{
    "info": {
        "author": "Alexandre Decan",
        "author_email": "",
        "bugtrack_url": null,
        "classifiers": [
            "Development Status :: 5 - Production/Stable",
            "Intended Audience :: Developers",
            "Intended Audience :: Education",
            "Intended Audience :: Information Technology",
            "Intended Audience :: Science/Research",
            "License :: OSI Approved :: GNU Lesser General Public License v3 (LGPLv3)",
            "Programming Language :: Python :: 2.7",
            "Programming Language :: Python :: 3.4",
            "Programming Language :: Python :: 3.5",
            "Programming Language :: Python :: 3.6",
            "Programming Language :: Python :: 3.7",
            "Topic :: Scientific/Engineering :: Mathematics"
        ],
        "description": "# Python data structure and operations for intervals\n\n[![Travis](https://travis-ci.org/AlexandreDecan/python-intervals.svg?branch=master)](https://travis-ci.org/AlexandreDecan/python-intervals)\n[![Coverage Status](https://coveralls.io/repos/github/AlexandreDecan/python-intervals/badge.svg?branch=master)](https://coveralls.io/github/AlexandreDecan/python-intervals?branch=master)\n[![PyPI](https://badge.fury.io/py/python-intervals.svg)](https://pypi.org/project/python-intervals)\n\n\nThis library provides data structure and operations for intervals in Python 2.7+ and Python 3.4+.\n\n  * [Features](#features)\n  * [Installation](#installation)\n  * [Documentation & usage](#documentation--usage)\n      * [Interval creation](#interval-creation)\n      * [Interval operations](#interval-operations)\n      * [Comparison operators](#comparison-operators)\n      * [Bounds of an interval](#bounds-of-an-interval)\n      * [Interval transformation](#interval-transformation)\n      * [Discrete iteration](#discrete-iteration)\n      * [Map intervals to data](#map-intervals-to-data)\n      * [Import & export intervals to strings](#import--export-intervals-to-strings)\n      * [Import & export intervals to Python built-in data types](#import--export-intervals-to-python-built-in-data-types)\n  * [Contributions](#contributions)\n  * [Licence](#licence)\n  * [Changelog](#changelog)\n\n\n## Features\n\n - Support intervals of any (comparable) objects.\n - Closed or open, finite or (semi-)infinite intervals.\n - Atomic intervals and interval sets are supported.\n - Automatic simplification of intervals.\n - Support comparison, transformation, intersection, union, complement, difference and containment.\n - Discrete iterations on the values of an interval.\n - Import and export intervals to strings and to Python built-in data types.\n - Dict-like structure to map intervals to data.\n\n\n## Installation\n\nYou can use `pip` to install it, as usual: `pip install python-intervals`.\n\nThis will install the latest available version from [PyPI](https://pypi.org/project/python-intervals).\nPrereleases are available from the *master* branch on [GitHub](https://github.com/AlexandreDecan/python-intervals).\n\nFor convenience, the library is contained within a single Python file, and can thus be easily integrated in other\nprojects without the need for an explicit dependency (hint: don't do that!).\n\n\n## Documentation & usage\n\n### Interval creation\n\nAssuming this library is imported using `import intervals as I`, intervals can be easily created using one of the\nfollowing helpers:\n\n```python\n>>> I.open(1, 2)\n(1,2)\n>>> I.closed(1, 2)\n[1,2]\n>>> I.openclosed(1, 2)\n(1,2]\n>>> I.closedopen(1, 2)\n[1,2)\n>>> I.singleton(1)\n[1]\n>>> I.empty()\n()\n\n```\n\nIntervals created with this library are `Interval` instances.\nAn `Interval` object is a disjunction of atomic intervals that represent single intervals (e.g. `[1,2]`) corresponding to `AtomicInterval` instances.\nExcept when atomic intervals are explicitly created or retrieved, only `Interval` instances are exposed.\n\nThe bounds of an interval can be any arbitrary values, as long as they are comparable:\n\n```python\n>>> I.closed(1.2, 2.4)\n[1.2,2.4]\n>>> I.closed('a', 'z')\n['a','z']\n>>> import datetime\n>>> I.closed(datetime.date(2011, 3, 15), datetime.date(2013, 10, 10))\n[datetime.date(2011, 3, 15),datetime.date(2013, 10, 10)]\n\n```\n\n\nInfinite and semi-infinite intervals are supported using `I.inf` and `-I.inf` as upper or lower bounds.\nThese two objects support comparison with any other object.\nWhen infinities are used as a lower or upper bound, the corresponding boundary is automatically converted to an open one.\n\n```python\n>>> I.inf > 'a', I.inf > 0, I.inf > True\n(True, True, True)\n>>> I.openclosed(-I.inf, 0)\n(-inf,0]\n>>> I.closed(-I.inf, I.inf)  # Automatically converted to an open interval\n(-inf,+inf)\n\n```\n\nEmpty intervals always resolve to `(I.inf, -I.inf)`, regardless of the provided bounds:\n\n```python\n>>> I.empty() == I.open(I.inf, -I.inf)\nTrue\n>>> I.closed(4, 3) == I.open(I.inf, -I.inf)\nTrue\n>>> I.openclosed('a', 'a') == I.open(I.inf, -I.inf)\nTrue\n\n```\n\nFor convenience, intervals are automatically simplified:\n```python\n>>> I.closed(0, 2) | I.closed(2, 4)\n[0,4]\n>>> I.closed(1, 2) | I.closed(3, 4) | I.closed(2, 3)\n[1,4]\n>>> I.empty() | I.closed(0, 1)\n[0,1]\n>>> I.closed(1, 2) | I.closed(2, 3) | I.closed(4, 5)\n[1,3] | [4,5]\n\n```\n\nNote that discrete intervals are **not** supported, e.g., combining `[0,1]` with `[2,3]` will **not** result\nin `[0,3]` even if there is no integer between `1` and `2`.\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Interval operations\n\nBoth `Interval` and `AtomicInterval` support following interval operations:\n\n - `x.is_empty()` tests if the interval is empty.\n   ```python\n   >>> I.closed(0, 1).is_empty()\n   False\n   >>> I.closed(0, 0).is_empty()\n   False\n   >>> I.openclosed(0, 0).is_empty()\n   True\n   >>> I.empty().is_empty()\n   True\n\n   ```\n\n - `x.intersection(other)` or `x & other` return the intersection of two intervals.\n   ```python\n   >>> I.closed(0, 2) & I.closed(1, 3)\n   [1,2]\n   >>> I.closed(0, 4) & I.open(2, 3)\n   (2,3)\n   >>> I.closed(0, 2) & I.closed(2, 3)\n   [2]\n   >>> I.closed(0, 2) & I.closed(3, 4)\n   ()\n\n   ```\n\n - `x.union(other)` or `x | other` return the union of two intervals.\n   ```python\n   >>> I.closed(0, 1) | I.closed(1, 2)\n   [0,2]\n   >>> I.closed(0, 1) | I.closed(2, 3)\n   [0,1] | [2,3]\n\n   ```\n\n - `x.complement(other)` or `~x` return the complement of the interval.\n   ```python\n   >>> ~I.closed(0, 1)\n   (-inf,0) | (1,+inf)\n   >>> ~(I.open(-I.inf, 0) | I.open(1, I.inf))\n   [0,1]\n   >>> ~I.open(-I.inf, I.inf)\n   ()\n\n   ```\n\n - `x.difference(other)` or `x - other` return the difference between `x` and `other`.\n   ```python\n   >>> I.closed(0,2) - I.closed(1,2)\n   [0,1)\n   >>> I.closed(0, 4) - I.closed(1, 2)\n   [0,1) | (2,4]\n\n   ```\n\n - `x.contains(other)` or `other in x` return True if given item is contained in the interval.\n Support `Interval`, `AtomicInterval` and arbitrary comparable values.\n   ```python\n   >>> 2 in I.closed(0, 2)\n   True\n   >>> 2 in I.open(0, 2)\n   False\n   >>> I.open(0, 1) in I.closed(0, 2)\n   True\n\n   ```\n\n - `x.overlaps(other)` tests if there is an overlap between two intervals.\n This method accepts a `adjacent` parameter which defaults to `False`.\n If `True`, it accepts adjacent intervals as well (e.g., [1, 2) and [2, 3] but not\n [1, 2) and (2, 3]).\n   ```python\n   >>> I.closed(1, 2).overlaps(I.closed(2, 3))\n   True\n   >>> I.closed(1, 2).overlaps(I.open(2, 3))\n   False\n   >>> I.closed(1, 2).overlaps(I.openclosed(2, 3), adjacent=True)\n   True\n   >>> I.closedopen(1, 2).overlaps(I.openclosed(2, 3), adjacent=True)\n   False\n\n   ```\n\n\nThe following methods are only available for `Interval` instances:\n\n - `x.enclosure()` returns the smallest interval that includes the current one.\n   ```python\n   >>> (I.closed(0, 1) | I.closed(2, 3)).enclosure()\n   [0,3]\n\n   ```\n\n - `x.to_atomic()` is equivalent to `x.enclosure()` but returns an `AtomicInterval` instead of an `Interval` object.\n\n - `x.is_atomic()` evaluates to `True` if interval is composed of a single (possibly empty) atomic interval.\n   ```python\n   >>> I.closed(0, 2).is_atomic()\n   True\n   >>> (I.closed(0, 1) | I.closed(1, 2)).is_atomic()\n   True\n   >>> (I.closed(0, 1) | I.closed(2, 3)).is_atomic()\n   False\n\n   ```\n\n\nIntervals can also be iterated to access the underlying `AtomicInterval` objects, sorted by their lower and upper bounds.\n\n```python\n>>> list(I.open(2, 3) | I.closed(0, 1) | I.closed(21, 24))\n[[0,1], (2,3), [21,24]]\n\n```\n\nThe `AtomicInterval` objects of an `Interval` can also be accessed using their indexes:\n\n```python\n>>> (I.open(2, 3) | I.closed(0, 1) | I.closed(21, 24))[0]\n[0,1]\n>>> (I.open(2, 3) | I.closed(0, 1) | I.closed(21, 24))[-2]\n(2,3)\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Comparison operators\n\nEquality between intervals can be checked with the classical `==` operator:\n\n```python\n>>> I.closed(0, 2) == I.closed(0, 1) | I.closed(1, 2)\nTrue\n>>> I.closed(0, 2) == I.closed(0, 2).to_atomic()\nTrue\n\n```\n\nMoreover, both `Interval` and `AtomicInterval` are comparable using e.g. `>`, `>=`, `<` or `<=`.\nThese comparison operators have a different behaviour than the usual ones.\nFor instance, `a < b` holds if `a` is entirely on the left of the lower bound of `b` and `a > b` holds if `a` is entirely\non the right of the upper bound of `b`.\n\n```python\n>>> I.closed(0, 1) < I.closed(2, 3)\nTrue\n>>> I.closed(0, 1) < I.closed(1, 2)\nFalse\n\n```\n\nSimilarly, `a <= b` holds if `a` is entirely on the left of the upper bound of `b`, and `a >= b`\nholds if `a` is entirely on the right of the lower bound of `b`.\n\n```python\n>>> I.closed(0, 1) <= I.closed(2, 3)\nTrue\n>>> I.closed(0, 2) <= I.closed(1, 3)\nTrue\n>>> I.closed(0, 3) <= I.closed(1, 2)\nFalse\n\n```\n\nIntervals can also be compared with single values. If `i` is an interval and `x` a value, then\n`x < i` holds if `x` is on the left of the lower bound of `i` and `x <= i` holds if `x` is on the\nleft of the upper bound of `i`. This behaviour is similar to the one that could be obtained by first\nconverting `x` to a singleton interval.\n\n```python\n>>> 5 < I.closed(0, 10)\nFalse\n>>> 5 <= I.closed(0, 10)\nTrue\n>>> I.closed(0, 10) < 5\nFalse\n>>> I.closed(0, 10) <= 5\nTrue\n\n```\n\n\nNote that all these semantics differ from classical comparison operators.\nAs a consequence, some intervals are never comparable in the classical sense, as illustrated hereafter:\n\n```python\n>>> I.closed(0, 4) <= I.closed(1, 2) or I.closed(0, 4) >= I.closed(1, 2)\nFalse\n>>> I.closed(0, 4) < I.closed(1, 2) or I.closed(0, 4) > I.closed(1, 2)\nFalse\n>>> I.empty() < I.empty()\nTrue\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Bounds of an interval\n\nThe left and right boundaries, and the lower and upper bounds of an `AtomicInterval` can be respectively accessed\nwith its `left`, `right`, `lower` and `upper` attributes.\nThe `left` and `right` bounds are either `I.CLOSED` (`True`) or `I.OPEN` (`False`).\n\n```python\n>> I.CLOSED, I.OPEN\nTrue, False\n>>> x = I.closedopen(0, 1).to_atomic()\n>>> x.left, x.lower, x.upper, x.right\n(True, 0, 1, False)\n\n```\n\nSimilarly, the bounds of an `Interval` instance can be accessed with its `left`, `right`,\n`lower` and `upper` attributes. In that case, `left` and `lower` refer to the lower bound of its enclosure,\nwhile `right` and `upper` refer to the upper bound of its enclosure:\n\n```python\n>>> x = I.open(0, 1) | I.closed(3, 4)\n>>> x.left, x.lower, x.upper, x.right\n(False, 0, 4, True)\n\n```\n\nOne can easily check for some interval properties based on the bounds of an interval:\n\n```python\n>>> x = I.openclosed(-I.inf, 0)\n>>> # Check that interval is left/right closed\n>>> x.left == I.CLOSED, x.right == I.CLOSED\n(False, True)\n>>> # Check that interval is left/right bounded\n>>> x.lower == -I.inf, x.upper == I.inf\n(True, False)\n>>> # Check for singleton\n>>> x.lower == x.upper\nFalse\n\n```\n\n\nBoth `Interval` and `AtomicInterval` instances are immutable but provide a `replace` method that\ncan be used to create a new instance based on the current one. This method accepts four optional\nparameters `left`, `lower`, `upper`, and `right`:\n\n```python\n>>> i = I.closed(0, 2).to_atomic()\n>>> i.replace(I.OPEN, -1, 3, I.CLOSED)\n(-1,3]\n>>> i.replace(lower=1, right=I.OPEN)\n[1,2)\n\n```\n\nFunctions can be passed instead of values. If a function is passed, it is called with the current corresponding\nvalue except if the corresponding bound is an infinity and parameter `ignore_inf` if set to `False`.\n\n```python\n>>> I.closed(0, 2).replace(upper=lambda x: 2 * x)\n[0,4]\n>>> i = I.closedopen(0, I.inf)\n>>> i.replace(upper=lambda x: 10)  # No change, infinity is ignored\n[0,+inf)\n>>> i.replace(upper=lambda x: 10, ignore_inf=False)  # Infinity is not ignored\n[0,10)\n\n```\n\nWhen `replace` is applied on an `Interval` that is not atomic, it is extended and/or restricted such that\nits enclosure satisfies the new bounds.\n\n```python\n>>> i = I.openclosed(0, 1) | I.closed(5, 10)\n>>> i.replace(I.CLOSED, -1, 8, I.OPEN)\n[-1,1] | [5,8)\n>>> i.replace(lower=4)\n(4,10]\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Interval transformation\n\nTo apply an arbitrary transformation on an interval, `Interval` instances expose an `apply` method.\nThis method accepts a function that will be applied on each of the underlying atomic intervals to perform the desired transformation.\nThe function is expected to return an `AtomicInterval`, an `Interval` or a 4-uple `(left, lower, upper, right)`.\n\n```python\n>>> i = I.closed(2, 3) | I.open(4, 5)\n>>> # Increment bound values\n>>> i.apply(lambda x: (x.left, x.lower + 1, x.upper + 1, x.right))\n[3,4] | (5,6)\n>>> # Invert bounds\n>>> i.apply(lambda x: (not x.left, x.lower, x.upper, not x.right))\n(2,3) | [4,5]\n\n```\n\nThe `apply` method is very powerful when used in combination with `replace`.\nBecause the latter allows functions to be passed as parameters and can ignore infinities, it can be\nconveniently used to transform intervals in presence of infinities.\n\n```python\n>>> i = I.openclosed(-I.inf, 0) | I.closed(3, 4) | I.closedopen(8, I.inf)\n>>> # Increment bound values\n>>> i.apply(lambda x: x.replace(upper=lambda v: v + 1))\n(-inf,1] | [3,5] | [8,+inf)\n>>> # Intervals are still automatically simplified\n>>> i.apply(lambda x: x.replace(lower=lambda v: v * 2))\n(-inf,0] | [16,+inf)\n>>> # Invert bounds\n>>> i.apply(lambda x: x.replace(left=lambda v: not v, right=lambda v: not v))\n(-inf,0) | (3,4) | (8,+inf)\n>>> # Replace infinities with -10 and 10\n>>> conv = lambda v: -10 if v == -I.inf else (10 if v == I.inf else v)\n>>> i.apply(lambda x: x.replace(lower=conv, upper=conv, ignore_inf=False))\n(-10,0] | [3,4] | [8,10)\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Discrete iteration\n\nThe `iterate` function takes an interval or atomic interval, and returns a generator to iterate over\nthe values of an interval. Obviously, as intervals are continuous, it is required to specify the increment\n `incr` between consecutive values. The iteration then starts from the lower bound and ends on the upper one,\ngiven they are not excluded by the interval:\n\n```python\n>>> list(I.iterate(I.closed(0, 3), incr=1))\n[0, 1, 2, 3]\n>>> list(I.iterate(I.closed(0, 3), incr=2))\n[0, 2]\n>>> list(I.iterate(I.open(0, 3), incr=2))\n[2]\n\n```\n\nWhen an interval represents an union of atomic intervals, `iterate` consecutively iterates on all atomic\nintervals, starting from each lower bound and ending on each upper one:\n\n```python\n>>> list(I.iterate(I.singleton(0) | I.singleton(1) | I.singleton(5), incr=2))  # Won't be [0]\n[0, 1, 5]\n>>> list(I.iterate(I.closed(0, 2) | I.closed(5, 6), incr=3))  # Won't be [0, 6]\n[0, 5]\n\n```\n\nIteration can be performed in reverse order by specifying `reverse=True`. In that case, `incr` will be\nsubtracted instead of being added, implying that `incr` must always be a \"positive\" value:\n\n```python\n>>> list(I.iterate(I.closed(0, 3), incr=1, reverse=True))  # Not incr=-1\n[3, 2, 1, 0]\n>>> list(I.iterate(I.closed(0, 3), incr=2, reverse=True))  # Not incr=-2\n[3, 1]\n\n```\n\nAgain, this library does not make any assumption about the objects being used in an interval, as long as they\nare comparable. However, it is not always possible to provide a meaningful value for `incr` (e.g., what would\nbe the step between two consecutive characters?). In these cases, a callable can be passed instead of a value.\nThis callable will be called with the current value, and is expected to return the next possible value.\n\n```python\n>>> list(I.iterate(I.closed('a', 'd'), incr=lambda d: chr(ord(d) + 1)))\n['a', 'b', 'c', 'd']\n>>> # Notice the reversed order, mind the \"- 1\"\n>>> list(I.iterate(I.closed('a', 'd'), incr=lambda d: chr(ord(d) - 1), reverse=True))\n['d', 'c', 'b', 'a']\n\n```\n\nBy default, the iteration always starts on the lower bound (unless `reverse=True`) of each atomic interval.\nThe `base` parameter can be used to change this behaviour, by specifying how the initial value to start\nthe iteration on must be computed. This parameter accepts a callable that will be called with the lower\nbound (unless `reverse=True`) for each underlying atomic interval, and that must return the first value to\nconsider instead of the lower bound.\n\nThis can be helpful to deal with (semi-)infinite intervals, or to *align* the generated values of\nthe iterator:\n\n```python\n>>> # Restrict values of a (semi-)infinite interval\n>>> list(I.iterate(I.openclosed(-I.inf, 2), incr=1, base=lambda x: max(0, x)))\n[0, 1, 2]\n>>> # Align on integers\n>>> list(I.iterate(I.closed(0.3, 4.9), incr=1, base=int))\n[1, 2, 3, 4]\n\n```\n\nThe `base` parameter can be used to change how `iterate` applies on unions of atomic interval, by\nspecifying a function that returns a single value, as illustrated next:\n\n```python\n>>> interval = I.closed(0, 1) | I.closed(2, 4) | I.closed(5, 6)\n>>> list(I.iterate(interval, incr=3))  # Won't be [0, 3, 6]\n[0, 2, 5]\n>>> list(I.iterate(interval, incr=3, base=lambda x: 0))\n[0, 3, 6]\n\n```\n\nNotice that this approach can be extremely inefficient in terms of performance when the intervals\nare \"far apart\" each other.\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Map intervals to data\n\nThe library provides an `IntervalDict` class, a `dict`-like data structure to store and query data\nalong with intervals. Any value can be stored in such data structure as long as it supports\nequality.\n\n\n```python\n>>> d = I.IntervalDict()\n>>> d[I.closed(0, 3)] = 'banana'\n>>> d[4] = 'apple'\n>>> d\n{[0,3]: 'banana', [4]: 'apple'}\n\n```\n\nWhen a value is defined for an interval that overlaps an existing one, it is automatically updated\nto take the new value into account:\n\n```python\n>>> d[I.closed(2, 4)] = 'orange'\n>>> d\n{[0,2): 'banana', [2,4]: 'orange'}\n\n```\n\nAn `IntervalDict` can be queried using single values or intervals. If a single value is used as a\nkey, its behaviour corresponds to the one of a classical `dict`:\n\n```python\n>>> d[2]\n'orange'\n>>> d[5]  # Key does not exist\nTraceback (most recent call last):\n ...\nKeyError: 5\n>>> d.get(5, default=0)\n0\n\n```\n\nWhen an interval is used as a key, a new `IntervalDict` containing the values\nfor that interval is returned:\n\n```python\n>>> d[~I.empty()]  # Get all values, similar to d.copy()\n{[0,2): 'banana', [2,4]: 'orange'}\n>>> d[I.closed(1, 3)]\n{[1,2): 'banana', [2,3]: 'orange'}\n>>> d[I.closed(-2, 1)]\n{[0,1]: 'banana'}\n>>> d[I.closed(-2, -1)]\n{}\n\n```\n\nBy using `.get`, a default value (defaulting to `None`) can be specified.\nThis value is used to \"fill the gaps\" if the queried interval is not completely\ncovered by the `IntervalDict`:\n\n```python\n>>> d.get(I.closed(-2, 1), default='peach')\n{[-2,0): 'peach', [0,1]: 'banana'}\n>>> d.get(I.closed(-2, -1), default='peach')\n{[-2,-1]: 'peach'}\n>>> d.get(I.singleton(1), default='peach')  # Key is covered, default is not used\n{[1]: 'banana'}\n\n```\n\nFor convenience, an `IntervalDict` provides a way to look for specific data values.\nThe `.find` method always returns a (possibly empty) `Interval` instance for which given\nvalue is defined:\n\n```python\n>>> d.find('banana')\n[0,2)\n>>> d.find('orange')\n[2,4]\n>>> d.find('carrot')\n()\n\n```\n\nThe active domain of an `IntervalDict` can be retrieved with its `.domain` method.\nThis method always returns a single `Interval` instance, where `.keys` returns a list\nof disjoint intervals, one for each stored value.\n\n```python\n>>> d.domain()\n[0,4]\n>>> d.keys()\n[[0,2), [2,4]]\n>>> d.values()\n['banana', 'orange']\n>>> d.items()\n[([0,2), 'banana'), ([2,4], 'orange')]\n\n```\n\nTwo `IntervalDict` instances can be combined together using the `.combine` method.\nThis method returns a new `IntervalDict` whose keys and values are taken from the two\nsource `IntervalDict`. Values corresponding to non-intersecting keys are simply copied,\nwhile values corresponding to intersecting keys are combined together using the provided\nfunction, as illustrated hereafter:\n\n```python\n>>> d1 = I.IntervalDict({I.closed(0, 2): 'banana'})\n>>> d2 = I.IntervalDict({I.closed(1, 3): 'orange'})\n>>> concat = lambda x, y: x + '/' + y\n>>> d1.combine(d2, how=concat)\n{[0,1): 'banana', [1,2]: 'banana/orange', (2,3]: 'orange'}\n\n```\n\nResulting keys always correspond to an outer join. Other joins can be easily simulated\nby querying the resulting `IntervalDict` as follows:\n\n```python\n>>> d = d1.combine(d2, how=concat)\n>>> d[d1.domain()]  # Left join\n{[0,1): 'banana', [1,2]: 'banana/orange'}\n>>> d[d2.domain()]  # Right join\n{[1,2]: 'banana/orange', (2,3]: 'orange'}\n>>> d[d1.domain() & d2.domain()]  # Inner join\n{[1,2]: 'banana/orange'}\n\n```\n\nFinally, similarly to a `dict`, an `IntervalDict` also supports `len`, `in` and `del`, and defines\n`.clear`, `.copy`, `.update`, `.pop`, `.popitem`, and `.setdefault`.\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Import & export intervals to strings\n\nIntervals can be exported to string, either using `repr` (as illustrated above) or with the `to_string` function.\n\n```python\n>>> I.to_string(I.closedopen(0, 1))\n'[0,1)'\n\n```\n\nThis function accepts both `Interval` and `AtomicInterval` instances.\nThe way string representations are built can be easily parametrized using the various parameters supported by\n`to_string`:\n\n```python\n>>> params = {\n...   'disj': ' or ',\n...   'sep': ' - ',\n...   'left_closed': '<',\n...   'right_closed': '>',\n...   'left_open': '..',\n...   'right_open': '..',\n...   'pinf': '+oo',\n...   'ninf': '-oo',\n...   'conv': lambda v: '\"{}\"'.format(v),\n... }\n>>> x = I.openclosed(0, 1) | I.closed(2, I.inf)\n>>> I.to_string(x, **params)\n'..\"0\" - \"1\"> or <\"2\" - +oo..'\n\n```\n\nSimilarly, intervals can be created from a string using the `from_string` function.\nA conversion function (`conv` parameter) has to be provided to convert a bound (as string) to a value.\n\n```python\n>>> I.from_string('[0, 1]', conv=int) == I.closed(0, 1)\nTrue\n>>> I.from_string('[1.2]', conv=float) == I.singleton(1.2)\nTrue\n>>> converter = lambda s: datetime.datetime.strptime(s, '%Y/%m/%d')\n>>> I.from_string('[2011/03/15, 2013/10/10]', conv=converter)\n[datetime.datetime(2011, 3, 15, 0, 0),datetime.datetime(2013, 10, 10, 0, 0)]\n\n```\n\nSimilarly to `to_string`, function `from_string` can be parametrized to deal with more elaborated inputs.\nNotice that as `from_string` expects regular expression patterns, we need to escape some characters.\n\n```python\n>>> s = '..\"0\" - \"1\"> or <\"2\" - +oo..'\n>>> params = {\n...   'disj': ' or ',\n...   'sep': ' - ',\n...   'left_closed': '<',\n...   'right_closed': '>',\n...   'left_open': r'\\.\\.',  # from_string expects regular expression patterns\n...   'right_open': r'\\.\\.',  # from_string expects regular expression patterns\n...   'pinf': r'\\+oo',  # from_string expects regular expression patterns\n...   'ninf': '-oo',\n...   'conv': lambda v: int(v[1:-1]),\n... }\n>>> I.from_string(s, **params)\n(0,1] | [2,+inf)\n\n```\n\nWhen a bound contains a comma or has a representation that cannot be automatically parsed with `from_string`,\nthe `bound` parameter can be used to specify the regular expression that should be used to match its representation.\n\n```python\n>>> s = '[(0, 1), (2, 3)]'  # Bounds are expected to be tuples\n>>> I.from_string(s, conv=eval, bound=r'\\(.+?\\)')\n[(0, 1),(2, 3)]\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n### Import & export intervals to Python built-in data types\n\nIntervals can also be exported to a list of 4-uples with `to_data`, e.g., to support JSON serialization.\n\n```python\n>>> x = I.openclosed(0, 1) | I.closedopen(2, I.inf)\n>>> I.to_data(x)\n[(False, 0, 1, True), (True, 2, inf, False)]\n\n```\n\nThe function to convert bounds can be specified with the `conv` parameter.\nThe values that must be used to represent positive and negative infinities can be specified with\n`pinf` and `ninf`. They default to `float('inf')` and `float('-inf')` respectively.\n\n```python\n>>> x = I.closed(datetime.date(2011, 3, 15), datetime.date(2013, 10, 10))\n>>> I.to_data(x, conv=lambda v: (v.year, v.month, v.day))\n[(True, (2011, 3, 15), (2013, 10, 10), True)]\n\n```\n\nIntervals can be imported from such a list of 4-uples with `from_data`.\nThe same set of parameters can be used to specify how bounds and infinities are converted.\n\n```python\n>>> x = [(True, (2011, 3, 15), (2013, 10, 10), False)]\n>>> I.from_data(x, conv=lambda v: datetime.date(*v))\n[datetime.date(2011, 3, 15),datetime.date(2013, 10, 10))\n\n```\n\n\n[&uparrow; back to top](#python-data-structure-and-operations-for-intervals)\n## Contributions\n\nContributions are very welcome!\nFeel free to report bugs or suggest new features using GitHub issues and/or pull requests.\n\n\n## Licence\n\nDistributed under [LGPLv3 - GNU Lesser General Public License, version 3](https://github.com/AlexandreDecan/python-intervals/blob/master/LICENSE.txt).\n\nYou can cite this library using:\n\n```\n@software{python-intervals,\n  author = {Decan, Alexandre},\n  title = {python-intervals: Python data structure and operations for intervals},\n  url = {https://github.com/AlexandreDecan/python-intervals},\n}\n```\n\n\n## Changelog\n\nThis library adheres to a [semantic versioning](https://semver.org) scheme.\n\n\n**1.10.0** (2019-09-26)\n\n - `IntervalDict` has a `.combine` method to merge its keys and values with another `IntervalDict`.\n\n\n**1.9.0** (2019-09-13)\n\n - Discrete iteration on the values of an interval with `iterate`.\n - Map intervals to data with the dict-like `IntervalDict` structure.\n - Faster comparisons between arbitrary values and intervals.\n - Deprecate `permissive` in `.overlaps` in favour of `adjacent`.\n - Fix `.union` when intervals share a bound, one inclusive and one exclusive ([#12](https://github.com/AlexandreDecan/python-intervals/issues/12)).\n - Fix `.overlaps` when intervals share a lower bound, and one interval is contained within the other one ([#13](https://github.com/AlexandreDecan/python-intervals/issues/13)).\n\n\n**1.8.0** (2018-12-15)\n\n - Intervals have a `.left`, `.lower`, `.upper`, and `.right` attribute that refer to its enclosure.\n - Intervals have a `.replace` method to create new intervals based on the current one. This method accepts both values and functions.\n - Intervals have an `.apply` method to apply a function on the underlying atomic intervals.\n - Intervals can be compared with single values as well.\n - `I.empty()` returns the same instance to save memory.\n - Infinities are singleton objects.\n - Set `len(I.empty()) = 1` and `I.empty()[0] == I.empty().to_atomic()` for consistency.\n\n\n**1.7.0** (2018-12-06)\n\n - Import from and export to Python built-in data types (a list of 4-uples) with `from_data` and `to_data` ([#6](https://github.com/AlexandreDecan/python-intervals/issues/6)).\n - Add examples for arbitrary interval transformations.\n\n\n**1.6.0** (2018-08-29)\n\n - Add support for customized infinity representation in `to_string` and `from_string` ([#3](https://github.com/AlexandreDecan/python-intervals/issues/3)).\n\n\n**1.5.4** (2018-07-29)\n\n - Fix `.overlaps` ([#2](https://github.com/AlexandreDecan/python-intervals/issues/2)).\n\n\n**1.5.3** (2018-06-21)\n\n - Fix invalid `repr` for atomic singleton intervals.\n\n\n**1.5.2** (2018-06-15)\n\n - Fix invalid comparisons when both `Interval` and `AtomicInterval` are compared.\n\n\n**1.5.1** (2018-04-25)\n\n - Fix [#1](https://github.com/AlexandreDecan/python-intervals/issues/1) by making empty intervals always resolving to `(I.inf, -I.inf)`.\n\n\n**1.5.0** (2018-04-17)\n\n - `Interval.__init__` accepts `Interval` instances in addition to `AtomicInterval` ones.\n\n\n**1.4.0** (2018-04-17)\n\n - Function `I.to_string` to export an interval to a string, with many options to customize the representation.\n - Function `I.from_string` to create an interval from a string, with many options to customize the parsing.\n\n\n**1.3.2** (2018-04-13)\n\n - Support for Python 2.7.\n\n\n**1.3.1** (2018-04-12)\n\n - Define `__slots__` to lower memory usage, and to speed up attribute access.\n - Define `Interval.__rand__` (and other magic methods) to support `Interval` from `AtomicInterval` instead of\n having a dedicated piece of code in `AtomicInterval`.\n - Fix `__all__`.\n - More tests to cover all comparisons.\n\n\n**1.3.0** (2018-04-04)\n\n - Meaningful `<=` and `>=` comparisons for intervals.\n\n\n**1.2.0** (2018-04-04)\n\n - `Interval` supports indexing to retrieve the underlying `AtomicInterval` objects.\n\n\n**1.1.0** (2018-04-04)\n\n - Both `AtomicInterval` and `Interval` are fully comparable.\n - Add `singleton(x)` to create a singleton interval [x].\n - Add `empty()` to create an empty interval.\n - Add `Interval.enclosure()` that returns the smallest interval that includes the current one.\n - Interval simplification is in O(n) instead of O(n*m).\n - `AtomicInterval` objects in an `Interval` are sorted by lower and upper bounds.\n\n\n**1.0.4** (2018-04-03)\n\n - All operations of `AtomicInterval` (except overlaps) accept `Interval`.\n - Raise `TypeError` instead of `ValueError` if type is not supported (coherent with `NotImplemented`).\n\n\n**1.0.3** (2018-04-03)\n\n - Initial working release on PyPi.\n\n\n**1.0.0** (2018-04-03)\n\n - Initial release.\n\n\n",
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