diff options
Diffstat (limited to 'theories/Numbers/Natural/SpecViaZ')
| -rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 23 |
1 files changed, 5 insertions, 18 deletions
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 578cb6256..596603b6f 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -39,16 +39,7 @@ Definition NZadd := N.add. Definition NZsub := N.sub. Definition NZmul := N.mul. -Theorem NZeq_equiv : equiv N.t N.eq. -Proof. -repeat split; repeat red; intros; auto; congruence. -Qed. - -Add Relation N.t N.eq - 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 N.eq. Add Morphism NZsucc with signature N.eq ==> N.eq as NZsucc_wd. Proof. @@ -297,14 +288,10 @@ Definition recursion (A : Type) (a : A) (f : N.t -> A -> A) (n : N.t) := Nrect (fun _ => A) a (fun n a => f (N.of_N n) a) (N.to_N n). Implicit Arguments recursion [A]. -Theorem recursion_wd : -forall (A : Type) (Aeq : relation A), - forall a a' : A, Aeq a a' -> - forall f f' : N.t -> A -> A, fun2_eq N.eq Aeq Aeq f f' -> - forall x x' : N.t, x == x' -> - Aeq (recursion a f x) (recursion a' f' x'). +Instance recursion_wd (A : Type) (Aeq : relation A) : + Proper (Aeq ==> (N.eq==>Aeq==>Aeq) ==> N.eq ==> Aeq) (@recursion A). Proof. -unfold fun2_wd, N.eq, fun2_eq. +unfold N.eq. intros A Aeq a a' Eaa' f f' Eff' x x' Exx'. unfold recursion. unfold N.to_N. @@ -312,7 +299,7 @@ rewrite <- Exx'; clear x' Exx'. replace (Zabs_N [x]) with (N_of_nat (Zabs_nat [x])). induction (Zabs_nat [x]). simpl; auto. -rewrite N_of_S, 2 Nrect_step; auto. +rewrite N_of_S, 2 Nrect_step; auto. apply Eff'; auto. destruct [x]; simpl; auto. change (nat_of_P p) with (nat_of_N (Npos p)); apply N_of_nat_of_N. change (nat_of_P p) with (nat_of_N (Npos p)); apply N_of_nat_of_N. |