src/HOL/GCD.thy
 changeset 32040 830141c9e528 parent 32036 8a9228872fbd parent 31996 1d93369079c4 child 32045 efc5f4806cd5
```     1.1 --- a/src/HOL/GCD.thy	Tue Jul 14 17:17:37 2009 +0200
1.2 +++ b/src/HOL/GCD.thy	Tue Jul 14 17:18:51 2009 +0200
1.3 @@ -37,7 +37,7 @@
1.4
1.5  subsection {* gcd *}
1.6
1.7 -class gcd = one +
1.8 +class gcd = zero + one + dvd +
1.9
1.10  fixes
1.11    gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
1.12 @@ -540,15 +540,15 @@
1.13
1.14  (* to do: add the three variations of these, and for ints? *)
1.15
1.16 -lemma finite_divisors_nat:
1.17 -  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
1.18 +lemma finite_divisors_nat[simp]:
1.19 +  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
1.20  proof-
1.21    have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
1.22    from finite_subset[OF _ this] show ?thesis using assms
1.23      by(bestsimp intro!:dvd_imp_le)
1.24  qed
1.25
1.26 -lemma finite_divisors_int:
1.27 +lemma finite_divisors_int[simp]:
1.28    assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
1.29  proof-
1.30    have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
1.31 @@ -557,10 +557,25 @@
1.32      by(bestsimp intro!:dvd_imp_le_int)
1.33  qed
1.34
1.35 +lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
1.36 +apply(rule antisym)
1.37 + apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
1.38 +apply simp
1.39 +done
1.40 +
1.41 +lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
1.42 +apply(rule antisym)
1.43 + apply(rule Max_le_iff[THEN iffD2])
1.44 +   apply simp
1.45 +  apply fastsimp
1.46 + apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
1.47 +apply simp
1.48 +done
1.49 +
1.50  lemma gcd_is_Max_divisors_nat:
1.51    "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
1.52  apply(rule Max_eqI[THEN sym])
1.53 -  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
1.54 +  apply (metis finite_Collect_conjI finite_divisors_nat)
1.55   apply simp
1.56   apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
1.57  apply simp
1.58 @@ -569,7 +584,7 @@
1.59  lemma gcd_is_Max_divisors_int:
1.60    "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
1.61  apply(rule Max_eqI[THEN sym])
1.62 -  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
1.63 +  apply (metis finite_Collect_conjI finite_divisors_int)
1.64   apply simp
1.65   apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
1.66  apply simp
1.67 @@ -1442,31 +1457,61 @@
1.68  lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
1.69  by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
1.70
1.71 +lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
1.72 +by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
1.73 +
1.74 +lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
1.75 +by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
1.76 +
1.77 +lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
1.78 +by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
1.79 +
1.80 +lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
1.81 +by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
1.82
1.83  lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
1.84 -apply(rule lcm_unique_nat[THEN iffD1])
1.85 -apply (metis dvd.order_trans lcm_unique_nat)
1.86 -done
1.87 +by(rule lcm_unique_nat[THEN iffD1])(metis dvd.order_trans lcm_unique_nat)
1.88
1.89  lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
1.90 -apply(rule lcm_unique_int[THEN iffD1])
1.91 -apply (metis dvd_trans lcm_unique_int)
1.92 -done
1.93 +by(rule lcm_unique_int[THEN iffD1])(metis dvd_trans lcm_unique_int)
1.94
1.95 -lemmas lcm_left_commute_nat =
1.96 -  mk_left_commute[of lcm, OF lcm_assoc_nat lcm_commute_nat]
1.97 -
1.98 -lemmas lcm_left_commute_int =
1.99 -  mk_left_commute[of lcm, OF lcm_assoc_int lcm_commute_int]
1.100 +lemmas lcm_left_commute_nat = mk_left_commute[of lcm, OF lcm_assoc_nat lcm_commute_nat]
1.101 +lemmas lcm_left_commute_int = mk_left_commute[of lcm, OF lcm_assoc_int lcm_commute_int]
1.102
1.103  lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
1.104  lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
1.105
1.106 +lemma fun_left_comm_idem_gcd_nat: "fun_left_comm_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
1.107 +proof qed (auto simp add: gcd_ac_nat)
1.108 +
1.109 +lemma fun_left_comm_idem_gcd_int: "fun_left_comm_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
1.110 +proof qed (auto simp add: gcd_ac_int)
1.111 +
1.112 +lemma fun_left_comm_idem_lcm_nat: "fun_left_comm_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
1.113 +proof qed (auto simp add: lcm_ac_nat)
1.114 +
1.115 +lemma fun_left_comm_idem_lcm_int: "fun_left_comm_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
1.116 +proof qed (auto simp add: lcm_ac_int)
1.117 +
1.118 +
1.119 +(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
1.120 +
1.121 +lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
1.122 +by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
1.123 +
1.124 +lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
1.125 +by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
1.126 +
1.127 +lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
1.128 +by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
1.129 +
1.130 +lemma lcm_1_iff_int[simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
1.131 +by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
1.132 +
1.133
1.134  subsection {* Primes *}
1.135
1.136 -(* Is there a better way to handle these, rather than making them
1.137 -   elim rules? *)
1.138 +(* FIXME Is there a better way to handle these, rather than making them elim rules? *)
1.139
1.140  lemma prime_ge_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
1.141    by (unfold prime_nat_def, auto)
1.142 @@ -1490,7 +1535,7 @@
1.143    by (unfold prime_nat_def, auto)
1.144
1.145  lemma prime_ge_0_int [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
1.146 -  by (unfold prime_int_def prime_nat_def, auto)
1.147 +  by (unfold prime_int_def prime_nat_def) auto
1.148
1.149  lemma prime_gt_0_int [elim]: "prime (p::int) \<Longrightarrow> p > 0"
1.150    by (unfold prime_int_def prime_nat_def, auto)
1.151 @@ -1504,8 +1549,6 @@
1.152  lemma prime_ge_2_int [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
1.153    by (unfold prime_int_def prime_nat_def, auto)
1.154
1.155 -thm prime_nat_def;
1.156 -thm prime_nat_def [transferred];
1.157
1.158  lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
1.159      m = 1 \<or> m = p))"
1.160 @@ -1566,35 +1609,13 @@
1.161  lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
1.162      EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
1.163    unfolding prime_nat_def dvd_def apply auto
1.164 -  apply (subgoal_tac "k > 1")
1.165 -  apply force
1.166 -  apply (subgoal_tac "k ~= 0")
1.167 -  apply force
1.168 -  apply (rule notI)
1.169 -  apply force
1.170 -done
1.171 +  by(metis mult_commute linorder_neq_iff linorder_not_le mult_1 n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
1.172
1.173 -(* there's a lot of messing around with signs of products here --
1.174 -   could this be made more automatic? *)
1.175  lemma not_prime_eq_prod_int: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
1.176      EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
1.177    unfolding prime_int_altdef dvd_def
1.178    apply auto
1.179 -  apply (rule_tac x = m in exI)
1.180 -  apply (rule_tac x = k in exI)
1.181 -  apply (auto simp add: mult_compare_simps)
1.182 -  apply (subgoal_tac "k > 0")
1.183 -  apply arith
1.184 -  apply (case_tac "k <= 0")
1.185 -  apply (subgoal_tac "m * k <= 0")
1.186 -  apply force
1.187 -  apply (subst zero_compare_simps(8))
1.188 -  apply auto
1.189 -  apply (subgoal_tac "m * k <= 0")
1.190 -  apply force
1.191 -  apply (subst zero_compare_simps(8))
1.192 -  apply auto
1.193 -done
1.194 +  by(metis div_mult_self1_is_id div_mult_self2_is_id int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos zless_le)
1.195
1.196  lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
1.197      n > 0 --> (p dvd x^n --> p dvd x)"
```