diff options
Diffstat (limited to 'theories/Arith/Wf_nat.v')
| -rw-r--r-- | theories/Arith/Wf_nat.v | 16 |
1 files changed, 8 insertions, 8 deletions
diff --git a/theories/Arith/Wf_nat.v b/theories/Arith/Wf_nat.v index 6ad640eb..5bc5d2a5 100644 --- a/theories/Arith/Wf_nat.v +++ b/theories/Arith/Wf_nat.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Wf_nat.v 11072 2008-06-08 16:13:37Z herbelin $ i*) +(*i $Id$ i*) (** Well-founded relations and natural numbers *) @@ -46,9 +46,9 @@ Defined. (** It is possible to directly prove the induction principle going back to primitive recursion on natural numbers ([induction_ltof1]) or to use the previous lemmas to extract a program with a fixpoint - ([induction_ltof2]) + ([induction_ltof2]) -the ML-like program for [induction_ltof1] is : +the ML-like program for [induction_ltof1] is : [[ let induction_ltof1 f F a = let rec indrec n k = @@ -58,7 +58,7 @@ let induction_ltof1 f F a = in indrec (f a + 1) a ]] -the ML-like program for [induction_ltof2] is : +the ML-like program for [induction_ltof2] is : [[ let induction_ltof2 F a = indrec a where rec indrec a = F a indrec;; @@ -78,7 +78,7 @@ Proof. unfold ltof in |- *; intros b ltfafb. apply IHn. apply lt_le_trans with (f a); auto with arith. -Defined. +Defined. Theorem induction_gtof1 : forall P:A -> Set, @@ -262,7 +262,7 @@ Unset Implicit Arguments. (** [n]th iteration of the function [f] *) -Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) {struct n} : A := +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) @@ -271,8 +271,8 @@ Fixpoint iter_nat (n:nat) (A:Type) (f:A -> A) (x:A) {struct n} : A := 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. +Proof. simple induction n; [ simpl in |- *; auto with arith - | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. + | intros; simpl in |- *; apply f_equal with (f := f); apply H ]. Qed. |