(** Notations and conventions *)

Local Notation "[ ]" := nil.Local Notation "[ a ; .. ; b ]" := (a :: .. (b :: []) ..).Fixpoint merge l1 l2 :=
  let fix merge_aux l2 :=
  match l1, l2 with
  | [], _ => l2
  | _, [] => l1
  | a1::l1', a2::l2' =>
      if a1 <=? a2 then a1 :: merge l1' l2 else a2 :: merge_aux l2'
  end
  in merge_aux l2.(** We implement mergesort using an explicit stack of pending mergings.
    Pending merging are represented like a binary number where digits are
    either None (denoting 0) or Some list to merge (denoting 1). The n-th
    digit represents the pending list to be merged at level n, if any.
    Merging a list to a stack is like adding 1 to the binary number
    represented by the stack but the carry is propagated by merging the
    lists. In practice, when used in mergesort, the n-th digit, if non 0,
    carries a list of length 2^n. For instance, adding singleton list
    [3] to the stack Some [4]::Some [2;6]::None::Some [1;3;5;5]
    reduces to propagate the carry [3;4] (resulting of the merge of [3]
    and [4]) to the list Some [2;6]::None::Some [1;3;5;5], which reduces
    to propagating the carry [2;3;4;6] (resulting of the merge of [3;4] and
    [2;6]) to the list None::Some [1;3;5;5], which locally produces
    Some [2;3;4;6]::Some [1;3;5;5], i.e. which produces the final result
    None::None::Some [2;3;4;6]::Some [1;3;5;5].

    For instance, here is how [6;2;3;1;5] is sorted:

<<
       operation             stack                list
       iter_merge            []                   [6;2;3;1;5]
    =  append_list_to_stack  [ + [6]]             [2;3;1;5]
    -> iter_merge            [[6]]                [2;3;1;5]
    =  append_list_to_stack  [[6] + [2]]          [3;1;5]
    =  append_list_to_stack  [ + [2;6];]          [3;1;5]
    -> iter_merge            [[2;6];]             [3;1;5]
    =  append_list_to_stack  [[2;6]; + [3]]       [1;5]
    -> merge_list            [[2;6];[3]]          [1;5]
    =  append_list_to_stack  [[2;6];[3] + [1]     [5]
    =  append_list_to_stack  [[2;6] + [1;3];]     [5]
    =  append_list_to_stack  [ + [1;2;3;6];;]     [5]
    -> merge_list            [[1;2;3;6];;]        [5]
    =  append_list_to_stack  [[1;2;3;6];; + [5]]  []
    -> merge_stack           [[1;2;3;6];;[5]]
    =                                             [1;2;3;5;6]
>>
    The complexity of the algorithm is n*log n, since there are
    2^(p-1) mergings to do of length 2, 2^(p-2) of length 4, ..., 2^0
    of length 2^p for a list of length 2^p. The algorithm does not need
    explicitly cutting the list in 2 parts at each step since it the
    successive accumulation of fragments on the stack which ensures
    that lists are merged on a dichotomic basis.
*)

Fixpoint merge_list_to_stack stack l :=
  match stack with
  | [] => [Some l]
  | None :: stack' => Some l :: stack'
  | Some l' :: stack' => None :: merge_list_to_stack stack' (merge l' l)
  end.Fixpoint merge_stack stack :=
  match stack with
  | [] => []
  | None :: stack' => merge_stack stack'
  | Some l :: stack' => merge l (merge_stack stack')
  end.Fixpoint iter_merge stack l :=
  match l with
  | [] => merge_stack stack
  | a::l' => iter_merge (merge_list_to_stack stack [a]) l'
  end.Definition sort := iter_merge [].(** The proof of correctness *)

Local Notation Sorted := (LocallySorted leb) (only parsing).Fixpoint SortedStack stack :=
  match stack with
  | [] => True
  | None :: stack' => SortedStack stack'
  | Some l :: stack' => Sorted l /\ SortedStack stack'
  end.Local Ltac invert H := inversion H; subst; clear H.Fixpoint flatten_stack (stack : list (option (list t))) :=
  match stack with
  | [] => []
  | None :: stack' => flatten_stack stack'
  | Some l :: stack' => l ++ flatten_stack stack'
  end.

(* Specifications: *)
Theorem Sorted_merge : forall l1 l2, Sorted l1 -> Sorted l2 -> Sorted (merge l1 l2).
Theorem Permuted_merge : forall l1 l2, Permutation (l1++l2) (merge l1 l2).
Theorem Sorted_merge_list_to_stack : forall stack l, SortedStack stack -> Sorted l -> SortedStack (merge_list_to_stack stack l).
Theorem Permuted_merge_list_to_stack : forall stack l, Permutation (l ++ flatten_stack stack) (flatten_stack (merge_list_to_stack stack l)).
Theorem Sorted_merge_stack : forall stack, SortedStack stack -> Sorted (merge_stack stack).
Theorem Permuted_merge_stack : forall stack, Permutation (flatten_stack stack) (merge_stack stack).
Theorem Sorted_iter_merge : forall stack l, SortedStack stack -> Sorted (iter_merge stack l).
Theorem Permuted_iter_merge : forall l stack, Permutation (flatten_stack stack ++ l) (iter_merge stack l).
Theorem LocallySorted_sort : forall l, Sorted (sort l).