src/HOL/Power.thy
 changeset 30240 5b25fee0362c parent 29608 564ea783ace8 child 30242 aea5d7fa7ef5
```     1.1 --- a/src/HOL/Power.thy	Wed Mar 04 10:43:39 2009 +0100
1.2 +++ b/src/HOL/Power.thy	Wed Mar 04 10:45:52 2009 +0100
1.3 @@ -31,7 +31,7 @@
1.4    by (induct n) (simp_all add: power_Suc)
1.5
1.6  lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
1.7 -  by (simp add: power_Suc)
1.8 +  unfolding One_nat_def by (simp add: power_Suc)
1.9
1.10  lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
1.11    by (induct n) (simp_all add: power_Suc mult_assoc)
1.12 @@ -143,11 +143,13 @@
1.13  done
1.14
1.15  lemma power_eq_0_iff [simp]:
1.16 -  "(a^n = 0) = (a = (0::'a::{ring_1_no_zero_divisors,recpower}) & n>0)"
1.17 +  "(a^n = 0) \<longleftrightarrow>
1.18 +   (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
1.19  apply (induct "n")
1.20 -apply (auto simp add: power_Suc zero_neq_one [THEN not_sym])
1.21 +apply (auto simp add: power_Suc zero_neq_one [THEN not_sym] no_zero_divisors)
1.22  done
1.23
1.24 +
1.25  lemma field_power_not_zero:
1.26    "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
1.27  by force
1.28 @@ -324,6 +326,24 @@
1.29    shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
1.30  by (cases n, simp_all, rule power_inject_base)
1.31
1.32 +text {* The divides relation *}
1.33 +
1.34 +lemma le_imp_power_dvd:
1.35 +  fixes a :: "'a::{comm_semiring_1,recpower}"
1.36 +  assumes "m \<le> n" shows "a^m dvd a^n"
1.37 +proof
1.38 +  have "a^n = a^(m + (n - m))"
1.39 +    using `m \<le> n` by simp
1.40 +  also have "\<dots> = a^m * a^(n - m)"
1.41 +    by (rule power_add)
1.42 +  finally show "a^n = a^m * a^(n - m)" .
1.43 +qed
1.44 +
1.45 +lemma power_le_dvd:
1.46 +  fixes a b :: "'a::{comm_semiring_1,recpower}"
1.47 +  shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
1.48 +  by (rule dvd_trans [OF le_imp_power_dvd])
1.49 +
1.50
1.51  subsection{*Exponentiation for the Natural Numbers*}
1.52
1.53 @@ -346,12 +366,19 @@
1.54    "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
1.55  by (induct n, simp_all add: power_Suc of_nat_mult)
1.56
1.57 -lemma nat_one_le_power [simp]: "1 \<le> i ==> Suc 0 \<le> i^n"
1.58 -by (insert one_le_power [of i n], simp)
1.59 +lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
1.60 +by (rule one_le_power [of i n, unfolded One_nat_def])
1.61
1.62  lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
1.63  by (induct "n", auto)
1.64
1.65 +lemma nat_power_eq_Suc_0_iff [simp]:
1.66 +  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
1.67 +by (induct_tac m, auto)
1.68 +
1.69 +lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
1.70 +by simp
1.71 +
1.72  text{*Valid for the naturals, but what if @{text"0<i<1"}?
1.73  Premises cannot be weakened: consider the case where @{term "i=0"},
1.74  @{term "m=1"} and @{term "n=0"}.*}
1.75 @@ -425,4 +452,3 @@
1.76  *}
1.77
1.78  end
1.79 -
```