src/HOL/Old_Number_Theory/Int2.thy
 changeset 41541 1fa4725c4656 parent 38159 e9b4835a54ee child 44766 d4d33a4d7548
```     1.1 --- a/src/HOL/Old_Number_Theory/Int2.thy	Thu Jan 13 21:50:13 2011 +0100
1.2 +++ b/src/HOL/Old_Number_Theory/Int2.thy	Thu Jan 13 23:50:16 2011 +0100
1.3 @@ -43,38 +43,39 @@
1.4    apply (force simp del:dvd_mult)
1.5    done
1.6
1.7 -lemma div_prop1: "[| 0 < z; (x::int) < y * z |] ==> x div z < y"
1.8 +lemma div_prop1:
1.9 +  assumes "0 < z" and "(x::int) < y * z"
1.10 +  shows "x div z < y"
1.11  proof -
1.12 -  assume "0 < z" then have modth: "x mod z \<ge> 0" by simp
1.13 +  from `0 < z` have modth: "x mod z \<ge> 0" by simp
1.14    have "(x div z) * z \<le> (x div z) * z" by simp
1.15    then have "(x div z) * z \<le> (x div z) * z + x mod z" using modth by arith
1.16    also have "\<dots> = x"
1.17      by (auto simp add: zmod_zdiv_equality [symmetric] zmult_ac)
1.18 -  also assume  "x < y * z"
1.19 +  also note `x < y * z`
1.20    finally show ?thesis
1.21 -    by (auto simp add: prems mult_less_cancel_right, insert prems, arith)
1.22 +    apply (auto simp add: mult_less_cancel_right)
1.23 +    using assms apply arith
1.24 +    done
1.25  qed
1.26
1.27 -lemma div_prop2: "[| 0 < z; (x::int) < (y * z) + z |] ==> x div z \<le> y"
1.28 +lemma div_prop2:
1.29 +  assumes "0 < z" and "(x::int) < (y * z) + z"
1.30 +  shows "x div z \<le> y"
1.31  proof -
1.32 -  assume "0 < z" and "x < (y * z) + z"
1.33 -  then have "x < (y + 1) * z" by (auto simp add: int_distrib)
1.34 +  from assms have "x < (y + 1) * z" by (auto simp add: int_distrib)
1.35    then have "x div z < y + 1"
1.36 -    apply -
1.37      apply (rule_tac y = "y + 1" in div_prop1)
1.38 -    apply (auto simp add: prems)
1.39 +    apply (auto simp add: `0 < z`)
1.40      done
1.41    then show ?thesis by auto
1.42  qed
1.43
1.44 -lemma zdiv_leq_prop: "[| 0 < y |] ==> y * (x div y) \<le> (x::int)"
1.45 +lemma zdiv_leq_prop: assumes "0 < y" shows "y * (x div y) \<le> (x::int)"
1.46  proof-
1.47 -  assume "0 < y"
1.48    from zmod_zdiv_equality have "x = y * (x div y) + x mod y" by auto
1.49 -  moreover have "0 \<le> x mod y"
1.50 -    by (auto simp add: prems pos_mod_sign)
1.51 -  ultimately show ?thesis
1.52 -    by arith
1.53 +  moreover have "0 \<le> x mod y" by (auto simp add: assms)
1.54 +  ultimately show ?thesis by arith
1.55  qed
1.56
1.57
1.58 @@ -87,7 +88,7 @@
1.59    by (auto simp add: zcong_def)
1.60
1.61  lemma zcong_shift: "[a = b] (mod m) ==> [a + c = b + c] (mod m)"
1.64
1.65  lemma zcong_zpower: "[x = y](mod m) ==> [x^z = y^z](mod m)"
1.66    by (induct z) (auto simp add: zcong_zmult)
1.67 @@ -126,11 +127,12 @@
1.68      x < m; y < m |] ==> x = y"
1.69    by (metis zcong_not zcong_sym zless_linear)
1.70
1.71 -lemma zcong_neg_1_impl_ne_1: "[| 2 < p; [x = -1] (mod p) |] ==>
1.72 -    ~([x = 1] (mod p))"
1.73 +lemma zcong_neg_1_impl_ne_1:
1.74 +  assumes "2 < p" and "[x = -1] (mod p)"
1.75 +  shows "~([x = 1] (mod p))"
1.76  proof
1.77 -  assume "2 < p" and "[x = 1] (mod p)" and "[x = -1] (mod p)"
1.78 -  then have "[1 = -1] (mod p)"
1.79 +  assume "[x = 1] (mod p)"
1.80 +  with assms have "[1 = -1] (mod p)"
1.81      apply (auto simp add: zcong_sym)
1.82      apply (drule zcong_trans, auto)
1.83      done
1.84 @@ -140,7 +142,7 @@
1.85      by auto
1.86    then have "p dvd 2"
1.87      by (auto simp add: dvd_def zcong_def)
1.88 -  with prems show False
1.89 +  with `2 < p` show False
1.90      by (auto simp add: zdvd_not_zless)
1.91  qed
1.92
1.93 @@ -181,15 +183,15 @@
1.94  lemma MultInv_prop2: "[| 2 < p; zprime p; ~([x = 0](mod p)) |] ==>
1.95    [(x * (MultInv p x)) = 1] (mod p)"
1.96  proof (simp add: MultInv_def zcong_eq_zdvd_prop)
1.97 -  assume "2 < p" and "zprime p" and "~ p dvd x"
1.98 +  assume 1: "2 < p" and 2: "zprime p" and 3: "~ p dvd x"
1.99    have "x * x ^ nat (p - 2) = x ^ (nat (p - 2) + 1)"
1.100      by auto
1.101 -  also from prems have "nat (p - 2) + 1 = nat (p - 2 + 1)"
1.102 +  also from 1 have "nat (p - 2) + 1 = nat (p - 2 + 1)"