diff --git a/docs/marimo/pattern_rewrites.py b/docs/marimo/pattern_rewrites.py
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+++ b/docs/marimo/pattern_rewrites.py
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+# /// script
+# requires-python = ">=3.12"
+# dependencies = [
+#     "xdsl==0.27.0",
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.10.17"
+app = marimo.App()
+
+
+@app.cell(hide_code=True)
+def _():
+    import marimo as mo
+
+    from xdsl.utils import marimo as xmo
+
+    from xdsl.parser import Parser
+    from xdsl.context import MLContext
+
+    from xdsl.dialects.builtin import Builtin
+    from xdsl.dialects.arith import Arith
+    from xdsl.dialects.func import Func
+
+    from xdsl.pattern_rewriter import (
+        PatternRewriter,
+        RewritePattern,
+        GreedyRewritePatternApplier,
+        PatternRewriteWalker,
+    )
+    from xdsl.dialects.arith import AddiOp, ConstantOp, MuliOp
+    from xdsl.dialects.builtin import ModuleOp
+    from xdsl.dialects.func import FuncOp
+
+    return (
+        AddiOp,
+        Arith,
+        Builtin,
+        ConstantOp,
+        Func,
+        FuncOp,
+        GreedyRewritePatternApplier,
+        MLContext,
+        ModuleOp,
+        MuliOp,
+        Parser,
+        PatternRewriteWalker,
+        PatternRewriter,
+        RewritePattern,
+        mo,
+        xmo,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(r"""# Pattern Rewrites""")
+    return
+
+
+@app.cell(hide_code=True)
+def _(Parser, ctx):
+    _before_text = """\
+    func.func @my_func(%x : index) -> index {
+      %c0 = arith.constant 0 : index
+      %c2 = arith.constant 2 : index
+      %also_x = arith.addi %c0, %x : index
+      %two_x = arith.muli %also_x, %c2 : index
+      func.return %two_x : index
+    }
+    """
+    before_module = Parser(ctx, _before_text).parse_module()
+    None
+    return (before_module,)
+
+
+@app.cell(hide_code=True)
+def _(Parser, ctx):
+    _after_text = """\
+    func.func @my_func(%x : index) -> index {
+      %two_x = arith.addi %x, %x : index
+      func.return %two_x : index
+    }
+    """
+    after_module = Parser(ctx, _after_text).parse_module()
+    None
+    return (after_module,)
+
+
+@app.cell(hide_code=True)
+def _(after_module, before_module, mo, xmo):
+    mo.md(
+        f"""
+    Most of the work in compilers is rewriting IR.
+    While some transformations have to take modules into account, some can be done locally, such as rewriting an addition with zero to just use the non-zero input.
+    These kinds of rewrites are called pattern rewrites, as they apply if some local pattern is matched.
+
+    We can use xDSL's pattern rewriting API to rewrite this function:
+
+    {xmo.module_html(before_module)}
+
+    Into this one:
+
+    {xmo.module_html(after_module)}
+    """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Rationale
+
+        A pattern rewrite is a compiler transformation that matches a DAG in the IR, and replace it with another DAG. For instance, simplifying `x + 0` to `x` is a common optimization that is represented as a pattern rewrite.
+
+        As xDSL and MLIR allow the definition of higher-level dialects, pattern rewrites are very common, and can be used to both write high-level optimizations, and lowerings from a high-level dialect to a lower-level one. A general rationale for pattern rewrites in MLIR can be found [here](https://mlir.llvm.org/docs/Rationale/RationaleGenericDAGRewriter/).
+
+        Pattern rewrites are applied step by step on the IR. For instance, using the rewrite `x + 0 -> x`, the IR `x + 0 + 0` will be progressively transformed to `x + 0`, then `x using this single rewrite. Different application ordering and variations exist for rewrite patterns, which will be covered by Matthias Springer at 4pm on day 3 of the winter school.
+
+        ## Defining a pattern rewrite
+
+        Each Pattern rewrite is a class that inherit from `PatternRewrite`. It defines a single `match_and_rewrite` method, that is called to apply a pattern on an operation. The method will either return without any modification of the IR, or will modify the IR using the `Rewriter`. The most common operation to call on the `Rewriter` is `replace_matched_op`, which will replace the matched operation with a sequence of other operations, and replace the result values of the matched operation with new values.
+
+        Here are two examples of pattern for `x + 0 -> x` and `x * 2 -> x + x`:
+        """
+    )
+    return
+
+
+@app.cell
+def _(
+    AddiOp,
+    ConstantOp,
+    CostantOp,
+    MuliOp,
+    Operation,
+    RewritePattern,
+    Rewriter,
+):
+    # The rewrite x + 0 -> x
+    class AddZeroPattern(RewritePattern):
+        def match_and_rewrite(self, op: Operation, rewriter: Rewriter):
+            # Match an `arith.addi`
+            if not isinstance(op, AddiOp):
+                return
+
+            # Check if the right hand side is a constant
+            if not isinstance(cst := op.rhs.owner, ConstantOp):
+                return
+
+            # Check if the constant is 0
+            if cst.value.value.data != 0:
+                return
+
+            # Replace `x + 0` with `x`
+            # The first argument is the new operations to insert
+            # The second argument is the replacement for the results of the matched
+            # operation (op). As we replace `x + 0` with `x`, we replace the
+            # `arith.addi` results with its first argument.
+            rewriter.replace_matched_op([], new_results=[op.lhs])
+
+    # The rewrite x * 2 -> x + x
+    class MulTwoPattern(RewritePattern):
+        def match_and_rewrite(self, op: Operation, rewriter: Rewriter):
+            # Match an `arith.addi`
+            if not isinstance(op, MuliOp):
+                return
+            x = op.lhs
+
+            # Check if the right hand side is a constant
+            if not isinstance(cst := op.rhs.owner, CostantOp):
+                return
+
+            # Check if the constant is 2
+            if cst.value.value.data != 2:
+                return
+
+            # Replace `x * 2` with `x + x`
+            # The results of the `arith.muli` operation are by default the results of the
+            # last operation added, so here the `arith.addi`
+            add = AddiOp(x, x)
+            rewriter.replace_matched_op([add])
+
+    return AddZeroPattern, MulTwoPattern
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Applying rewrite patterns
+
+        There are two steps to apply rewrite patterns in xDSL.
+
+        * First, combining multiple rewrite patterns in a single one, giving priorities between patterns.
+        * Then, walking the IR and applying the pattern recursively on all operations until no more pattern can be applied on any operation:
+
+        Here is an example on how to apply rewrites on all operations in a module:
+        """
+    )
+    return
+
+
+@app.cell
+def _(
+    AddZeroPattern,
+    GreedyRewritePatternApplier,
+    ModuleOp,
+    MulTwoPattern,
+    Operation,
+    PatternRewriteWalker,
+):
+    def apply_all_rewrites(module: ModuleOp, rewrites: list[Operation]):
+        merged_pattern = GreedyRewritePatternApplier(AddZeroPattern(), MulTwoPattern())
+        walker = PatternRewriteWalker(merged_pattern)
+        walker.rewrite_module(module)
+
+    return (apply_all_rewrites,)
+
+
+@app.cell(hide_code=True)
+def _(Arith, Builtin, Func, MLContext):
+    ctx = MLContext()
+    ctx.load_dialect(Builtin)
+    ctx.load_dialect(Arith)
+    ctx.load_dialect(Func)
+    return (ctx,)
+
+
+if __name__ == "__main__":
+    app.run()