diff options
Diffstat (limited to 'theories/ZArith/Zmisc.v')
| -rw-r--r-- | theories/ZArith/Zmisc.v | 88 |
1 files changed, 44 insertions, 44 deletions
diff --git a/theories/ZArith/Zmisc.v b/theories/ZArith/Zmisc.v index 8246e324..d01cada6 100644 --- a/theories/ZArith/Zmisc.v +++ b/theories/ZArith/Zmisc.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Zmisc.v 5920 2004-07-16 20:01:26Z herbelin $ i*) +(*i $Id: Zmisc.v 9245 2006-10-17 12:53:34Z notin $ i*) Require Import BinInt. Require Import Zcompare. @@ -20,78 +20,78 @@ Open Local Scope Z_scope. (** [n]th iteration of the function [f] *) Fixpoint iter_nat (n:nat) (A:Set) (f:A -> A) (x:A) {struct n} : A := match n with - | O => x - | S n' => f (iter_nat n' A f x) + | O => x + | S n' => f (iter_nat n' A f x) end. Fixpoint iter_pos (n:positive) (A:Set) (f:A -> A) (x:A) {struct n} : A := match n with - | xH => f x - | xO n' => iter_pos n' A f (iter_pos n' A f x) - | xI n' => f (iter_pos n' A f (iter_pos n' A f x)) + | xH => f x + | xO n' => iter_pos n' A f (iter_pos n' A f x) + | xI n' => f (iter_pos n' A f (iter_pos n' A f x)) end. Definition iter (n:Z) (A:Set) (f:A -> A) (x:A) := match n with - | Z0 => x - | Zpos p => iter_pos p A f x - | Zneg p => x + | Z0 => x + | Zpos p => iter_pos p A f x + | Zneg p => x end. Theorem iter_nat_plus : - forall (n m:nat) (A:Set) (f:A -> A) (x:A), - iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x). + forall (n m:nat) (A:Set) (f:A -> A) (x:A), + iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x). Proof. -simple induction n; - [ simpl in |- *; auto with arith - | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. + simple induction n; + [ simpl in |- *; auto with arith + | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. Qed. Theorem iter_nat_of_P : - forall (p:positive) (A:Set) (f:A -> A) (x:A), - iter_pos p A f x = iter_nat (nat_of_P p) A f x. + forall (p:positive) (A:Set) (f:A -> A) (x:A), + iter_pos p A f x = iter_nat (nat_of_P p) A f x. Proof. -intro n; induction n as [p H| p H| ]; - [ intros; simpl in |- *; rewrite (H A f x); - rewrite (H A f (iter_nat (nat_of_P p) A f x)); - rewrite (ZL6 p); symmetry in |- *; apply f_equal with (f := f); - apply iter_nat_plus - | intros; unfold nat_of_P in |- *; simpl in |- *; rewrite (H A f x); - rewrite (H A f (iter_nat (nat_of_P p) A f x)); - rewrite (ZL6 p); symmetry in |- *; apply iter_nat_plus - | simpl in |- *; auto with arith ]. + intro n; induction n as [p H| p H| ]; + [ intros; simpl in |- *; rewrite (H A f x); + rewrite (H A f (iter_nat (nat_of_P p) A f x)); + rewrite (ZL6 p); symmetry in |- *; apply f_equal with (f := f); + apply iter_nat_plus + | intros; unfold nat_of_P in |- *; simpl in |- *; rewrite (H A f x); + rewrite (H A f (iter_nat (nat_of_P p) A f x)); + rewrite (ZL6 p); symmetry in |- *; apply iter_nat_plus + | simpl in |- *; auto with arith ]. Qed. Theorem iter_pos_plus : - forall (p q:positive) (A:Set) (f:A -> A) (x:A), - iter_pos (p + q) A f x = iter_pos p A f (iter_pos q A f x). + forall (p q:positive) (A:Set) (f:A -> A) (x:A), + iter_pos (p + q) A f x = iter_pos p A f (iter_pos q A f x). Proof. -intros n m; intros. -rewrite (iter_nat_of_P m A f x). -rewrite (iter_nat_of_P n A f (iter_nat (nat_of_P m) A f x)). -rewrite (iter_nat_of_P (n + m) A f x). -rewrite (nat_of_P_plus_morphism n m). -apply iter_nat_plus. + intros n m; intros. + rewrite (iter_nat_of_P m A f x). + rewrite (iter_nat_of_P n A f (iter_nat (nat_of_P m) A f x)). + rewrite (iter_nat_of_P (n + m) A f x). + rewrite (nat_of_P_plus_morphism n m). + apply iter_nat_plus. Qed. (** Preservation of invariants : if [f : A->A] preserves the invariant [Inv], then the iterates of [f] also preserve it. *) Theorem iter_nat_invariant : - forall (n:nat) (A:Set) (f:A -> A) (Inv:A -> Prop), - (forall x:A, Inv x -> Inv (f x)) -> - forall x:A, Inv x -> Inv (iter_nat n A f x). + forall (n:nat) (A:Set) (f:A -> A) (Inv:A -> Prop), + (forall x:A, Inv x -> Inv (f x)) -> + forall x:A, Inv x -> Inv (iter_nat n A f x). Proof. -simple induction n; intros; - [ trivial with arith - | simpl in |- *; apply H0 with (x := iter_nat n0 A f x); apply H; - trivial with arith ]. + simple induction n; intros; + [ trivial with arith + | simpl in |- *; apply H0 with (x := iter_nat n0 A f x); apply H; + trivial with arith ]. Qed. Theorem iter_pos_invariant : - forall (p:positive) (A:Set) (f:A -> A) (Inv:A -> Prop), - (forall x:A, Inv x -> Inv (f x)) -> - forall x:A, Inv x -> Inv (iter_pos p A f x). + forall (p:positive) (A:Set) (f:A -> A) (Inv:A -> Prop), + (forall x:A, Inv x -> Inv (f x)) -> + forall x:A, Inv x -> Inv (iter_pos p A f x). Proof. -intros; rewrite iter_nat_of_P; apply iter_nat_invariant; trivial with arith. + intros; rewrite iter_nat_of_P; apply iter_nat_invariant; trivial with arith. Qed. |