lean2/doc/lua/lua.md

# Lua API documentation

We the [Lua](http://www.lua.org) script language to customize and
extend [Lean](http://leanprover.net). This
[link](http://www.lua.org/docs.html) is a good starting point for
learning Lua. In this document, we focus on the Lean specific APIs.
Most of Lean components are exposed in the Lua API.

**Remark:** the script [md2lua.sh](md2lua.sh) extracts the Lua code
examples from the `*.md` documentation files in this directory.

## Hierarchical names

In Lean, we use _hierarchical names_ for identifying configuration
options, constants, and kernel objects. A hierarchical name is
essentially a list of strings and integers.
The following example demonstrates how to create and manipulate
hierarchical names using the Lua API.

```lua
local x = name("x")    -- create a simple hierarchical name
local y = name("y")
-- In Lua, 'assert(p)' succeeds if 'p' does not evaluate to false (or nil)
assert(x == name("x")) -- test if 'x' is equal to 'name("x")'
assert(x ~= y)         -- '~=' is the not equal operator in Lua
assert(x ~= "x")

assert(is_name(x)) -- test whether argument is a hierarchical name or not.
assert(not is_name("x"))

local x1 = name(x, 1) -- x1 is a name composed of the string "x" and number 1
assert(tostring(x1) == "x::1")
assert(x1 ~= name("x::1"))  -- x1 is not equal to the string x::1
assert(x1 == name("x", 1))

local o = name("lean", "pp", "colors")
-- The previous construct is syntax sugar for the following expression
assert(o == name(name(name("lean"), "pp"), "colors"))

assert(x < y) -- '<' is a total order on hierarchical names

local h = x:hash() -- retrieve the hash code for 'x'
assert(h ~= y:hash())
```

## Lua tables

Tables are the only mutable, non-atomic type of data in Lua.  Tables
are used to implement mappings, arrays, lists, objects, etc. Here is a
small examples demonstrating how to use Lua tables:

```lua
local t = {}    -- create an empty table
t["x"]  = 10    -- insert the entry "x" -> 10
t.x     = 20    -- syntax-sugar for t["x"] = 20
t["y"]  = 30    -- insert the entry "y" -> 30
assert(t["x"] == 20)
local p = { x = 10, y = 20 } -- create a table with two entries
assert(p.x == 10)
assert(p["x"] == 10)
assert(p.y == 20)
assert(p["y"] == 20)
```

Arrays are implemented by indexing tables with integers.
The arrays don't have a fixed size and grow dynamically.
The arrays in Lua usually start at index 1. The Lua libraries
use this convention. The following example initializes
an array with 100 elements.

```lua
local a = {}    -- new array
for i=1, 100 do
    a[i] = 0
end
local b = {2, 4, 6, 8, 10} -- create an array with 5 elements
assert(#b == 5)    -- array has 5 elements
assert(b[1] == 2)
assert(b[2] == 4)
```
In Lua, we cannot provide custom hash and equality functions to tables.
So, we cannot effectively use Lua tables to implement mappings where
the keys are Lean hierarchical names, expressions, etc.
The following example demonstrates the issue.

```lua
local m  = {} -- create empty table
local a  = name("a")
m[a]     = 10 -- map the hierarchical name 'a' to 10
assert(m[a] == 10)
local a1 = name("a")
assert(a == a1)      -- 'a' and 'a1' are the same hierarchical name
assert(m[a1] == nil) -- 'a1' is not a key in the mapping 'm'
assert(m[a]  == 10)
-- 'a' and 'a1' are two instances of the same object
-- Lua tables assume that different instances are not equal
```

## Splay maps

In Lean, we provide splay maps for implementing mappings where the keys are
Lean objects such as hierarchical names. A splay map is implemented using
a [splay tree](http://en.wikipedia.org/wiki/Splay_tree), a self-adjusting binary
search tree. We can also use Lua atomic data types
as keys in splay maps. However, we should not mix different types in the
same splay map. The Lean splay map assumes that `<` is a total order on the
keys inserted in the map.

```lua
local m = splay_map() -- create an empty splay map
assert(is_splay_map(m))
assert(#m == 0)
local a  = name("a", 1)
local a1 = name("a", 1)
m:insert(a, 10)          -- add the entry 'a::1' -> 10
assert(m:find(a1) == 10)
m:insert(name("b"), 20)
assert(#m == 2)
m:erase(a)               -- remove entry with key 'a::1'
assert(m:find(a) == nil)
assert(not m:contains(a))
assert(#m == 1)
m:insert(name("c"), 30)
m:for_each(              -- traverse the entries in the splay map
    function(k, v)       -- executing the given function
        print(k, v)
    end
)
local m2 = m:copy()      -- the splay maps are copied in constant time
assert(#m2 == #m)
m2:insert(name("b", 2), 40)
assert(#m2 == #m + 1)
```

## Multiple precision numerals

We expose [GMP](http://gmplib.org/) (GNU Multiple precision arithmetic
library) in Lua. Internally, Lean uses multiple precision numerals for
representing expressing containing numerals.
In Lua, we can create multiple precision integers (_mpz_) and multiple
precision rationals (_mpq_). The following example demonstrates how to
use `mpz` and `mpq` numerals.

```lua
local ten = mpz(10) -- create the mpz numeral 10.
assert(is_mpz(ten)) -- test whether 'ten' is a mpz numeral or not
assert(not is_mpz(10))
local big = mpz("100000000000000000000000") -- create a mpz numeral from a string
-- The operators +, -, *, /, ^, <, <=, >, >=, ==, ~=
-- The operators +, -, *, /, ^ accept mixed mpz and Lua native types
assert(ten + 1 == mpz(11))
-- In Lua, objects of different types are always different
-- So, the mpz number 10 is different from the native Lua numeral 10
assert(mpz(10) ~= 10)
assert(mpz(3) / 2 == mpz(1))
-- The second argument of ^ must be a non-negative Lua numeral
assert(mpz(10) ^ 100 > mpz("1000000000000000000000000"))
assert(mpz(3) * mpz("1000000000000000000000") == mpz("3000000000000000000000"))
assert(mpz(4) + "10" == mpz(14))
local q1 = mpq(10) -- create the mpq numeral 10
local q2 = q1 / 3  -- create the mpq numeral 10/3
assert(q2 == mpq("10/3"))
local q3 = mpq(0.25) -- create the mpq numeral 1/4
assert(q3 == mpq(1)/4)
assert(is_mpq(q3)) -- test whether 'q3' is a mpq numeral or not
assert(not is_mpq(mpz(10))) -- mpz numerals are not mpq
assert(ten == mpz(10))
local q4 = mpq(ten) -- convert the mpz numeral 'ten' into a mpq numeral
assert(is_mpq(q4))
assert(mpq(1)/3 + mpq(2)/3 == mpq(1))
assert(mpq(0.5)^5 == mpq("1/32"))
assert(mpq(1)/2 > mpq("1/3"))
```

## S-expressions

In Lean, we use Lisp-like non-mutable S-expressions as a basis for
building configuration options, statistics, formatting objects, and
other structured objects. S-expressions can be atomic values (nil, strings,
hierarchical names, integers, doubles, Booleans, and multiple precision
integers and rationals), or pairs (aka _cons-cell_).
The following example demonstrates how to create S-expressions using Lua.

```lua
local s = sexpr(1, 2) -- Create a pair containing the atomic values 1 and 2
assert(is_sexpr(s))   -- 's' is a pair
assert(s:is_cons())   -- 's' is a cons-cell/pair
assert(s:head():is_atom())   -- the 'head' is an atomic S-expression
assert(s:head() == sexpr(1)) -- the 'head' of 's' is the atomic S-expression 1
assert(s:tail() == sexpr(2)) -- the 'head' of 's' is the atomic S-expression 2
s = sexpr(1, 2, 3, nil) -- Create a 'list' containing 1, 2 and 3
assert(s:length() == 3)
assert(s:head() == sexpr(1))
assert(s:tail() == sexpr(2, 3, nil))
assert(s:head():is_int())      -- the 'head' is an integer
assert(s:head():to_int() == 1) -- return the integer stored in the 'head' of 's'
local h, t = s:fields()        -- return the 'head' and 'tail' of s
assert(h == sexpr(1))
```

The following example demonstrates how to test the kind of and extract
the value stored in atomic S-expressions.

```lua
assert(sexpr(1):is_int())
assert(sexpr(1):to_int() == 1)
assert(sexpr(true):is_bool())
assert(sexpr(false):to_bool() == false)
assert(sexpr("hello"):is_string())
assert(sexpr("hello"):to_string() == "hello")
assert(sexpr(name("n", 1)):is_name())
assert(sexpr(name("n", 1)):to_name() == name("n", 1))
assert(sexpr(mpz(10)):is_mpz())
assert(sexpr(mpz(10)):to_mpz() == mpz(10))
assert(sexpr(mpq(3)/2):is_mpq())
assert(sexpr(mpq(3)/2):to_mpq() == mpq(6)/4)
```
We can also use the method `fields` to extract the value stored
in atomic S-expressions. It is more convenient than using
the `to_*` methods.

```lua
assert(sexpr(10):fields() == 10)
assert(sexpr("hello"):fields() == "hello")
```

The `to_*` methods raise an error is the argument does not match
the expected type. Remark: in Lua, we catch errors using
the builtin function [`pcall`](http://pgl.yoyo.org/luai/i/pcall) (aka _protected call_).

```lua
local s = sexpr(10)
local ok, ex = pcall(function() s:to_string() end)
assert(not ok)
-- 'ex' is a Lean exception
assert(is_exception(ex))
```

We say an S-expression `s` is a _list_ iff `s` is a pair, and the
`tail` is nil or a list. So, every _list_ is a pair, but not every
pair is a list.

```lua
assert(sexpr(1, 2):is_cons()) -- The S-expression is a pair
assert(not sexpr(1, 2):is_list()) -- This pair is not a list
assert(sexpr(1, nil):is_list())   -- List with one element
assert(sexpr(1, 2, nil):is_list()) -- List with two elements
-- The expression sexpr(1, 2, nil) is syntax-sugar
-- for sexpr(1, sexpr(2, nil))
assert(sexpr(1, 2, nil) == sexpr(1, sexpr(2, nil)))
-- The methond 'length' returns the length of the list
assert(sexpr(1, 2, nil):length() == 2)
```

We can encode trees using S-expressions. The following example
demonstrates how to write a simple recursive function that
collects all leaves (aka atomic values) stored in a S-expression
tree.

```lua
function collect(S)
    -- We store the result in a Lua table
    local result = {}
    function loop(S)
        if S:is_cons() then
            loop(S:head())
            loop(S:tail())
        elseif not S:is_nil() then
            result[#result + 1] = S:fields()
        end
    end
    loop(S)
    return result
end
-- Create a simple tree
local tree = sexpr(sexpr(1, 5), sexpr(sexpr(4, 3), 5))
local leaves = collect(tree) -- store the leaves in a Lua table
assert(#leaves == 5)
assert(leaves[1] == 1)
assert(leaves[2] == 5)
assert(leaves[3] == 4)
```

## Options

Lean components can be configured used _options_ objects.  The options
can be explicitly provided or read from a global variable. An options
object is a non-mutable value based on S-expressions.
An options object is essentially a list of pairs, where each pair
is a hierarchical name and a value. The following example demonstrates
how to create options objects using Lua.

```lua
-- Create an options set containing the entries
--   pp::colors  -> false,
--   pp::unicode -> false
local o1 = options(name("pp", "colors"), false, name("pp", "unicode"), false)
assert(is_options(o1))
print(o1)
-- The same example using syntax-sugar for hierarchical names.
-- The Lean-Lua API will automatically convert Lua arrays into hierarchical names.
local o2 = options({"pp", "colors"}, false, {"pp", "unicode"}, false)
print(o2)
-- An error is raised if the option is not known.
local ok, ex = pcall(function() options({"pp", "foo"}, true) end)
assert(not ok)
assert(ex:what():find("unknown option 'pp::foo'"))
```

Options objects are non-mutable values. The method `update` returns a new
updates options object.

```lua
local o1 = options() -- create the empty options
assert(o1:empty())
local o2 = o1:update({'pp', 'colors'}, true)
assert(o1:empty())
assert(not o2:empty())
assert(o2:size() == 1)
assert(o2:get({'pp', 'colors'}) == true)
assert(o2:get{'pp', 'colors'} == true)
assert(o2:contains{'pp', 'colors'})
assert(not o2:contains{'pp', 'unicode'})
-- We can provide a default value for 'get'.
-- The default value is used if the options object does
-- not contain the key.
assert(o2:get({'pp', 'width'}) == 0)
assert(o2:get({'pp', 'width'}, 10) == 10)
assert(o2:get({'pp', 'width'}, 20) == 20)
local o3 = o2:update({'pp', 'width'}, 100)
assert(o3:get({'pp', 'width'}) == 100)
assert(o3:get({'pp', 'width'}, 1) == 100)
assert(o3:get({'pp', 'width'}, 20) == 100)
```

The functions `get_options` and `set_options` get/set the global
options object. For example, the global options object is used
by the `print` method.

```lua
local o = options({'pp', 'unicode'}, false)
set_options(o)
assert(get_options():contains{'pp', 'unicode'})
```