lambdapi 2.2.1 2.2.0 2.1.0 2.0.0 1.0

Proof assistant for the λΠ-calculus modulo rewriting
IN THIS PACKAGE
Module .
`module L = List`
`include module type of struct include L end`
`type 'a t = 'a list = `
 `| []` `| :: of 'a * 'a list`

An alias for the type of lists.

`val length : 'a list -> int`

Return the length (number of elements) of the given list.

`val compare_lengths : 'a list -> 'b list -> int`

Compare the lengths of two lists. `compare_lengths l1 l2` is equivalent to `compare (length l1) (length l2)`, except that the computation stops after reaching the end of the shortest list.

• since 4.05.0
`val compare_length_with : 'a list -> int -> int`

Compare the length of a list to an integer. `compare_length_with l len` is equivalent to `compare (length l) len`, except that the computation stops after at most `len` iterations on the list.

• since 4.05.0
`val cons : 'a -> 'a list -> 'a list`

`cons x xs` is `x :: xs`

• since 4.03.0 (4.05.0 in ListLabels)
`val hd : 'a list -> 'a`

Return the first element of the given list.

• raises Failure

if the list is empty.

`val tl : 'a list -> 'a list`

Return the given list without its first element.

• raises Failure

if the list is empty.

`val nth : 'a list -> int -> 'a`

Return the `n`-th element of the given list. The first element (head of the list) is at position 0.

• raises Failure

if the list is too short.

• raises Invalid_argument

if `n` is negative.

`val nth_opt : 'a list -> int -> 'a option`

Return the `n`-th element of the given list. The first element (head of the list) is at position 0. Return `None` if the list is too short.

• raises Invalid_argument

if `n` is negative.

• since 4.05
`val rev : 'a list -> 'a list`

List reversal.

`val append : 'a list -> 'a list -> 'a list`

Concatenate two lists. Same function as the infix operator `@`. Not tail-recursive (length of the first argument). The `@` operator is not tail-recursive either.

`val rev_append : 'a list -> 'a list -> 'a list`

`rev_append l1 l2` reverses `l1` and concatenates it with `l2`. This is equivalent to `(``rev`` l1) @ l2`, but `rev_append` is tail-recursive and more efficient.

`val concat : 'a list list -> 'a list`

Concatenate a list of lists. The elements of the argument are all concatenated together (in the same order) to give the result. Not tail-recursive (length of the argument + length of the longest sub-list).

`val flatten : 'a list list -> 'a list`

Same as `concat`. Not tail-recursive (length of the argument + length of the longest sub-list).

Comparison

`val equal : ( 'a -> 'a -> bool ) -> 'a list -> 'a list -> bool`

`equal eq [a1; ...; an] [b1; ..; bm]` holds when the two input lists have the same length, and for each pair of elements `ai`, `bi` at the same position we have `eq ai bi`.

Note: the `eq` function may be called even if the lists have different length. If you know your equality function is costly, you may want to check `compare_lengths` first.

• since 4.12.0
`val compare : ( 'a -> 'a -> int ) -> 'a list -> 'a list -> int`

`compare cmp [a1; ...; an] [b1; ...; bm]` performs a lexicographic comparison of the two input lists, using the same `'a -> 'a -> int` interface as `Stdlib.compare`:

• `a1 :: l1` is smaller than `a2 :: l2` (negative result) if `a1` is smaller than `a2`, or if they are equal (0 result) and `l1` is smaller than `l2`
• the empty list `[]` is strictly smaller than non-empty lists

Note: the `cmp` function will be called even if the lists have different lengths.

• since 4.12.0

Iterators

`val iter : ( 'a -> unit ) -> 'a list -> unit`

`iter f [a1; ...; an]` applies function `f` in turn to `a1; ...; an`. It is equivalent to `begin f a1; f a2; ...; f an; () end`.

`val iteri : ( int -> 'a -> unit ) -> 'a list -> unit`

Same as `iter`, but the function is applied to the index of the element as first argument (counting from 0), and the element itself as second argument.

• since 4.00.0
`val map : ( 'a -> 'b ) -> 'a list -> 'b list`

`map f [a1; ...; an]` applies function `f` to `a1, ..., an`, and builds the list `[f a1; ...; f an]` with the results returned by `f`. Not tail-recursive.

`val mapi : ( int -> 'a -> 'b ) -> 'a list -> 'b list`

Same as `map`, but the function is applied to the index of the element as first argument (counting from 0), and the element itself as second argument. Not tail-recursive.

• since 4.00.0
`val rev_map : ( 'a -> 'b ) -> 'a list -> 'b list`

`rev_map f l` gives the same result as `rev`` (``map`` f l)`, but is tail-recursive and more efficient.

`val fold_left_map : ( 'a -> 'b -> 'a * 'c ) -> 'a -> 'b list -> 'a * 'c list`

`fold_left_map` is a combination of `fold_left` and `map` that threads an accumulator through calls to `f`.

• since 4.11.0
`val fold_left : ( 'a -> 'b -> 'a ) -> 'a -> 'b list -> 'a`

`fold_left f init [b1; ...; bn]` is `f (... (f (f init b1) b2) ...) bn`.

`val fold_right : ( 'a -> 'b -> 'b ) -> 'a list -> 'b -> 'b`

`fold_right f [a1; ...; an] init` is `f a1 (f a2 (... (f an init) ...))`. Not tail-recursive.

Iterators on two lists

`val iter2 : ( 'a -> 'b -> unit ) -> 'a list -> 'b list -> unit`

`iter2 f [a1; ...; an] [b1; ...; bn]` calls in turn `f a1 b1; ...; f an bn`.

• raises Invalid_argument

if the two lists are determined to have different lengths.

`val map2 : ( 'a -> 'b -> 'c ) -> 'a list -> 'b list -> 'c list`

`map2 f [a1; ...; an] [b1; ...; bn]` is `[f a1 b1; ...; f an bn]`.

• raises Invalid_argument

if the two lists are determined to have different lengths. Not tail-recursive.

`val rev_map2 : ( 'a -> 'b -> 'c ) -> 'a list -> 'b list -> 'c list`

`rev_map2 f l1 l2` gives the same result as `rev`` (``map2`` f l1 l2)`, but is tail-recursive and more efficient.

`val fold_left2 : ( 'a -> 'b -> 'c -> 'a ) -> 'a -> 'b list -> 'c list -> 'a`

`fold_left2 f init [a1; ...; an] [b1; ...; bn]` is `f (... (f (f init a1 b1) a2 b2) ...) an bn`.

• raises Invalid_argument

if the two lists are determined to have different lengths.

`val fold_right2 : ( 'a -> 'b -> 'c -> 'c ) -> 'a list -> 'b list -> 'c -> 'c`

`fold_right2 f [a1; ...; an] [b1; ...; bn] init` is `f a1 b1 (f a2 b2 (... (f an bn init) ...))`.

• raises Invalid_argument

if the two lists are determined to have different lengths. Not tail-recursive.

List scanning

`val for_all : ( 'a -> bool ) -> 'a list -> bool`

`for_all f [a1; ...; an]` checks if all elements of the list satisfy the predicate `f`. That is, it returns `(f a1) && (f a2) && ... && (f an)` for a non-empty list and `true` if the list is empty.

`val exists : ( 'a -> bool ) -> 'a list -> bool`

`exists f [a1; ...; an]` checks if at least one element of the list satisfies the predicate `f`. That is, it returns `(f a1) || (f a2) || ... || (f an)` for a non-empty list and `false` if the list is empty.

`val for_all2 : ( 'a -> 'b -> bool ) -> 'a list -> 'b list -> bool`

Same as `for_all`, but for a two-argument predicate.

• raises Invalid_argument

if the two lists are determined to have different lengths.

`val exists2 : ( 'a -> 'b -> bool ) -> 'a list -> 'b list -> bool`

Same as `exists`, but for a two-argument predicate.

• raises Invalid_argument

if the two lists are determined to have different lengths.

`val mem : 'a -> 'a list -> bool`

`mem a set` is true if and only if `a` is equal to an element of `set`.

`val memq : 'a -> 'a list -> bool`

Same as `mem`, but uses physical equality instead of structural equality to compare list elements.

List searching

`val find : ( 'a -> bool ) -> 'a list -> 'a`

`find f l` returns the first element of the list `l` that satisfies the predicate `f`.

• raises Not_found

if there is no value that satisfies `f` in the list `l`.

`val find_opt : ( 'a -> bool ) -> 'a list -> 'a option`

`find f l` returns the first element of the list `l` that satisfies the predicate `f`. Returns `None` if there is no value that satisfies `f` in the list `l`.

• since 4.05
`val filter : ( 'a -> bool ) -> 'a list -> 'a list`

`filter f l` returns all the elements of the list `l` that satisfy the predicate `f`. The order of the elements in the input list is preserved.

`val find_all : ( 'a -> bool ) -> 'a list -> 'a list`

`find_all` is another name for `filter`.

`val filteri : ( int -> 'a -> bool ) -> 'a list -> 'a list`

Same as `filter`, but the predicate is applied to the index of the element as first argument (counting from 0), and the element itself as second argument.

• since 4.11.0
`val partition : ( 'a -> bool ) -> 'a list -> 'a list * 'a list`

`partition f l` returns a pair of lists `(l1, l2)`, where `l1` is the list of all the elements of `l` that satisfy the predicate `f`, and `l2` is the list of all the elements of `l` that do not satisfy `f`. The order of the elements in the input list is preserved.

```val partition_map : ( 'a -> ( 'b, 'c ) Either.t ) -> 'a list -> 'b list * 'c list```

`partition_map f l` returns a pair of lists `(l1, l2)` such that, for each element `x` of the input list `l`:

• if `f x` is `Left y1`, then `y1` is in `l1`, and
• if `f x` is `Right y2`, then `y2` is in `l2`.

The output elements are included in `l1` and `l2` in the same relative order as the corresponding input elements in `l`.

In particular, `partition_map (fun x -> if f x then Left x else Right x) l` is equivalent to `partition f l`.

• since 4.12.0

Association lists

`val assoc : 'a -> ('a * 'b) list -> 'b`

`assoc a l` returns the value associated with key `a` in the list of pairs `l`. That is, `assoc a [ ...; (a,b); ...] = b` if `(a,b)` is the leftmost binding of `a` in list `l`.

• raises Not_found

if there is no value associated with `a` in the list `l`.

`val assoc_opt : 'a -> ('a * 'b) list -> 'b option`

`assoc_opt a l` returns the value associated with key `a` in the list of pairs `l`. That is, `assoc_opt a [ ...; (a,b); ...] = Some b` if `(a,b)` is the leftmost binding of `a` in list `l`. Returns `None` if there is no value associated with `a` in the list `l`.

• since 4.05
`val assq : 'a -> ('a * 'b) list -> 'b`

Same as `assoc`, but uses physical equality instead of structural equality to compare keys.

`val assq_opt : 'a -> ('a * 'b) list -> 'b option`

Same as `assoc_opt`, but uses physical equality instead of structural equality to compare keys.

• since 4.05.0
`val mem_assoc : 'a -> ('a * 'b) list -> bool`

Same as `assoc`, but simply return `true` if a binding exists, and `false` if no bindings exist for the given key.

`val mem_assq : 'a -> ('a * 'b) list -> bool`

Same as `mem_assoc`, but uses physical equality instead of structural equality to compare keys.

`val remove_assoc : 'a -> ('a * 'b) list -> ('a * 'b) list`

`remove_assoc a l` returns the list of pairs `l` without the first pair with key `a`, if any. Not tail-recursive.

`val remove_assq : 'a -> ('a * 'b) list -> ('a * 'b) list`

Same as `remove_assoc`, but uses physical equality instead of structural equality to compare keys. Not tail-recursive.

Lists of pairs

`val combine : 'a list -> 'b list -> ('a * 'b) list`

Transform a pair of lists into a list of pairs: `combine [a1; ...; an] [b1; ...; bn]` is `[(a1,b1); ...; (an,bn)]`.

• raises Invalid_argument

if the two lists have different lengths. Not tail-recursive.

Sorting

`val sort : ( 'a -> 'a -> int ) -> 'a list -> 'a list`

Sort a list in increasing order according to a comparison function. The comparison function must return 0 if its arguments compare as equal, a positive integer if the first is greater, and a negative integer if the first is smaller (see Array.sort for a complete specification). For example, `Stdlib.compare` is a suitable comparison function. The resulting list is sorted in increasing order. `sort` is guaranteed to run in constant heap space (in addition to the size of the result list) and logarithmic stack space.

The current implementation uses Merge Sort. It runs in constant heap space and logarithmic stack space.

`val stable_sort : ( 'a -> 'a -> int ) -> 'a list -> 'a list`

Same as `sort`, but the sorting algorithm is guaranteed to be stable (i.e. elements that compare equal are kept in their original order).

The current implementation uses Merge Sort. It runs in constant heap space and logarithmic stack space.

`val fast_sort : ( 'a -> 'a -> int ) -> 'a list -> 'a list`

Same as `sort` or `stable_sort`, whichever is faster on typical input.

`val sort_uniq : ( 'a -> 'a -> int ) -> 'a list -> 'a list`

Same as `sort`, but also remove duplicates.

• since 4.02.0 (4.03.0 in ListLabels)
`val merge : ( 'a -> 'a -> int ) -> 'a list -> 'a list -> 'a list`

Merge two lists: Assuming that `l1` and `l2` are sorted according to the comparison function `cmp`, `merge cmp l1 l2` will return a sorted list containing all the elements of `l1` and `l2`. If several elements compare equal, the elements of `l1` will be before the elements of `l2`. Not tail-recursive (sum of the lengths of the arguments).

Lists and Sequences

`val to_seq : 'a list -> 'a Seq.t`

Iterate on the list.

• since 4.07
`val of_seq : 'a Seq.t -> 'a list`

Create a list from a sequence.

• since 4.07
`val zip : 'a list -> 'b list -> ('a * 'b) list`
`val unzip : ('a * 'b) list -> 'a list * 'b list`
`val pp : 'a Base.pp -> unit Base.outfmt -> 'a list Base.pp`

`pp elt sep ppf l` prints the list `l` on the formatter `ppf` using `sep` as separator, and `elt` for printing the elements.

`val cmp : 'a Base.cmp -> 'a list Base.cmp`

Total order on lists.

`val eq : 'a Base.eq -> 'a list Base.eq`

`eq eq_elt l1 l2` tests the equality of `l1` and `l2`, comparing their elements with `eq_elt`.

`val find_map : ( 'a -> 'b option ) -> 'c t -> 'd option`

`find_map f l` applies `f` to the elements of `l` in order, and returns the first result of the form `Some v`, or `None` if none exist.

• since 4.10.0
`val filter_map : ( 'a -> 'b option ) -> 'a list -> 'b list`

`filter_map f l` applies `f` to every element of `l`, filters out the `None` elements and returns the list of the arguments of the `Some` elements.

• since 4.08.0
`val concat_map : ( 'a -> 'b list ) -> 'a t -> 'b list`

`concat_map f l` gives the same result as `concat`` (``map`` f l)`. Tail-recursive.

• since 4.10.0
`val filter_rev_map : ( 'a -> 'b option ) -> 'a list -> 'b list`

`filter_rev_map f l` is equivalent to `filter_map f (List.rev l)`, but it only traverses the list once and is tail-recursive.

`val filteri_map : ( int -> 'a -> 'b option ) -> 'a list -> 'b list`

`filteri_map f l` applies `f` element wise on `l` and keeps `x` such that for `e` in `l`, `f e = Some(x)`.

`val cut : 'a list -> int -> 'a list * 'a list`

`cut l k` returns a pair of lists `(l1, l2)` such that `l1` has length `min (List.length l) k` and `l1 @ l2` is equal to `l`.

`val add_array2 : 'a array -> 'b array -> ('a * 'b) list -> ('a * 'b) list`

`add_array a1 a2 l` returns a list containing the elements of `l`, and the (corresponding) elements of `a1` and `a2`. Note that `a1` and `a2` should have the same lenght otherwise `Invalid_argument` is raised.

`val same_length : 'a list -> 'b list -> bool`

`same_length l1 l2` returns `true` whenever `l1` and `l2` are lists of the same length. The function stops as soon as possible.

`val max : ?cmp:( 'a -> 'a -> int ) -> 'a list -> 'a`

`max ?cmp l` finds the max of list `l` with compare function `?cmp` defaulting to `Stdlib.compare`.

• raises Invalid_argument

if `l` is empty.

`val assoc_eq : 'a Base.eq -> 'a -> ('a * 'b) list -> 'b`

`assoc_eq e k l` is `List.assoc k l` with equality function `e`.

• raises Not_found

if `k` is not a key of `l`.

`val remove_phys_dups : 'a list -> 'a0 t`

`remove_phys_dups l` uniqify list `l` keeping only the last element, using physical equality.

`val destruct : 'a list -> int -> 'a list * 'a * 'a list`

`destruct l i` returns a triple `(left_rev, e, right)` where `e` is the `i`-th element of `l`, `left_rev` is the reversed prefix of `l` up to its `i`-th element (excluded), and `right` is the remaining suffix of `l` (starting at its `i+1`-th element).

• raises Invalid_argument

when `i < 0`.

• raises Not_found

when `i ≥ length v`.

`val reconstruct : 'a list -> 'a list -> 'a list -> 'a list`

`reconstruct left_rev l right` concatenates (reversed) `left_rev`, `l` and `right`. This function will typically be used in combination with `destruct` to insert a sublist `l` in the place of the element at the specified position in the specified list.

`val init : int -> ( int -> 'a ) -> 'a list`

`init n f` creates a list with `f 0` up to `f n` as its elements. Note that `Invalid_argument` is raised if `n` is negative.

`val mem_sorted : 'a Base.cmp -> 'a -> 'a list -> bool`

`mem_sorted cmp x l` tells whether `x` is in `l` assuming that `l` is sorted wrt `cmp`.

`val insert : 'a Base.cmp -> 'a -> 'a list -> 'a list`

`insert cmp x l` inserts `x` in the list `l` assuming that `l` is sorted in increasing order wrt `cmp`.

`val insert_uniq : 'a Base.cmp -> 'a -> 'a list -> 'a list`

`insert cmp x l` inserts `x` in the list `l` assuming that `l` is sorted in increasing order wrt `cmp`, but only if `x` does not occur in `l`.

`val split_last : 'a list -> 'a list * 'a`

`split_last l` returns `(l',x)` if `l = append l' [x]`, and

• raises Invalid_argument

otherwise.

`val rev_mapi : ( int -> 'a -> 'b ) -> 'c t -> 'd t`

`rev_mapi f [x1;..;xn]` returns `f (n-1) xn; ..; f 0 x1`.

`val swap : int -> 'a list -> 'a list`

`swap i xs` put the i-th element (counted from 0) of `xs` at the head.

• raises Invalid_argument

if the i-th element does not exist.

`val fold_left_while : ( 'a -> 'b -> 'c ) -> ( 'd -> bool ) -> 'e -> 'f t -> 'g`

`fold_left_while f cond a [b1 b2 ..]` computes (f..(f (f a b1) b2)..bm) where `cond` is true for b1..bm and false for b_m+1 or bm is last element

`val remove_first : ( 'a -> bool ) -> 'a list -> 'a list`

`remove_first f l` removes from `l` the first element satisfying `f`.

`val remove_heads : int -> 'a t -> 'a t`

`remove_heads n xs` remove the min(n,length xs) elements of `xs`.

`val split : ( 'a -> bool ) -> 'a list -> 'a list * 'a * 'a list`

`split f l` returns the tuple `(l1,x,l2)` such that `x` is the first element of `l` satisying `f`, `l1` is the sub-list of `l` preceding `x`, and `l2` is the sub-list of `l` following `x`.

• raises Not_found

if there is no element of `l` satisying `f`.

`val iter_head_tail : ( 'a -> 'a list -> unit ) -> 'a0 list -> unit`

`iter_head_tail f l` iterates `f` on all pairs (head, tail) of `l`.