Library prosa.util.unit_growth

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat.
Require Import prosa.util.tactics prosa.util.notation.

We say that a function f is a unit growth function iff for any time instant t it holds that f (t + 1) ≤ f t + 1.
Definition unit_growth_function (f : nat → nat) :=
  ∀ t, f (t + 1) ≤ f t + 1.

In this section, we prove a few useful lemmas about unit growth functions.
Section Lemmas.

Let f be any unit growth function over natural numbers.
  Variable f : nat → nat.
  Hypothesis H_unit_growth_function : unit_growth_function f.

In the following section, we prove a result similar to the intermediate value theorem for continuous functions.
  Section ExistsIntermediateValue.

Consider any interval x1, x2.
    Variable x1 x2 : nat.
    Hypothesis H_is_interval : x1 ≤ x2.

Let t be any value such that f x1 ≤ y < f x2.
    Variable y : nat.
    Hypothesis H_between : f x1 ≤ y < f x2.

Then, we prove that there exists an intermediate point x_mid such that f x_mid = y.
    Lemma exists_intermediate_point :
      ∃ x_mid, x1 ≤ x_mid < x2 ∧ f x_mid = y.
    Proof.
      rename H_is_interval into INT, H_unit_growth_function into UNIT, H_between into BETWEEN.
      move: x2 INT BETWEEN; clear x2.
      suff DELTA:
        ∀ delta,
          f x1 ≤ y < f (x1 + delta) →
          ∃ x_mid, x1 ≤ x_mid < x1 + delta ∧ f x_mid = y.
      { move ⇒ x2 LE /andP [GEy LTy].
        (* apply DELTA. *)
        specialize (DELTA (x2 - x1)); feed DELTA.
        { by apply/andP; split; last by rewrite addnBA // addKn. }
        by rewrite subnKC in DELTA.
      }
      induction delta.
      { rewrite addn0; move ⇒ /andP [GE0 LT0].
        by apply (leq_ltn_trans GE0) in LT0; rewrite ltnn in LT0.
      }
      { move ⇒ /andP [GT LT].
        specialize (UNIT (x1 + delta)); rewrite leq_eqVlt in UNIT.
        have LE: y ≤ f (x1 + delta).
        { move: UNIT ⇒ /orP [/eqP EQ | UNIT]; first by rewrite !addn1 in EQ; rewrite addnS EQ ltnS in LT.
          rewrite [X in _ < X]addn1 ltnS in UNIT.
          apply: (leq_trans _ UNIT).
          by rewrite addn1 -addnS ltnW.
        } clear UNIT LT.
        rewrite leq_eqVlt in LE.
        move: LE ⇒ /orP [/eqP EQy | LT].
        { ∃ (x1 + delta); split; last by rewrite EQy.
          by apply/andP; split; [apply leq_addr | rewrite addnS].
        }
        { feed (IHdelta); first by apply/andP; split.
          move: IHdelta ⇒ [x_mid [/andP [GE0 LT0] EQ0]].
          ∃ x_mid; split; last by done.
          apply/andP; split; first by done.
          by apply: (leq_trans LT0); rewrite addnS.
        }
      }
    Qed.

  End ExistsIntermediateValue.

In this section, we, again, prove an analogue of the intermediate value theorem, but for predicates over natural numbers.
  Section ExistsIntermediateValuePredicates.

Let P be any predicate on natural numbers.
    Variable P : nat → bool.

Consider a time interval [t1,t2] such that ...
    Variables t1 t2 : nat.
    Hypothesis H_t1_le_t2 : t1 ≤ t2.

... P doesn't hold for t1 ...
    Hypothesis H_not_P_at_t1 : ~~ P t1.

... but holds for t2.
    Hypothesis H_P_at_t2 : P t2.

Then we prove that within time interval [t1,t2] there exists time instant t such that t is the first time instant when P holds.
    Lemma exists_first_intermediate_point :
      ∃ t, (t1 < t ≤ t2) ∧ (∀ x, t1 ≤ x < t → ~~ P x) ∧ P t.
    Proof.
      have EX: ∃ x, P x && (t1 < x ≤ t2).
      { ∃ t2.
        apply/andP; split; first by done.
        apply/andP; split; last by done.
        move: H_t1_le_t2; rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ | NEQ1]; last by done.
        by exfalso; subst t2; move: H_not_P_at_t1 ⇒ /negP NPt1.
      }
      have MIN := ex_minnP EX.
      move: MIN ⇒ [x /andP [Px /andP [LT1 LT2]] MIN]; clear EX.
      ∃ x; repeat split; [ apply/andP; split | | ]; try done.
      move ⇒ y /andP [NEQ1 NEQ2]; apply/negPn; intros Py.
      feed (MIN y).
      { apply/andP; split; first by done.
        apply/andP; split.
        - move: NEQ1. rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ | NEQ1]; last by done.
          by exfalso; subst y; move: H_not_P_at_t1 ⇒ /negP NPt1.
        - by apply ltnW, leq_trans with x.
      }
      by move: NEQ2; rewrite ltnNge; move ⇒ /negP NEQ2.
    Qed.

  End ExistsIntermediateValuePredicates.

End Lemmas.