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Diffstat (limited to 'theories/Numbers/Integer/BigInts/Zeqmod.v')
-rw-r--r-- | theories/Numbers/Integer/BigInts/Zeqmod.v | 48 |
1 files changed, 48 insertions, 0 deletions
diff --git a/theories/Numbers/Integer/BigInts/Zeqmod.v b/theories/Numbers/Integer/BigInts/Zeqmod.v new file mode 100644 index 000000000..ca3286211 --- /dev/null +++ b/theories/Numbers/Integer/BigInts/Zeqmod.v @@ -0,0 +1,48 @@ +Require Import ZArith. +Require Import ZAux. + +Open Local Scope Z_scope. +Notation "x == y '[' 'mod' z ']'" := ((x mod z) = (y mod z)) + (at level 70, no associativity) : Z_scope. + +Theorem Zeqmod_refl : forall p n : Z, n == n [mod p]. +Proof. +reflexivity. +Qed. + +Theorem Zeqmod_symm : forall p n m : Z, n == m [mod p] -> m == n [mod p]. +Proof. +now symmetry. +Qed. + +Theorem Zeqmod_trans : + forall p n m k : Z, n == m [mod p] -> m == k [mod p] -> n == k [mod p]. +Proof. +intros p n m k H1 H2; now transitivity (m mod p). +Qed. + +Theorem Zplus_eqmod_compat_l : + forall p n m k : Z, 0 < p -> (n == m [mod p] <-> k + n == k + m [mod p]). +intros p n m k H1. +assert (LR : forall n' m' k' : Z, n' == m' [mod p] -> k' + n' == k' + m' [mod p]). +intros n' m' k' H2. +pattern ((k' + n') mod p); rewrite Zmod_plus; [| assumption]. +pattern ((k' + m') mod p); rewrite Zmod_plus; [| assumption]. +rewrite H2. apply Zeqmod_refl. +split; [apply LR |]. +intro H2. apply (LR (k + n) (k + m) (-k)) in H2. +do 2 rewrite Zplus_assoc in H2. rewrite Zplus_opp_l in H2. +now do 2 rewrite Zplus_0_l in H2. +Qed. + +Theorem Zplus_eqmod_compat_r : + forall p n m k : Z, 0 < p -> (n == m [mod p] <-> n + k == m + k [mod p]). +intros p n m k; rewrite (Zplus_comm n k); rewrite (Zplus_comm m k); +apply Zplus_eqmod_compat_l. +Qed. + +(* + Local Variables: + tags-file-name: "~/coq/trunk/theories/Numbers/TAGS" + End: +*) |