Library prosa.util.superadditivity

In this section, we define and prove facts about superadditivity and superadditive functions. The definition of superadditivity presented here slightly differs from the standard one (f a + f b f (a + b) for any a and b), but it is proven to be equivalent to it.
Section Superadditivity.

First, we define subadditivity as a point-wise property; i.e., f is subadditive at h if standard subadditivity holds for any pair (a,b) that sums to h.
  Definition superadditive_at f h :=
     a b,
      a + b = h
      f a + f b f h.

Second, we define the concept of partial subadditivity until a certain horizon h. This definition is useful when dealing with finite sequences.
  Definition superadditive_until f h :=
      x < h
      superadditive_at f x.

Finally, give a definition of subadditive function: f is subadditive when it is subadditive at any point h.
  Definition superadditive f :=
      superadditive_at f h.

In this section, we show that the proposed definition of subadditivity is equivalent to the standard one.
First, we give a standard definition of subadditivity.
    Definition superadditive_standard f :=
       a b,
        f a + f b f (a + b).

Then, we prove that the two definitions are implied by each other.
    Lemma superadditive_standard_equivalence :
        superadditive f superadditive_standard f.
      - moveSUPER a b.
        by apply SUPER.
      - moveSUPER h a b AB.
        rewrite -AB.
        by apply SUPER.

  End EquivalenceWithStandardDefinition.

In the following section, we prove some useful facts about superadditivity.
  Section Facts.

Consider a function f.
    Variable f : nat nat.

First, we show that if f is superadditive in zero, then its value in zero must also be zero.
    Lemma superadditive_first_zero:
      superadditive_at f 0
      f 0 = 0.
      destruct (f 0) eqn:Fx; first by done.
      specialize (SUPER 0 0 (addn0 0)).
      contradict SUPER.
      apply /negP; rewrite -ltnNge.
      by lia.

In this section, we show some of the properties of superadditive functions.
    Section SuperadditiveFunctions.

Assume that f is superadditive.
      Hypothesis h_superadditive : superadditive f.

First, we show that f must also be monotone.
      Lemma superadditive_monotone:
        monotone leq f.
        movex y LEQ.
        apply leq_trans with (f x + f (y - x)).
        - by lia.
        - apply h_superadditive.
          by lia.

Next, we prove that moving any factor m outside of the arguments of f leads to a smaller or equal number.
      Lemma superadditive_leq_mul:
         n m,
          m × f n f (m × n).
        moven m.
        elim: m⇒ [| m IHm]; first by lia.
        rewrite !mulSnr.
        apply leq_trans with (f (m × n) + f n).
        - by rewrite leq_add2r.
        - by apply h_superadditive.

In the next section, we show that any superadditive function that is not the zero constant function (i.e., f x = 0 for any x) is forced to grow beyond any finite limit.
      Section NonZeroSuperadditiveFunctions.

Assume that f is not the zero constant function ...
        Hypothesis h_non_zero: n, f n > 0.

... then, f will eventually grow larger than any number.
        Lemma superadditive_unbounded:
           t, n', t f n'.
          move: h_non_zero ⇒ [n LT_n].
           (t × n).
          apply leq_trans with (t × f n).
          - by apply leq_pmulr.
          - by apply superadditive_leq_mul.

      End NonZeroSuperadditiveFunctions.

    End SuperadditiveFunctions.

  End Facts.

In this section, we present the define and prove facts about the minimal superadditive extension of superadditive functions. Given a prefix of a function, there are many ways to continue the function in order to maintain superadditivity. Among these possible extrapolations, there always exists a minimal one.
Consider a function f.
    Variable f : nat nat.

First, we define what it means to find the minimal extension once a horizon is specified.
    Section Definitions.

Consider a horizon h..
      Variable h : nat.

Then, the minimal superadditive extension will be the maximum sum over the pairs that sum to h. Note that, in this formula, there are two important edge cases: both h=0 and h=1, the sequence of valid sums will be empty, so their maximum will be 0. In both cases, the extrapolation is nonetheless correct.
      Definition minimal_superadditive_extension :=
        max0 [seq f a + f (h-a) | a <- index_iota 1 h].

    End Definitions.

In the following section, we prove some facts about the minimal superadditive extension. Note that we currently do not prove that the implemented extension is minimal. However, we plan to add this fact in the future. The following discussion provides useful information on the subject, including its connection with Network Calculus:
    Section Facts.

Consider a horizon h ...
      Variable h : nat.

... and assume that we know f to be superadditive until h.
      Hypothesis h_superadditive_until : superadditive_until f h.

Moreover, consider a second function, f', which is equivalent to f in all of its points except for h, in which its value is exactly the superadditive extension of f in h.
      Variable f' : nat nat.
      Hypothesis h_f'_min_extension : t,
          f' t =
          if t == h
          then minimal_superadditive_extension h
          else f t.

First, we prove that f' is superadditive also in h.
      Theorem minimal_extension_superadditive_at_horizon :
        superadditive_at f' h.
        movea b SUM.
        rewrite !h_f'_min_extension.
        rewrite -SUM.
        destruct a as [|a'] eqn:EQa; destruct b as [|b'] eqn:EQb ⇒ //=.
        { rewrite add0n eq_refl superadditive_first_zero; first by rewrite add0n.
          by apply h_superadditive_until; lia. }
        { rewrite addn0 eq_refl superadditive_first_zero; first by rewrite addn0.
          by apply h_superadditive_until; lia. }
        { rewrite -!EQa -!EQb eq_refl //=.
          rewrite -{1}(addn0 a) eqn_add2l {1}EQb //=.
          rewrite -{1}(add0n b) eqn_add2r {2}EQa //=.
          rewrite /minimal_superadditive_extension.
          apply in_max0_le.
          apply /mapP; a.
          - by rewrite mem_iota; lia.
          - by have → : a + b - a = b by lia. }

And finally, we prove that f' is superadditive until h.+1.
      Lemma minimal_extension_superadditive_until :
        superadditive_until f' h.+1.
        movet LEQh a b SUM.
        destruct (ltngtP t h) as [LT | GT | EQ].
        - rewrite !h_f'_min_extension.
          rewrite !ltn_eqF; try lia.
          by apply h_superadditive_until.
        - by lia.
        - rewrite EQ in SUM; rewrite EQ.
          by apply minimal_extension_superadditive_at_horizon.

    End Facts.

  End MinimalExtensionOfSuperadditiveFunctions.

End Superadditivity.