# 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.

(* ...is 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.
Proof.
rename H_is_interval into INT, H_step_function into STEP, 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.
{ movex2 LE /andP [GEy LTy].
exploit (DELTA (x2 - x1));
first by apply/andP; split; last 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).
} 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.
}
}
Qed.

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.
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.
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.
Qed.

End ExistsIntermediateValuePredicates.

End StepFunction.