diff --git a/hypothesis-python/RELEASE.rst b/hypothesis-python/RELEASE.rst
new file mode 100644
index 0000000000..0eef32929a
--- /dev/null
+++ b/hypothesis-python/RELEASE.rst
@@ -0,0 +1,11 @@
+RELEASE_TYPE: minor
+
+This release automatically rewrites some simple filters, such as
+``integers().filter(lambda x: x > 9)`` to the more efficient
+``integers(min_value=10)``, based on the AST of the predicate.
+
+We continue to recommend using the efficient form directly wherever
+possible, but this should be useful for e.g. :pypi:`pandera` "``Checks``"
+where you already have a simple predicate and translating manually
+is really annoying.  See :issue:`2701` for ideas about floats and
+simple text strategies.
diff --git a/hypothesis-python/src/hypothesis/internal/filtering.py b/hypothesis-python/src/hypothesis/internal/filtering.py
index e2f8b02107..01520a7622 100644
--- a/hypothesis-python/src/hypothesis/internal/filtering.py
+++ b/hypothesis-python/src/hypothesis/internal/filtering.py
@@ -27,6 +27,8 @@
 See https://github.com/HypothesisWorks/hypothesis/issues/2701 for details.
 """
 
+import ast
+import inspect
 import math
 import operator
 from decimal import Decimal
@@ -35,6 +37,7 @@
 from typing import Any, Callable, Dict, NamedTuple, Optional, TypeVar
 
 from hypothesis.internal.compat import ceil, floor
+from hypothesis.internal.reflection import extract_lambda_source
 
 Ex = TypeVar("Ex")
 Predicate = Callable[[Ex], bool]
@@ -49,7 +52,7 @@ class ConstructivePredicate(NamedTuple):
         -> {"min_value": 0"}, None
 
         integers().filter(lambda x: x >= 0 and x % 7)
-        -> {"min_value": 0"}, lambda x: x % 7
+        -> {"min_value": 0}, lambda x: x % 7
 
     At least in principle - for now we usually return the predicate unchanged
     if needed.
@@ -66,6 +69,123 @@ def unchanged(cls, predicate):
         return cls({}, predicate)
 
 
+ARG = object()
+
+
+def convert(node: ast.AST, argname: str) -> object:
+    if isinstance(node, ast.Name):
+        if node.id != argname:
+            raise ValueError("Non-local variable")
+        return ARG
+    return ast.literal_eval(node)
+
+
+def comp_to_kwargs(x: ast.AST, op: ast.AST, y: ast.AST, *, argname: str) -> dict:
+    a = convert(x, argname)
+    b = convert(y, argname)
+    num = (int, float)
+    if not (a is ARG and isinstance(b, num)) and not (isinstance(a, num) and b is ARG):
+        # It would be possible to work out if comparisons between two literals
+        # are always true or false, but it's too rare to be worth the complexity.
+        # (and we can't even do `arg == arg`, because what if it's NaN?)
+        raise ValueError("Can't analyse this comparison")
+
+    if isinstance(op, ast.Lt):
+        if a is ARG:
+            return {"max_value": b, "exclude_max": True}
+        return {"min_value": a, "exclude_min": True}
+    elif isinstance(op, ast.LtE):
+        if a is ARG:
+            return {"max_value": b}
+        return {"min_value": a}
+    elif isinstance(op, ast.Eq):
+        if a is ARG:
+            return {"min_value": b, "max_value": b}
+        return {"min_value": a, "max_value": a}
+    elif isinstance(op, ast.GtE):
+        if a is ARG:
+            return {"min_value": b}
+        return {"max_value": a}
+    elif isinstance(op, ast.Gt):
+        if a is ARG:
+            return {"min_value": b, "exclude_min": True}
+        return {"max_value": a, "exclude_max": True}
+    raise ValueError("Unhandled comparison operator")  # e.g. ast.Ne
+
+
+def merge_preds(*con_predicates: ConstructivePredicate) -> ConstructivePredicate:
+    # This function is just kinda messy.  Unfortunately the neatest way
+    # to do this is just to roll out each case and handle them in turn.
+    base = {
+        "min_value": -math.inf,
+        "max_value": math.inf,
+        "exclude_min": False,
+        "exclude_max": False,
+    }
+    predicate = None
+    for kw, p in con_predicates:
+        predicate = p or predicate
+        if "min_value" in kw:
+            if kw["min_value"] > base["min_value"]:
+                base["exclude_min"] = kw.get("exclude_min", False)
+                base["min_value"] = kw["min_value"]
+            elif kw["min_value"] == base["min_value"]:
+                base["exclude_min"] |= kw.get("exclude_min", False)
+        if "max_value" in kw:
+            if kw["max_value"] < base["max_value"]:
+                base["exclude_max"] = kw.get("exclude_max", False)
+                base["max_value"] = kw["max_value"]
+            elif kw["max_value"] == base["max_value"]:
+                base["exclude_max"] |= kw.get("exclude_max", False)
+
+    if not base["exclude_min"]:
+        del base["exclude_min"]
+        if base["min_value"] == -math.inf:
+            del base["min_value"]
+    if not base["exclude_max"]:
+        del base["exclude_max"]
+        if base["max_value"] == math.inf:
+            del base["max_value"]
+    return ConstructivePredicate(base, predicate)
+
+
+def numeric_bounds_from_ast(
+    tree: ast.AST, argname: str, fallback: ConstructivePredicate
+) -> ConstructivePredicate:
+    """Take an AST; return a ConstructivePredicate.
+
+    >>> lambda x: x >= 0
+    {"min_value": 0}, None
+    >>> lambda x: x < 10
+    {"max_value": 10, "exclude_max": True}, None
+    >>> lambda x: x >= y
+    {}, lambda x: x >= y
+
+    See also https://greentreesnakes.readthedocs.io/en/latest/
+    """
+    if isinstance(tree, ast.Compare):
+        ops = tree.ops
+        vals = tree.comparators
+        comparisons = [(tree.left, ops[0], vals[0])]
+        for i, (op, val) in enumerate(zip(ops[1:], vals[1:]), start=1):
+            comparisons.append((vals[i - 1], op, val))
+        bounds = []
+        for comp in comparisons:
+            try:
+                kwargs = comp_to_kwargs(*comp, argname=argname)
+                bounds.append(ConstructivePredicate(kwargs, None))
+            except ValueError:
+                bounds.append(fallback)
+        return merge_preds(*bounds)
+
+    if isinstance(tree, ast.BoolOp) and isinstance(tree.op, ast.And):
+        return merge_preds(
+            *[numeric_bounds_from_ast(node, argname, fallback) for node in tree.values]
+        )
+
+    return fallback
+
+
 UNSATISFIABLE = ConstructivePredicate.unchanged(lambda _: False)
 
 
@@ -76,6 +196,7 @@ def get_numeric_predicate_bounds(predicate: Predicate) -> ConstructivePredicate:
     all the values are representable in the types that we're planning to generate
     so that the strategy validation doesn't complain.
     """
+    unchanged = ConstructivePredicate.unchanged(predicate)
     if (
         isinstance(predicate, partial)
         and len(predicate.args) == 1
@@ -87,7 +208,7 @@ def get_numeric_predicate_bounds(predicate: Predicate) -> ConstructivePredicate:
             or not isinstance(arg, (int, float, Fraction, Decimal))
             or math.isnan(arg)
         ):
-            return ConstructivePredicate.unchanged(predicate)
+            return unchanged
         options = {
             # We're talking about op(arg, x) - the reverse of our usual intuition!
             operator.lt: {"min_value": arg, "exclude_min": True},  # lambda x: arg < x
@@ -99,9 +220,38 @@ def get_numeric_predicate_bounds(predicate: Predicate) -> ConstructivePredicate:
         if predicate.func in options:
             return ConstructivePredicate(options[predicate.func], None)
 
-    # TODO: handle lambdas by AST analysis
+    # This section is a little complicated, but stepping through with comments should
+    # help to clarify it.  We start by finding the source code for our predicate and
+    # parsing it to an abstract syntax tree; if this fails for any reason we bail out
+    # and fall back to standard rejection sampling (a running theme).
+    try:
+        if predicate.__name__ == "<lambda>":
+            source = extract_lambda_source(predicate)
+        else:
+            source = inspect.getsource(predicate)
+        tree: ast.AST = ast.parse(source)
+    except Exception:
+        return unchanged
+
+    # Dig down to the relevant subtree - our tree is probably a Module containing
+    # either a FunctionDef, or an Expr which in turn contains a lambda definition.
+    while isinstance(tree, ast.Module) and len(tree.body) == 1:
+        tree = tree.body[0]
+    while isinstance(tree, ast.Expr):
+        tree = tree.value
 
-    return ConstructivePredicate.unchanged(predicate)
+    if isinstance(tree, ast.Lambda) and len(tree.args.args) == 1:
+        return numeric_bounds_from_ast(tree.body, tree.args.args[0].arg, unchanged)
+    elif isinstance(tree, ast.FunctionDef) and len(tree.args.args) == 1:
+        if len(tree.body) != 1 or not isinstance(tree.body[0], ast.Return):
+            # If the body of the function is anything but `return <expr>`,
+            # i.e. as simple as a lambda, we can't process it (yet).
+            return unchanged
+        argname = tree.args.args[0].arg
+        body = tree.body[0].value
+        assert isinstance(body, ast.AST)
+        return numeric_bounds_from_ast(body, argname, unchanged)
+    return unchanged
 
 
 def get_integer_predicate_bounds(predicate: Predicate) -> ConstructivePredicate:
diff --git a/hypothesis-python/tests/cover/test_filter_rewriting.py b/hypothesis-python/tests/cover/test_filter_rewriting.py
index 4183037225..4ca67a55ed 100644
--- a/hypothesis-python/tests/cover/test_filter_rewriting.py
+++ b/hypothesis-python/tests/cover/test_filter_rewriting.py
@@ -22,6 +22,7 @@
 
 from hypothesis import given, strategies as st
 from hypothesis.errors import Unsatisfiable
+from hypothesis.internal.reflection import get_pretty_function_description
 from hypothesis.strategies._internal.lazy import LazyStrategy
 from hypothesis.strategies._internal.numbers import IntegersStrategy
 from hypothesis.strategies._internal.strategies import FilteredStrategy
@@ -56,7 +57,28 @@
         (st.integers(), partial(operator.eq, 3), 3, 3),
         (st.integers(), partial(operator.ge, 3), None, 3),
         (st.integers(), partial(operator.gt, 3), None, 2),
+        # Simple lambdas
+        (st.integers(), lambda x: x < 3, None, 2),
+        (st.integers(), lambda x: x <= 3, None, 3),
+        (st.integers(), lambda x: x == 3, 3, 3),
+        (st.integers(), lambda x: x >= 3, 3, None),
+        (st.integers(), lambda x: x > 3, 4, None),
+        # Simple lambdas, reverse comparison
+        (st.integers(), lambda x: 3 > x, None, 2),
+        (st.integers(), lambda x: 3 >= x, None, 3),
+        (st.integers(), lambda x: 3 == x, 3, 3),
+        (st.integers(), lambda x: 3 <= x, 3, None),
+        (st.integers(), lambda x: 3 < x, 4, None),
+        # More complicated lambdas
+        (st.integers(), lambda x: 0 < x < 5, 1, 4),
+        (st.integers(), lambda x: 0 < x >= 1, 1, None),
+        (st.integers(), lambda x: 1 > x <= 0, None, 0),
+        (st.integers(), lambda x: x > 0 and x > 0, 1, None),
+        (st.integers(), lambda x: x < 1 and x < 1, None, 0),
+        (st.integers(), lambda x: x > 1 and x > 0, 2, None),
+        (st.integers(), lambda x: x < 1 and x < 2, None, 0),
     ],
+    ids=get_pretty_function_description,
 )
 @given(data=st.data())
 def test_filter_rewriting(data, strategy, predicate, start, end):
@@ -115,6 +137,9 @@ def mod2(x):
     return x % 2
 
 
+Y = 2 ** 20
+
+
 @given(
     data=st.data(),
     predicates=st.permutations(
@@ -124,6 +149,8 @@ def mod2(x):
             partial(operator.ge, 4),
             partial(operator.gt, 5),
             mod2,
+            lambda x: x > 2 or x % 7,
+            lambda x: 0 < x <= Y,
         ]
     ),
 )
@@ -142,4 +169,42 @@ def test_rewrite_filter_chains_with_some_unhandled(data, predicates):
     unwrapped = s.wrapped_strategy
     assert isinstance(unwrapped, FilteredStrategy)
     assert isinstance(unwrapped.filtered_strategy, IntegersStrategy)
-    assert unwrapped.flat_conditions == (mod2,)
+    for pred in unwrapped.flat_conditions:
+        assert pred is mod2 or pred.__name__ == "<lambda>"
+
+
+class NotAFunction:
+    def __call__(self, bar):
+        return True
+
+
+lambda_without_source = eval("lambda x: x > 2", {}, {})
+
+
+@pytest.mark.parametrize(
+    "start, end, predicate",
+    [
+        (1, 4, lambda x: 0 < x < 5 and x % 7),
+        (0, 9, lambda x: 0 <= x < 10 and x % 3),
+        (1, None, lambda x: 0 < x <= Y),
+        (None, None, lambda x: x == x),
+        (None, None, lambda x: 1 == 1),
+        (None, None, lambda x: 1 <= 2),
+        (None, None, lambda x: x != 0),
+        (None, None, NotAFunction()),
+        (None, None, lambda_without_source),
+        (None, None, lambda x, y=2: x >= 0),
+    ],
+)
+@given(data=st.data())
+def test_rewriting_partially_understood_filters(data, start, end, predicate):
+    s = st.integers().filter(predicate).wrapped_strategy
+
+    assert isinstance(s, FilteredStrategy)
+    assert isinstance(s.filtered_strategy, IntegersStrategy)
+    assert s.filtered_strategy.start == start
+    assert s.filtered_strategy.end == end
+    assert s.flat_conditions == (predicate,)
+
+    value = data.draw(s)
+    assert predicate(value)
diff --git a/hypothesis-python/tests/cover/test_filtered_strategy.py b/hypothesis-python/tests/cover/test_filtered_strategy.py
index 97ea7bbf9d..428b2a2086 100644
--- a/hypothesis-python/tests/cover/test_filtered_strategy.py
+++ b/hypothesis-python/tests/cover/test_filtered_strategy.py
@@ -19,7 +19,8 @@
 
 
 def test_filter_iterations_are_marked_as_discarded():
-    x = st.integers(0, 255).filter(lambda x: x == 0)
+    variable_equal_to_zero = 0  # non-local references disables filter-rewriting
+    x = st.integers(0, 255).filter(lambda x: x == variable_equal_to_zero)
 
     data = ConjectureData.for_buffer([2, 1, 0])