Integral domains

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@14299 85f007b7-540e-0410-9357-904b9bb8a0f7

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diff --git a/plugins/setoid_ring/Integral_domain.v b/plugins/setoid_ring/Integral_domain.v
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+Require Export Cring.
+
+
+(* Definition of integral domains: commutative ring without zero divisor *)
+
+Class Integral_domain {R : Type}`{Rcr:Cring R} := {
+ integral_domain_product:
+   forall x y, x * y == 0 -> x == 0 \/ y == 0;
+ integral_domain_one_zero: not (1 == 0)}.
+
+Section integral_domain.
+
+Context {R:Type}`{Rid:Integral_domain R}.
+
+Lemma integral_domain_minus_one_zero: ~ - (1:R) == 0.
+red;intro. apply integral_domain_one_zero. 
+assert (0 == - (0:R)). cring.
+rewrite H0. rewrite <- H. cring.
+Qed.
+
+
+Definition pow (r : R) (n : nat) := Ring_theory.pow_N 1 mul r (N_of_nat n).
+
+Lemma pow_not_zero: forall p n, pow p n == 0 -> p == 0.
+induction n. unfold pow; simpl. intros. absurd (1 == 0). 
+simpl. apply integral_domain_one_zero.
+ trivial. setoid_replace (pow p (S n)) with (p * (pow p n)).
+intros. 
+case (integral_domain_product p (pow p n) H). trivial. trivial. 
+unfold pow; simpl. 
+clear IHn. induction n; simpl; try cring. 
+ rewrite Ring_theory.pow_pos_Psucc. cring. apply ring_setoid.
+apply ring_mult_comp.
+apply cring_mul_comm.
+apply ring_mul_assoc.
+Qed.
+
+Lemma Rintegral_domain_pow:
+  forall c p r, ~c == 0 -> c * (pow p r) == ring0 -> p == ring0.
+intros. case (integral_domain_product c (pow p r) H0). intros; absurd (c == ring0); auto. 
+intros. apply pow_not_zero with r. trivial. Qed.   
+
+End integral_domain.
+