CoRN.reals.fast.LazyNat

Require Export BinPos.

Lazy Nat

This s a lazified version of the natural number that allow one to delay computation until demanded. This is useful for large natural numbers (often upper bounds) where only a small number of terms are actually need for compuation.

Inductive LazyNat : Set :=
| LazyO : LazyNat
| LazyS : (unit → LazyNat) → LazyNat.

Successor

Definition lazyS (n:LazyNat) : LazyNat := LazyS (fun _ ⇒ n).

Convert a nat to a lazy nat

Fixpoint LazifyNat (n : nat) {struct n} : LazyNat :=
  match n with
  | O ⇒ LazyO
  | S p ⇒ LazyS (fun _ ⇒ (LazifyNat p))
  end.

Predecessor

Definition LazyPred (n:LazyNat) : LazyNat :=
match n with
| LazyO ⇒ LazyO
| LazyS n' ⇒ (n' tt)
end.

Lemma LazifyPred : ∀ n, LazifyNat (pred n) = LazyPred (LazifyNat n).
Proof.
induction n; reflexivity.
Qed.

Addition

Fixpoint LazyPlus (n m : LazyNat) {struct n} : LazyNat :=
  match n with
  | LazyO ⇒ m
  | LazyS p ⇒ LazyS (fun _ ⇒ LazyPlus (p tt) m)
  end.

Lemma LazifyPlus : ∀ n m, (LazifyNat (n + m) = LazyPlus (LazifyNat n) (LazifyNat m))%nat.
Proof.
induction n.
  reflexivity.
simpl.
intros m.
rewrite IHn.
reflexivity.
Qed.

Multiplication

Fixpoint Pmult_LazyNat (x:positive) (pow2:LazyNat) {struct x} : LazyNat :=
  match x with
    | xI x' ⇒ (LazyPlus pow2 (Pmult_LazyNat x' (LazyPlus pow2 pow2)))%nat
    | xO x' ⇒ Pmult_LazyNat x' (LazyPlus pow2 pow2)%nat
    | xH ⇒ pow2
  end.

Lemma LazifyPmult_LazyNat : ∀ x pow2, LazifyNat (Pmult_nat x pow2) = Pmult_LazyNat x (LazifyNat pow2).
Proof.
induction x; simpl; intros pow2; repeat (rewrite LazifyPlus||rewrite IHx); reflexivity.
Qed.

Convert a positive to a lazy nat. This is the most common way of generating lazy nats.

Definition LazyNat_of_P (x:positive) := Pmult_LazyNat x (LazyS (fun _ ⇒ LazyO)).

Lemma LazifyNat_of_P : ∀ x, LazifyNat (nat_of_P x) = LazyNat_of_P x.
Proof.
intros x.
refine (LazifyPmult_LazyNat _ _).
Qed.

Fixpoint Pplus_LazyNat (p:positive)(n:LazyNat) {struct n} : positive :=
match n with
| LazyO ⇒ p
| (LazyS n') ⇒ (Pplus_LazyNat (Psucc p) (n' tt))
end.