diff options
| author |  glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
|---|---|---|
| committer | glondu <glondu@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2009-09-17 15:58:14 +0000 |
| commit | 61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch) | |
| tree | 961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/ZArith/Zmisc.v | |
| parent | 6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (diff) | |
Delete trailing whitespaces in all *.{v,ml*} files
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12337 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/ZArith/Zmisc.v')
| -rw-r--r-- | theories/ZArith/Zmisc.v | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/theories/ZArith/Zmisc.v b/theories/ZArith/Zmisc.v index 34e76b8ac..93ac74d54 100644 --- a/theories/ZArith/Zmisc.v +++ b/theories/ZArith/Zmisc.v @@ -37,14 +37,14 @@ Definition iter (n:Z) (A:Type) (f:A -> A) (x:A) := Theorem iter_nat_of_P : forall (p:positive) (A:Type) (f:A -> A) (x:A), iter_pos p A f x = iter_nat (nat_of_P p) A f x. -Proof. +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 (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 (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. @@ -59,7 +59,7 @@ Qed. Theorem iter_pos_plus : forall (p q:positive) (A:Type) (f:A -> A) (x:A), iter_pos (p + q) A f x = iter_pos p A f (iter_pos q A f x). -Proof. +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)). @@ -68,14 +68,14 @@ Proof. apply iter_nat_plus. Qed. -(** Preservation of invariants : if [f : A->A] preserves the invariant [Inv], +(** 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:Type) (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. +Proof. simple induction n; intros; [ trivial with arith | simpl in |- *; apply H0 with (x := iter_nat n0 A f x); apply H; @@ -86,6 +86,6 @@ Theorem iter_pos_invariant : forall (p:positive) (A:Type) (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. +Proof. intros; rewrite iter_nat_of_P; apply iter_nat_invariant; trivial with arith. Qed. |