Library Coq.Sorting.Mergesort

A modular implementation of mergesort (the complexity is O(n.log n) in the length of the list)

Notations and conventions

The main module defining mergesort on a given boolean order <=?. We require minimal hypotheses : this boolean order should only be total: forall x y, (x<=?y) \/ (y<=?x). Transitivity is not mandatory, but without it one can only prove LocallySorted and not StronglySorted.

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.

The proof of correctness

An example