diff options
Diffstat (limited to 'theories/Numbers/Natural/Peano/NPeano.v')
| -rw-r--r-- | theories/Numbers/Natural/Peano/NPeano.v | 23 |
1 files changed, 7 insertions, 16 deletions
diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 8d9e17fb09..38951218d5 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -28,14 +28,7 @@ Definition NZadd := plus. Definition NZsub := minus. Definition NZmul := mult. -Theorem NZeq_equiv : equiv nat NZeq. -Proof (eq_equiv nat). - -Add Relation nat NZeq - reflexivity proved by (proj1 NZeq_equiv) - symmetry proved by (proj2 (proj2 NZeq_equiv)) - transitivity proved by (proj1 (proj2 NZeq_equiv)) -as NZeq_rel. +Instance NZeq_equiv : Equivalence NZeq. (* If we say "Add Relation nat (@eq nat)" instead of "Add Relation nat NZeq" then the theorem generated for succ_wd below is forall x, succ x = succ x, @@ -189,13 +182,11 @@ Proof. reflexivity. Qed. -Theorem recursion_wd : forall (A : Type) (Aeq : relation A), - forall a a' : A, Aeq a a' -> - forall f f' : nat -> A -> A, fun2_eq (@eq nat) Aeq Aeq f f' -> - forall n n' : nat, n = n' -> - Aeq (recursion a f n) (recursion a' f' n'). +Instance recursion_wd (A : Type) (Aeq : relation A) : + Proper (Aeq ==> (eq==>Aeq==>Aeq) ==> eq ==> Aeq) (@recursion A). Proof. -unfold fun2_eq; induction n; intros n' Enn'; rewrite <- Enn' in *; simpl; auto. +intros A Aeq a a' Ha f f' Hf n n' Hn. subst n'. +induction n; simpl; auto. apply Hf; auto. Qed. Theorem recursion_0 : @@ -206,10 +197,10 @@ Qed. Theorem recursion_succ : forall (A : Type) (Aeq : relation A) (a : A) (f : nat -> A -> A), - Aeq a a -> fun2_wd (@eq nat) Aeq Aeq f -> + Aeq a a -> Proper (eq==>Aeq==>Aeq) f -> forall n : nat, Aeq (recursion a f (S n)) (f n (recursion a f n)). Proof. -induction n; simpl; auto. +unfold Proper, respectful in *; induction n; simpl; auto. Qed. End NPeanoAxiomsMod. |