Coq setoid 20110129
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- 5. Inductive Z’ : Set := | Z’pos : nat -‐> Z’ | Z’zero : Z’ | Z’neg : nat -‐> Z’. Inductive Z’ : Set := | mkZ’ : nat -‐> nat -‐> Z.
- 6. Require Import Setoid. Require Import Relation_Definitions. Require Import Arith. Require Import Compare_dec. Require Import Omega. Inductive Z' : Set := | mkZ' : nat -‐> nat -‐> Z'. Definition Z'eq (z z':Z') : Prop := let (a,b) := z in let (c,d) := z' in a + d = b + c. Notation "a =Z= b" := (Z'eq a b) (at level 70).
- 7. (* Z’eq Z’ *) Lemma Z'eq_refl : reflexive Z' Z'eq. Lemma Z'eq_sym : symmetric Z' Z'eq. Lemma Z'eq_trans : transitive Z' Z'eq. (* Z’ Z’eq Setoid *) Add Parametric Relation : Z' Z'eq reflexivity proved by Z'eq_refl symmetry proved by Z'eq_sym transitivity proved by Z'eq_trans as Z'_rel.
- 8. (* Z’ *) Definition Z'plus (z z':Z') : Z' := let (a,b) := z in let (c,d) := z' in mkZ' (a+c) (b+d). (* Z’plus Z’ Z’eq well-‐defined x =Z= y -‐> x0 =Z= y0 -‐> Z'plus x x0 =Z= Z'plus y y0 . *) Add Parametric Morphism : Z'plus with signature Z'eq ==> Z'eq ==> Z'eq as Z'_plus_mor. Proof. Qed.
- 9. (* *) Lemma Z'eq_plus_id_l : forall z a, Z'plus (mkZ' a a) z =Z= z. Lemma Z'eq_plus_comm : forall z z', Z'plus z z' =Z= Z'plus z' z. (* rewrite setoid_rewrite *) Lemma Z'eq_plus_id_r : forall z a, Z'plus z (mkZ' a a) =Z= z. Proof. intros. setoid_rewrite (Z'eq_plus_comm z (mkZ' a a)). apply Z'eq_plus_id_l. Qed. (* tactic setoid_reflexivity, setoid_rewrite, setoid_replace *)
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