1.1 --- a/src/HOL/Power.thy Tue May 11 14:00:02 2004 +0200
1.2 +++ b/src/HOL/Power.thy Tue May 11 20:11:08 2004 +0200
1.3 @@ -11,7 +11,7 @@
1.4
1.5 subsection{*Powers for Arbitrary (Semi)Rings*}
1.6
1.7 -axclass ringpower \<subseteq> semiring, power
1.8 +axclass ringpower \<subseteq> comm_semiring_1_cancel, power
1.9 power_0 [simp]: "a ^ 0 = 1"
1.10 power_Suc: "a ^ (Suc n) = a * (a ^ n)"
1.11
1.12 @@ -46,31 +46,31 @@
1.13 done
1.14
1.15 lemma zero_less_power:
1.16 - "0 < (a::'a::{ordered_semiring,ringpower}) ==> 0 < a^n"
1.17 + "0 < (a::'a::{ordered_semidom,ringpower}) ==> 0 < a^n"
1.18 apply (induct_tac "n")
1.19 apply (simp_all add: power_Suc zero_less_one mult_pos)
1.20 done
1.21
1.22 lemma zero_le_power:
1.23 - "0 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 0 \<le> a^n"
1.24 + "0 \<le> (a::'a::{ordered_semidom,ringpower}) ==> 0 \<le> a^n"
1.25 apply (simp add: order_le_less)
1.26 apply (erule disjE)
1.27 apply (simp_all add: zero_less_power zero_less_one power_0_left)
1.28 done
1.29
1.30 lemma one_le_power:
1.31 - "1 \<le> (a::'a::{ordered_semiring,ringpower}) ==> 1 \<le> a^n"
1.32 + "1 \<le> (a::'a::{ordered_semidom,ringpower}) ==> 1 \<le> a^n"
1.33 apply (induct_tac "n")
1.34 apply (simp_all add: power_Suc)
1.35 apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
1.36 apply (simp_all add: zero_le_one order_trans [OF zero_le_one])
1.37 done
1.38
1.39 -lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semiring)"
1.40 +lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
1.41 by (simp add: order_trans [OF zero_le_one order_less_imp_le])
1.42
1.43 lemma power_gt1_lemma:
1.44 - assumes gt1: "1 < (a::'a::{ordered_semiring,ringpower})"
1.45 + assumes gt1: "1 < (a::'a::{ordered_semidom,ringpower})"
1.46 shows "1 < a * a^n"
1.47 proof -
1.48 have "1*1 < a*1" using gt1 by simp
1.49 @@ -81,11 +81,11 @@
1.50 qed
1.51
1.52 lemma power_gt1:
1.53 - "1 < (a::'a::{ordered_semiring,ringpower}) ==> 1 < a ^ (Suc n)"
1.54 + "1 < (a::'a::{ordered_semidom,ringpower}) ==> 1 < a ^ (Suc n)"
1.55 by (simp add: power_gt1_lemma power_Suc)
1.56
1.57 lemma power_le_imp_le_exp:
1.58 - assumes gt1: "(1::'a::{ringpower,ordered_semiring}) < a"
1.59 + assumes gt1: "(1::'a::{ringpower,ordered_semidom}) < a"
1.60 shows "!!n. a^m \<le> a^n ==> m \<le> n"
1.61 proof (induct m)
1.62 case 0
1.63 @@ -109,26 +109,26 @@
1.64
1.65 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
1.66 lemma power_inject_exp [simp]:
1.67 - "1 < (a::'a::{ordered_semiring,ringpower}) ==> (a^m = a^n) = (m=n)"
1.68 + "1 < (a::'a::{ordered_semidom,ringpower}) ==> (a^m = a^n) = (m=n)"
1.69 by (force simp add: order_antisym power_le_imp_le_exp)
1.70
1.71 text{*Can relax the first premise to @{term "0<a"} in the case of the
1.72 natural numbers.*}
1.73 lemma power_less_imp_less_exp:
1.74 - "[| (1::'a::{ringpower,ordered_semiring}) < a; a^m < a^n |] ==> m < n"
1.75 + "[| (1::'a::{ringpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
1.76 by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
1.77 power_le_imp_le_exp)
1.78
1.79
1.80 lemma power_mono:
1.81 - "[|a \<le> b; (0::'a::{ringpower,ordered_semiring}) \<le> a|] ==> a^n \<le> b^n"
1.82 + "[|a \<le> b; (0::'a::{ringpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
1.83 apply (induct_tac "n")
1.84 apply (simp_all add: power_Suc)
1.85 apply (auto intro: mult_mono zero_le_power order_trans [of 0 a b])
1.86 done
1.87
1.88 lemma power_strict_mono [rule_format]:
1.89 - "[|a < b; (0::'a::{ringpower,ordered_semiring}) \<le> a|]
1.90 + "[|a < b; (0::'a::{ringpower,ordered_semidom}) \<le> a|]
1.91 ==> 0 < n --> a^n < b^n"
1.92 apply (induct_tac "n")
1.93 apply (auto simp add: mult_strict_mono zero_le_power power_Suc
1.94 @@ -136,7 +136,7 @@
1.95 done
1.96
1.97 lemma power_eq_0_iff [simp]:
1.98 - "(a^n = 0) = (a = (0::'a::{ordered_ring,ringpower}) & 0<n)"
1.99 + "(a^n = 0) = (a = (0::'a::{ordered_idom,ringpower}) & 0<n)"
1.100 apply (induct_tac "n")
1.101 apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
1.102 done
1.103 @@ -174,13 +174,13 @@
1.104 apply assumption
1.105 done
1.106
1.107 -lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_ring,ringpower}) ^ n"
1.108 +lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,ringpower}) ^ n"
1.109 apply (induct_tac "n")
1.110 apply (auto simp add: power_Suc abs_mult)
1.111 done
1.112
1.113 lemma zero_less_power_abs_iff [simp]:
1.114 - "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_ring,ringpower}) | n=0)"
1.115 + "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,ringpower}) | n=0)"
1.116 proof (induct "n")
1.117 case 0
1.118 show ?case by (simp add: zero_less_one)
1.119 @@ -190,12 +190,12 @@
1.120 qed
1.121
1.122 lemma zero_le_power_abs [simp]:
1.123 - "(0::'a::{ordered_ring,ringpower}) \<le> (abs a)^n"
1.124 + "(0::'a::{ordered_idom,ringpower}) \<le> (abs a)^n"
1.125 apply (induct_tac "n")
1.126 apply (auto simp add: zero_le_one zero_le_power)
1.127 done
1.128
1.129 -lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring,ringpower}) ^ n"
1.130 +lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{comm_ring_1,ringpower}) ^ n"
1.131 proof -
1.132 have "-a = (- 1) * a" by (simp add: minus_mult_left [symmetric])
1.133 thus ?thesis by (simp only: power_mult_distrib)
1.134 @@ -203,14 +203,14 @@
1.135
1.136 text{*Lemma for @{text power_strict_decreasing}*}
1.137 lemma power_Suc_less:
1.138 - "[|(0::'a::{ordered_semiring,ringpower}) < a; a < 1|]
1.139 + "[|(0::'a::{ordered_semidom,ringpower}) < a; a < 1|]
1.140 ==> a * a^n < a^n"
1.141 apply (induct_tac n)
1.142 apply (auto simp add: power_Suc mult_strict_left_mono)
1.143 done
1.144
1.145 lemma power_strict_decreasing:
1.146 - "[|n < N; 0 < a; a < (1::'a::{ordered_semiring,ringpower})|]
1.147 + "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,ringpower})|]
1.148 ==> a^N < a^n"
1.149 apply (erule rev_mp)
1.150 apply (induct_tac "N")
1.151 @@ -223,7 +223,7 @@
1.152
1.153 text{*Proof resembles that of @{text power_strict_decreasing}*}
1.154 lemma power_decreasing:
1.155 - "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semiring,ringpower})|]
1.156 + "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,ringpower})|]
1.157 ==> a^N \<le> a^n"
1.158 apply (erule rev_mp)
1.159 apply (induct_tac "N")
1.160 @@ -235,13 +235,13 @@
1.161 done
1.162
1.163 lemma power_Suc_less_one:
1.164 - "[| 0 < a; a < (1::'a::{ordered_semiring,ringpower}) |] ==> a ^ Suc n < 1"
1.165 + "[| 0 < a; a < (1::'a::{ordered_semidom,ringpower}) |] ==> a ^ Suc n < 1"
1.166 apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
1.167 done
1.168
1.169 text{*Proof again resembles that of @{text power_strict_decreasing}*}
1.170 lemma power_increasing:
1.171 - "[|n \<le> N; (1::'a::{ordered_semiring,ringpower}) \<le> a|] ==> a^n \<le> a^N"
1.172 + "[|n \<le> N; (1::'a::{ordered_semidom,ringpower}) \<le> a|] ==> a^n \<le> a^N"
1.173 apply (erule rev_mp)
1.174 apply (induct_tac "N")
1.175 apply (auto simp add: power_Suc le_Suc_eq)
1.176 @@ -253,13 +253,13 @@
1.177
1.178 text{*Lemma for @{text power_strict_increasing}*}
1.179 lemma power_less_power_Suc:
1.180 - "(1::'a::{ordered_semiring,ringpower}) < a ==> a^n < a * a^n"
1.181 + "(1::'a::{ordered_semidom,ringpower}) < a ==> a^n < a * a^n"
1.182 apply (induct_tac n)
1.183 apply (auto simp add: power_Suc mult_strict_left_mono order_less_trans [OF zero_less_one])
1.184 done
1.185
1.186 lemma power_strict_increasing:
1.187 - "[|n < N; (1::'a::{ordered_semiring,ringpower}) < a|] ==> a^n < a^N"
1.188 + "[|n < N; (1::'a::{ordered_semidom,ringpower}) < a|] ==> a^n < a^N"
1.189 apply (erule rev_mp)
1.190 apply (induct_tac "N")
1.191 apply (auto simp add: power_less_power_Suc power_Suc less_Suc_eq)
1.192 @@ -272,7 +272,7 @@
1.193
1.194 lemma power_le_imp_le_base:
1.195 assumes le: "a ^ Suc n \<le> b ^ Suc n"
1.196 - and xnonneg: "(0::'a::{ordered_semiring,ringpower}) \<le> a"
1.197 + and xnonneg: "(0::'a::{ordered_semidom,ringpower}) \<le> a"
1.198 and ynonneg: "0 \<le> b"
1.199 shows "a \<le> b"
1.200 proof (rule ccontr)
1.201 @@ -286,7 +286,7 @@
1.202
1.203 lemma power_inject_base:
1.204 "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
1.205 - ==> a = (b::'a::{ordered_semiring,ringpower})"
1.206 + ==> a = (b::'a::{ordered_semidom,ringpower})"
1.207 by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
1.208
1.209