diff options
Diffstat (limited to 'theories/ZArith/Zlogarithm.v')
| -rw-r--r-- | theories/ZArith/Zlogarithm.v | 42 |
1 files changed, 21 insertions, 21 deletions
diff --git a/theories/ZArith/Zlogarithm.v b/theories/ZArith/Zlogarithm.v index 3711ea021..09323ebd4 100644 --- a/theories/ZArith/Zlogarithm.v +++ b/theories/ZArith/Zlogarithm.v @@ -77,7 +77,7 @@ Section Log_pos. (* Log of positive integers *) forall x:positive, 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 |- *; + simple induction x; intros; simpl; [ elim H; intros Hp HR; clear H; split; [ auto with zarith | rewrite two_p_S with (x := Z.succ (log_inf p)) by (apply Z.le_le_succ_r; trivial); @@ -90,7 +90,7 @@ Section Log_pos. (* Log of positive integers *) rewrite two_p_S by trivial; 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 |- *; + | unfold two_power_pos; unfold shift_pos; simpl; omega ]. Qed. @@ -103,7 +103,7 @@ Section Log_pos. (* Log of positive integers *) Lemma log_sup_correct1 : forall p:positive, 0 <= log_sup p. Proof. - simple induction p; intros; simpl in |- *; auto with zarith. + simple induction p; intros; simpl; auto with zarith. Qed. (** For every [p], either [p] is a power of two and [(log_inf p)=(log_sup p)] @@ -115,16 +115,16 @@ Section Log_pos. (* Log of positive integers *) else log_sup p = Z.succ (log_inf p). Proof. simple induction p; intros; - [ elim H; right; simpl in |- *; + [ elim H; right; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite BinInt.Pos2Z.inj_xI; unfold Z.succ; omega | elim H; clear H; intro Hif; - [ left; simpl in |- *; + [ left; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite (two_p_S (log_sup p0) (log_sup_correct1 p0)); rewrite <- (proj1 Hif); rewrite <- (proj2 Hif); auto - | right; simpl in |- *; + | right; simpl; rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0)); rewrite BinInt.Pos2Z.inj_xO; unfold Z.succ; omega ] @@ -146,12 +146,12 @@ Section Log_pos. (* Log of positive integers *) Lemma log_inf_le_log_sup : forall p:positive, log_inf p <= log_sup p. Proof. - simple induction p; simpl in |- *; intros; omega. + simple induction p; simpl; intros; omega. Qed. 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. + simple induction p; simpl; intros; omega. Qed. (** Now it's possible to specify and build the [Log] rounded to the nearest *) @@ -167,7 +167,7 @@ Section Log_pos. (* Log of positive integers *) Theorem log_near_correct1 : forall p:positive, 0 <= log_near p. Proof. - simple induction p; simpl in |- *; intros; + simple induction p; simpl; intros; [ elim p0; auto with zarith | elim p0; auto with zarith | trivial with zarith ]. @@ -182,9 +182,9 @@ Section Log_pos. (* Log of positive integers *) Proof. simple induction p. intros p0 [Einf| Esup]. - simpl in |- *. rewrite Einf. + simpl. rewrite Einf. case p0; [ left | left | right ]; reflexivity. - simpl in |- *; rewrite Esup. + simpl; rewrite Esup. elim (log_sup_log_inf p0). generalize (log_inf_le_log_sup p0). generalize (log_sup_le_Slog_inf p0). @@ -192,10 +192,10 @@ Section Log_pos. (* Log of positive integers *) intros; omega. case p0; intros; auto with zarith. intros p0 [Einf| Esup]. - simpl in |- *. + simpl. repeat rewrite Einf. case p0; intros; auto with zarith. - simpl in |- *. + simpl. repeat rewrite Esup. case p0; intros; auto with zarith. auto. @@ -216,7 +216,7 @@ Section divers. Lemma ZERO_le_N_digits : forall x:Z, 0 <= N_digits x. Proof. - simple induction x; simpl in |- *; + simple induction x; simpl; [ apply Z.le_refl | exact log_inf_correct1 | exact log_inf_correct1 ]. Qed. @@ -245,21 +245,21 @@ Section divers. Proof. split; [ elim p; - [ simpl in |- *; tauto - | simpl in |- *; intros; generalize (H H0); intro H1; elim H1; + [ simpl; tauto + | simpl; intros; generalize (H H0); intro H1; elim H1; intros y0 Hy0; exists (S y0); rewrite Hy0; reflexivity | intro; exists 0%nat; reflexivity ] - | intros; elim H; intros; rewrite H0; elim x; intros; simpl in |- *; trivial ]. + | intros; elim H; intros; rewrite H0; elim x; intros; simpl; trivial ]. Qed. Lemma Is_power_or : forall p:positive, Is_power p \/ ~ Is_power p. Proof. simple induction p; - [ intros; right; simpl in |- *; tauto + [ intros; right; simpl; tauto | intros; elim H; - [ intros; left; simpl in |- *; exact H0 - | intros; right; simpl in |- *; exact H0 ] - | left; simpl in |- *; trivial ]. + [ intros; left; simpl; exact H0 + | intros; right; simpl; exact H0 ] + | left; simpl; trivial ]. Qed. End divers. |