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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: Zpower.v,v 1.2.2.1 2004/07/16 19:31:44 herbelin Exp $ i*)
+
+Require ZArith_base.
+Require Omega.
+Require Zcomplements.
+V7only [Import Z_scope.].
+Open Local Scope Z_scope.
+
+Section section1.
+
+(** [Zpower_nat z n] is the n-th power of [z] when [n] is an unary
+ integer (type [nat]) and [z] a signed integer (type [Z]) *)
+
+Definition Zpower_nat :=
+ [z:Z][n:nat] (iter_nat n Z ([x:Z]` z * x `) `1`).
+
+(** [Zpower_nat_is_exp] says [Zpower_nat] is a morphism for
+ [plus : nat->nat] and [Zmult : Z->Z] *)
+
+Lemma Zpower_nat_is_exp :
+ (n,m:nat)(z:Z)
+ `(Zpower_nat z (plus n m)) = (Zpower_nat z n)*(Zpower_nat z m)`.
+
+Intros; Elim n;
+[ Simpl; Elim (Zpower_nat z m); Auto with zarith
+| Unfold Zpower_nat; Intros; Simpl; Rewrite H;
+ Apply Zmult_assoc].
+Qed.
+
+(** [Zpower_pos z n] is the n-th power of [z] when [n] is an binary
+ integer (type [positive]) and [z] a signed integer (type [Z]) *)
+
+Definition Zpower_pos :=
+ [z:Z][n:positive] (iter_pos n Z ([x:Z]`z * x`) `1`).
+
+(** This theorem shows that powers of unary and binary integers
+ are the same thing, modulo the function convert : [positive -> nat] *)
+
+Theorem Zpower_pos_nat :
+ (z:Z)(p:positive)(Zpower_pos z p) = (Zpower_nat z (convert p)).
+
+Intros; Unfold Zpower_pos; Unfold Zpower_nat; Apply iter_convert.
+Qed.
+
+(** Using the theorem [Zpower_pos_nat] and the lemma [Zpower_nat_is_exp] we
+ deduce that the function [[n:positive](Zpower_pos z n)] is a morphism
+ for [add : positive->positive] and [Zmult : Z->Z] *)
+
+Theorem Zpower_pos_is_exp :
+ (n,m:positive)(z:Z)
+ ` (Zpower_pos z (add n m)) = (Zpower_pos z n)*(Zpower_pos z m)`.
+
+Intros.
+Rewrite -> (Zpower_pos_nat z n).
+Rewrite -> (Zpower_pos_nat z m).
+Rewrite -> (Zpower_pos_nat z (add n m)).
+Rewrite -> (convert_add n m).
+Apply Zpower_nat_is_exp.
+Qed.
+
+Definition Zpower :=
+ [x,y:Z]Cases y of
+ (POS p) => (Zpower_pos x p)
+ | ZERO => `1`
+ | (NEG p) => `0`
+ end.
+
+V8Infix "^" Zpower : Z_scope.
+
+Hints Immediate Zpower_nat_is_exp : zarith.
+Hints Immediate Zpower_pos_is_exp : zarith.
+Hints Unfold Zpower_pos : zarith.
+Hints Unfold Zpower_nat : zarith.
+
+Lemma Zpower_exp : (x:Z)(n,m:Z)
+ `n >= 0` -> `m >= 0` -> `(Zpower x (n+m))=(Zpower x n)*(Zpower x m)`.
+NewDestruct n; NewDestruct m; Auto with zarith.
+Simpl; Intros; Apply Zred_factor0.
+Simpl; Auto with zarith.
+Intros; Compute in H0; Absurd INFERIEUR=INFERIEUR; Auto with zarith.
+Intros; Compute in H0; Absurd INFERIEUR=INFERIEUR; Auto with zarith.
+Qed.
+
+End section1.
+
+(* Exporting notation "^" *)
+
+V8Infix "^" Zpower : Z_scope.
+
+Hints Immediate Zpower_nat_is_exp : zarith.
+Hints Immediate Zpower_pos_is_exp : zarith.
+Hints Unfold Zpower_pos : zarith.
+Hints Unfold Zpower_nat : zarith.
+
+Section Powers_of_2.
+
+(** For the powers of two, that will be widely used, a more direct
+ calculus is possible. We will also prove some properties such
+ as [(x:positive) x < 2^x] that are true for all integers bigger
+ than 2 but more difficult to prove and useless. *)
+
+(** [shift n m] computes [2^n * m], or [m] shifted by [n] positions *)
+
+Definition shift_nat :=
+ [n:nat][z:positive](iter_nat n positive xO z).
+Definition shift_pos :=
+ [n:positive][z:positive](iter_pos n positive xO z).
+Definition shift :=
+ [n:Z][z:positive]
+ Cases n of
+ ZERO => z
+ | (POS p) => (iter_pos p positive xO z)
+ | (NEG p) => z
+ end.
+
+Definition two_power_nat := [n:nat] (POS (shift_nat n xH)).
+Definition two_power_pos := [x:positive] (POS (shift_pos x xH)).
+
+Lemma two_power_nat_S :
+ (n:nat)` (two_power_nat (S n)) = 2*(two_power_nat n)`.
+Intro; Simpl; Apply refl_equal.
+Qed.
+
+Lemma shift_nat_plus :
+ (n,m:nat)(x:positive)
+ (shift_nat (plus n m) x)=(shift_nat n (shift_nat m x)).
+
+Intros; Unfold shift_nat; Apply iter_nat_plus.
+Qed.
+
+Theorem shift_nat_correct :
+ (n:nat)(x:positive)(POS (shift_nat n x))=`(Zpower_nat 2 n)*(POS x)`.
+
+Unfold shift_nat; Induction n;
+[ Simpl; Trivial with zarith
+| Intros; Replace (Zpower_nat `2` (S n0)) with `2 * (Zpower_nat 2 n0)`;
+[ Rewrite <- Zmult_assoc; Rewrite <- (H x); Simpl; Reflexivity
+| Auto with zarith ]
+].
+Qed.
+
+Theorem two_power_nat_correct :
+ (n:nat)(two_power_nat n)=(Zpower_nat `2` n).
+
+Intro n.
+Unfold two_power_nat.
+Rewrite -> (shift_nat_correct n).
+Omega.
+Qed.
+
+(** Second we show that [two_power_pos] and [two_power_nat] are the same *)
+Lemma shift_pos_nat : (p:positive)(x:positive)
+ (shift_pos p x)=(shift_nat (convert p) x).
+
+Unfold shift_pos.
+Unfold shift_nat.
+Intros; Apply iter_convert.
+Qed.
+
+Lemma two_power_pos_nat :
+ (p:positive) (two_power_pos p)=(two_power_nat (convert p)).
+
+Intro; Unfold two_power_pos; Unfold two_power_nat.
+Apply f_equal with f:=POS.
+Apply shift_pos_nat.
+Qed.
+
+(** Then we deduce that [two_power_pos] is also correct *)
+
+Theorem shift_pos_correct :
+ (p,x:positive) ` (POS (shift_pos p x)) = (Zpower_pos 2 p) * (POS x)`.
+
+Intros.
+Rewrite -> (shift_pos_nat p x).
+Rewrite -> (Zpower_pos_nat `2` p).
+Apply shift_nat_correct.
+Qed.
+
+Theorem two_power_pos_correct :
+ (x:positive) (two_power_pos x)=(Zpower_pos `2` x).
+
+Intro.
+Rewrite -> two_power_pos_nat.
+Rewrite -> Zpower_pos_nat.
+Apply two_power_nat_correct.
+Qed.
+
+(** Some consequences *)
+
+Theorem two_power_pos_is_exp :
+ (x,y:positive) (two_power_pos (add x y))
+ =(Zmult (two_power_pos x) (two_power_pos y)).
+Intros.
+Rewrite -> (two_power_pos_correct (add x y)).
+Rewrite -> (two_power_pos_correct x).
+Rewrite -> (two_power_pos_correct y).
+Apply Zpower_pos_is_exp.
+Qed.
+
+(** The exponentiation [z -> 2^z] for [z] a signed integer.
+ For convenience, we assume that [2^z = 0] for all [z < 0]
+ We could also define a inductive type [Log_result] with
+ 3 contructors [ Zero | Pos positive -> | minus_infty]
+ but it's more complexe and not so useful. *)
+
+Definition two_p :=
+ [x:Z]Cases x of
+ ZERO => `1`
+ | (POS y) => (two_power_pos y)
+ | (NEG y) => `0`
+ end.
+
+Theorem two_p_is_exp :
+ (x,y:Z) ` 0 <= x` -> ` 0 <= y` ->
+ ` (two_p (x+y)) = (two_p x)*(two_p y)`.
+Induction x;
+[ Induction y; Simpl; Auto with zarith
+| Induction y;
+ [ Unfold two_p; Rewrite -> (Zmult_sym (two_power_pos p) `1`);
+ Rewrite -> (Zmult_one (two_power_pos p)); Auto with zarith
+ | Unfold Zplus; Unfold two_p;
+ Intros; Apply two_power_pos_is_exp
+ | Intros; Unfold Zle in H0; Unfold Zcompare in H0;
+ Absurd SUPERIEUR=SUPERIEUR; Trivial with zarith
+ ]
+| Induction y;
+ [ Simpl; Auto with zarith
+ | Intros; Unfold Zle in H; Unfold Zcompare in H;
+ Absurd (SUPERIEUR=SUPERIEUR); Trivial with zarith
+ | Intros; Unfold Zle in H; Unfold Zcompare in H;
+ Absurd (SUPERIEUR=SUPERIEUR); Trivial with zarith
+ ]
+].
+Qed.
+
+Lemma two_p_gt_ZERO : (x:Z) ` 0 <= x` -> ` (two_p x) > 0`.
+Induction x; Intros;
+[ Simpl; Omega
+| Simpl; Unfold two_power_pos; Apply POS_gt_ZERO
+| Absurd ` 0 <= (NEG p)`;
+ [ Simpl; Unfold Zle; Unfold Zcompare;
+ Do 2 Unfold not; Auto with zarith
+ | Assumption ]
+].
+Qed.
+
+Lemma two_p_S : (x:Z) ` 0 <= x` ->
+ `(two_p (Zs x)) = 2 * (two_p x)`.
+Intros; Unfold Zs.
+Rewrite (two_p_is_exp x `1` H (ZERO_le_POS xH)).
+Apply Zmult_sym.
+Qed.
+
+Lemma two_p_pred :
+ (x:Z)` 0 <= x` -> ` (two_p (Zpred x)) < (two_p x)`.
+Intros; Apply natlike_ind
+with P:=[x:Z]` (two_p (Zpred x)) < (two_p x)`;
+[ Simpl; Unfold Zlt; Auto with zarith
+| Intros; Elim (Zle_lt_or_eq `0` x0 H0);
+ [ Intros;
+ Replace (two_p (Zpred (Zs x0)))
+ with (two_p (Zs (Zpred x0)));
+ [ Rewrite -> (two_p_S (Zpred x0));
+ [ Rewrite -> (two_p_S x0);
+ [ Omega
+ | Assumption]
+ | Apply Zlt_ZERO_pred_le_ZERO; Assumption]
+ | Rewrite <- (Zs_pred x0); Rewrite <- (Zpred_Sn x0); Trivial with zarith]
+ | Intro Hx0; Rewrite <- Hx0; Simpl; Unfold Zlt; Auto with zarith]
+| Assumption].
+Qed.
+
+Lemma Zlt_lt_double : (x,y:Z) ` 0 <= x < y` -> ` x < 2*y`.
+Intros; Omega. Qed.
+
+End Powers_of_2.
+
+Hints Resolve two_p_gt_ZERO : zarith.
+Hints Immediate two_p_pred two_p_S : zarith.
+
+Section power_div_with_rest.
+
+(** Division by a power of two.
+ To [n:Z] and [p:positive], [q],[r] are associated such that
+ [n = 2^p.q + r] and [0 <= r < 2^p] *)
+
+(** Invariant: [d*q + r = d'*q + r /\ d' = 2*d /\ 0<= r < d /\ 0 <= r' < d'] *)
+Definition Zdiv_rest_aux :=
+ [qrd:(Z*Z)*Z]
+ let (qr,d)=qrd in let (q,r)=qr in
+ (Cases q of
+ ZERO => ` (0, r)`
+ | (POS xH) => ` (0, d + r)`
+ | (POS (xI n)) => ` ((POS n), d + r)`
+ | (POS (xO n)) => ` ((POS n), r)`
+ | (NEG xH) => ` (-1, d + r)`
+ | (NEG (xI n)) => ` ((NEG n) - 1, d + r)`
+ | (NEG (xO n)) => ` ((NEG n), r)`
+ end, ` 2*d`).
+
+Definition Zdiv_rest :=
+ [x:Z][p:positive]let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in qr.
+
+Lemma Zdiv_rest_correct1 :
+ (x:Z)(p:positive)
+ let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in d=(two_power_pos p).
+
+Intros x p;
+Rewrite (iter_convert p ? Zdiv_rest_aux ((x,`0`),`1`));
+Rewrite (two_power_pos_nat p);
+Elim (convert p); Simpl;
+[ Trivial with zarith
+| Intro n; Rewrite (two_power_nat_S n);
+ Unfold 2 Zdiv_rest_aux;
+ Elim (iter_nat n (Z*Z)*Z Zdiv_rest_aux ((x,`0`),`1`));
+ NewDestruct a; Intros; Apply f_equal with f:=[z:Z]`2*z`; Assumption ].
+Qed.
+
+Lemma Zdiv_rest_correct2 :
+ (x:Z)(p:positive)
+ let (qr,d)=(iter_pos p ? Zdiv_rest_aux ((x,`0`),`1`)) in
+ let (q,r)=qr in
+ ` x=q*d + r` /\ ` 0 <= r < d`.
+
+Intros; Apply iter_pos_invariant with
+ f:=Zdiv_rest_aux
+ Inv:=[qrd:(Z*Z)*Z]let (qr,d)=qrd in let (q,r)=qr in
+ ` x=q*d + r` /\ ` 0 <= r < d`;
+[ Intro x0; Elim x0; Intro y0; Elim y0;
+ Intros q r d; Unfold Zdiv_rest_aux;
+ Elim q;
+ [ Omega
+ | NewDestruct p0;
+ [ Rewrite POS_xI; Intro; Elim H; Intros; Split;
+ [ Rewrite H0; Rewrite Zplus_assoc;
+ Rewrite Zmult_plus_distr_l;
+ Rewrite Zmult_1_n; Rewrite Zmult_assoc;
+ Rewrite (Zmult_sym (POS p0) `2`); Apply refl_equal
+ | Omega ]
+ | Rewrite POS_xO; Intro; Elim H; Intros; Split;
+ [ Rewrite H0;
+ Rewrite Zmult_assoc; Rewrite (Zmult_sym (POS p0) `2`);
+ Apply refl_equal
+ | Omega ]
+ | Omega ]
+ | NewDestruct p0;
+ [ Rewrite NEG_xI; Unfold Zminus; Intro; Elim H; Intros; Split;
+ [ Rewrite H0; Rewrite Zplus_assoc;
+ Apply f_equal with f:=[z:Z]`z+r`;
+ Do 2 (Rewrite Zmult_plus_distr_l);
+ Rewrite Zmult_assoc;
+ Rewrite (Zmult_sym (NEG p0) `2`);
+ Rewrite <- Zplus_assoc;
+ Apply f_equal with f:=[z:Z]`2 * (NEG p0) * d + z`;
+ Omega
+ | Omega ]
+ | Rewrite NEG_xO; Unfold Zminus; Intro; Elim H; Intros; Split;
+ [ Rewrite H0;
+ Rewrite Zmult_assoc; Rewrite (Zmult_sym (NEG p0) `2`);
+ Apply refl_equal
+ | Omega ]
+ | Omega ] ]
+| Omega].
+Qed.
+
+Inductive Set Zdiv_rest_proofs[x:Z; p:positive] :=
+ Zdiv_rest_proof : (q:Z)(r:Z)
+ `x = q * (two_power_pos p) + r`
+ -> `0 <= r`
+ -> `r < (two_power_pos p)`
+ -> (Zdiv_rest_proofs x p).
+
+Lemma Zdiv_rest_correct :
+ (x:Z)(p:positive)(Zdiv_rest_proofs x p).
+Intros x p.
+Generalize (Zdiv_rest_correct1 x p); Generalize (Zdiv_rest_correct2 x p).
+Elim (iter_pos p (Z*Z)*Z Zdiv_rest_aux ((x,`0`),`1`)).
+Induction a.
+Intros.
+Elim H; Intros H1 H2; Clear H.
+Rewrite -> H0 in H1; Rewrite -> H0 in H2;
+Elim H2; Intros;
+Apply Zdiv_rest_proof with q:=a0 r:=b; Assumption.
+Qed.
+
+End power_div_with_rest.
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