.constant 2 : index
+  %seed = arith.muli %arg0, %c2 : index
+  %0 = affine.apply affine_map<(d0) -> (d0 * 16)>(%seed)
+  // 2 * 16 = 32.
+  // CHECK: divisibility = "udiv = 32, sdiv = 32"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @affine_apply_mul_then_floordiv
+func.func @affine_apply_mul_then_floordiv(%arg0 : index) -> index {
+  %0 = affine.apply affine_map<(d0) -> (d0 * 16)>(%arg0)
+  %1 = affine.apply affine_map<(d0) -> (d0 floordiv 4)>(%0)
+  // 16 floordiv 4 = 4.
+  // CHECK: divisibility = "udiv = 4, sdiv = 4"
+  %2 = "test.int_divisibility"(%1) : (index) -> index
+  return %2 : index
+}
+
+// -----
+
+// CHECK-LABEL: @affine_apply_mod_zero
+func.func @affine_apply_mod_zero(%arg0 : index) -> index {
+  %0 = affine.apply affine_map<(d0) -> (d0 * 16)>(%arg0)
+  %1 = affine.apply affine_map<(d0) -> (d0 mod 16)>(%0)
+  // 16 % 16 == 0, so x mod 16 is always 0 -> divisibility 0 (lattice top).
+  // CHECK: divisibility = "udiv = 0, sdiv = 0"
+  %2 = "test.int_divisibility"(%1) : (index) -> index
+  return %2 : index
+}
+
+// -----
+
+// CHECK-LABEL: @affine_apply_constant
+func.func @affine_apply_constant() -> index {
+  %0 = affine.apply affine_map<() -> (64)>()
+  // CHECK: divisibility = "udiv = 64, sdiv = 64"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
+
+// -----
+
+// CHECK-LABEL: @scf_for_constant_step
+func.func @scf_for_constant_step() {
+  %c0 = arith.constant 0 : index
+  %c64 = arith.constant 64 : index
+  %c8 = arith.constant 8 : index
+  scf.for %iv = %c0 to %c64 step %c8 {
+    // CHECK: divisibility = "udiv = 8, sdiv = 8"
+    %0 = "test.int_divisibility"(%iv) : (index) -> index
+  }
+  return
+}
+
+// -----
+
+// CHECK-LABEL: @scf_for_nontrivial_gcd
+func.func @scf_for_nontrivial_gcd() {
+  %c12 = arith.constant 12 : index
+  %c100 = arith.constant 100 : index
+  %c18 = arith.constant 18 : index
+  scf.for %iv = %c12 to %c100 step %c18 {
+    // gcd(12, 18) = 6.
+    // CHECK: divisibility = "udiv = 6, sdiv = 6"
+    %0 = "test.int_divisibility"(%iv) : (index) -> index
+  }
+  return
+}
+
+// -----
+
+// CHECK-LABEL: @scf_for_coprime
+func.func @scf_for_coprime() {
+  %c15 = arith.constant 15 : index
+  %c100 = arith.constant 100 : index
+  %c8 = arith.constant 8 : index
+  scf.for %iv = %c15 to %c100 step %c8 {
+    // gcd(15, 8) = 1.
+    // CHECK: divisibility = "udiv = 1, sdiv = 1"
+    %0 = "test.int_divisibility"(%iv) : (index) -> index
+  }
+  return
+}
+
+// -----
+
+// CHECK-LABEL: @affine_apply_mul_plus_const
+func.func @affine_apply_mul_plus_const(%arg0 : index) -> index {
+  %c4 = arith.constant 4 : index
+  %seed = arith.muli %arg0, %c4 : index
+  %0 = affine.apply affine_map<(d0) -> (d0 * 8 + 16)>(%seed)
+  // seed has udiv = 4, multiplied by 8 -> 32, then +16. gcd(32,16) = 16.
+  // CHECK: divisibility = "udiv = 16, sdiv = 16"
+  %1 = "test.int_divisibility"(%0) : (index) -> index
+  return %1 : index
+}
diff --git a/mlir/test/lib/Analysis/CMakeLists.txt b/mlir/test/lib/Analysis/CMakeLists.txt
index c37671ade37b3..d86af5017f24b 100644
--- a/mlir/test/lib/Analysis/CMakeLists.txt
+++ b/mlir/test/lib/Analysis/CMakeLists.txt
@@ -15,6 +15,7 @@ add_mlir_library(MLIRTestAnalysis
   DataFlow/TestDeadCodeAnalysis.cpp
   DataFlow/TestDenseBackwardDataFlowAnalysis.cpp
   DataFlow/TestDenseForwardDataFlowAnalysis.cpp
+  DataFlow/TestIntegerDivisibilityAnalysis.cpp
   DataFlow/TestLivenessAnalysis.cpp
   DataFlow/TestSparseBackwardDataFlowAnalysis.cpp
   DataFlow/TestStridedMetadataRangeAnalysis.cpp
@@ -27,6 +28,7 @@ add_mlir_library(MLIRTestAnalysis
 mlir_target_link_libraries(MLIRTestAnalysis PUBLIC
   MLIRAffineDialect
   MLIRAnalysis
+  MLIRArithDialect
   MLIRFunctionInterfaces
   MLIRMemRefDialect
   MLIRPass
diff --git a/mlir/test/lib/Analysis/DataFlow/TestIntegerDivisibilityAnalysis.cpp b/mlir/test/lib/Analysis/DataFlow/TestIntegerDivisibilityAnalysis.cpp
new file mode 100644
index 0000000000000..dba8dfd6d37d8
--- /dev/null
+++ b/mlir/test/lib/Analysis/DataFlow/TestIntegerDivisibilityAnalysis.cpp
@@ -0,0 +1,97 @@
+//===- TestIntegerDivisibilityAnalysis.cpp - Test int divisibility --------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Analysis/DataFlow/ConstantPropagationAnalysis.h"
+#include "mlir/Analysis/DataFlow/DeadCodeAnalysis.h"
+#include "mlir/Analysis/DataFlow/IntegerDivisibilityAnalysis.h"
+#include "mlir/Analysis/DataFlowFramework.h"
+#include "mlir/Dialect/Affine/IR/AffineOps.h"
+#include "mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h"
+#include "mlir/Dialect/Arith/IR/Arith.h"
+#include "mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h"
+#include "mlir/IR/BuiltinAttributes.h"
+#include "mlir/IR/DialectRegistry.h"
+#include "mlir/IR/Operation.h"
+#include "mlir/Pass/Pass.h"
+#include "mlir/Pass/PassRegistry.h"
+#include "llvm/ADT/SmallVector.h"
+#include "llvm/Support/raw_ostream.h"
+
+using namespace mlir;
+using namespace mlir::dataflow;
+
+namespace {
+struct TestIntegerDivisibilityAnalysisPass
+    : public PassWrapper<TestIntegerDivisibilityAnalysisPass, OperationPass<>> {
+  MLIR_DEFINE_EXPLICIT_INTERNAL_INLINE_TYPE_ID(
+      TestIntegerDivisibilityAnalysisPass)
+
+  StringRef getArgument() const override {
+    return "test-int-divisibility-analysis";
+  }
+  StringRef getDescription() const override {
+    return "Test integer divisibility analysis by annotating "
+           "'test.int_divisibility' ops with the divisibility of their "
+           "operand.";
+  }
+
+  void getDependentDialects(DialectRegistry &registry) const override {
+    registry.insert<arith::ArithDialect, affine::AffineDialect>();
+    arith::registerInferIntDivisibilityOpInterfaceExternalModels(registry);
+    affine::registerInferIntDivisibilityOpInterfaceExternalModels(registry);
+  }
+
+  void runOnOperation() override {
+    Operation *rootOp = getOperation();
+    MLIRContext *context = &getContext();
+
+    // The pass is rooted on `test.int_divisibility` ops, which are expected
+    // to have a single operand for which to annotate divisibility information.
+    SmallVector<std::pair<Operation *, Value>> queryOps;
+    rootOp->walk([&](Operation *op) {
+      if (op->getName().getStringRef() == "test.int_divisibility" &&
+          op->getNumOperands() == 1)
+        queryOps.emplace_back(op, op->getOperand(0));
+    });
+
+    DataFlowSolver solver;
+    // DeadCodeAnalysis is the base analysis that allows the solver to traverse
+    // control flow. It is required by IntegerDivisibilityAnalysis.
+    solver.load<DeadCodeAnalysis>();
+    // SparseConstantPropagation allows the solver to call
+    // visitNonControlFlowArguments and analyze arguments like loop induction
+    // variables.
+    solver.load<SparseConstantPropagation>();
+    solver.load<IntegerDivisibilityAnalysis>();
+    if (failed(solver.initializeAndRun(rootOp)))
+      return signalPassFailure();
+
+    for (auto &[op, value] : queryOps) {
+      const auto *lattice =
+          solver.lookupState<IntegerDivisibilityLattice>(value);
+      if (!lattice || lattice->getValue().isUninitialized()) {
+        op->setAttr("divisibility", StringAttr::get(context, "uninitialized"));
+        continue;
+      }
+
+      // Format for the divisibility information is "udiv = X, sdiv = Y".
+      const auto &div = lattice->getValue().getValue();
+      std::string result;
+      llvm::raw_string_ostream os(result);
+      os << "udiv = " << div.udiv() << ", sdiv = " << div.sdiv();
+      op->setAttr("divisibility", StringAttr::get(context, result));
+    }
+  }
+};
+} // end anonymous namespace
+
+namespace mlir::test {
+void registerTestIntegerDivisibilityAnalysisPass() {
+  PassRegistration<TestIntegerDivisibilityAnalysisPass>();
+}
+} // end namespace mlir::test
diff --git a/mlir/tools/mlir-opt/mlir-opt.cpp b/mlir/tools/mlir-opt/mlir-opt.cpp
index c4754b3a08551..13c0934f34656 100644
--- a/mlir/tools/mlir-opt/mlir-opt.cpp
+++ b/mlir/tools/mlir-opt/mlir-opt.cpp
@@ -104,6 +104,7 @@ void registerTestComposeSubView();
 void registerTestMultiBuffering();
 void registerTestIRVisitorsPass();
 void registerTestGenericIRVisitorsPass();
+void registerTestIntegerDivisibilityAnalysisPass();
 void registerTestInterfaces();
 void registerTestIRVisitorsPass();
 void registerTestLastModifiedPass();
@@ -253,6 +254,7 @@ static void registerTestPasses() {
   mlir::test::registerTestMultiBuffering();
   mlir::test::registerTestIRVisitorsPass();
   mlir::test::registerTestGenericIRVisitorsPass();
+  mlir::test::registerTestIntegerDivisibilityAnalysisPass();
   mlir::test::registerTestInterfaces();
   mlir::test::registerTestIrdlTestDialectConversionPass();
   mlir::test::registerTestIRVisitorsPass();

>From 434d45b656f45a87032f4c212e491dffa1a17b76 Mon Sep 17 00:00:00 2001
From: Alan Li <me at alanli.org>
Date: Wed, 13 May 2026 11:46:42 -0700
Subject: [PATCH 7/7] [MLIR] Register divisibility analysis external models in
 RegisterAllDialects

Wires up arith::registerInferIntDivisibilityOpInterfaceExternalModels and
affine::registerInferIntDivisibilityOpInterfaceExternalModels in the
global dialect registry, so any caller that uses registerAllDialects
picks up divisibility inference on arith + affine ops automatically.

This is the last commit of the port -- IntegerDivisibilityAnalysis is now
self-contained and consumable by downstream passes (e.g., the MemRef and
Vector narrow-type emulation patterns) by passing a configured
DataFlowSolver* into the populate function.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply at anthropic.com>
---
 mlir/lib/RegisterAllDialects.cpp | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/mlir/lib/RegisterAllDialects.cpp b/mlir/lib/RegisterAllDialects.cpp
index 01a7401db4710..d9819a63f289b 100644
--- a/mlir/lib/RegisterAllDialects.cpp
+++ b/mlir/lib/RegisterAllDialects.cpp
@@ -15,8 +15,10 @@
 
 #include "mlir/Dialect/AMDGPU/IR/AMDGPUDialect.h"
 #include "mlir/Dialect/Affine/IR/AffineOps.h"
+#include "mlir/Dialect/Affine/IR/InferIntDivisibilityOpInterfaceImpl.h"
 #include "mlir/Dialect/Affine/IR/ValueBoundsOpInterfaceImpl.h"
 #include "mlir/Dialect/Arith/IR/Arith.h"
+#include "mlir/Dialect/Arith/IR/InferIntDivisibilityOpInterfaceImpl.h"
 #include "mlir/Dialect/Arith/IR/ValueBoundsOpInterfaceImpl.h"
 #include "mlir/Dialect/Arith/Transforms/BufferDeallocationOpInterfaceImpl.h"
 #include "mlir/Dialect/Arith/Transforms/BufferViewFlowOpInterfaceImpl.h"
@@ -159,10 +161,12 @@ void mlir::registerAllDialects(DialectRegistry &registry) {
   // clang-format on
 
   // Register all external models.
+  affine::registerInferIntDivisibilityOpInterfaceExternalModels(registry);
   affine::registerValueBoundsOpInterfaceExternalModels(registry);
   arith::registerBufferDeallocationOpInterfaceExternalModels(registry);
   arith::registerBufferizableOpInterfaceExternalModels(registry);
   arith::registerBufferViewFlowOpInterfaceExternalModels(registry);
+  arith::registerInferIntDivisibilityOpInterfaceExternalModels(registry);
   arith::registerShardingInterfaceExternalModels(registry);
   arith::registerValueBoundsOpInterfaceExternalModels(registry);
   bufferization::func_ext::registerBufferizableOpInterfaceExternalModels(



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