diff --git a/src/HOL/Groups_List.thy b/src/HOL/Groups_List.thy
--- a/src/HOL/Groups_List.thy
+++ b/src/HOL/Groups_List.thy
@@ -1,432 +1,444 @@
 (* Author: Tobias Nipkow, TU Muenchen *)
 
 section \<open>Sum and product over lists\<close>
 
 theory Groups_List
 imports List
 begin
 
 locale monoid_list = monoid
 begin
 
 definition F :: "'a list \<Rightarrow> 'a"
 where
   eq_foldr [code]: "F xs = foldr f xs \<^bold>1"
 
 lemma Nil [simp]:
   "F [] = \<^bold>1"
   by (simp add: eq_foldr)
 
 lemma Cons [simp]:
   "F (x # xs) = x \<^bold>* F xs"
   by (simp add: eq_foldr)
 
 lemma append [simp]:
   "F (xs @ ys) = F xs \<^bold>* F ys"
   by (induct xs) (simp_all add: assoc)
 
 end
 
 locale comm_monoid_list = comm_monoid + monoid_list
 begin
 
 lemma rev [simp]:
   "F (rev xs) = F xs"
   by (simp add: eq_foldr foldr_fold  fold_rev fun_eq_iff assoc left_commute)
 
 end
 
 locale comm_monoid_list_set = list: comm_monoid_list + set: comm_monoid_set
 begin
 
 lemma distinct_set_conv_list:
   "distinct xs \<Longrightarrow> set.F g (set xs) = list.F (map g xs)"
   by (induct xs) simp_all
 
 lemma set_conv_list [code]:
   "set.F g (set xs) = list.F (map g (remdups xs))"
   by (simp add: distinct_set_conv_list [symmetric])
 
 end
 
 
 subsection \<open>List summation\<close>
 
 context monoid_add
 begin
 
 sublocale sum_list: monoid_list plus 0
 defines
   sum_list = sum_list.F ..
 
 end
 
 context comm_monoid_add
 begin
 
 sublocale sum_list: comm_monoid_list plus 0
 rewrites
   "monoid_list.F plus 0 = sum_list"
 proof -
   show "comm_monoid_list plus 0" ..
   then interpret sum_list: comm_monoid_list plus 0 .
   from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
 qed
 
 sublocale sum: comm_monoid_list_set plus 0
 rewrites
   "monoid_list.F plus 0 = sum_list"
   and "comm_monoid_set.F plus 0 = sum"
 proof -
   show "comm_monoid_list_set plus 0" ..
   then interpret sum: comm_monoid_list_set plus 0 .
   from sum_list_def show "monoid_list.F plus 0 = sum_list" by simp
   from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)
 qed
 
 end
 
 text \<open>Some syntactic sugar for summing a function over a list:\<close>
 syntax (ASCII)
   "_sum_list" :: "pttrn => 'a list => 'b => 'b"    ("(3SUM _<-_. _)" [0, 51, 10] 10)
 syntax
   "_sum_list" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Sum>_\<leftarrow>_. _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Sum>x\<leftarrow>xs. b" == "CONST sum_list (CONST map (\<lambda>x. b) xs)"
 
+context
+  includes lifting_syntax
+begin
+
+lemma sum_list_transfer [transfer_rule]:
+  "(list_all2 A ===> A) sum_list sum_list"
+    if [transfer_rule]: "A 0 0" "(A ===> A ===> A) (+) (+)"
+  unfolding sum_list.eq_foldr [abs_def]
+  by transfer_prover
+
+end
+
 text \<open>TODO duplicates\<close>
 lemmas sum_list_simps = sum_list.Nil sum_list.Cons
 lemmas sum_list_append = sum_list.append
 lemmas sum_list_rev = sum_list.rev
 
 lemma (in monoid_add) fold_plus_sum_list_rev:
   "fold plus xs = plus (sum_list (rev xs))"
 proof
   fix x
   have "fold plus xs x = sum_list (rev xs @ [x])"
     by (simp add: foldr_conv_fold sum_list.eq_foldr)
   also have "\<dots> = sum_list (rev xs) + x"
     by simp
   finally show "fold plus xs x = sum_list (rev xs) + x"
     .
 qed
 
 lemma (in comm_monoid_add) sum_list_map_remove1:
   "x \<in> set xs \<Longrightarrow> sum_list (map f xs) = f x + sum_list (map f (remove1 x xs))"
   by (induct xs) (auto simp add: ac_simps)
 
 lemma (in monoid_add) size_list_conv_sum_list:
   "size_list f xs = sum_list (map f xs) + size xs"
   by (induct xs) auto
 
 lemma (in monoid_add) length_concat:
   "length (concat xss) = sum_list (map length xss)"
   by (induct xss) simp_all
 
 lemma (in monoid_add) length_product_lists:
   "length (product_lists xss) = foldr (*) (map length xss) 1"
 proof (induct xss)
   case (Cons xs xss) then show ?case by (induct xs) (auto simp: length_concat o_def)
 qed simp
 
 lemma (in monoid_add) sum_list_map_filter:
   assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<not> P x \<Longrightarrow> f x = 0"
   shows "sum_list (map f (filter P xs)) = sum_list (map f xs)"
   using assms by (induct xs) auto
 
 lemma sum_list_filter_le_nat:
   fixes f :: "'a \<Rightarrow> nat"
   shows "sum_list (map f (filter P xs)) \<le> sum_list (map f xs)"
 by(induction xs; simp)
 
 lemma (in comm_monoid_add) distinct_sum_list_conv_Sum:
   "distinct xs \<Longrightarrow> sum_list xs = Sum (set xs)"
   by (induct xs) simp_all
 
 lemma sum_list_upt[simp]:
   "m \<le> n \<Longrightarrow> sum_list [m..<n] = \<Sum> {m..<n}"
 by(simp add: distinct_sum_list_conv_Sum)
 
 context ordered_comm_monoid_add
 begin
 
 lemma sum_list_nonneg: "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> 0 \<le> sum_list xs"
 by (induction xs) auto
 
 lemma sum_list_nonpos: "(\<And>x. x \<in> set xs \<Longrightarrow> x \<le> 0) \<Longrightarrow> sum_list xs \<le> 0"
 by (induction xs) (auto simp: add_nonpos_nonpos)
 
 lemma sum_list_nonneg_eq_0_iff:
   "(\<And>x. x \<in> set xs \<Longrightarrow> 0 \<le> x) \<Longrightarrow> sum_list xs = 0 \<longleftrightarrow> (\<forall>x\<in> set xs. x = 0)"
 by (induction xs) (simp_all add: add_nonneg_eq_0_iff sum_list_nonneg)
 
 end
 
 context canonically_ordered_monoid_add
 begin
 
 lemma sum_list_eq_0_iff [simp]:
   "sum_list ns = 0 \<longleftrightarrow> (\<forall>n \<in> set ns. n = 0)"
 by (simp add: sum_list_nonneg_eq_0_iff)
 
 lemma member_le_sum_list:
   "x \<in> set xs \<Longrightarrow> x \<le> sum_list xs"
 by (induction xs) (auto simp: add_increasing add_increasing2)
 
 lemma elem_le_sum_list:
   "k < size ns \<Longrightarrow> ns ! k \<le> sum_list (ns)"
 by (rule member_le_sum_list) simp
 
 end
 
 lemma (in ordered_cancel_comm_monoid_diff) sum_list_update:
   "k < size xs \<Longrightarrow> sum_list (xs[k := x]) = sum_list xs + x - xs ! k"
 apply(induction xs arbitrary:k)
  apply (auto simp: add_ac split: nat.split)
 apply(drule elem_le_sum_list)
 by (simp add: local.add_diff_assoc local.add_increasing)
 
 lemma (in monoid_add) sum_list_triv:
   "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
   by (induct xs) (simp_all add: distrib_right)
 
 lemma (in monoid_add) sum_list_0 [simp]:
   "(\<Sum>x\<leftarrow>xs. 0) = 0"
   by (induct xs) (simp_all add: distrib_right)
 
 text\<open>For non-Abelian groups \<open>xs\<close> needs to be reversed on one side:\<close>
 lemma (in ab_group_add) uminus_sum_list_map:
   "- sum_list (map f xs) = sum_list (map (uminus \<circ> f) xs)"
   by (induct xs) simp_all
 
 lemma (in comm_monoid_add) sum_list_addf:
   "(\<Sum>x\<leftarrow>xs. f x + g x) = sum_list (map f xs) + sum_list (map g xs)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in ab_group_add) sum_list_subtractf:
   "(\<Sum>x\<leftarrow>xs. f x - g x) = sum_list (map f xs) - sum_list (map g xs)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in semiring_0) sum_list_const_mult:
   "(\<Sum>x\<leftarrow>xs. c * f x) = c * (\<Sum>x\<leftarrow>xs. f x)"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in semiring_0) sum_list_mult_const:
   "(\<Sum>x\<leftarrow>xs. f x * c) = (\<Sum>x\<leftarrow>xs. f x) * c"
   by (induct xs) (simp_all add: algebra_simps)
 
 lemma (in ordered_ab_group_add_abs) sum_list_abs:
   "\<bar>sum_list xs\<bar> \<le> sum_list (map abs xs)"
   by (induct xs) (simp_all add: order_trans [OF abs_triangle_ineq])
 
 lemma sum_list_mono:
   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, ordered_ab_semigroup_add}"
   shows "(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x) \<Longrightarrow> (\<Sum>x\<leftarrow>xs. f x) \<le> (\<Sum>x\<leftarrow>xs. g x)"
 by (induct xs) (simp, simp add: add_mono)
 
 lemma sum_list_strict_mono:
   fixes f g :: "'a \<Rightarrow> 'b::{monoid_add, strict_ordered_ab_semigroup_add}"
   shows "\<lbrakk> xs \<noteq> [];  \<And>x. x \<in> set xs \<Longrightarrow> f x < g x \<rbrakk>
     \<Longrightarrow> sum_list (map f xs) < sum_list (map g xs)"
 proof (induction xs)
   case Nil thus ?case by simp
 next
   case C: (Cons _ xs)
   show ?case
   proof (cases xs)
     case Nil thus ?thesis using C.prems by simp
   next
     case Cons thus ?thesis using C by(simp add: add_strict_mono)
   qed
 qed
 
 lemma (in monoid_add) sum_list_distinct_conv_sum_set:
   "distinct xs \<Longrightarrow> sum_list (map f xs) = sum f (set xs)"
   by (induct xs) simp_all
 
 lemma (in monoid_add) interv_sum_list_conv_sum_set_nat:
   "sum_list (map f [m..<n]) = sum f (set [m..<n])"
   by (simp add: sum_list_distinct_conv_sum_set)
 
 lemma (in monoid_add) interv_sum_list_conv_sum_set_int:
   "sum_list (map f [k..l]) = sum f (set [k..l])"
   by (simp add: sum_list_distinct_conv_sum_set)
 
 text \<open>General equivalence between \<^const>\<open>sum_list\<close> and \<^const>\<open>sum\<close>\<close>
 lemma (in monoid_add) sum_list_sum_nth:
   "sum_list xs = (\<Sum> i = 0 ..< length xs. xs ! i)"
   using interv_sum_list_conv_sum_set_nat [of "(!) xs" 0 "length xs"] by (simp add: map_nth)
 
 lemma sum_list_map_eq_sum_count:
   "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) (set xs)"
 proof(induction xs)
   case (Cons x xs)
   show ?case (is "?l = ?r")
   proof cases
     assume "x \<in> set xs"
     have "?l = f x + (\<Sum>x\<in>set xs. count_list xs x * f x)" by (simp add: Cons.IH)
     also have "set xs = insert x (set xs - {x})" using \<open>x \<in> set xs\<close>by blast
     also have "f x + (\<Sum>x\<in>insert x (set xs - {x}). count_list xs x * f x) = ?r"
       by (simp add: sum.insert_remove eq_commute)
     finally show ?thesis .
   next
     assume "x \<notin> set xs"
     hence "\<And>xa. xa \<in> set xs \<Longrightarrow> x \<noteq> xa" by blast
     thus ?thesis by (simp add: Cons.IH \<open>x \<notin> set xs\<close>)
   qed
 qed simp
 
 lemma sum_list_map_eq_sum_count2:
 assumes "set xs \<subseteq> X" "finite X"
 shows "sum_list (map f xs) = sum (\<lambda>x. count_list xs x * f x) X"
 proof-
   let ?F = "\<lambda>x. count_list xs x * f x"
   have "sum ?F X = sum ?F (set xs \<union> (X - set xs))"
     using Un_absorb1[OF assms(1)] by(simp)
   also have "\<dots> = sum ?F (set xs)"
     using assms(2)
     by(simp add: sum.union_disjoint[OF _ _ Diff_disjoint] del: Un_Diff_cancel)
   finally show ?thesis by(simp add:sum_list_map_eq_sum_count)
 qed
 
 lemma sum_list_nonneg:
     "(\<And>x. x \<in> set xs \<Longrightarrow> (x :: 'a :: ordered_comm_monoid_add) \<ge> 0) \<Longrightarrow> sum_list xs \<ge> 0"
   by (induction xs) simp_all
 
 lemma sum_list_Suc:
   "sum_list (map (\<lambda>x. Suc(f x)) xs) = sum_list (map f xs) + length xs"
 by(induction xs; simp)
 
 lemma (in monoid_add) sum_list_map_filter':
   "sum_list (map f (filter P xs)) = sum_list (map (\<lambda>x. if P x then f x else 0) xs)"
   by (induction xs) simp_all
 
 text \<open>Summation of a strictly ascending sequence with length \<open>n\<close>
   can be upper-bounded by summation over \<open>{0..<n}\<close>.\<close>
 
 lemma sorted_wrt_less_sum_mono_lowerbound:
   fixes f :: "nat \<Rightarrow> ('b::ordered_comm_monoid_add)"
   assumes mono: "\<And>x y. x\<le>y \<Longrightarrow> f x \<le> f y"
   shows "sorted_wrt (<) ns \<Longrightarrow>
     (\<Sum>i\<in>{0..<length ns}. f i) \<le> (\<Sum>i\<leftarrow>ns. f i)"
 proof (induction ns rule: rev_induct)
   case Nil
   then show ?case by simp
 next
   case (snoc n ns)
   have "sum f {0..<length (ns @ [n])}
       = sum f {0..<length ns} + f (length ns)"
     by simp
   also have "sum f {0..<length ns} \<le> sum_list (map f ns)"
     using snoc by (auto simp: sorted_wrt_append)
   also have "length ns \<le> n"
     using sorted_wrt_less_idx[OF snoc.prems(1), of "length ns"] by auto
   finally have "sum f {0..<length (ns @ [n])} \<le> sum_list (map f ns) + f n"
     using mono add_mono by blast
   thus ?case by simp
 qed
 
 
 subsection \<open>Further facts about \<^const>\<open>List.n_lists\<close>\<close>
 
 lemma length_n_lists: "length (List.n_lists n xs) = length xs ^ n"
   by (induct n) (auto simp add: comp_def length_concat sum_list_triv)
 
 lemma distinct_n_lists:
   assumes "distinct xs"
   shows "distinct (List.n_lists n xs)"
 proof (rule card_distinct)
   from assms have card_length: "card (set xs) = length xs" by (rule distinct_card)
   have "card (set (List.n_lists n xs)) = card (set xs) ^ n"
   proof (induct n)
     case 0 then show ?case by simp
   next
     case (Suc n)
     moreover have "card (\<Union>ys\<in>set (List.n_lists n xs). (\<lambda>y. y # ys) ` set xs)
       = (\<Sum>ys\<in>set (List.n_lists n xs). card ((\<lambda>y. y # ys) ` set xs))"
       by (rule card_UN_disjoint) auto
     moreover have "\<And>ys. card ((\<lambda>y. y # ys) ` set xs) = card (set xs)"
       by (rule card_image) (simp add: inj_on_def)
     ultimately show ?case by auto
   qed
   also have "\<dots> = length xs ^ n" by (simp add: card_length)
   finally show "card (set (List.n_lists n xs)) = length (List.n_lists n xs)"
     by (simp add: length_n_lists)
 qed
 
 
 subsection \<open>Tools setup\<close>
 
 lemmas sum_code = sum.set_conv_list
 
 lemma sum_set_upto_conv_sum_list_int [code_unfold]:
   "sum f (set [i..j::int]) = sum_list (map f [i..j])"
   by (simp add: interv_sum_list_conv_sum_set_int)
 
 lemma sum_set_upt_conv_sum_list_nat [code_unfold]:
   "sum f (set [m..<n]) = sum_list (map f [m..<n])"
   by (simp add: interv_sum_list_conv_sum_set_nat)
 
-context
-  includes lifting_syntax
-begin
-
-lemma sum_list_transfer [transfer_rule]:
-  "(list_all2 A ===> A) sum_list sum_list"
-    if [transfer_rule]: "A 0 0" "(A ===> A ===> A) (+) (+)"
-  unfolding sum_list.eq_foldr [abs_def]
-  by transfer_prover
-
-end
-
 
 subsection \<open>List product\<close>
 
 context monoid_mult
 begin
 
 sublocale prod_list: monoid_list times 1
 defines
   prod_list = prod_list.F ..
 
 end
 
 context comm_monoid_mult
 begin
 
 sublocale prod_list: comm_monoid_list times 1
 rewrites
   "monoid_list.F times 1 = prod_list"
 proof -
   show "comm_monoid_list times 1" ..
   then interpret prod_list: comm_monoid_list times 1 .
   from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
 qed
 
 sublocale prod: comm_monoid_list_set times 1
 rewrites
   "monoid_list.F times 1 = prod_list"
   and "comm_monoid_set.F times 1 = prod"
 proof -
   show "comm_monoid_list_set times 1" ..
   then interpret prod: comm_monoid_list_set times 1 .
   from prod_list_def show "monoid_list.F times 1 = prod_list" by simp
   from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)
 qed
 
 end
 
-lemma prod_list_zero_iff:
-  "prod_list xs = 0 \<longleftrightarrow> (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) \<in> set xs"
-  by (induction xs) simp_all
-
 text \<open>Some syntactic sugar:\<close>
 
 syntax (ASCII)
   "_prod_list" :: "pttrn => 'a list => 'b => 'b"    ("(3PROD _<-_. _)" [0, 51, 10] 10)
 syntax
   "_prod_list" :: "pttrn => 'a list => 'b => 'b"    ("(3\<Prod>_\<leftarrow>_. _)" [0, 51, 10] 10)
 translations \<comment> \<open>Beware of argument permutation!\<close>
   "\<Prod>x\<leftarrow>xs. b" \<rightleftharpoons> "CONST prod_list (CONST map (\<lambda>x. b) xs)"
 
+context
+  includes lifting_syntax
+begin
+
+lemma prod_list_transfer [transfer_rule]:
+  "(list_all2 A ===> A) prod_list prod_list"
+    if [transfer_rule]: "A 1 1" "(A ===> A ===> A) (*) (*)"
+  unfolding prod_list.eq_foldr [abs_def]
+  by transfer_prover
+
 end
+
+lemma prod_list_zero_iff:
+  "prod_list xs = 0 \<longleftrightarrow> (0 :: 'a :: {semiring_no_zero_divisors, semiring_1}) \<in> set xs"
+  by (induction xs) simp_all
+
+end
diff --git a/src/HOL/Power.thy b/src/HOL/Power.thy
--- a/src/HOL/Power.thy
+++ b/src/HOL/Power.thy
@@ -1,1002 +1,1015 @@
 (*  Title:      HOL/Power.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1997  University of Cambridge
 *)
 
 section \<open>Exponentiation\<close>
 
 theory Power
   imports Num
 begin
 
 subsection \<open>Powers for Arbitrary Monoids\<close>
 
 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)
 
 text \<open>Special syntax for squares.\<close>
 abbreviation power2 :: "'a \<Rightarrow> 'a"  ("(_\<^sup>2)" [1000] 999)
   where "x\<^sup>2 \<equiv> x ^ 2"
 
 end
 
+context
+  includes lifting_syntax
+begin
+
+lemma power_transfer [transfer_rule]:
+  \<open>(R ===> (=) ===> R) (^) (^)\<close>
+    if [transfer_rule]: \<open>R 1 1\<close>
+      \<open>(R ===> R ===> R) (*) (*)\<close>
+    for R :: \<open>'a::power \<Rightarrow> 'b::power \<Rightarrow> bool\<close>
+  by (simp only: power_def [abs_def]) transfer_prover
+
+end
+
 context monoid_mult
 begin
 
 subclass power .
 
 lemma power_one [simp]: "1 ^ n = 1"
   by (induct n) simp_all
 
 lemma power_one_right [simp]: "a ^ 1 = a"
   by simp
 
 lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
   by simp
 
 lemma power_commutes: "a ^ n * a = a * a ^ n"
   by (induct n) (simp_all add: mult.assoc)
 
 lemma power_Suc2: "a ^ Suc n = a ^ n * a"
   by (simp add: power_commutes)
 
 lemma power_add: "a ^ (m + n) = a ^ m * a ^ n"
   by (induct m) (simp_all add: algebra_simps)
 
 lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"
   by (induct n) (simp_all add: power_add)
 
 lemma power_even_eq: "a ^ (2 * n) = (a ^ n)\<^sup>2"
   by (subst mult.commute) (simp add: power_mult)
 
 lemma power_odd_eq: "a ^ Suc (2*n) = a * (a ^ n)\<^sup>2"
   by (simp add: power_even_eq)
 
 lemma power_numeral_even: "z ^ numeral (Num.Bit0 w) = (let w = z ^ (numeral w) in w * w)"
   by (simp only: numeral_Bit0 power_add Let_def)
 
 lemma power_numeral_odd: "z ^ numeral (Num.Bit1 w) = (let w = z ^ (numeral w) in z * w * w)"
   by (simp only: numeral_Bit1 One_nat_def add_Suc_right add_0_right
       power_Suc power_add Let_def mult.assoc)
 
 lemma power2_eq_square: "a\<^sup>2 = a * a"
   by (simp add: numeral_2_eq_2)
 
 lemma power3_eq_cube: "a ^ 3 = a * a * a"
   by (simp add: numeral_3_eq_3 mult.assoc)
 
 lemma power4_eq_xxxx: "x^4 = x * x * x * x"
   by (simp add: mult.assoc power_numeral_even)
 
 lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
 proof (induct "f x" arbitrary: f)
   case 0
   then show ?case by (simp add: fun_eq_iff)
 next
   case (Suc n)
   define g where "g x = f x - 1" for x
   with Suc have "n = g x" by simp
   with Suc have "times x ^^ g x = times (x ^ g x)" by simp
   moreover from Suc g_def have "f x = g x + 1" by simp
   ultimately show ?case
     by (simp add: power_add funpow_add fun_eq_iff mult.assoc)
 qed
 
 lemma power_commuting_commutes:
   assumes "x * y = y * x"
   shows "x ^ n * y = y * x ^n"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   have "x ^ Suc n * y = x ^ n * y * x"
     by (subst power_Suc2) (simp add: assms ac_simps)
   also have "\<dots> = y * x ^ Suc n"
     by (simp only: Suc power_Suc2) (simp add: ac_simps)
   finally show ?case .
 qed
 
 lemma power_minus_mult: "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
   by (simp add: power_commutes split: nat_diff_split)
 
 lemma left_right_inverse_power:
   assumes "x * y = 1"
   shows   "x ^ n * y ^ n = 1"
 proof (induct n)
   case (Suc n)
   moreover have "x ^ Suc n * y ^ Suc n = x^n * (x * y) * y^n"
     by (simp add: power_Suc2[symmetric] mult.assoc[symmetric])
   ultimately show ?case by (simp add: assms)
 qed simp
 
 end
 
 context comm_monoid_mult
 begin
 
 lemma power_mult_distrib [algebra_simps, algebra_split_simps, field_simps, field_split_simps, divide_simps]:
   "(a * b) ^ n = (a ^ n) * (b ^ n)"
   by (induction n) (simp_all add: ac_simps)
 
 end
 
 text \<open>Extract constant factors from powers.\<close>
 declare power_mult_distrib [where a = "numeral w" for w, simp]
 declare power_mult_distrib [where b = "numeral w" for w, simp]
 
 lemma power_add_numeral [simp]: "a^numeral m * a^numeral n = a^numeral (m + n)"
   for a :: "'a::monoid_mult"
   by (simp add: power_add [symmetric])
 
 lemma power_add_numeral2 [simp]: "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
   for a :: "'a::monoid_mult"
   by (simp add: mult.assoc [symmetric])
 
 lemma power_mult_numeral [simp]: "(a^numeral m)^numeral n = a^numeral (m * n)"
   for a :: "'a::monoid_mult"
   by (simp only: numeral_mult power_mult)
 
 context semiring_numeral
 begin
 
 lemma numeral_sqr: "numeral (Num.sqr k) = numeral k * numeral k"
   by (simp only: sqr_conv_mult numeral_mult)
 
 lemma numeral_pow: "numeral (Num.pow k l) = numeral k ^ numeral l"
   by (induct l)
     (simp_all only: numeral_class.numeral.simps pow.simps
       numeral_sqr numeral_mult power_add power_one_right)
 
 lemma power_numeral [simp]: "numeral k ^ numeral l = numeral (Num.pow k l)"
   by (rule numeral_pow [symmetric])
 
 end
 
 context semiring_1
 begin
 
 lemma of_nat_power [simp]: "of_nat (m ^ n) = of_nat m ^ n"
   by (induct n) simp_all
 
 lemma zero_power: "0 < n \<Longrightarrow> 0 ^ n = 0"
   by (cases n) simp_all
 
 lemma power_zero_numeral [simp]: "0 ^ numeral k = 0"
   by (simp add: numeral_eq_Suc)
 
 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
   by (rule power_zero_numeral)
 
 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
   by (rule power_one)
 
 lemma power_0_Suc [simp]: "0 ^ Suc n = 0"
   by simp
 
 text \<open>It looks plausible as a simprule, but its effect can be strange.\<close>
 lemma power_0_left: "0 ^ n = (if n = 0 then 1 else 0)"
   by (cases n) simp_all
 
 end
 
 context semiring_char_0 begin
 
 lemma numeral_power_eq_of_nat_cancel_iff [simp]:
   "numeral x ^ n = of_nat y \<longleftrightarrow> numeral x ^ n = y"
   using of_nat_eq_iff by fastforce
 
 lemma real_of_nat_eq_numeral_power_cancel_iff [simp]:
   "of_nat y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
   using numeral_power_eq_of_nat_cancel_iff [of x n y] by (metis (mono_tags))
 
 lemma of_nat_eq_of_nat_power_cancel_iff[simp]: "(of_nat b) ^ w = of_nat x \<longleftrightarrow> b ^ w = x"
   by (metis of_nat_power of_nat_eq_iff)
 
 lemma of_nat_power_eq_of_nat_cancel_iff[simp]: "of_nat x = (of_nat b) ^ w \<longleftrightarrow> x = b ^ w"
   by (metis of_nat_eq_of_nat_power_cancel_iff)
 
 end
 
 context comm_semiring_1
 begin
 
 text \<open>The divides relation.\<close>
 
 lemma le_imp_power_dvd:
   assumes "m \<le> n"
   shows "a ^ m dvd a ^ n"
 proof
   from assms have "a ^ n = a ^ (m + (n - m))" 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 dvd_power [simp]:
   fixes n :: nat
   assumes "n > 0 \<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 semiring_1_no_zero_divisors
 begin
 
 subclass power .
 
 lemma power_eq_0_iff [simp]: "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
   by (induct n) auto
 
 lemma power_not_zero: "a \<noteq> 0 \<Longrightarrow> a ^ n \<noteq> 0"
   by (induct n) auto
 
 lemma zero_eq_power2 [simp]: "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
   unfolding power2_eq_square by simp
 
 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
 
 lemma power_minus': "NO_MATCH 1 x \<Longrightarrow> (-x) ^ n = (-1)^n * x ^ n"
   by (rule power_minus)
 
 lemma power_minus_Bit0: "(- x) ^ numeral (Num.Bit0 k) = x ^ numeral (Num.Bit0 k)"
   by (induct k, simp_all only: numeral_class.numeral.simps power_add
     power_one_right mult_minus_left mult_minus_right minus_minus)
 
 lemma power_minus_Bit1: "(- x) ^ numeral (Num.Bit1 k) = - (x ^ numeral (Num.Bit1 k))"
   by (simp only: eval_nat_numeral(3) power_Suc power_minus_Bit0 mult_minus_left)
 
 lemma power2_minus [simp]: "(- a)\<^sup>2 = a\<^sup>2"
   by (fact power_minus_Bit0)
 
 lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
 proof (induct n)
   case 0
   show ?case by simp
 next
   case (Suc n)
   then show ?case by (simp add: power_add power2_eq_square)
 qed
 
 lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
   by simp
 
 lemma power_minus_even [simp]: "(-a) ^ (2*n) = a ^ (2*n)"
   by (simp add: power_minus [of a])
 
 end
 
 context ring_1_no_zero_divisors
 begin
 
 lemma power2_eq_1_iff: "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
   using square_eq_1_iff [of a] by (simp add: power2_eq_square)
 
 end
 
 context idom
 begin
 
 lemma power2_eq_iff: "x\<^sup>2 = y\<^sup>2 \<longleftrightarrow> x = y \<or> x = - y"
   unfolding power2_eq_square by (rule square_eq_iff)
 
 end
 
 context semidom_divide
 begin
 
 lemma power_diff:
   "a ^ (m - n) = (a ^ m) div (a ^ n)" if "a \<noteq> 0" and "n \<le> m"
 proof -
   define q where "q = m - n"
   with \<open>n \<le> m\<close> have "m = q + n" by simp
   with \<open>a \<noteq> 0\<close> q_def show ?thesis
     by (simp add: power_add)
 qed
 
 end
 
 context algebraic_semidom
 begin
 
 lemma div_power: "b dvd a \<Longrightarrow> (a div b) ^ n = a ^ n div b ^ n"
   by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
 
 lemma is_unit_power_iff: "is_unit (a ^ n) \<longleftrightarrow> is_unit a \<or> n = 0"
   by (induct n) (auto simp add: is_unit_mult_iff)
 
 lemma dvd_power_iff:
   assumes "x \<noteq> 0"
   shows   "x ^ m dvd x ^ n \<longleftrightarrow> is_unit x \<or> m \<le> n"
 proof
   assume *: "x ^ m dvd x ^ n"
   {
     assume "m > n"
     note *
     also have "x ^ n = x ^ n * 1" by simp
     also from \<open>m > n\<close> have "m = n + (m - n)" by simp
     also have "x ^ \<dots> = x ^ n * x ^ (m - n)" by (rule power_add)
     finally have "x ^ (m - n) dvd 1"
       by (subst (asm) dvd_times_left_cancel_iff) (insert assms, simp_all)
     with \<open>m > n\<close> have "is_unit x" by (simp add: is_unit_power_iff)
   }
   thus "is_unit x \<or> m \<le> n" by force
 qed (auto intro: unit_imp_dvd simp: is_unit_power_iff le_imp_power_dvd)
 
 
 end
 
 context normalization_semidom
 begin
 
 lemma normalize_power: "normalize (a ^ n) = normalize a ^ n"
   by (induct n) (simp_all add: normalize_mult)
 
 lemma unit_factor_power: "unit_factor (a ^ n) = unit_factor a ^ n"
   by (induct n) (simp_all add: unit_factor_mult)
 
 end
 
 context division_ring
 begin
 
 text \<open>Perhaps these should be simprules.\<close>
 lemma power_inverse [field_simps, field_split_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
 proof (cases "a = 0")
   case True
   then show ?thesis by (simp add: power_0_left)
 next
   case False
   then have "inverse (a ^ n) = inverse a ^ n"
     by (induct n) (simp_all add: nonzero_inverse_mult_distrib power_commutes)
   then show ?thesis by simp
 qed
 
 lemma power_one_over [field_simps, field_split_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
   using power_inverse [of a] by (simp add: divide_inverse)
 
 end
 
 context field
 begin
 
 lemma power_divide [field_simps, field_split_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
   by (induct n) simp_all
 
 end
 
 
 subsection \<open>Exponentiation on ordered types\<close>
 
 context linordered_semidom
 begin
 
 lemma zero_less_power [simp]: "0 < a \<Longrightarrow> 0 < a ^ n"
   by (induct n) simp_all
 
 lemma zero_le_power [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> a ^ n"
   by (induct n) simp_all
 
 lemma power_mono: "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 one_le_power [simp]: "1 \<le> a \<Longrightarrow> 1 \<le> a ^ n"
   using power_mono [of 1 a n] by simp
 
 lemma power_le_one: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ n \<le> 1"
   using power_mono [of a 1 n] by simp
 
 lemma power_gt1_lemma:
   assumes gt1: "1 < a"
   shows "1 < a * a ^ n"
 proof -
   from gt1 have "0 \<le> a"
     by (fact order_trans [OF zero_le_one less_imp_le])
   from gt1 have "1 * 1 < a * 1" by simp
   also from gt1 have "\<dots> \<le> a * a ^ n"
     by (simp only: mult_mono \<open>0 \<le> a\<close> one_le_power order_less_imp_le zero_le_one order_refl)
   finally show ?thesis by simp
 qed
 
 lemma power_gt1: "1 < a \<Longrightarrow> 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"
   shows "a ^ m \<le> a ^ n \<Longrightarrow> m \<le> n"
 proof (induct m arbitrary: n)
   case 0
   show ?case by simp
 next
   case (Suc m)
   show ?case
   proof (cases n)
     case 0
     with Suc have "a * a ^ m \<le> 1" by simp
     with gt1 show ?thesis
       by (force simp only: power_gt1_lemma not_less [symmetric])
   next
     case (Suc n)
     with Suc.prems Suc.hyps show ?thesis
       by (force dest: mult_left_le_imp_le simp add: less_trans [OF zero_less_one gt1])
   qed
 qed
 
 lemma of_nat_zero_less_power_iff [simp]: "of_nat x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
   by (induct n) auto
 
 text \<open>Surely we can strengthen this? It holds for \<open>0<a<1\<close> too.\<close>
 lemma power_inject_exp [simp]: "1 < a \<Longrightarrow> a ^ m = a ^ n \<longleftrightarrow> m = n"
   by (force simp add: order_antisym power_le_imp_le_exp)
 
 text \<open>
   Can relax the first premise to \<^term>\<open>0<a\<close> in the case of the
   natural numbers.
 \<close>
 lemma power_less_imp_less_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_strict_mono [rule_format]: "a < b \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 < n \<longrightarrow> a ^ n < b ^ n"
   by (induct n) (auto simp: mult_strict_mono le_less_trans [of 0 a b])
 
 lemma power_mono_iff [simp]:
   shows "\<lbrakk>a \<ge> 0; b \<ge> 0; n>0\<rbrakk> \<Longrightarrow> a ^ n \<le> b ^ n \<longleftrightarrow> a \<le> b"
   using power_mono [of a b] power_strict_mono [of b a] not_le by auto
 
 text\<open>Lemma for \<open>power_strict_decreasing\<close>\<close>
 lemma power_Suc_less: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a * a ^ n < a ^ n"
   by (induct n) (auto simp: mult_strict_left_mono)
 
 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 \<open>Proof resembles that of \<open>power_strict_decreasing\<close>.\<close>
 lemma power_decreasing: "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")
      apply simp
     apply (rule mult_mono)
        apply auto
     done
 qed
 
 lemma power_decreasing_iff [simp]: "\<lbrakk>0 < b; b < 1\<rbrakk> \<Longrightarrow> b ^ m \<le> b ^ n \<longleftrightarrow> n \<le> m"
   using power_strict_decreasing [of m n b]
   by (auto intro: power_decreasing ccontr)
 
 lemma power_strict_decreasing_iff [simp]: "\<lbrakk>0 < b; b < 1\<rbrakk> \<Longrightarrow> b ^ m < b ^ n \<longleftrightarrow> n < m"
   using power_decreasing_iff [of b m n] unfolding le_less
   by (auto dest: power_strict_decreasing le_neq_implies_less)
 
 lemma power_Suc_less_one: "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
   using power_strict_decreasing [of 0 "Suc n" a] by simp
 
 text \<open>Proof again resembles that of \<open>power_strict_decreasing\<close>.\<close>
 lemma power_increasing: "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")
      apply simp
     apply (rule mult_mono)
        apply (auto simp add: order_trans [OF zero_le_one])
     done
 qed
 
 text \<open>Lemma for \<open>power_strict_increasing\<close>.\<close>
 lemma power_less_power_Suc: "1 < a \<Longrightarrow> a ^ n < a * a ^ n"
   by (induct n) (auto simp: mult_strict_left_mono less_trans [OF zero_less_one])
 
 lemma power_strict_increasing: "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")
      apply 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 \<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 \<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 "0 \<le> b"
   shows "a \<le> b"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   then have "b < a" by (simp only: linorder_not_le)
   then have "b ^ Suc n < a ^ Suc n"
     by (simp only: assms(2) power_strict_mono)
   with le show False
     by (simp add: linorder_not_less [symmetric])
 qed
 
 lemma power_less_imp_less_base:
   assumes less: "a ^ n < b ^ n"
   assumes nonneg: "0 \<le> b"
   shows "a < b"
 proof (rule contrapos_pp [OF less])
   assume "\<not> ?thesis"
   then have "b \<le> a" by (simp only: linorder_not_less)
   from this nonneg have "b ^ n \<le> a ^ n" by (rule power_mono)
   then show "\<not> a ^ n < b ^ n" by (simp only: linorder_not_less)
 qed
 
 lemma power_inject_base: "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: "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)
 
 lemma power_eq_iff_eq_base: "0 < n \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a ^ n = b ^ n \<longleftrightarrow> a = b"
   using power_eq_imp_eq_base [of a n b] by auto
 
 lemma power2_le_imp_le: "x\<^sup>2 \<le> y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
   unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)
 
 lemma power2_less_imp_less: "x\<^sup>2 < y\<^sup>2 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
   by (rule power_less_imp_less_base)
 
 lemma power2_eq_imp_eq: "x\<^sup>2 = y\<^sup>2 \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
   unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp
 
 lemma power_Suc_le_self: "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
   using power_decreasing [of 1 "Suc n" a] by simp
 
 lemma power2_eq_iff_nonneg [simp]:
   assumes "0 \<le> x" "0 \<le> y"
   shows "(x ^ 2 = y ^ 2) \<longleftrightarrow> x = y"
 using assms power2_eq_imp_eq by blast
 
 lemma of_nat_less_numeral_power_cancel_iff[simp]:
   "of_nat x < numeral i ^ n \<longleftrightarrow> x < numeral i ^ n"
   using of_nat_less_iff[of x "numeral i ^ n", unfolded of_nat_numeral of_nat_power] .
 
 lemma of_nat_le_numeral_power_cancel_iff[simp]:
   "of_nat x \<le> numeral i ^ n \<longleftrightarrow> x \<le> numeral i ^ n"
   using of_nat_le_iff[of x "numeral i ^ n", unfolded of_nat_numeral of_nat_power] .
 
 lemma numeral_power_less_of_nat_cancel_iff[simp]:
   "numeral i ^ n < of_nat x \<longleftrightarrow> numeral i ^ n < x"
   using of_nat_less_iff[of "numeral i ^ n" x, unfolded of_nat_numeral of_nat_power] .
 
 lemma numeral_power_le_of_nat_cancel_iff[simp]:
   "numeral i ^ n \<le> of_nat x \<longleftrightarrow> numeral i ^ n \<le> x"
   using of_nat_le_iff[of "numeral i ^ n" x, unfolded of_nat_numeral of_nat_power] .
 
 lemma of_nat_le_of_nat_power_cancel_iff[simp]: "(of_nat b) ^ w \<le> of_nat x \<longleftrightarrow> b ^ w \<le> x"
   by (metis of_nat_le_iff of_nat_power)
 
 lemma of_nat_power_le_of_nat_cancel_iff[simp]: "of_nat x \<le> (of_nat b) ^ w \<longleftrightarrow> x \<le> b ^ w"
   by (metis of_nat_le_iff of_nat_power)
 
 lemma of_nat_less_of_nat_power_cancel_iff[simp]: "(of_nat b) ^ w < of_nat x \<longleftrightarrow> b ^ w < x"
   by (metis of_nat_less_iff of_nat_power)
 
 lemma of_nat_power_less_of_nat_cancel_iff[simp]: "of_nat x < (of_nat b) ^ w \<longleftrightarrow> x < b ^ w"
   by (metis of_nat_less_iff of_nat_power)
 
 end
 
 
 text \<open>Some @{typ nat}-specific lemmas:\<close>
 
 lemma mono_ge2_power_minus_self:
   assumes "k \<ge> 2" shows "mono (\<lambda>m. k ^ m - m)"
 unfolding mono_iff_le_Suc
 proof
   fix n
   have "k ^ n < k ^ Suc n" using power_strict_increasing_iff[of k "n" "Suc n"] assms by linarith
   thus "k ^ n - n \<le> k ^ Suc n - Suc n" by linarith
 qed
 
 lemma self_le_ge2_pow[simp]:
   assumes "k \<ge> 2" shows "m \<le> k ^ m"
 proof (induction m)
   case 0 show ?case by simp
 next
   case (Suc m)
   hence "Suc m \<le> Suc (k ^ m)" by simp
   also have "... \<le> k^m + k^m" using one_le_power[of k m] assms by linarith
   also have "... \<le> k * k^m" by (metis mult_2 mult_le_mono1[OF assms])
   finally show ?case by simp
 qed
 
 lemma diff_le_diff_pow[simp]:
   assumes "k \<ge> 2" shows "m - n \<le> k ^ m - k ^ n"
 proof (cases "n \<le> m")
   case True
   thus ?thesis
     using monoD[OF mono_ge2_power_minus_self[OF assms] True] self_le_ge2_pow[OF assms, of m]
     by (simp add: le_diff_conv le_diff_conv2)
 qed auto
 
 
 context linordered_ring_strict
 begin
 
 lemma sum_squares_eq_zero_iff: "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   by (simp add: add_nonneg_eq_0_iff)
 
 lemma sum_squares_le_zero_iff: "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)
 
 lemma sum_squares_gt_zero_iff: "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
   by (simp add: not_le [symmetric] sum_squares_le_zero_iff)
 
 end
 
 context linordered_idom
 begin
 
 lemma zero_le_power2 [simp]: "0 \<le> a\<^sup>2"
   by (simp add: power2_eq_square)
 
 lemma zero_less_power2 [simp]: "0 < a\<^sup>2 \<longleftrightarrow> a \<noteq> 0"
   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)
 
 lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
   by (force simp add: power2_eq_square mult_less_0_iff)
 
 lemma power_abs: "\<bar>a ^ n\<bar> = \<bar>a\<bar> ^ n" \<comment> \<open>FIXME simp?\<close>
   by (induct n) (simp_all add: abs_mult)
 
 lemma power_sgn [simp]: "sgn (a ^ n) = sgn a ^ n"
   by (induct n) (simp_all add: sgn_mult)
 
 lemma abs_power_minus [simp]: "\<bar>(- a) ^ n\<bar> = \<bar>a ^ n\<bar>"
   by (simp add: power_abs)
 
 lemma zero_less_power_abs_iff [simp]: "0 < \<bar>a\<bar> ^ n \<longleftrightarrow> a \<noteq> 0 \<or> n = 0"
 proof (induct n)
   case 0
   show ?case by simp
 next
   case Suc
   then show ?case by (auto simp: zero_less_mult_iff)
 qed
 
 lemma zero_le_power_abs [simp]: "0 \<le> \<bar>a\<bar> ^ n"
   by (rule zero_le_power [OF abs_ge_zero])
 
 lemma power2_less_eq_zero_iff [simp]: "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
   by (simp add: le_less)
 
 lemma abs_power2 [simp]: "\<bar>a\<^sup>2\<bar> = a\<^sup>2"
   by (simp add: power2_eq_square)
 
 lemma power2_abs [simp]: "\<bar>a\<bar>\<^sup>2 = a\<^sup>2"
   by (simp add: power2_eq_square)
 
 lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2 * n) < 0"
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)"
     by (simp add: ac_simps power_add power2_eq_square)
   then show ?case
     by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
 qed
 
 lemma odd_0_le_power_imp_0_le: "0 \<le> a ^ Suc (2 * n) \<Longrightarrow> 0 \<le> a"
   using odd_power_less_zero [of a n]
   by (force simp add: linorder_not_less [symmetric])
 
 lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2 * n)"
 proof (induct n)
   case 0
   show ?case by simp
 next
   case (Suc n)
   have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)"
     by (simp add: ac_simps power_add power2_eq_square)
   then show ?case
     by (simp add: Suc zero_le_mult_iff)
 qed
 
 lemma sum_power2_ge_zero: "0 \<le> x\<^sup>2 + y\<^sup>2"
   by (intro add_nonneg_nonneg zero_le_power2)
 
 lemma not_sum_power2_lt_zero: "\<not> x\<^sup>2 + y\<^sup>2 < 0"
   unfolding not_less by (rule sum_power2_ge_zero)
 
 lemma sum_power2_eq_zero_iff: "x\<^sup>2 + y\<^sup>2 = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   unfolding power2_eq_square by (simp add: add_nonneg_eq_0_iff)
 
 lemma sum_power2_le_zero_iff: "x\<^sup>2 + y\<^sup>2 \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   by (simp add: le_less sum_power2_eq_zero_iff not_sum_power2_lt_zero)
 
 lemma sum_power2_gt_zero_iff: "0 < x\<^sup>2 + y\<^sup>2 \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
   unfolding not_le [symmetric] by (simp add: sum_power2_le_zero_iff)
 
 lemma abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> x\<^sup>2 \<le> y\<^sup>2"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume ?lhs
   then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono) simp
   then show ?rhs by simp
 next
   assume ?rhs
   then show ?lhs
     by (auto intro!: power2_le_imp_le [OF _ abs_ge_zero])
 qed
 
 lemma abs_square_le_1:"x\<^sup>2 \<le> 1 \<longleftrightarrow> \<bar>x\<bar> \<le> 1"
   using abs_le_square_iff [of x 1] by simp
 
 lemma abs_square_eq_1: "x\<^sup>2 = 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
   by (auto simp add: abs_if power2_eq_1_iff)
 
 lemma abs_square_less_1: "x\<^sup>2 < 1 \<longleftrightarrow> \<bar>x\<bar> < 1"
   using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)
 
 lemma square_le_1:
   assumes "- 1 \<le> x" "x \<le> 1"
   shows "x\<^sup>2 \<le> 1"
     using assms
     by (metis add.inverse_inverse linear mult_le_one neg_equal_0_iff_equal neg_le_iff_le power2_eq_square power_minus_Bit0)
 
 end
 
 
 subsection \<open>Miscellaneous rules\<close>
 
 lemma (in linordered_semidom) self_le_power: "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
   using power_increasing [of 1 n a] power_one_right [of a] by auto
 
 lemma (in power) power_eq_if: "p ^ m = (if m=0 then 1 else p * (p ^ (m - 1)))"
   unfolding One_nat_def by (cases m) simp_all
 
 lemma (in comm_semiring_1) power2_sum: "(x + y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 + 2 * x * y"
   by (simp add: algebra_simps power2_eq_square mult_2_right)
 
 context comm_ring_1
 begin
 
 lemma power2_diff: "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
   by (simp add: algebra_simps power2_eq_square mult_2_right)
 
 lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>2"
   by (simp add: algebra_simps power2_eq_square)
 
 lemma minus_power_mult_self: "(- a) ^ n * (- a) ^ n = a ^ (2 * n)"
   by (simp add: power_mult_distrib [symmetric])
     (simp add: power2_eq_square [symmetric] power_mult [symmetric])
 
 lemma minus_one_mult_self [simp]: "(- 1) ^ n * (- 1) ^ n = 1"
   using minus_power_mult_self [of 1 n] by simp
 
 lemma left_minus_one_mult_self [simp]: "(- 1) ^ n * ((- 1) ^ n * a) = a"
   by (simp add: mult.assoc [symmetric])
 
 end
 
 text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
 
 lemmas zero_compare_simps =
   add_strict_increasing add_strict_increasing2 add_increasing
   zero_le_mult_iff zero_le_divide_iff
   zero_less_mult_iff zero_less_divide_iff
   mult_le_0_iff divide_le_0_iff
   mult_less_0_iff divide_less_0_iff
   zero_le_power2 power2_less_0
 
 
 subsection \<open>Exponentiation for the Natural Numbers\<close>
 
 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_zero_less_power_iff [simp]: "x ^ n > 0 \<longleftrightarrow> x > 0 \<or> n = 0"
   for x :: nat
   by (induct n) auto
 
 lemma nat_power_eq_Suc_0_iff [simp]: "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
 
 text \<open>
   Valid for the naturals, but what if \<open>0 < i < 1\<close>? Premises cannot be
   weakened: consider the case where \<open>i = 0\<close>, \<open>m = 1\<close> and \<open>n = 0\<close>.
 \<close>
 
 lemma nat_power_less_imp_less:
   fixes i :: nat
   assumes nonneg: "0 < i"
   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
 next
   case False
   with nonneg have "1 < i" by auto
   from power_strict_increasing_iff [OF this] less show ?thesis ..
 qed
 
 lemma power_dvd_imp_le: "i ^ m dvd i ^ n \<Longrightarrow> 1 < i \<Longrightarrow> m \<le> n"
   for i m n :: nat
   apply (rule power_le_imp_le_exp)
    apply assumption
   apply (erule dvd_imp_le)
   apply simp
   done
 
 lemma dvd_power_iff_le:
   fixes k::nat
   shows "2 \<le> k \<Longrightarrow> ((k ^ m) dvd (k ^ n) \<longleftrightarrow> m \<le> n)"
   using le_imp_power_dvd power_dvd_imp_le by force
 
 lemma power2_nat_le_eq_le: "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
   for m n :: nat
   by (auto intro: power2_le_imp_le power_mono)
 
 lemma power2_nat_le_imp_le:
   fixes m n :: nat
   assumes "m\<^sup>2 \<le> n"
   shows "m \<le> n"
 proof (cases m)
   case 0
   then show ?thesis by simp
 next
   case (Suc k)
   show ?thesis
   proof (rule ccontr)
     assume "\<not> ?thesis"
     then have "n < m" by simp
     with assms Suc show False
       by (simp add: power2_eq_square)
   qed
 qed
 
 lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
 shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
 proof(induction k)
   case 0 thus ?case by simp
 next
   case (Suc k)
   show ?case
   proof cases
     assume "k=0"
     hence "?P (Suc k) 0" using assms by simp
     thus ?case ..
   next
     assume "k\<noteq>0"
     with Suc obtain n where IH: "?P k n" by auto
     show ?case
     proof (cases "k = b^(n+1) - 1")
       case True
       hence "?P (Suc k) (n+1)" using assms
         by (simp add: power_less_power_Suc)
       thus ?thesis ..
     next
       case False
       hence "?P (Suc k) n" using IH by auto
       thus ?thesis ..
     qed
   qed
 qed
 
 lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "k \<ge> 2"
 shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
 proof -
   have "1 \<le> k - 1" using assms(2) by arith
   from ex_power_ivl1[OF assms(1) this]
   obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
   hence "b^n < k \<and> k \<le> b^(n+1)" using assms by auto
   thus ?thesis ..
 qed
 
 
 subsubsection \<open>Cardinality of the Powerset\<close>
 
 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
   unfolding UNIV_bool by simp
 
 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
 proof (induct rule: finite_induct)
   case empty
   show ?case by simp
 next
   case (insert x A)
   from \<open>x \<notin> A\<close> have disjoint: "Pow A \<inter> insert x ` Pow A = {}" by blast
   from \<open>x \<notin> A\<close> have inj_on: "inj_on (insert x) (Pow A)"
     unfolding inj_on_def by auto
 
   have "card (Pow (insert x A)) = card (Pow A \<union> insert x ` Pow A)"
     by (simp only: Pow_insert)
   also have "\<dots> = card (Pow A) + card (insert x ` Pow A)"
     by (rule card_Un_disjoint) (use \<open>finite A\<close> disjoint in simp_all)
   also from inj_on have "card (insert x ` Pow A) = card (Pow A)"
     by (rule card_image)
   also have "\<dots> + \<dots> = 2 * \<dots>" by (simp add: mult_2)
   also from insert(3) have "\<dots> = 2 ^ Suc (card A)" by simp
   also from insert(1,2) have "Suc (card A) = card (insert x A)"
     by (rule card_insert_disjoint [symmetric])
   finally show ?case .
 qed
 
 
 subsection \<open>Code generator tweak\<close>
 
 code_identifier
   code_module Power \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 end