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-rw-r--r--theories/ZArith/Zlogarithm.v60
1 files changed, 29 insertions, 31 deletions
diff --git a/theories/ZArith/Zlogarithm.v b/theories/ZArith/Zlogarithm.v
index 30948ca7a..3711ea021 100644
--- a/theories/ZArith/Zlogarithm.v
+++ b/theories/ZArith/Zlogarithm.v
@@ -34,22 +34,22 @@ Section Log_pos. (* Log of positive integers *)
Fixpoint log_inf (p:positive) : Z :=
match p with
| xH => 0 (* 1 *)
- | xO q => Zsucc (log_inf q) (* 2n *)
- | xI q => Zsucc (log_inf q) (* 2n+1 *)
+ | xO q => Z.succ (log_inf q) (* 2n *)
+ | xI q => Z.succ (log_inf q) (* 2n+1 *)
end.
Fixpoint log_sup (p:positive) : Z :=
match p with
| xH => 0 (* 1 *)
- | xO n => Zsucc (log_sup n) (* 2n *)
- | xI n => Zsucc (Zsucc (log_inf n)) (* 2n+1 *)
+ | xO n => Z.succ (log_sup n) (* 2n *)
+ | xI n => Z.succ (Z.succ (log_inf n)) (* 2n+1 *)
end.
Hint Unfold log_inf log_sup.
Lemma Psize_log_inf : forall p, Zpos (Pos.size p) = Z.succ (log_inf p).
Proof.
- induction p; simpl; now rewrite <- ?Z.succ_Zpos, ?IHp.
+ induction p; simpl; now rewrite ?Pos2Z.inj_succ, ?IHp.
Qed.
Lemma Zlog2_log_inf : forall p, Z.log2 (Zpos p) = log_inf p.
@@ -71,24 +71,24 @@ Section Log_pos. (* Log of positive integers *)
(** Then we give the specifications of [log_inf] and [log_sup]
and prove their validity *)
- Hint Resolve Zle_trans: zarith.
+ Hint Resolve Z.le_trans: zarith.
Theorem log_inf_correct :
forall x:positive,
- 0 <= log_inf x /\ two_p (log_inf x) <= Zpos x < two_p (Zsucc (log_inf x)).
+ 0 <= log_inf x /\ two_p (log_inf x) <= Zpos x < two_p (Z.succ (log_inf x)).
Proof.
simple induction x; intros; simpl in |- *;
[ elim H; intros Hp HR; clear H; split;
[ auto with zarith
- | rewrite two_p_S with (x := Zsucc (log_inf p)) by (apply Zle_le_succ; trivial);
+ | rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial);
rewrite two_p_S by trivial;
- rewrite two_p_S in HR by trivial; rewrite (BinInt.Zpos_xI p);
+ rewrite two_p_S in HR by trivial; rewrite (BinInt.Pos2Z.inj_xI p);
omega ]
| elim H; intros Hp HR; clear H; split;
[ auto with zarith
- | rewrite two_p_S with (x := Zsucc (log_inf p)) by (apply Zle_le_succ; trivial);
+ | rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial);
rewrite two_p_S by trivial;
- rewrite two_p_S in HR by trivial; rewrite (BinInt.Zpos_xO p);
+ rewrite two_p_S in HR by trivial; rewrite (BinInt.Pos2Z.inj_xO p);
omega ]
| unfold two_power_pos in |- *; unfold shift_pos in |- *; simpl in |- *;
omega ].
@@ -112,12 +112,12 @@ Section Log_pos. (* Log of positive integers *)
Theorem log_sup_log_inf :
forall p:positive,
IF Zpos p = two_p (log_inf p) then Zpos p = two_p (log_sup p)
- else log_sup p = Zsucc (log_inf p).
+ else log_sup p = Z.succ (log_inf p).
Proof.
simple induction p; intros;
[ elim H; right; simpl in |- *;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
- rewrite BinInt.Zpos_xI; unfold Zsucc in |- *; omega
+ rewrite BinInt.Pos2Z.inj_xI; unfold Z.succ; omega
| elim H; clear H; intro Hif;
[ left; simpl in |- *;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
@@ -126,21 +126,21 @@ Section Log_pos. (* Log of positive integers *)
auto
| right; simpl in |- *;
rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
- rewrite BinInt.Zpos_xO; unfold Zsucc in |- *;
+ rewrite BinInt.Pos2Z.inj_xO; unfold Z.succ;
omega ]
| left; auto ].
Qed.
Theorem log_sup_correct2 :
- forall x:positive, two_p (Zpred (log_sup x)) < Zpos x <= two_p (log_sup x).
+ forall x:positive, two_p (Z.pred (log_sup x)) < Zpos x <= two_p (log_sup x).
Proof.
intro.
elim (log_sup_log_inf x).
(* x is a power of two and [log_sup = log_inf] *)
intros [E1 E2]; rewrite E2.
- split; [ apply two_p_pred; apply log_sup_correct1 | apply Zle_refl ].
+ split; [ apply two_p_pred; apply log_sup_correct1 | apply Z.le_refl ].
intros [E1 E2]; rewrite E2.
- rewrite <- (Zpred_succ (log_inf x)).
+ rewrite (Z.pred_succ (log_inf x)).
generalize (log_inf_correct2 x); omega.
Qed.
@@ -149,7 +149,7 @@ Section Log_pos. (* Log of positive integers *)
simple induction p; simpl in |- *; intros; omega.
Qed.
- Lemma log_sup_le_Slog_inf : forall p:positive, log_sup p <= Zsucc (log_inf p).
+ Lemma log_sup_le_Slog_inf : forall p:positive, log_sup p <= Z.succ (log_inf p).
Proof.
simple induction p; simpl in |- *; intros; omega.
Qed.
@@ -161,8 +161,8 @@ Section Log_pos. (* Log of positive integers *)
| xH => 0
| xO xH => 1
| xI xH => 2
- | xO y => Zsucc (log_near y)
- | xI y => Zsucc (log_near y)
+ | xO y => Z.succ (log_near y)
+ | xI y => Z.succ (log_near y)
end.
Theorem log_near_correct1 : forall p:positive, 0 <= log_near p.
@@ -171,12 +171,10 @@ Section Log_pos. (* Log of positive integers *)
[ elim p0; auto with zarith
| elim p0; auto with zarith
| trivial with zarith ].
- intros; apply Zle_le_succ.
- generalize H0; elim p1; intros; simpl in |- *;
- [ assumption | assumption | apply Zorder.Zle_0_pos ].
- intros; apply Zle_le_succ.
- generalize H0; elim p1; intros; simpl in |- *;
- [ assumption | assumption | apply Zorder.Zle_0_pos ].
+ intros; apply Z.le_le_succ_r.
+ generalize H0; now elim p1.
+ intros; apply Z.le_le_succ_r.
+ generalize H0; now elim p1.
Qed.
Theorem log_near_correct2 :
@@ -219,19 +217,19 @@ Section divers.
Lemma ZERO_le_N_digits : forall x:Z, 0 <= N_digits x.
Proof.
simple induction x; simpl in |- *;
- [ apply Zle_refl | exact log_inf_correct1 | exact log_inf_correct1 ].
+ [ apply Z.le_refl | exact log_inf_correct1 | exact log_inf_correct1 ].
Qed.
- Lemma log_inf_shift_nat : forall n:nat, log_inf (shift_nat n 1) = Z_of_nat n.
+ Lemma log_inf_shift_nat : forall n:nat, log_inf (shift_nat n 1) = Z.of_nat n.
Proof.
simple induction n; intros;
- [ try trivial | rewrite Znat.inj_S; rewrite <- H; reflexivity ].
+ [ try trivial | rewrite Nat2Z.inj_succ; rewrite <- H; reflexivity ].
Qed.
- Lemma log_sup_shift_nat : forall n:nat, log_sup (shift_nat n 1) = Z_of_nat n.
+ Lemma log_sup_shift_nat : forall n:nat, log_sup (shift_nat n 1) = Z.of_nat n.
Proof.
simple induction n; intros;
- [ try trivial | rewrite Znat.inj_S; rewrite <- H; reflexivity ].
+ [ try trivial | rewrite Nat2Z.inj_succ; rewrite <- H; reflexivity ].
Qed.
(** [Is_power p] means that p is a power of two *)
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-14 15:05:39 +0000