lean2/doc/lua/lua.md

Lua API documentation

We the Lua script language to customize and extend Lean. This link 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 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.

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:

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.

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.

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, 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.

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 (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.

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"))