Library prosa.util.step_function

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

Section StepFunction.

  Section Defs.

    (* We say that a function f... *)
    Variable f: nat nat.

    (* a step function iff the following holds. *)
    Definition is_step_function :=
       t, f (t + 1) f t + 1.

  End Defs.

  Section Lemmas.

    (* Let f be any step function over natural numbers. *)
    Variable f: nat nat.
    Hypothesis H_step_function: is_step_function f.

    (* In this 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.
        rename H_is_interval into INT, H_step_function into STEP, H_between into BETWEEN.
        move: x2 INT BETWEEN; clear x2.
        suff DELTA:
            f x1 y < f (x1 + delta)
             x_mid, x1 x_mid < x1 + delta f x_mid = y.
        { movex2 LE /andP [GEy LTy].
          exploit (DELTA (x2 - x1));
            first by apply/andP; split; last by rewrite addnBA // addKn.
            by rewrite addnBA // addKn.
        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 (STEP (x1 + delta)); rewrite leq_eqVlt in STEP.
          have LE: y f (x1 + delta).
          { move: STEP ⇒ /orP [/eqP EQ | STEP];
              first by rewrite !addn1 in EQ; rewrite addnS EQ ltnS in LT.
            rewrite [X in _ < X]addn1 ltnS in STEP.
            apply: (leq_trans _ STEP).
              by rewrite addn1 -addnS ltnW.
          } clear STEP LT.
          rewrite leq_eqVlt in LE.
          move: LE ⇒ /orP [/eqP EQy | LT].
          { (x1 + delta); split; last by rewrite EQy.
              by apply/andP; split; [by apply leq_addr | by 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.

    End ExistsIntermediateValue.

  End Lemmas.

  (* In this section, we prove an analogue of the intermediate
     value theorem, but for predicates of 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.
      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.
      movey /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.

  End ExistsIntermediateValuePredicates.

End StepFunction.