Module Sort (Import X:Orders.TotalLeBool').
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.

Theorem Sorted_merge_list_to_stack : forall stack l,
SortedStack stack -> Sorted l -> SortedStack (merge_list_to_stack stack l).