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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Zlogarithm.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*)
(**********************************************************************)
(** The integer logarithms with base 2.
There are three logarithms,
depending on the rounding of the real 2-based logarithm:
- [Log_inf]: [y = (Log_inf x) iff 2^y <= x < 2^(y+1)]
i.e. [Log_inf x] is the biggest integer that is smaller than [Log x]
- [Log_sup]: [y = (Log_sup x) iff 2^(y-1) < x <= 2^y]
i.e. [Log_inf x] is the smallest integer that is bigger than [Log x]
- [Log_nearest]: [y= (Log_nearest x) iff 2^(y-1/2) < x <= 2^(y+1/2)]
i.e. [Log_nearest x] is the integer nearest from [Log x] *)
Require ZArith_base.
Require Omega.
Require Zcomplements.
Require Zpower.
V7only [Import Z_scope.].
Open Local Scope Z_scope.
Section Log_pos. (* Log of positive integers *)
(** First we build [log_inf] and [log_sup] *)
Fixpoint log_inf [p:positive] : Z :=
Cases p of
xH => `0` (* 1 *)
| (xO q) => (Zs (log_inf q)) (* 2n *)
| (xI q) => (Zs (log_inf q)) (* 2n+1 *)
end.
Fixpoint log_sup [p:positive] : Z :=
Cases p of
xH => `0` (* 1 *)
| (xO n) => (Zs (log_sup n)) (* 2n *)
| (xI n) => (Zs (Zs (log_inf n))) (* 2n+1 *)
end.
Hints Unfold log_inf log_sup.
(** Then we give the specifications of [log_inf] and [log_sup]
and prove their validity *)
(*i Hints Resolve ZERO_le_S : zarith. i*)
Hints Resolve Zle_trans : zarith.
Theorem log_inf_correct : (x:positive) ` 0 <= (log_inf x)` /\
` (two_p (log_inf x)) <= (POS x) < (two_p (Zs (log_inf x)))`.
Induction x; Intros; Simpl;
[ Elim H; Intros Hp HR; Clear H; Split;
[ Auto with zarith
| Conditional (Apply Zle_le_S; Trivial) Rewrite two_p_S with x:=(Zs (log_inf p));
Conditional Trivial Rewrite two_p_S;
Conditional Trivial Rewrite two_p_S in HR;
Rewrite (POS_xI p); Omega ]
| Elim H; Intros Hp HR; Clear H; Split;
[ Auto with zarith
| Conditional (Apply Zle_le_S; Trivial) Rewrite two_p_S with x:=(Zs (log_inf p));
Conditional Trivial Rewrite two_p_S;
Conditional Trivial Rewrite two_p_S in HR;
Rewrite (POS_xO p); Omega ]
| Unfold two_power_pos; Unfold shift_pos; Simpl; Omega
].
Qed.
Definition log_inf_correct1 :=
[p:positive](proj1 ? ? (log_inf_correct p)).
Definition log_inf_correct2 :=
[p:positive](proj2 ? ? (log_inf_correct p)).
Opaque log_inf_correct1 log_inf_correct2.
Hints Resolve log_inf_correct1 log_inf_correct2 : zarith.
Lemma log_sup_correct1 : (p:positive)` 0 <= (log_sup p)`.
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)]
either [(log_sup p)=(log_inf p)+1] *)
Theorem log_sup_log_inf : (p:positive)
IF (POS p)=(two_p (log_inf p))
then (POS p)=(two_p (log_sup p))
else ` (log_sup p)=(Zs (log_inf p))`.
Induction p; Intros;
[ Elim H; Right; Simpl;
Rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
Rewrite POS_xI; Unfold Zs; Omega
| Elim H; Clear H; Intro Hif;
[ 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;
Rewrite (two_p_S (log_inf p0) (log_inf_correct1 p0));
Rewrite POS_xO; Unfold Zs; Omega ]
| Left; Auto ].
Qed.
Theorem log_sup_correct2 : (x:positive)
` (two_p (Zpred (log_sup x))) < (POS x) <= (two_p (log_sup x))`.
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_n ].
Intros (E1,E2); Rewrite E2.
Rewrite <- (Zpred_Sn (log_inf x)).
Generalize (log_inf_correct2 x); Omega.
Qed.
Lemma log_inf_le_log_sup :
(p:positive) `(log_inf p) <= (log_sup p)`.
Induction p; Simpl; Intros; Omega.
Qed.
Lemma log_sup_le_Slog_inf :
(p:positive) `(log_sup p) <= (Zs (log_inf p))`.
Induction p; Simpl; Intros; Omega.
Qed.
(** Now it's possible to specify and build the [Log] rounded to the nearest *)
Fixpoint log_near[x:positive] : Z :=
Cases x of
xH => `0`
| (xO xH) => `1`
| (xI xH) => `2`
| (xO y) => (Zs (log_near y))
| (xI y) => (Zs (log_near y))
end.
Theorem log_near_correct1 : (p:positive)` 0 <= (log_near p)`.
Induction p; Simpl; Intros;
[Elim p0; Auto with zarith | Elim p0; Auto with zarith | Trivial with zarith ].
Intros; Apply Zle_le_S.
Generalize H0; Elim p1; Intros; Simpl;
[ Assumption | Assumption | Apply ZERO_le_POS ].
Intros; Apply Zle_le_S.
Generalize H0; Elim p1; Intros; Simpl;
[ Assumption | Assumption | Apply ZERO_le_POS ].
Qed.
Theorem log_near_correct2: (p:positive)
(log_near p)=(log_inf p)
\/(log_near p)=(log_sup p).
Induction p.
Intros p0 [Einf|Esup].
Simpl. Rewrite Einf.
Case p0; [Left | Left | Right]; Reflexivity.
Simpl; Rewrite Esup.
Elim (log_sup_log_inf p0).
Generalize (log_inf_le_log_sup p0).
Generalize (log_sup_le_Slog_inf p0).
Case p0; Auto with zarith.
Intros; Omega.
Case p0; Intros; Auto with zarith.
Intros p0 [Einf|Esup].
Simpl.
Repeat Rewrite Einf.
Case p0; Intros; Auto with zarith.
Simpl.
Repeat Rewrite Esup.
Case p0; Intros; Auto with zarith.
Auto.
Qed.
(*i******************
Theorem log_near_correct: (p:positive)
`| (two_p (log_near p)) - (POS p) | <= (POS p)-(two_p (log_inf p))`
/\`| (two_p (log_near p)) - (POS p) | <= (two_p (log_sup p))-(POS p)`.
Intro.
Induction p.
Intros p0 [(Einf1,Einf2)|(Esup1,Esup2)].
Unfold log_near log_inf log_sup. Fold log_near log_inf log_sup.
Rewrite Einf1.
Repeat Rewrite two_p_S.
Case p0; [Left | Left | Right].
Split.
Simpl.
Rewrite E1; Case p0; Try Reflexivity.
Compute.
Unfold log_near; Fold log_near.
Unfold log_inf; Fold log_inf.
Repeat Rewrite E1.
Split.
**********************************i*)
End Log_pos.
Section divers.
(** Number of significative digits. *)
Definition N_digits :=
[x:Z]Cases x of
(POS p) => (log_inf p)
| (NEG p) => (log_inf p)
| ZERO => `0`
end.
Lemma ZERO_le_N_digits : (x:Z) ` 0 <= (N_digits x)`.
Induction x; Simpl;
[ Apply Zle_n
| Exact log_inf_correct1
| Exact log_inf_correct1].
Qed.
Lemma log_inf_shift_nat :
(n:nat)(log_inf (shift_nat n xH))=(inject_nat n).
Induction n; Intros;
[ Try Trivial
| Rewrite -> inj_S; Rewrite <- H; Reflexivity].
Qed.
Lemma log_sup_shift_nat :
(n:nat)(log_sup (shift_nat n xH))=(inject_nat n).
Induction n; Intros;
[ Try Trivial
| Rewrite -> inj_S; Rewrite <- H; Reflexivity].
Qed.
(** [Is_power p] means that p is a power of two *)
Fixpoint Is_power[p:positive] : Prop :=
Cases p of
xH => True
| (xO q) => (Is_power q)
| (xI q) => False
end.
Lemma Is_power_correct :
(p:positive) (Is_power p) <-> (Ex [y:nat](p=(shift_nat y xH))).
Split;
[ Elim p;
[ Simpl; Tauto
| Simpl; Intros; Generalize (H H0); Intro H1; Elim H1; Intros y0 Hy0;
Exists (S y0); Rewrite Hy0; Reflexivity
| Intro; Exists O; Reflexivity]
| Intros; Elim H; Intros; Rewrite -> H0; Elim x; Intros; Simpl; Trivial].
Qed.
Lemma Is_power_or : (p:positive) (Is_power p)\/~(Is_power p).
Induction p;
[ Intros; Right; Simpl; Tauto
| Intros; Elim H;
[ Intros; Left; Simpl; Exact H0
| Intros; Right; Simpl; Exact H0]
| Left; Simpl; Trivial].
Qed.
End divers.
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