author chaieb Mon Jul 21 13:36:44 2008 +0200 (2008-07-21) changeset 27667 62500b980749 parent 27666 1436d81d1294 child 27668 6eb20b2cecf8
Added theorems zmod_eq_dvd_iff and nat_mod_eq_iff previously in Pocklington.thy --- relevant for algebra
 src/HOL/IntDiv.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/IntDiv.thy	Mon Jul 21 13:36:39 2008 +0200
1.2 +++ b/src/HOL/IntDiv.thy	Mon Jul 21 13:36:44 2008 +0200
1.3 @@ -1513,6 +1513,51 @@
1.4
1.5  end
1.6
1.7 +lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
1.8 +proof
1.9 +  assume H: "x mod n = y mod n"
1.10 +  hence "x mod n - y mod n = 0" by simp
1.11 +  hence "(x mod n - y mod n) mod n = 0" by simp
1.12 +  hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])
1.13 +  thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
1.14 +next
1.15 +  assume H: "n dvd x - y"
1.16 +  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
1.17 +  hence "x = n*k + y" by simp
1.18 +  hence "x mod n = (n*k + y) mod n" by simp
1.19 +  thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)
1.20 +qed
1.21 +
1.22 +lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
1.23 +  shows "\<exists>q. x = y + n * q"
1.24 +proof-
1.25 +  from xy have th: "int x - int y = int (x - y)" by simp
1.26 +  from xyn have "int x mod int n = int y mod int n"
1.27 +    by (simp add: zmod_int[symmetric])
1.28 +  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
1.29 +  hence "n dvd x - y" by (simp add: th zdvd_int)
1.30 +  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
1.31 +qed
1.32 +
1.33 +lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
1.34 +  (is "?lhs = ?rhs")
1.35 +proof
1.36 +  assume H: "x mod n = y mod n"
1.37 +  {assume xy: "x \<le> y"
1.38 +    from H have th: "y mod n = x mod n" by simp
1.39 +    from nat_mod_eq_lemma[OF th xy] have ?rhs
1.40 +      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
1.41 +  moreover
1.42 +  {assume xy: "y \<le> x"
1.43 +    from nat_mod_eq_lemma[OF H xy] have ?rhs
1.44 +      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
1.45 +  ultimately  show ?rhs using linear[of x y] by blast
1.46 +next
1.47 +  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
1.48 +  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
1.49 +  thus  ?lhs by simp
1.50 +qed
1.51 +
1.52  code_modulename SML
1.53    IntDiv Integer
1.54
```