diff options
Diffstat (limited to 'theories/Arith/Wf_nat.v')
-rw-r--r-- | theories/Arith/Wf_nat.v | 30 |
1 files changed, 3 insertions, 27 deletions
diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index d3d7b5835..b4468dd1b 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -258,30 +258,6 @@ Qed. Unset Implicit Arguments. -(** [n]th iteration of the function [f] *) - -Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) : A := - match n with - | O => x - | S n' => f (iter_nat n' A f x) - end. - -Theorem iter_nat_plus : - forall (n m:nat) (A:Type) (f:A -> A) (x:A), - iter_nat (n + m) A f x = iter_nat n A f (iter_nat m A f x). -Proof. - induction n; simpl; trivial. - intros. now f_equal. -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: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. - induction n; simpl; trivial. - intros A f Inv Hf x Hx. apply Hf, IHn; trivial. -Qed. +Notation iter_nat := @nat_iter (only parsing). +Notation iter_nat_plus := @nat_iter_plus (only parsing). +Notation iter_nat_invariant := @nat_iter_invariant (only parsing). |