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diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/__init__.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/__init__.py
new file mode 100644
index 0000000000000000000000000000000000000000..bb4c5aa8afe6fd818e136ec0797b7429e2da76cf
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/__init__.py
@@ -0,0 +1,50 @@
+""" Rewrite Rules
+
+DISCLAIMER: This module is experimental. The interface is subject to change.
+
+A rule is a function that transforms one expression into another
+
+    Rule :: Expr -> Expr
+
+A strategy is a function that says how a rule should be applied to a syntax
+tree. In general strategies take rules and produce a new rule
+
+    Strategy :: [Rules], Other-stuff -> Rule
+
+This allows developers to separate a mathematical transformation from the
+algorithmic details of applying that transformation. The goal is to separate
+the work of mathematical programming from algorithmic programming.
+
+Submodules
+
+strategies.rl         - some fundamental rules
+strategies.core       - generic non-SymPy specific strategies
+strategies.traverse   - strategies that traverse a SymPy tree
+strategies.tools      - some conglomerate strategies that do depend on SymPy
+"""
+
+from . import rl
+from . import traverse
+from .rl import rm_id, unpack, flatten, sort, glom, distribute, rebuild
+from .util import new
+from .core import (
+    condition, debug, chain, null_safe, do_one, exhaust, minimize, tryit)
+from .tools import canon, typed
+from . import branch
+
+__all__ = [
+    'rl',
+
+    'traverse',
+
+    'rm_id', 'unpack', 'flatten', 'sort', 'glom', 'distribute', 'rebuild',
+
+    'new',
+
+    'condition', 'debug', 'chain', 'null_safe', 'do_one', 'exhaust',
+    'minimize', 'tryit',
+
+    'canon', 'typed',
+
+    'branch',
+]
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diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/__init__.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/__init__.py
new file mode 100644
index 0000000000000000000000000000000000000000..fec5afe84a58f3d887a8c762692a3673a2b6d4c8
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/__init__.py
@@ -0,0 +1,14 @@
+from . import traverse
+from .core import (
+    condition, debug, multiplex, exhaust, notempty,
+    chain, onaction, sfilter, yieldify, do_one, identity)
+from .tools import canon
+
+__all__ = [
+    'traverse',
+
+    'condition', 'debug', 'multiplex', 'exhaust', 'notempty', 'chain',
+    'onaction', 'sfilter', 'yieldify', 'do_one', 'identity',
+
+    'canon',
+]
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/core.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/core.py
new file mode 100644
index 0000000000000000000000000000000000000000..2dabaef69b60d994799f71414699223f84e1809b
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/core.py
@@ -0,0 +1,116 @@
+""" Generic SymPy-Independent Strategies """
+
+
+def identity(x):
+    yield x
+
+
+def exhaust(brule):
+    """ Apply a branching rule repeatedly until it has no effect """
+    def exhaust_brl(expr):
+        seen = {expr}
+        for nexpr in brule(expr):
+            if nexpr not in seen:
+                seen.add(nexpr)
+                yield from exhaust_brl(nexpr)
+        if seen == {expr}:
+            yield expr
+    return exhaust_brl
+
+
+def onaction(brule, fn):
+    def onaction_brl(expr):
+        for result in brule(expr):
+            if result != expr:
+                fn(brule, expr, result)
+            yield result
+    return onaction_brl
+
+
+def debug(brule, file=None):
+    """ Print the input and output expressions at each rule application """
+    if not file:
+        from sys import stdout
+        file = stdout
+
+    def write(brl, expr, result):
+        file.write("Rule: %s\n" % brl.__name__)
+        file.write("In: %s\nOut: %s\n\n" % (expr, result))
+
+    return onaction(brule, write)
+
+
+def multiplex(*brules):
+    """ Multiplex many branching rules into one """
+    def multiplex_brl(expr):
+        seen = set()
+        for brl in brules:
+            for nexpr in brl(expr):
+                if nexpr not in seen:
+                    seen.add(nexpr)
+                    yield nexpr
+    return multiplex_brl
+
+
+def condition(cond, brule):
+    """ Only apply branching rule if condition is true """
+    def conditioned_brl(expr):
+        if cond(expr):
+            yield from brule(expr)
+        else:
+            pass
+    return conditioned_brl
+
+
+def sfilter(pred, brule):
+    """ Yield only those results which satisfy the predicate """
+    def filtered_brl(expr):
+        yield from filter(pred, brule(expr))
+    return filtered_brl
+
+
+def notempty(brule):
+    def notempty_brl(expr):
+        yielded = False
+        for nexpr in brule(expr):
+            yielded = True
+            yield nexpr
+        if not yielded:
+            yield expr
+    return notempty_brl
+
+
+def do_one(*brules):
+    """ Execute one of the branching rules """
+    def do_one_brl(expr):
+        yielded = False
+        for brl in brules:
+            for nexpr in brl(expr):
+                yielded = True
+                yield nexpr
+            if yielded:
+                return
+    return do_one_brl
+
+
+def chain(*brules):
+    """
+    Compose a sequence of brules so that they apply to the expr sequentially
+    """
+    def chain_brl(expr):
+        if not brules:
+            yield expr
+            return
+
+        head, tail = brules[0], brules[1:]
+        for nexpr in head(expr):
+            yield from chain(*tail)(nexpr)
+
+    return chain_brl
+
+
+def yieldify(rl):
+    """ Turn a rule into a branching rule """
+    def brl(expr):
+        yield rl(expr)
+    return brl
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/__init__.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/__init__.py
new file mode 100644
index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_core.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_core.py
new file mode 100644
index 0000000000000000000000000000000000000000..ac620b0afb6dbadc4d97b29ddbb341cd920b6588
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_core.py
@@ -0,0 +1,117 @@
+from sympy.strategies.branch.core import (
+    exhaust, debug, multiplex, condition, notempty, chain, onaction, sfilter,
+    yieldify, do_one, identity)
+
+
+def posdec(x):
+    if x > 0:
+        yield x - 1
+    else:
+        yield x
+
+
+def branch5(x):
+    if 0 < x < 5:
+        yield x - 1
+    elif 5 < x < 10:
+        yield x + 1
+    elif x == 5:
+        yield x + 1
+        yield x - 1
+    else:
+        yield x
+
+
+def even(x):
+    return x % 2 == 0
+
+
+def inc(x):
+    yield x + 1
+
+
+def one_to_n(n):
+    yield from range(n)
+
+
+def test_exhaust():
+    brl = exhaust(branch5)
+    assert set(brl(3)) == {0}
+    assert set(brl(7)) == {10}
+    assert set(brl(5)) == {0, 10}
+
+
+def test_debug():
+    from io import StringIO
+    file = StringIO()
+    rl = debug(posdec, file)
+    list(rl(5))
+    log = file.getvalue()
+    file.close()
+
+    assert posdec.__name__ in log
+    assert '5' in log
+    assert '4' in log
+
+
+def test_multiplex():
+    brl = multiplex(posdec, branch5)
+    assert set(brl(3)) == {2}
+    assert set(brl(7)) == {6, 8}
+    assert set(brl(5)) == {4, 6}
+
+
+def test_condition():
+    brl = condition(even, branch5)
+    assert set(brl(4)) == set(branch5(4))
+    assert set(brl(5)) == set()
+
+
+def test_sfilter():
+    brl = sfilter(even, one_to_n)
+    assert set(brl(10)) == {0, 2, 4, 6, 8}
+
+
+def test_notempty():
+    def ident_if_even(x):
+        if even(x):
+            yield x
+
+    brl = notempty(ident_if_even)
+    assert set(brl(4)) == {4}
+    assert set(brl(5)) == {5}
+
+
+def test_chain():
+    assert list(chain()(2)) == [2]  # identity
+    assert list(chain(inc, inc)(2)) == [4]
+    assert list(chain(branch5, inc)(4)) == [4]
+    assert set(chain(branch5, inc)(5)) == {5, 7}
+    assert list(chain(inc, branch5)(5)) == [7]
+
+
+def test_onaction():
+    L = []
+
+    def record(fn, input, output):
+        L.append((input, output))
+
+    list(onaction(inc, record)(2))
+    assert L == [(2, 3)]
+
+    list(onaction(identity, record)(2))
+    assert L == [(2, 3)]
+
+
+def test_yieldify():
+    yinc = yieldify(lambda x: x + 1)
+    assert list(yinc(3)) == [4]
+
+
+def test_do_one():
+    def bad(expr):
+        raise ValueError
+
+    assert list(do_one(inc)(3)) == [4]
+    assert list(do_one(inc, bad)(3)) == [4]
+    assert list(do_one(inc, posdec)(3)) == [4]
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_tools.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_tools.py
new file mode 100644
index 0000000000000000000000000000000000000000..c2bd224030c337f0a000d94f6e7e65f3b8bd118f
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_tools.py
@@ -0,0 +1,42 @@
+from sympy.strategies.branch.tools import canon
+from sympy.core.basic import Basic
+from sympy.core.numbers import Integer
+from sympy.core.singleton import S
+
+
+def posdec(x):
+    if isinstance(x, Integer) and x > 0:
+        yield x - 1
+    else:
+        yield x
+
+
+def branch5(x):
+    if isinstance(x, Integer):
+        if 0 < x < 5:
+            yield x - 1
+        elif 5 < x < 10:
+            yield x + 1
+        elif x == 5:
+            yield x + 1
+            yield x - 1
+        else:
+            yield x
+
+
+def test_zero_ints():
+    expr = Basic(S(2), Basic(S(5), S(3)), S(8))
+    expected = {Basic(S(0), Basic(S(0), S(0)), S(0))}
+
+    brl = canon(posdec)
+    assert set(brl(expr)) == expected
+
+
+def test_split5():
+    expr = Basic(S(2), Basic(S(5), S(3)), S(8))
+    expected = {
+        Basic(S(0), Basic(S(0), S(0)), S(10)),
+        Basic(S(0), Basic(S(10), S(0)), S(10))}
+
+    brl = canon(branch5)
+    assert set(brl(expr)) == expected
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_traverse.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_traverse.py
new file mode 100644
index 0000000000000000000000000000000000000000..e051928210981223004de28b8c617d0438e11ac6
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tests/test_traverse.py
@@ -0,0 +1,53 @@
+from sympy.core.basic import Basic
+from sympy.core.numbers import Integer
+from sympy.core.singleton import S
+from sympy.strategies.branch.traverse import top_down, sall
+from sympy.strategies.branch.core import do_one, identity
+
+
+def inc(x):
+    if isinstance(x, Integer):
+        yield x + 1
+
+
+def test_top_down_easy():
+    expr = Basic(S(1), S(2))
+    expected = Basic(S(2), S(3))
+    brl = top_down(inc)
+
+    assert set(brl(expr)) == {expected}
+
+
+def test_top_down_big_tree():
+    expr = Basic(S(1), Basic(S(2)), Basic(S(3), Basic(S(4)), S(5)))
+    expected = Basic(S(2), Basic(S(3)), Basic(S(4), Basic(S(5)), S(6)))
+    brl = top_down(inc)
+
+    assert set(brl(expr)) == {expected}
+
+
+def test_top_down_harder_function():
+    def split5(x):
+        if x == 5:
+            yield x - 1
+            yield x + 1
+
+    expr = Basic(Basic(S(5), S(6)), S(1))
+    expected = {Basic(Basic(S(4), S(6)), S(1)), Basic(Basic(S(6), S(6)), S(1))}
+    brl = top_down(split5)
+
+    assert set(brl(expr)) == expected
+
+
+def test_sall():
+    expr = Basic(S(1), S(2))
+    expected = Basic(S(2), S(3))
+    brl = sall(inc)
+
+    assert list(brl(expr)) == [expected]
+
+    expr = Basic(S(1), S(2), Basic(S(3), S(4)))
+    expected = Basic(S(2), S(3), Basic(S(3), S(4)))
+    brl = sall(do_one(inc, identity))
+
+    assert list(brl(expr)) == [expected]
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tools.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tools.py
new file mode 100644
index 0000000000000000000000000000000000000000..a6c9097323a7962080ae4497ead976818e386518
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/tools.py
@@ -0,0 +1,12 @@
+from .core import exhaust, multiplex
+from .traverse import top_down
+
+
+def canon(*rules):
+    """ Strategy for canonicalization
+
+    Apply each branching rule in a top-down fashion through the tree.
+    Multiplex through all branching rule traversals
+    Keep doing this until there is no change.
+    """
+    return exhaust(multiplex(*map(top_down, rules)))
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/traverse.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/traverse.py
new file mode 100644
index 0000000000000000000000000000000000000000..28b7098dbda401fc0f0b6d27988d8c37e2f231ae
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/branch/traverse.py
@@ -0,0 +1,25 @@
+""" Branching Strategies to Traverse a Tree """
+from itertools import product
+from sympy.strategies.util import basic_fns
+from .core import chain, identity, do_one
+
+
+def top_down(brule, fns=basic_fns):
+    """ Apply a rule down a tree running it on the top nodes first """
+    return chain(do_one(brule, identity),
+                 lambda expr: sall(top_down(brule, fns), fns)(expr))
+
+
+def sall(brule, fns=basic_fns):
+    """ Strategic all - apply rule to args """
+    op, new, children, leaf = map(fns.get, ('op', 'new', 'children', 'leaf'))
+
+    def all_rl(expr):
+        if leaf(expr):
+            yield expr
+        else:
+            myop = op(expr)
+            argss = product(*map(brule, children(expr)))
+            for args in argss:
+                yield new(myop, *args)
+    return all_rl
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/core.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/core.py
new file mode 100644
index 0000000000000000000000000000000000000000..75b75cb5f2e0693eea98a7b1c9b3e7f036ec26f6
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/core.py
@@ -0,0 +1,151 @@
+""" Generic SymPy-Independent Strategies """
+from __future__ import annotations
+from collections.abc import Callable, Mapping
+from typing import TypeVar
+from sys import stdout
+
+
+_S = TypeVar('_S')
+_T = TypeVar('_T')
+
+
+def identity(x: _T) -> _T:
+    return x
+
+
+def exhaust(rule: Callable[[_T], _T]) -> Callable[[_T], _T]:
+    """ Apply a rule repeatedly until it has no effect """
+    def exhaustive_rl(expr: _T) -> _T:
+        new, old = rule(expr), expr
+        while new != old:
+            new, old = rule(new), new
+        return new
+    return exhaustive_rl
+
+
+def memoize(rule: Callable[[_S], _T]) -> Callable[[_S], _T]:
+    """Memoized version of a rule
+
+    Notes
+    =====
+
+    This cache can grow infinitely, so it is not recommended to use this
+    than ``functools.lru_cache`` unless you need very heavy computation.
+    """
+    cache: dict[_S, _T] = {}
+
+    def memoized_rl(expr: _S) -> _T:
+        if expr in cache:
+            return cache[expr]
+        else:
+            result = rule(expr)
+            cache[expr] = result
+            return result
+    return memoized_rl
+
+
+def condition(
+    cond: Callable[[_T], bool], rule: Callable[[_T], _T]
+) -> Callable[[_T], _T]:
+    """ Only apply rule if condition is true """
+    def conditioned_rl(expr: _T) -> _T:
+        if cond(expr):
+            return rule(expr)
+        return expr
+    return conditioned_rl
+
+
+def chain(*rules: Callable[[_T], _T]) -> Callable[[_T], _T]:
+    """
+    Compose a sequence of rules so that they apply to the expr sequentially
+    """
+    def chain_rl(expr: _T) -> _T:
+        for rule in rules:
+            expr = rule(expr)
+        return expr
+    return chain_rl
+
+
+def debug(rule, file=None):
+    """ Print out before and after expressions each time rule is used """
+    if file is None:
+        file = stdout
+
+    def debug_rl(*args, **kwargs):
+        expr = args[0]
+        result = rule(*args, **kwargs)
+        if result != expr:
+            file.write("Rule: %s\n" % rule.__name__)
+            file.write("In:   %s\nOut:  %s\n\n" % (expr, result))
+        return result
+    return debug_rl
+
+
+def null_safe(rule: Callable[[_T], _T | None]) -> Callable[[_T], _T]:
+    """ Return original expr if rule returns None """
+    def null_safe_rl(expr: _T) -> _T:
+        result = rule(expr)
+        if result is None:
+            return expr
+        return result
+    return null_safe_rl
+
+
+def tryit(rule: Callable[[_T], _T], exception) -> Callable[[_T], _T]:
+    """ Return original expr if rule raises exception """
+    def try_rl(expr: _T) -> _T:
+        try:
+            return rule(expr)
+        except exception:
+            return expr
+    return try_rl
+
+
+def do_one(*rules: Callable[[_T], _T]) -> Callable[[_T], _T]:
+    """ Try each of the rules until one works. Then stop. """
+    def do_one_rl(expr: _T) -> _T:
+        for rl in rules:
+            result = rl(expr)
+            if result != expr:
+                return result
+        return expr
+    return do_one_rl
+
+
+def switch(
+    key: Callable[[_S], _T],
+    ruledict: Mapping[_T, Callable[[_S], _S]]
+) -> Callable[[_S], _S]:
+    """ Select a rule based on the result of key called on the function """
+    def switch_rl(expr: _S) -> _S:
+        rl = ruledict.get(key(expr), identity)
+        return rl(expr)
+    return switch_rl
+
+
+# XXX Untyped default argument for minimize function
+# where python requires SupportsRichComparison type
+def _identity(x):
+    return x
+
+
+def minimize(
+    *rules: Callable[[_S], _T],
+    objective=_identity
+) -> Callable[[_S], _T]:
+    """ Select result of rules that minimizes objective
+
+    >>> from sympy.strategies import minimize
+    >>> inc = lambda x: x + 1
+    >>> dec = lambda x: x - 1
+    >>> rl = minimize(inc, dec)
+    >>> rl(4)
+    3
+
+    >>> rl = minimize(inc, dec, objective=lambda x: -x)  # maximize
+    >>> rl(4)
+    5
+    """
+    def minrule(expr: _S) -> _T:
+        return min([rule(expr) for rule in rules], key=objective)
+    return minrule
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/rl.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/rl.py
new file mode 100644
index 0000000000000000000000000000000000000000..c002f8b9aeb73015c1246b44ab0c11710c4d5a72
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/rl.py
@@ -0,0 +1,176 @@
+""" Generic Rules for SymPy
+
+This file assumes knowledge of Basic and little else.
+"""
+from sympy.utilities.iterables import sift
+from .util import new
+
+
+# Functions that create rules
+def rm_id(isid, new=new):
+    """ Create a rule to remove identities.
+
+    isid - fn :: x -> Bool  --- whether or not this element is an identity.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies import rm_id
+    >>> from sympy import Basic, S
+    >>> remove_zeros = rm_id(lambda x: x==0)
+    >>> remove_zeros(Basic(S(1), S(0), S(2)))
+    Basic(1, 2)
+    >>> remove_zeros(Basic(S(0), S(0))) # If only identites then we keep one
+    Basic(0)
+
+    See Also:
+        unpack
+    """
+    def ident_remove(expr):
+        """ Remove identities """
+        ids = list(map(isid, expr.args))
+        if sum(ids) == 0:           # No identities. Common case
+            return expr
+        elif sum(ids) != len(ids):  # there is at least one non-identity
+            return new(expr.__class__,
+                       *[arg for arg, x in zip(expr.args, ids) if not x])
+        else:
+            return new(expr.__class__, expr.args[0])
+
+    return ident_remove
+
+
+def glom(key, count, combine):
+    """ Create a rule to conglomerate identical args.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies import glom
+    >>> from sympy import Add
+    >>> from sympy.abc import x
+
+    >>> key     = lambda x: x.as_coeff_Mul()[1]
+    >>> count   = lambda x: x.as_coeff_Mul()[0]
+    >>> combine = lambda cnt, arg: cnt * arg
+    >>> rl = glom(key, count, combine)
+
+    >>> rl(Add(x, -x, 3*x, 2, 3, evaluate=False))
+    3*x + 5
+
+    Wait, how are key, count and combine supposed to work?
+
+    >>> key(2*x)
+    x
+    >>> count(2*x)
+    2
+    >>> combine(2, x)
+    2*x
+    """
+    def conglomerate(expr):
+        """ Conglomerate together identical args x + x -> 2x """
+        groups = sift(expr.args, key)
+        counts = {k: sum(map(count, args)) for k, args in groups.items()}
+        newargs = [combine(cnt, mat) for mat, cnt in counts.items()]
+        if set(newargs) != set(expr.args):
+            return new(type(expr), *newargs)
+        else:
+            return expr
+
+    return conglomerate
+
+
+def sort(key, new=new):
+    """ Create a rule to sort by a key function.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies import sort
+    >>> from sympy import Basic, S
+    >>> sort_rl = sort(str)
+    >>> sort_rl(Basic(S(3), S(1), S(2)))
+    Basic(1, 2, 3)
+    """
+
+    def sort_rl(expr):
+        return new(expr.__class__, *sorted(expr.args, key=key))
+    return sort_rl
+
+
+def distribute(A, B):
+    """ Turns an A containing Bs into a B of As
+
+    where A, B are container types
+
+    >>> from sympy.strategies import distribute
+    >>> from sympy import Add, Mul, symbols
+    >>> x, y = symbols('x,y')
+    >>> dist = distribute(Mul, Add)
+    >>> expr = Mul(2, x+y, evaluate=False)
+    >>> expr
+    2*(x + y)
+    >>> dist(expr)
+    2*x + 2*y
+    """
+
+    def distribute_rl(expr):
+        for i, arg in enumerate(expr.args):
+            if isinstance(arg, B):
+                first, b, tail = expr.args[:i], expr.args[i], expr.args[i + 1:]
+                return B(*[A(*(first + (arg,) + tail)) for arg in b.args])
+        return expr
+    return distribute_rl
+
+
+def subs(a, b):
+    """ Replace expressions exactly """
+    def subs_rl(expr):
+        if expr == a:
+            return b
+        else:
+            return expr
+    return subs_rl
+
+
+# Functions that are rules
+def unpack(expr):
+    """ Rule to unpack singleton args
+
+    >>> from sympy.strategies import unpack
+    >>> from sympy import Basic, S
+    >>> unpack(Basic(S(2)))
+    2
+    """
+    if len(expr.args) == 1:
+        return expr.args[0]
+    else:
+        return expr
+
+
+def flatten(expr, new=new):
+    """ Flatten T(a, b, T(c, d), T2(e)) to T(a, b, c, d, T2(e)) """
+    cls = expr.__class__
+    args = []
+    for arg in expr.args:
+        if arg.__class__ == cls:
+            args.extend(arg.args)
+        else:
+            args.append(arg)
+    return new(expr.__class__, *args)
+
+
+def rebuild(expr):
+    """ Rebuild a SymPy tree.
+
+    Explanation
+    ===========
+
+    This function recursively calls constructors in the expression tree.
+    This forces canonicalization and removes ugliness introduced by the use of
+    Basic.__new__
+    """
+    if expr.is_Atom:
+        return expr
+    else:
+        return expr.func(*list(map(rebuild, expr.args)))
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tools.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tools.py
new file mode 100644
index 0000000000000000000000000000000000000000..e6a94c16db57206d7c83c8a5e13930c4cffdde47
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tools.py
@@ -0,0 +1,53 @@
+from . import rl
+from .core import do_one, exhaust, switch
+from .traverse import top_down
+
+
+def subs(d, **kwargs):
+    """ Full simultaneous exact substitution.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies.tools import subs
+    >>> from sympy import Basic, S
+    >>> mapping = {S(1): S(4), S(4): S(1), Basic(S(5)): Basic(S(6), S(7))}
+    >>> expr = Basic(S(1), Basic(S(2), S(3)), Basic(S(4), Basic(S(5))))
+    >>> subs(mapping)(expr)
+    Basic(4, Basic(2, 3), Basic(1, Basic(6, 7)))
+    """
+    if d:
+        return top_down(do_one(*map(rl.subs, *zip(*d.items()))), **kwargs)
+    else:
+        return lambda x: x
+
+
+def canon(*rules, **kwargs):
+    """ Strategy for canonicalization.
+
+    Explanation
+    ===========
+
+    Apply each rule in a bottom_up fashion through the tree.
+    Do each one in turn.
+    Keep doing this until there is no change.
+    """
+    return exhaust(top_down(exhaust(do_one(*rules)), **kwargs))
+
+
+def typed(ruletypes):
+    """ Apply rules based on the expression type
+
+    inputs:
+        ruletypes -- a dict mapping {Type: rule}
+
+    Examples
+    ========
+
+    >>> from sympy.strategies import rm_id, typed
+    >>> from sympy import Add, Mul
+    >>> rm_zeros = rm_id(lambda x: x==0)
+    >>> rm_ones  = rm_id(lambda x: x==1)
+    >>> remove_idents = typed({Add: rm_zeros, Mul: rm_ones})
+    """
+    return switch(type, ruletypes)
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/traverse.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/traverse.py
new file mode 100644
index 0000000000000000000000000000000000000000..869361f443742b5b7346c9c970f103b955e8473e
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/traverse.py
@@ -0,0 +1,37 @@
+"""Strategies to Traverse a Tree."""
+from sympy.strategies.util import basic_fns
+from sympy.strategies.core import chain, do_one
+
+
+def top_down(rule, fns=basic_fns):
+    """Apply a rule down a tree running it on the top nodes first."""
+    return chain(rule, lambda expr: sall(top_down(rule, fns), fns)(expr))
+
+
+def bottom_up(rule, fns=basic_fns):
+    """Apply a rule down a tree running it on the bottom nodes first."""
+    return chain(lambda expr: sall(bottom_up(rule, fns), fns)(expr), rule)
+
+
+def top_down_once(rule, fns=basic_fns):
+    """Apply a rule down a tree - stop on success."""
+    return do_one(rule, lambda expr: sall(top_down(rule, fns), fns)(expr))
+
+
+def bottom_up_once(rule, fns=basic_fns):
+    """Apply a rule up a tree - stop on success."""
+    return do_one(lambda expr: sall(bottom_up(rule, fns), fns)(expr), rule)
+
+
+def sall(rule, fns=basic_fns):
+    """Strategic all - apply rule to args."""
+    op, new, children, leaf = map(fns.get, ('op', 'new', 'children', 'leaf'))
+
+    def all_rl(expr):
+        if leaf(expr):
+            return expr
+        else:
+            args = map(rule, children(expr))
+            return new(op(expr), *args)
+
+    return all_rl
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tree.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tree.py
new file mode 100644
index 0000000000000000000000000000000000000000..c2006fde4fc5d09f3d38baae4d7335b4cbd971b7
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/tree.py
@@ -0,0 +1,139 @@
+from functools import partial
+from sympy.strategies import chain, minimize
+from sympy.strategies.core import identity
+import sympy.strategies.branch as branch
+from sympy.strategies.branch import yieldify
+
+
+def treeapply(tree, join, leaf=identity):
+    """ Apply functions onto recursive containers (tree).
+
+    Explanation
+    ===========
+
+    join - a dictionary mapping container types to functions
+      e.g. ``{list: minimize, tuple: chain}``
+
+    Keys are containers/iterables.  Values are functions [a] -> a.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies.tree import treeapply
+    >>> tree = [(3, 2), (4, 1)]
+    >>> treeapply(tree, {list: max, tuple: min})
+    2
+
+    >>> add = lambda *args: sum(args)
+    >>> def mul(*args):
+    ...     total = 1
+    ...     for arg in args:
+    ...         total *= arg
+    ...     return total
+    >>> treeapply(tree, {list: mul, tuple: add})
+    25
+    """
+    for typ in join:
+        if isinstance(tree, typ):
+            return join[typ](*map(partial(treeapply, join=join, leaf=leaf),
+                                  tree))
+    return leaf(tree)
+
+
+def greedy(tree, objective=identity, **kwargs):
+    """ Execute a strategic tree.  Select alternatives greedily
+
+    Trees
+    -----
+
+    Nodes in a tree can be either
+
+    function - a leaf
+    list     - a selection among operations
+    tuple    - a sequence of chained operations
+
+    Textual examples
+    ----------------
+
+    Text: Run f, then run g, e.g. ``lambda x: g(f(x))``
+    Code: ``(f, g)``
+
+    Text: Run either f or g, whichever minimizes the objective
+    Code: ``[f, g]``
+
+    Textx: Run either f or g, whichever is better, then run h
+    Code: ``([f, g], h)``
+
+    Text: Either expand then simplify or try factor then foosimp. Finally print
+    Code: ``([(expand, simplify), (factor, foosimp)], print)``
+
+    Objective
+    ---------
+
+    "Better" is determined by the objective keyword.  This function makes
+    choices to minimize the objective.  It defaults to the identity.
+
+    Examples
+    ========
+
+    >>> from sympy.strategies.tree import greedy
+    >>> inc    = lambda x: x + 1
+    >>> dec    = lambda x: x - 1
+    >>> double = lambda x: 2*x
+
+    >>> tree = [inc, (dec, double)] # either inc or dec-then-double
+    >>> fn = greedy(tree)
+    >>> fn(4)  # lowest value comes from the inc
+    5
+    >>> fn(1)  # lowest value comes from dec then double
+    0
+
+    This function selects between options in a tuple.  The result is chosen
+    that minimizes the objective function.
+
+    >>> fn = greedy(tree, objective=lambda x: -x)  # maximize
+    >>> fn(4)  # highest value comes from the dec then double
+    6
+    >>> fn(1)  # highest value comes from the inc
+    2
+
+    Greediness
+    ----------
+
+    This is a greedy algorithm.  In the example:
+
+        ([a, b], c)  # do either a or b, then do c
+
+    the choice between running ``a`` or ``b`` is made without foresight to c
+    """
+    optimize = partial(minimize, objective=objective)
+    return treeapply(tree, {list: optimize, tuple: chain}, **kwargs)
+
+
+def allresults(tree, leaf=yieldify):
+    """ Execute a strategic tree.  Return all possibilities.
+
+    Returns a lazy iterator of all possible results
+
+    Exhaustiveness
+    --------------
+
+    This is an exhaustive algorithm.  In the example
+
+        ([a, b], [c, d])
+
+    All of the results from
+
+        (a, c), (b, c), (a, d), (b, d)
+
+    are returned.  This can lead to combinatorial blowup.
+
+    See sympy.strategies.greedy for details on input
+    """
+    return treeapply(tree, {list: branch.multiplex, tuple: branch.chain},
+                     leaf=leaf)
+
+
+def brute(tree, objective=identity, **kwargs):
+    return lambda expr: min(tuple(allresults(tree, **kwargs)(expr)),
+                            key=objective)
diff --git a/env-llmeval/lib/python3.10/site-packages/sympy/strategies/util.py b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/util.py
new file mode 100644
index 0000000000000000000000000000000000000000..13aab5f6a49650c5ded9cd913c23c6682f18d40a
--- /dev/null
+++ b/env-llmeval/lib/python3.10/site-packages/sympy/strategies/util.py
@@ -0,0 +1,17 @@
+from sympy.core.basic import Basic
+
+new = Basic.__new__
+
+
+def assoc(d, k, v):
+    d = d.copy()
+    d[k] = v
+    return d
+
+
+basic_fns = {'op': type,
+             'new': Basic.__new__,
+             'leaf': lambda x: not isinstance(x, Basic) or x.is_Atom,
+             'children': lambda x: x.args}
+
+expr_fns = assoc(basic_fns, 'new', lambda op, *args: op(*args))