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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_succ. cring. apply ring_setoid.
apply ring_mult_comp.
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.
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