Polymorphic Lists

Basic theory of polymorphic lists

module List

  type list 'a = Nil | Cons 'a (list 'a)

  let predicate is_nil (l:list 'a)
    ensures { result <-> l = Nil }
  =
    match l with Nil -> true | Cons _ _ -> false end

end

Length of a list

module Length

  use int.Int
  use List

  let rec function length (l: list 'a) : int =
    match l with
    | Nil      -> 0
    | Cons _ r -> 1 + length r
    end

  lemma Length_nonnegative: forall l: list 'a. length l >= 0

  lemma Length_nil: forall l: list 'a. length l = 0 <-> l = Nil

end

Membership in a list

module Mem
  use List

  predicate mem (x: 'a) (l: list 'a) = match l with
    | Nil      -> false
    | Cons y r -> x = y \/ mem x r
    end

end

Quantifiers on lists

module Quant

  use List
  use Mem

  let rec function for_all (p: 'a -> bool) (l:list 'a) : bool
    ensures { result <-> forall x. mem x l -> p x }
  =
    match l with
    | Nil -> true
    | Cons x r -> p x && for_all p r
    end

  let rec function for_some (p: 'a -> bool) (l:list 'a) : bool
    ensures { result <-> exists x. mem x l /\ p x }
  =
    match l with
    | Nil -> false
    | Cons x r -> p x || for_some p r
    end

  let function mem (eq:'a -> 'a -> bool) (x:'a) (l:list 'a) : bool
    ensures  { result <-> exists y. mem y l /\ eq x y }
  =
    for_some (eq x) l

end

module Elements

  use set.Fset
  use List
  use Mem

  function elements (l: list 'a) : fset 'a =
    match l with
    | Nil -> empty
    | Cons x r -> add x (elements r)
    end

  lemma elements_mem:
    forall x: 'a, l: list 'a. mem x l <-> Fset.mem x (elements l)

end

Nth element of a list

module Nth

  use List
  use option.Option
  use int.Int

  let rec function nth (n: int) (l: list 'a) : option 'a =
    match l with
    | Nil      -> None
    | Cons x r -> if n = 0 then Some x else nth (n - 1) r
    end

end

module NthNoOpt

  use List
  use int.Int

  function nth (n: int) (l: list 'a) : 'a

  axiom nth_cons_0: forall x:'a, r:list 'a. nth 0 (Cons x r) = x
  axiom nth_cons_n: forall x:'a, r:list 'a, n:int.
    n > 0 -> nth n (Cons x r) = nth (n-1) r

end

module NthLength

  use int.Int
  use option.Option
  use List
  use export Nth
  use export Length

  lemma nth_none_1:
     forall l: list 'a, i: int. i < 0 -> nth i l = None

  lemma nth_none_2:
     forall l: list 'a, i: int. i >= length l -> nth i l = None

  lemma nth_none_3:
     forall l: list 'a, i: int. nth i l = None -> i < 0 \/ i >= length l

end

Head and tail

module HdTl

  use List
  use option.Option

  let function hd (l: list 'a) : option 'a = match l with
    | Nil      -> None
    | Cons h _ -> Some h
  end

  let function tl (l: list 'a) : option (list 'a) = match l with
    | Nil      -> None
    | Cons _ t -> Some t
  end

end

module HdTlNoOpt

  use List

  function hd (l: list 'a) : 'a

  axiom hd_cons: forall x:'a, r:list 'a. hd (Cons x r) = x

  function tl (l: list 'a) : list 'a

  axiom tl_cons: forall x:'a, r:list 'a. tl (Cons x r) = r

end

Relation between head, tail, and nth

module NthHdTl

  use int.Int
  use option.Option
  use List
  use Nth
  use HdTl

  lemma Nth_tl:
    forall l1 l2: list 'a. tl l1 = Some l2 ->
    forall i: int. i <> -1 -> nth i l2 = nth (i+1) l1

  lemma Nth0_head:
    forall l: list 'a. nth 0 l = hd l

end

Appending two lists

module Append

  use List

  let rec function (++) (l1 l2: list 'a) : list 'a =
    match l1 with
    | Nil -> l2
    | Cons x1 r1 -> Cons x1 (r1 ++ l2)
    end

  lemma Append_assoc:
    forall l1 [@induction] l2 l3: list 'a.
    l1 ++ (l2 ++ l3) = (l1 ++ l2) ++ l3

  lemma Append_l_nil:
    forall l: list 'a. l ++ Nil = l

  use Length
  use int.Int

  lemma Append_length:
    forall l1 [@induction] l2: list 'a. length (l1 ++ l2) = length l1 + length l2

  use Mem

  lemma mem_append:
    forall x: 'a, l1 [@induction] l2: list 'a.
    mem x (l1 ++ l2) <-> mem x l1 \/ mem x l2

  lemma mem_decomp:
    forall x: 'a, l: list 'a.
    mem x l -> exists l1 l2: list 'a. l = l1 ++ Cons x l2

end

module NthLengthAppend

  use int.Int
  use List
  use export NthLength
  use export Append

  lemma nth_append_1:
    forall l1 l2: list 'a, i: int.
    i < length l1 -> nth i (l1 ++ l2) = nth i l1

  lemma nth_append_2:
    forall l1 [@induction] l2: list 'a, i: int.
    length l1 <= i -> nth i (l1 ++ l2) = nth (i - length l1) l2

end

Reversing a list

module Reverse

  use List
  use Append

  let rec function reverse (l: list 'a) : list 'a =
    match l with
    | Nil      -> Nil
    | Cons x r -> reverse r ++ Cons x Nil
    end

  lemma reverse_append:
    forall l1 l2: list 'a, x: 'a.
    (reverse (Cons x l1)) ++ l2 = (reverse l1) ++ (Cons x l2)

  lemma reverse_cons:
    forall l: list 'a, x: 'a.
    reverse (Cons x l) = reverse l ++ Cons x Nil

  lemma cons_reverse:
    forall l: list 'a, x: 'a.
    Cons x (reverse l) = reverse (l ++ Cons x Nil)

  lemma reverse_reverse:
    forall l: list 'a. reverse (reverse l) = l

  use Mem

  lemma reverse_mem:
    forall l: list 'a, x: 'a. mem x l <-> mem x (reverse l)

  use Length

  lemma Reverse_length:
    forall l: list 'a. length (reverse l) = length l

end

Reverse append

module RevAppend

  use List

  let rec function rev_append (s t: list 'a) : list 'a =
    match s with
    | Cons x r -> rev_append r (Cons x t)
    | Nil -> t
    end

  use Append

  lemma rev_append_append_l:
    forall r [@induction] s t: list 'a.
      rev_append (r ++ s) t = rev_append s (rev_append r t)

  use int.Int
  use Length

  lemma rev_append_length:
    forall s [@induction] t: list 'a.
      length (rev_append s t) = length s + length t

  use Reverse

  lemma rev_append_def:
    forall r [@induction] s: list 'a. rev_append r s = reverse r ++ s

  lemma rev_append_append_r:
    forall r s t: list 'a.
      rev_append r (s ++ t) = rev_append (rev_append s r) t

end

Zip

module Combine

  use List

  let rec function combine (x: list 'a) (y: list 'b) : list ('a, 'b)
  = match x, y with
    | Cons x0 x, Cons y0 y -> Cons (x0, y0) (combine x y)
    | _ -> Nil
    end

end

Sorted lists for some order as parameter

module Sorted

  use List

  type t
  predicate le t t
  clone relations.Transitive with
    type t = t, predicate rel = le, axiom Trans

  inductive sorted (l: list t) =
    | Sorted_Nil:
        sorted Nil
    | Sorted_One:
        forall x: t. sorted (Cons x Nil)
    | Sorted_Two:
        forall x y: t, l: list t.
        le x y -> sorted (Cons y l) -> sorted (Cons x (Cons y l))

  use Mem

  lemma sorted_mem:
    forall x: t, l: list t.
    (forall y: t. mem y l -> le x y) /\ sorted l <-> sorted (Cons x l)

  use Append

  lemma sorted_append:
    forall  l1 [@induction] l2: list t.
    (sorted l1 /\ sorted l2 /\ (forall x y: t. mem x l1 -> mem y l2 -> le x y))
    <->
    sorted (l1 ++ l2)

end

Sorted lists of integers

module SortedInt

  use int.Int
  clone export Sorted with type t = int, predicate le = (<=), goal Transitive.Trans

end

module RevSorted

  type t
  predicate le t t
  clone relations.Transitive with
    type t = t, predicate rel = le, axiom Trans
  predicate ge (x y: t) = le y x

  use List

  clone Sorted as Incr with type t = t, predicate le = le, goal .
  clone Sorted as Decr with type t = t, predicate le = ge, goal .

  predicate compat (s t: list t) =
    match s, t with
    | Cons x _, Cons y _ -> le x y
    | _, _ -> true
    end

  use RevAppend

  lemma rev_append_sorted_incr:
    forall s [@induction] t: list t.
      Incr.sorted (rev_append s t) <->
        Decr.sorted s /\ Incr.sorted t /\ compat s t

  lemma rev_append_sorted_decr:
    forall s [@induction] t: list t.
      Decr.sorted (rev_append s t) <->
        Incr.sorted s /\ Decr.sorted t /\ compat t s

end

Number of occurrences in a list

module NumOcc

  use int.Int
  use List

  function num_occ (x: 'a) (l: list 'a) : int =
    match l with
    | Nil      -> 0
    | Cons y r -> (if x = y then 1 else 0) + num_occ x r
    end

  lemma Num_Occ_NonNeg: forall x:'a, l: list 'a. num_occ x l >= 0

  use Mem

  lemma Mem_Num_Occ :
    forall x: 'a, l: list 'a. mem x l <-> num_occ x l > 0

  use Append

  lemma Append_Num_Occ :
    forall x: 'a, l1 [@induction] l2: list 'a.
    num_occ x (l1 ++ l2) = num_occ x l1 + num_occ x l2

  use Reverse

  lemma reverse_num_occ :
    forall x: 'a, l: list 'a.
    num_occ x l = num_occ x (reverse l)

end

Permutation of lists

module Permut

  use NumOcc
  use List

  predicate permut (l1: list 'a) (l2: list 'a) =
    forall x: 'a. num_occ x l1 = num_occ x l2

  lemma Permut_refl: forall l: list 'a. permut l l

  lemma Permut_sym: forall l1 l2: list 'a. permut l1 l2 -> permut l2 l1

  lemma Permut_trans:
    forall l1 l2 l3: list 'a. permut l1 l2 -> permut l2 l3 -> permut l1 l3

  lemma Permut_cons:
    forall x: 'a, l1 l2: list 'a.
    permut l1 l2 -> permut (Cons x l1) (Cons x l2)

  lemma Permut_swap:
    forall x y: 'a, l: list 'a. permut (Cons x (Cons y l)) (Cons y (Cons x l))

  use Append

  lemma Permut_cons_append:
    forall x : 'a, l1 l2 : list 'a.
    permut (Cons x l1 ++ l2) (l1 ++ Cons x l2)

  lemma Permut_assoc:
    forall l1 l2 l3: list 'a.
    permut ((l1 ++ l2) ++ l3) (l1 ++ (l2 ++ l3))

  lemma Permut_append:
    forall l1 l2 k1 k2 : list 'a.
    permut l1 k1 -> permut l2 k2 -> permut (l1 ++ l2) (k1 ++ k2)

  lemma Permut_append_swap:
    forall l1 l2 : list 'a.
    permut (l1 ++ l2) (l2 ++ l1)

  use Mem

  lemma Permut_mem:
    forall x: 'a, l1 l2: list 'a. permut l1 l2 -> mem x l1 -> mem x l2

  use Length

  lemma Permut_length:
    forall l1 [@induction] l2: list 'a. permut l1 l2 -> length l1 = length l2

end

List with pairwise distinct elements

module Distinct

  use List
  use Mem

  inductive distinct (l: list 'a) =
    | distinct_zero: distinct (Nil: list 'a)
    | distinct_one : forall x:'a. distinct (Cons x Nil)
    | distinct_many:
        forall x:'a, l: list 'a.
        not (mem x l) -> distinct l -> distinct (Cons x l)

  use Append

  lemma distinct_append:
    forall l1 [@induction] l2: list 'a.
    distinct l1 -> distinct l2 -> (forall x:'a. mem x l1 -> not (mem x l2)) ->
    distinct (l1 ++ l2)

end

module Prefix

  use List
  use int.Int

  let rec function prefix (n: int) (l: list 'a) : list 'a =
    if n <= 0 then Nil else
    match l with
    | Nil -> Nil
    | Cons x r -> Cons x (prefix (n-1) r)
    end

end

module Sum

  use List
  use int.Int

  let rec function sum (l: list int) : int =
    match l with
    | Nil -> 0
    | Cons x r -> x + sum r
    end

end

(*
(** {2 Induction on lists} *)

module Induction

  use List

  (* type elt *)

  (* predicate p (list elt) *)

  axiom Induction:
    forall p: list 'a -> bool.
    p (Nil: list 'a) ->
    (forall x:'a. forall l:list 'a. p l -> p (Cons x l)) ->
    forall l:list 'a. p l

end
*)

Maps as lists of pairs

module Map

  use List

  function map (f: 'a -> 'b) (l: list 'a) : list 'b =
    match l with
    | Nil      -> Nil
    | Cons x r -> Cons (f x) (map f r)
    end
end

Generic recursors on lists

module FoldLeft

  use List

  function fold_left (f: 'b -> 'a -> 'b) (acc: 'b) (l: list 'a) : 'b =
    match l with
    | Nil      -> acc
    | Cons x r -> fold_left f (f acc x) r
    end

  use Append

  lemma fold_left_append:
    forall l1 [@induction] l2: list 'a, f: 'b -> 'a -> 'b, acc : 'b.
    fold_left f acc (l1++l2) = fold_left f (fold_left f acc l1) l2

end

module FoldRight

  use List

  function fold_right (f: 'a -> 'b -> 'b) (l: list 'a) (acc: 'b) : 'b =
    match l with
    | Nil      -> acc
    | Cons x r -> f x (fold_right f r acc)
    end

  use Append

  lemma fold_right_append:
    forall l1 [@induction] l2: list 'a, f: 'a -> 'b -> 'b, acc : 'b.
    fold_right f (l1++l2) acc = fold_right f l1 (fold_right f l2 acc)

end

Importation of all list theories in one

module ListRich

  use export List
  use export Length
  use export Mem
  use export Nth
  use export HdTl
  use export NthHdTl
  use export Append
  use export Reverse
  use export RevAppend
  use export NumOcc
  use export Permut

end