MultisetInsertion Sort Verified With Multisets

Our specification of sorted in Sort was based in part on permutations, which enabled us to express the idea that sorting rearranges list elements but does not add or remove any.
Another way to express that idea is to use multisets, aka bags. A set is like a list in which element order is irrelevant, and in which no duplicate elements are permitted. That is, an element can either be in the set or not in the set, but it can't be in the set multiple times. A multiset relaxes that restriction: an element can be in the multiset multiple times. The number of times the element occurs in the multiset is the element's multiplicity.
For example:
  • {1, 2} is a set, and is the same as set {2, 1}.
  • [1; 1; 2] is a list, and is different than list [2; 1; 1].
  • {1, 1, 2} is a multiset and the same as multiset {2, 1, 1}.
In this chapter we'll explore using multisets to specify sortedness.

Multisets

We will represent multisets as functions: if m is a multiset, then m n will be the multiplicity of n in m. Since we are sorting lists of natural numbers, the type of multisets would be nat nat. The input is the value, the output is its multiplicity. To help avoid confusion between those two uses of nat, we'll introduce a synonym, value.
Definition value := nat.

Definition multiset := value nat.
The empty multiset has multiplicity 0 for every value.
Definition empty : multiset :=
  fun x ⇒ 0.
Multiset singleton v contains only v, and exactly once.
Definition singleton (v: value) : multiset :=
  fun xif x =? v then 1 else 0.
The union of two multisets is their pointwise sum.
Definition union (a b : multiset) : multiset :=
  fun xa x + b x.

Exercise: 1 star, standard (union_assoc)

Prove that multiset union is associative.
To prove that one multiset equals another we use the axiom of functional extensionality, which was introduced in Logic. We begin the proof below by using Coq's tactic extensionality, which applies that axiom.
Lemma union_assoc: a b c : multiset,
   union a (union b c) = union (union a b) c.
Proof.
  intros.
  extensionality x.
  (* FILL IN HERE *) Admitted.

Exercise: 1 star, standard (union_comm)

Prove that multiset union is commutative.
Lemma union_comm: a b : multiset,
   union a b = union b a.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars, standard (union_swap)

Prove that the multisets in a nested union can be swapped. You do not need extensionality if you use the previous two lemmas.
Lemma union_swap : a b c : multiset,
    union a (union b c) = union b (union a c).
Proof.
  (* FILL IN HERE *) Admitted.
Note that this is not an efficient implementation of multisets. We wouldn't want to use it for programs with high performance requirements. But we are using multisets for specifications, not for programs. We don't intend to build large multisets, only to use them in verifying algorithms such as insertion sort. So this inefficiency is not a problem.

Specification of Sorting

A sorting algorithm must rearrange the elements into a list that is totally ordered. Using multisets, we can restate that as: the algorithm must produce a list with the same multiset of values, and this list must be totally ordered. Let's formalize that idea.
The contents of a list are the elements it contains, without any notion of order. Function contents extracts the contents of a list as a multiset.
Fixpoint contents (al: list value) : multiset :=
  match al with
  | []empty
  | a :: blunion (singleton a) (contents bl)
  end.
The insertion-sort program sort from Sort preserves the contents of a list. Here's an example of that:
Example sort_pi_same_contents:
  contents (sort [3;1;4;1;5;9;2;6;5;3;5]) = contents [3;1;4;1;5;9;2;6;5;3;5].
Proof.
  extensionality x.
  repeat (destruct x; try reflexivity).
  (* Why does this work? Try it step by step, without repeat. *)
Qed.
A sorting algorithm must preserve contents and totally order the list.
Definition is_a_sorting_algorithm' (f: list nat list nat) := al,
    contents al = contents (f al) sorted (f al).
That definition is similar to is_a_sorting_algorithm from Sort, except that we're now using contents instead of Permutation.

Verification of Insertion Sort

The following series of exercises will take you through a verification of insertion sort using multisets.

Exercise: 3 stars, standard (insert_contents)

Prove that insertion sort's insert function produces the same contents as merely prepending the inserted element to the front of the list.
Proceed by induction. You do not need extensionality if you make use of the above lemmas about union.
Lemma insert_contents: x l,
     contents (insert x l) = contents (x :: l).
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars, standard (sort_contents)

Prove that insertion sort preserves contents. Proceed by induction. Make use of insert_contents.
Theorem sort_contents: l,
    contents l = contents (sort l).
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 1 star, standard (insertion_sort_correct)

Finish the proof of correctness!
Theorem insertion_sort_correct :
  is_a_sorting_algorithm' sort.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 1 star, standard (permutations_vs_multiset)

Compare your proofs of insert_perm, sort_perm with your proofs of insert_contents, sort_contents. Which proofs are simpler?
  • easier with permutations
  • easier with multisets
  • about the same
Regardless of "difficulty", which do you prefer or find easier to think about?
  • permutations
  • multisets
Put an X in one box in each list.
(* Do not modify the following line: *)
Definition manual_grade_for_permutations_vs_multiset : option (nat×string) := None.

Equivalence of Permutation and Multiset Specifications

We have developed two specifications of sorting, one based on permutations (is_a_sorting_algorithm) and one based on multisets (is_a_sorting_algorithm'). These two specifications are actually equivalent, which will be the final theorem in this chapter.
One reason for that equivalence is that permutations and multisets are closely related. We'll begin by proving:

        Permutation al blcontents al = contents bl
The forward direction is relatively easy, but the backward direction is surprisingly difficult.

The Forward Direction

Exercise: 3 stars, standard (perm_contents)

The forward direction is the easier one. Proceed by induction on the evidence for Permutation al bl:
Lemma perm_contents: al bl : list nat,
    Permutation al bl contents al = contents bl.
Proof.
  (* FILL IN HERE *) Admitted.

The Backward Direction (Advanced)

The backward direction is surprisingly difficult. This proof approach is due to Zhong Sheng Hu. The first three lemmas are used to prove the fourth one. Don't forget that union, singleton, and empty must be explicitly unfolded to access their definitions.

Exercise: 2 stars, advanced (contents_nil_inv)

Lemma contents_nil_inv : l, ( x, 0 = contents l x) l = nil.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 3 stars, advanced (contents_cons_inv)

Lemma contents_cons_inv : l x n,
    S n = contents l x
     l1 l2,
      l = l1 ++ x :: l2
       contents (l1 ++ l2) x = n.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 2 stars, advanced (contents_insert_other)

Lemma contents_insert_other : l1 l2 x y,
    y x contents (l1 ++ x :: l2) y = contents (l1 ++ l2) y.
Proof.
  (* FILL IN HERE *) Admitted.

Exercise: 3 stars, advanced (contents_perm)

Lemma contents_perm: al bl,
    contents al = contents bl Permutation al bl.
Proof.
  intros al bl H0.
  assert (H: x, contents al x = contents bl x).
  { rewrite H0. auto. }
  clear H0.
  generalize dependent bl.
(* FILL IN HERE *) Admitted.

The Main Theorem

With both directions proved, we can establish the correspondence between multisets and permutations.

Exercise: 1 star, standard (same_contents_iff_perm)

Use contents_perm (even if you haven't proved it) and perm_contents to quickly prove the next theorem.
Theorem same_contents_iff_perm: al bl,
    contents al = contents bl Permutation al bl.
Proof.
  (* FILL IN HERE *) Admitted.
Therefore the two specifications are equivalent.

Exercise: 2 stars, standard (sort_specifications_equivalent)

Theorem sort_specifications_equivalent: sort,
    is_a_sorting_algorithm sort is_a_sorting_algorithm' sort.
Proof.
  (* FILL IN HERE *) Admitted.
That means we can verify sorting algorithms using either permutations or multisets, whichever we find more convenient.
(* 2021-08-11 15:15 *)