# HG changeset patch
# User haftmann
# Date 1240813357 -7200
# Node ID c2524d1235281a3097e1dd0e62b5262602713a0e
# Parent  e46639644fcd2ab7f9a05747d0d07c088eb86ba7# Parent  a1efb13fc5d8b8cf356676a7d9cdd60eacb3224f
merged

diff -r e46639644fcd -r c2524d123528 src/HOL/Power.thy
--- a/src/HOL/Power.thy	Sun Apr 26 23:41:18 2009 +0100
+++ b/src/HOL/Power.thy	Mon Apr 27 08:22:37 2009 +0200
@@ -11,85 +11,169 @@
 
 subsection {* Powers for Arbitrary Monoids *}
 
-class recpower = monoid_mult
+class power = one + times
 begin
 
 primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80) where
     power_0: "a ^ 0 = 1"
   | power_Suc: "a ^ Suc n = a * a ^ n"
 
+notation (latex output)
+  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
+notation (HTML output)
+  power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
+
 end
 
-lemma power_0_Suc [simp]: "(0::'a::{recpower,semiring_0}) ^ (Suc n) = 0"
-  by simp
+context monoid_mult
+begin
 
-text{*It looks plausible as a simprule, but its effect can be strange.*}
-lemma power_0_left: "0^n = (if n=0 then 1 else (0::'a::{recpower,semiring_0}))"
-  by (induct n) simp_all
+subclass power ..
 
-lemma power_one [simp]: "1^n = (1::'a::recpower)"
+lemma power_one [simp]:
+  "1 ^ n = 1"
   by (induct n) simp_all
 
-lemma power_one_right [simp]: "(a::'a::recpower) ^ 1 = a"
-  unfolding One_nat_def by simp
+lemma power_one_right [simp]:
+  "a ^ 1 = a * 1"
+  by simp
 
-lemma power_commutes: "(a::'a::recpower) ^ n * a = a * a ^ n"
+lemma power_commutes:
+  "a ^ n * a = a * a ^ n"
   by (induct n) (simp_all add: mult_assoc)
 
-lemma power_Suc2: "(a::'a::recpower) ^ Suc n = a ^ n * a"
+lemma power_Suc2:
+  "a ^ Suc n = a ^ n * a"
   by (simp add: power_commutes)
 
-lemma power_add: "(a::'a::recpower) ^ (m+n) = (a^m) * (a^n)"
-  by (induct m) (simp_all add: mult_ac)
+lemma power_add:
+  "a ^ (m + n) = a ^ m * a ^ n"
+  by (induct m) (simp_all add: algebra_simps)
 
-lemma power_mult: "(a::'a::recpower) ^ (m*n) = (a^m) ^ n"
+lemma power_mult:
+  "a ^ (m * n) = (a ^ m) ^ n"
   by (induct n) (simp_all add: power_add)
 
-lemma power_mult_distrib: "((a::'a::{recpower,comm_monoid_mult}) * b) ^ n = (a^n) * (b^n)"
+end
+
+context comm_monoid_mult
+begin
+
+lemma power_mult_distrib:
+  "(a * b) ^ n = (a ^ n) * (b ^ n)"
   by (induct n) (simp_all add: mult_ac)
 
-lemma zero_less_power[simp]:
-     "0 < (a::'a::{ordered_semidom,recpower}) ==> 0 < a^n"
-by (induct n) (simp_all add: mult_pos_pos)
+end
+
+context semiring_1
+begin
+
+lemma of_nat_power:
+  "of_nat (m ^ n) = of_nat m ^ n"
+  by (induct n) (simp_all add: of_nat_mult)
+
+end
+
+context comm_semiring_1
+begin
+
+text {* The divides relation *}
+
+lemma le_imp_power_dvd:
+  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
+proof
+  have "a ^ n = a ^ (m + (n - m))"
+    using `m \<le> n` by simp
+  also have "\<dots> = a ^ m * a ^ (n - m)"
+    by (rule power_add)
+  finally show "a ^ n = a ^ m * a ^ (n - m)" .
+qed
+
+lemma power_le_dvd:
+  "a ^ n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a ^ m dvd b"
+  by (rule dvd_trans [OF le_imp_power_dvd])
+
+lemma dvd_power_same:
+  "x dvd y \<Longrightarrow> x ^ n dvd y ^ n"
+  by (induct n) (auto simp add: mult_dvd_mono)
+
+lemma dvd_power_le:
+  "x dvd y \<Longrightarrow> m \<ge> n \<Longrightarrow> x ^ n dvd y ^ m"
+  by (rule power_le_dvd [OF dvd_power_same])
 
-lemma zero_le_power[simp]:
-     "0 \<le> (a::'a::{ordered_semidom,recpower}) ==> 0 \<le> a^n"
-by (induct n) (simp_all add: mult_nonneg_nonneg)
+lemma dvd_power [simp]:
+  assumes "n > (0::nat) \<or> x = 1"
+  shows "x dvd (x ^ n)"
+using assms proof
+  assume "0 < n"
+  then have "x ^ n = x ^ Suc (n - 1)" by simp
+  then show "x dvd (x ^ n)" by simp
+next
+  assume "x = 1"
+  then show "x dvd (x ^ n)" by simp
+qed
+
+end
+
+context ring_1
+begin
+
+lemma power_minus:
+  "(- a) ^ n = (- 1) ^ n * a ^ n"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case (Suc n) then show ?case
+    by (simp del: power_Suc add: power_Suc2 mult_assoc)
+qed
+
+end
+
+context ordered_semidom
+begin
+
+lemma zero_less_power [simp]:
+  "0 < a \<Longrightarrow> 0 < a ^ n"
+  by (induct n) (simp_all add: mult_pos_pos)
+
+lemma zero_le_power [simp]:
+  "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
+  by (induct n) (simp_all add: mult_nonneg_nonneg)
 
 lemma one_le_power[simp]:
-     "1 \<le> (a::'a::{ordered_semidom,recpower}) ==> 1 \<le> a^n"
-apply (induct "n")
-apply simp_all
-apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
-apply (simp_all add: order_trans [OF zero_le_one])
-done
-
-lemma gt1_imp_ge0: "1 < a ==> 0 \<le> (a::'a::ordered_semidom)"
-  by (simp add: order_trans [OF zero_le_one order_less_imp_le])
+  "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
+  apply (induct n)
+  apply simp_all
+  apply (rule order_trans [OF _ mult_mono [of 1 _ 1]])
+  apply (simp_all add: order_trans [OF zero_le_one])
+  done
 
 lemma power_gt1_lemma:
-  assumes gt1: "1 < (a::'a::{ordered_semidom,recpower})"
-  shows "1 < a * a^n"
+  assumes gt1: "1 < a"
+  shows "1 < a * a ^ n"
 proof -
-  have "1*1 < a*1" using gt1 by simp
-  also have "\<dots> \<le> a * a^n" using gt1
-    by (simp only: mult_mono gt1_imp_ge0 one_le_power order_less_imp_le
+  from gt1 have "0 \<le> a"
+    by (fact order_trans [OF zero_le_one less_imp_le])
+  have "1 * 1 < a * 1" using gt1 by simp
+  also have "\<dots> \<le> a * a ^ n" using gt1
+    by (simp only: mult_mono `0 \<le> a` one_le_power order_less_imp_le
         zero_le_one order_refl)
   finally show ?thesis by simp
 qed
 
-lemma one_less_power[simp]:
-  "\<lbrakk>1 < (a::'a::{ordered_semidom,recpower}); 0 < n\<rbrakk> \<Longrightarrow> 1 < a ^ n"
-by (cases n, simp_all add: power_gt1_lemma)
+lemma power_gt1:
+  "1 < a \<Longrightarrow> 1 < a ^ Suc n"
+  by (simp add: power_gt1_lemma)
 
-lemma power_gt1:
-     "1 < (a::'a::{ordered_semidom,recpower}) ==> 1 < a ^ (Suc n)"
-by (simp add: power_gt1_lemma)
+lemma one_less_power [simp]:
+  "1 < a \<Longrightarrow> 0 < n \<Longrightarrow> 1 < a ^ n"
+  by (cases n) (simp_all add: power_gt1_lemma)
 
 lemma power_le_imp_le_exp:
-  assumes gt1: "(1::'a::{recpower,ordered_semidom}) < a"
-  shows "!!n. a^m \<le> a^n ==> m \<le> n"
-proof (induct m)
+  assumes gt1: "1 < a"
+  shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
+proof (induct m arbitrary: n)
   case 0
   show ?case by simp
 next
@@ -97,212 +181,128 @@
   show ?case
   proof (cases n)
     case 0
-    from prems have "a * a^m \<le> 1" by simp
+    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
     with gt1 show ?thesis
       by (force simp only: power_gt1_lemma
-          linorder_not_less [symmetric])
+          not_less [symmetric])
   next
     case (Suc n)
-    from prems show ?thesis
+    with Suc.prems Suc.hyps show ?thesis
       by (force dest: mult_left_le_imp_le
-          simp add: order_less_trans [OF zero_less_one gt1])
+          simp add: less_trans [OF zero_less_one gt1])
   qed
 qed
 
 text{*Surely we can strengthen this? It holds for @{text "0<a<1"} too.*}
 lemma power_inject_exp [simp]:
-     "1 < (a::'a::{ordered_semidom,recpower}) ==> (a^m = a^n) = (m=n)"
+  "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
   by (force simp add: order_antisym power_le_imp_le_exp)
 
 text{*Can relax the first premise to @{term "0<a"} in the case of the
 natural numbers.*}
 lemma power_less_imp_less_exp:
-     "[| (1::'a::{recpower,ordered_semidom}) < a; a^m < a^n |] ==> m < n"
-by (simp add: order_less_le [of m n] order_less_le [of "a^m" "a^n"]
-              power_le_imp_le_exp)
-
+  "1 < a \<Longrightarrow> a ^ m < a ^ n \<Longrightarrow> m < n"
+  by (simp add: order_less_le [of m n] less_le [of "a^m" "a^n"]
+    power_le_imp_le_exp)
 
 lemma power_mono:
-     "[|a \<le> b; (0::'a::{recpower,ordered_semidom}) \<le> a|] ==> a^n \<le> b^n"
-apply (induct "n")
-apply simp_all
-apply (auto intro: mult_mono order_trans [of 0 a b])
-done
+  "a \<le> b \<Longrightarrow> 0 \<le> a \<Longrightarrow> a ^ n \<le> b ^ n"
+  by (induct n)
+    (auto intro: mult_mono order_trans [of 0 a b])
 
 lemma power_strict_mono [rule_format]:
-     "[|a < b; (0::'a::{recpower,ordered_semidom}) \<le> a|]
-      ==> 0 < n --> a^n < b^n"
-apply (induct "n")
-apply (auto simp add: mult_strict_mono order_le_less_trans [of 0 a b])
-done
-
-lemma power_eq_0_iff [simp]:
-  "(a^n = 0) \<longleftrightarrow>
-   (a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,recpower}) & n\<noteq>0)"
-apply (induct "n")
-apply (auto simp add: no_zero_divisors)
-done
-
-
-lemma field_power_not_zero:
-  "a \<noteq> (0::'a::{ring_1_no_zero_divisors,recpower}) ==> a^n \<noteq> 0"
-by force
-
-lemma nonzero_power_inverse:
-  fixes a :: "'a::{division_ring,recpower}"
-  shows "a \<noteq> 0 ==> inverse (a ^ n) = (inverse a) ^ n"
-apply (induct "n")
-apply (auto simp add: nonzero_inverse_mult_distrib power_commutes)
-done (* TODO: reorient or rename to nonzero_inverse_power *)
-
-text{*Perhaps these should be simprules.*}
-lemma power_inverse:
-  fixes a :: "'a::{division_ring,division_by_zero,recpower}"
-  shows "inverse (a ^ n) = (inverse a) ^ n"
-apply (cases "a = 0")
-apply (simp add: power_0_left)
-apply (simp add: nonzero_power_inverse)
-done (* TODO: reorient or rename to inverse_power *)
-
-lemma power_one_over: "1 / (a::'a::{field,division_by_zero,recpower})^n = 
-    (1 / a)^n"
-apply (simp add: divide_inverse)
-apply (rule power_inverse)
-done
-
-lemma nonzero_power_divide:
-    "b \<noteq> 0 ==> (a/b) ^ n = ((a::'a::{field,recpower}) ^ n) / (b ^ n)"
-by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
-
-lemma power_divide:
-    "(a/b) ^ n = ((a::'a::{field,division_by_zero,recpower}) ^ n / b ^ n)"
-apply (case_tac "b=0", simp add: power_0_left)
-apply (rule nonzero_power_divide)
-apply assumption
-done
-
-lemma power_abs: "abs(a ^ n) = abs(a::'a::{ordered_idom,recpower}) ^ n"
-apply (induct "n")
-apply (auto simp add: abs_mult)
-done
-
-lemma abs_power_minus [simp]:
-  fixes a:: "'a::{ordered_idom,recpower}" shows "abs((-a) ^ n) = abs(a ^ n)"
-  by (simp add: abs_minus_cancel power_abs) 
-
-lemma zero_less_power_abs_iff [simp,noatp]:
-     "(0 < (abs a)^n) = (a \<noteq> (0::'a::{ordered_idom,recpower}) | n=0)"
-proof (induct "n")
-  case 0
-    show ?case by simp
-next
-  case (Suc n)
-    show ?case by (auto simp add: prems zero_less_mult_iff)
-qed
-
-lemma zero_le_power_abs [simp]:
-     "(0::'a::{ordered_idom,recpower}) \<le> (abs a)^n"
-by (rule zero_le_power [OF abs_ge_zero])
-
-lemma power_minus: "(-a) ^ n = (- 1)^n * (a::'a::{ring_1,recpower}) ^ n"
-proof (induct n)
-  case 0 show ?case by simp
-next
-  case (Suc n) then show ?case
-    by (simp del: power_Suc add: power_Suc2 mult_assoc)
-qed
+  "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
+  by (induct n)
+   (auto simp add: mult_strict_mono le_less_trans [of 0 a b])
 
 text{*Lemma for @{text power_strict_decreasing}*}
 lemma power_Suc_less:
-     "[|(0::'a::{ordered_semidom,recpower}) < a; a < 1|]
-      ==> a * a^n < a^n"
-apply (induct n)
-apply (auto simp add: mult_strict_left_mono)
-done
+  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
+  by (induct n)
+    (auto simp add: mult_strict_left_mono)
 
-lemma power_strict_decreasing:
-     "[|n < N; 0 < a; a < (1::'a::{ordered_semidom,recpower})|]
-      ==> a^N < a^n"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: power_Suc_less less_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "a * a^m < 1 * a^n", simp)
-apply (rule mult_strict_mono)
-apply (auto simp add: order_less_imp_le)
-done
+lemma power_strict_decreasing [rule_format]:
+  "n < N \<Longrightarrow> 0 < a \<Longrightarrow> a < 1 \<longrightarrow> a ^ N < a ^ n"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: power_Suc_less less_Suc_eq)
+  apply (subgoal_tac "a * a^N < 1 * a^n")
+  apply simp
+  apply (rule mult_strict_mono) apply auto
+  done
+qed
 
 text{*Proof resembles that of @{text power_strict_decreasing}*}
-lemma power_decreasing:
-     "[|n \<le> N; 0 \<le> a; a \<le> (1::'a::{ordered_semidom,recpower})|]
-      ==> a^N \<le> a^n"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: le_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "a * a^m \<le> 1 * a^n", simp)
-apply (rule mult_mono)
-apply auto
-done
+lemma power_decreasing [rule_format]:
+  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: le_Suc_eq)
+  apply (subgoal_tac "a * a^N \<le> 1 * a^n", simp)
+  apply (rule mult_mono) apply auto
+  done
+qed
 
 lemma power_Suc_less_one:
-     "[| 0 < a; a < (1::'a::{ordered_semidom,recpower}) |] ==> a ^ Suc n < 1"
-apply (insert power_strict_decreasing [of 0 "Suc n" a], simp)
-done
+  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
+  using power_strict_decreasing [of 0 "Suc n" a] by simp
 
 text{*Proof again resembles that of @{text power_strict_decreasing}*}
-lemma power_increasing:
-     "[|n \<le> N; (1::'a::{ordered_semidom,recpower}) \<le> a|] ==> a^n \<le> a^N"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: le_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "1 * a^n \<le> a * a^m", simp)
-apply (rule mult_mono)
-apply (auto simp add: order_trans [OF zero_le_one])
-done
+lemma power_increasing [rule_format]:
+  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: le_Suc_eq)
+  apply (subgoal_tac "1 * a^n \<le> a * a^N", simp)
+  apply (rule mult_mono) apply (auto simp add: order_trans [OF zero_le_one])
+  done
+qed
 
 text{*Lemma for @{text power_strict_increasing}*}
 lemma power_less_power_Suc:
-     "(1::'a::{ordered_semidom,recpower}) < a ==> a^n < a * a^n"
-apply (induct n)
-apply (auto simp add: mult_strict_left_mono order_less_trans [OF zero_less_one])
-done
+  "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
+  by (induct n) (auto simp add: mult_strict_left_mono less_trans [OF zero_less_one])
 
-lemma power_strict_increasing:
-     "[|n < N; (1::'a::{ordered_semidom,recpower}) < a|] ==> a^n < a^N"
-apply (erule rev_mp)
-apply (induct "N")
-apply (auto simp add: power_less_power_Suc less_Suc_eq)
-apply (rename_tac m)
-apply (subgoal_tac "1 * a^n < a * a^m", simp)
-apply (rule mult_strict_mono)
-apply (auto simp add: order_less_trans [OF zero_less_one] order_less_imp_le)
-done
+lemma power_strict_increasing [rule_format]:
+  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
+proof (induct N)
+  case 0 then show ?case by simp
+next
+  case (Suc N) then show ?case 
+  apply (auto simp add: power_less_power_Suc less_Suc_eq)
+  apply (subgoal_tac "1 * a^n < a * a^N", simp)
+  apply (rule mult_strict_mono) apply (auto simp add: less_trans [OF zero_less_one] less_imp_le)
+  done
+qed
 
 lemma power_increasing_iff [simp]:
-  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x \<le> b ^ y) = (x \<le> y)"
-by (blast intro: power_le_imp_le_exp power_increasing order_less_imp_le) 
+  "1 < b \<Longrightarrow> b ^ x \<le> b ^ y \<longleftrightarrow> x \<le> y"
+  by (blast intro: power_le_imp_le_exp power_increasing less_imp_le)
 
 lemma power_strict_increasing_iff [simp]:
-  "1 < (b::'a::{ordered_semidom,recpower}) ==> (b ^ x < b ^ y) = (x < y)"
+  "1 < b \<Longrightarrow> b ^ x < b ^ y \<longleftrightarrow> x < y"
 by (blast intro: power_less_imp_less_exp power_strict_increasing) 
 
 lemma power_le_imp_le_base:
-assumes le: "a ^ Suc n \<le> b ^ Suc n"
-    and ynonneg: "(0::'a::{ordered_semidom,recpower}) \<le> b"
-shows "a \<le> b"
+  assumes le: "a ^ Suc n \<le> b ^ Suc n"
+    and ynonneg: "0 \<le> b"
+  shows "a \<le> b"
 proof (rule ccontr)
   assume "~ a \<le> b"
   then have "b < a" by (simp only: linorder_not_le)
   then have "b ^ Suc n < a ^ Suc n"
     by (simp only: prems power_strict_mono)
-  from le and this show "False"
+  from le and this show False
     by (simp add: linorder_not_less [symmetric])
 qed
 
 lemma power_less_imp_less_base:
-  fixes a b :: "'a::{ordered_semidom,recpower}"
   assumes less: "a ^ n < b ^ n"
   assumes nonneg: "0 \<le> b"
   shows "a < b"
@@ -310,83 +310,144 @@
   assume "~ a < b"
   hence "b \<le> a" by (simp only: linorder_not_less)
   hence "b ^ n \<le> a ^ n" using nonneg by (rule power_mono)
-  thus "~ a ^ n < b ^ n" by (simp only: linorder_not_less)
+  thus "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
 qed
 
 lemma power_inject_base:
-     "[| a ^ Suc n = b ^ Suc n; 0 \<le> a; 0 \<le> b |]
-      ==> a = (b::'a::{ordered_semidom,recpower})"
-by (blast intro: power_le_imp_le_base order_antisym order_eq_refl sym)
+  "a ^ Suc n = b ^ Suc n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a = b"
+by (blast intro: power_le_imp_le_base antisym eq_refl sym)
 
 lemma power_eq_imp_eq_base:
-  fixes a b :: "'a::{ordered_semidom,recpower}"
-  shows "\<lbrakk>a ^ n = b ^ n; 0 \<le> a; 0 \<le> b; 0 < n\<rbrakk> \<Longrightarrow> a = b"
-by (cases n, simp_all del: power_Suc, rule power_inject_base)
+  "a ^ n = b ^ n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 < n \<Longrightarrow> a = b"
+  by (cases n) (simp_all del: power_Suc, rule power_inject_base)
 
-text {* The divides relation *}
+end
+
+context ordered_idom
+begin
 
-lemma le_imp_power_dvd:
-  fixes a :: "'a::{comm_semiring_1,recpower}"
-  assumes "m \<le> n" shows "a^m dvd a^n"
-proof
-  have "a^n = a^(m + (n - m))"
-    using `m \<le> n` by simp
-  also have "\<dots> = a^m * a^(n - m)"
-    by (rule power_add)
-  finally show "a^n = a^m * a^(n - m)" .
+lemma power_abs:
+  "abs (a ^ n) = abs a ^ n"
+  by (induct n) (auto simp add: abs_mult)
+
+lemma abs_power_minus [simp]:
+  "abs ((-a) ^ n) = abs (a ^ n)"
+  by (simp add: abs_minus_cancel power_abs) 
+
+lemma zero_less_power_abs_iff [simp, noatp]:
+  "0 < abs a ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
+proof (induct n)
+  case 0 show ?case by simp
+next
+  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
 qed
 
-lemma power_le_dvd:
-  fixes a b :: "'a::{comm_semiring_1,recpower}"
-  shows "a^n dvd b \<Longrightarrow> m \<le> n \<Longrightarrow> a^m dvd b"
-  by (rule dvd_trans [OF le_imp_power_dvd])
+lemma zero_le_power_abs [simp]:
+  "0 \<le> abs a ^ n"
+  by (rule zero_le_power [OF abs_ge_zero])
+
+end
+
+context ring_1_no_zero_divisors
+begin
+
+lemma field_power_not_zero:
+  "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
+  by (induct n) auto
+
+end
+
+context division_ring
+begin
 
+text {* FIXME reorient or rename to @{text nonzero_inverse_power} *}
+lemma nonzero_power_inverse:
+  "a \<noteq> 0 \<Longrightarrow> inverse (a ^ n) = (inverse a) ^ n"
+  by (induct n)
+    (simp_all add: nonzero_inverse_mult_distrib power_commutes field_power_not_zero)
 
-lemma dvd_power_same:
-  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> x^n dvd y^n"
-by (induct n) (auto simp add: mult_dvd_mono)
+end
+
+context field
+begin
+
+lemma nonzero_power_divide:
+  "b \<noteq> 0 \<Longrightarrow> (a / b) ^ n = a ^ n / b ^ n"
+  by (simp add: divide_inverse power_mult_distrib nonzero_power_inverse)
+
+end
+
+lemma power_0_Suc [simp]:
+  "(0::'a::{power, semiring_0}) ^ Suc n = 0"
+  by simp
 
-lemma dvd_power_le:
-  "(x::'a::{comm_semiring_1,recpower}) dvd y \<Longrightarrow> m >= n \<Longrightarrow> x^n dvd y^m"
-by(rule power_le_dvd[OF dvd_power_same])
+text{*It looks plausible as a simprule, but its effect can be strange.*}
+lemma power_0_left:
+  "0 ^ n = (if n = 0 then 1 else (0::'a::{power, semiring_0}))"
+  by (induct n) simp_all
+
+lemma power_eq_0_iff [simp]:
+  "a ^ n = 0 \<longleftrightarrow>
+     a = (0::'a::{mult_zero,zero_neq_one,no_zero_divisors,power}) \<and> n \<noteq> 0"
+  by (induct n)
+    (auto simp add: no_zero_divisors elim: contrapos_pp)
+
+lemma power_diff:
+  fixes a :: "'a::field"
+  assumes nz: "a \<noteq> 0"
+  shows "n \<le> m \<Longrightarrow> a ^ (m - n) = a ^ m / a ^ n"
+  by (induct m n rule: diff_induct) (simp_all add: nz)
 
-lemma dvd_power [simp]:
-  "n > 0 | (x::'a::{comm_semiring_1,recpower}) = 1 \<Longrightarrow> x dvd x^n"
-apply (erule disjE)
- apply (subgoal_tac "x ^ n = x^(Suc (n - 1))")
-  apply (erule ssubst)
-  apply (subst power_Suc)
-  apply auto
+text{*Perhaps these should be simprules.*}
+lemma power_inverse:
+  fixes a :: "'a::{division_ring,division_by_zero,power}"
+  shows "inverse (a ^ n) = (inverse a) ^ n"
+apply (cases "a = 0")
+apply (simp add: power_0_left)
+apply (simp add: nonzero_power_inverse)
+done (* TODO: reorient or rename to inverse_power *)
+
+lemma power_one_over:
+  "1 / (a::'a::{field,division_by_zero, power}) ^ n =  (1 / a) ^ n"
+  by (simp add: divide_inverse) (rule power_inverse)
+
+lemma power_divide:
+  "(a / b) ^ n = (a::'a::{field,division_by_zero}) ^ n / b ^ n"
+apply (cases "b = 0")
+apply (simp add: power_0_left)
+apply (rule nonzero_power_divide)
+apply assumption
 done
 
+class recpower = monoid_mult
+
 
 subsection {* Exponentiation for the Natural Numbers *}
 
 instance nat :: recpower ..
 
-lemma of_nat_power:
-  "of_nat (m ^ n) = (of_nat m::'a::{semiring_1,recpower}) ^ n"
-by (induct n, simp_all add: of_nat_mult)
+lemma nat_one_le_power [simp]:
+  "Suc 0 \<le> i \<Longrightarrow> Suc 0 \<le> i ^ n"
+  by (rule one_le_power [of i n, unfolded One_nat_def])
 
-lemma nat_one_le_power [simp]: "Suc 0 \<le> i ==> Suc 0 \<le> i^n"
-by (rule one_le_power [of i n, unfolded One_nat_def])
-
-lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
-by (induct "n", auto)
+lemma nat_zero_less_power_iff [simp]:
+  "x ^ n > 0 \<longleftrightarrow> x > (0::nat) \<or> n = 0"
+  by (induct n) auto
 
 lemma nat_power_eq_Suc_0_iff [simp]: 
-  "((x::nat)^m = Suc 0) = (m = 0 | x = Suc 0)"
-by (induct m, auto)
+  "x ^ m = Suc 0 \<longleftrightarrow> m = 0 \<or> x = Suc 0"
+  by (induct m) auto
 
-lemma power_Suc_0[simp]: "(Suc 0)^n = Suc 0"
-by simp
+lemma power_Suc_0 [simp]:
+  "Suc 0 ^ n = Suc 0"
+  by simp
 
 text{*Valid for the naturals, but what if @{text"0<i<1"}?
 Premises cannot be weakened: consider the case where @{term "i=0"},
 @{term "m=1"} and @{term "n=0"}.*}
 lemma nat_power_less_imp_less:
   assumes nonneg: "0 < (i\<Colon>nat)"
-  assumes less: "i^m < i^n"
+  assumes less: "i ^ m < i ^ n"
   shows "m < n"
 proof (cases "i = 1")
   case True with less power_one [where 'a = nat] show ?thesis by simp
@@ -395,10 +456,4 @@
   from power_strict_increasing_iff [OF this] less show ?thesis ..
 qed
 
-lemma power_diff:
-  assumes nz: "a ~= 0"
-  shows "n <= m ==> (a::'a::{recpower, field}) ^ (m-n) = (a^m) / (a^n)"
-  by (induct m n rule: diff_induct)
-    (simp_all add: nonzero_mult_divide_cancel_left nz)
-
 end