# HG changeset patch
# User wenzelm
# Date 1470859536 -7200
# Node ID f90e3926e627a957ce98a2a1bdd99053c5c15e34
# Parent  4453cfb745e5b4c66d0c7ab2e26fcb0871f7aef9
misc tuning and modernization;

diff -r 4453cfb745e5 -r f90e3926e627 src/HOL/Fun_Def.thy
--- a/src/HOL/Fun_Def.thy	Wed Aug 10 22:05:00 2016 +0200
+++ b/src/HOL/Fun_Def.thy	Wed Aug 10 22:05:36 2016 +0200
@@ -5,32 +5,29 @@
 section \<open>Function Definitions and Termination Proofs\<close>
 
 theory Fun_Def
-imports Basic_BNF_LFPs Partial_Function SAT
-keywords
-  "function" "termination" :: thy_goal and
-  "fun" "fun_cases" :: thy_decl
+  imports Basic_BNF_LFPs Partial_Function SAT
+  keywords
+    "function" "termination" :: thy_goal and
+    "fun" "fun_cases" :: thy_decl
 begin
 
 subsection \<open>Definitions with default value\<close>
 
-definition
-  THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
-  "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
+definition THE_default :: "'a \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
+  where "THE_default d P = (if (\<exists>!x. P x) then (THE x. P x) else d)"
 
 lemma THE_defaultI': "\<exists>!x. P x \<Longrightarrow> P (THE_default d P)"
   by (simp add: theI' THE_default_def)
 
-lemma THE_default1_equality:
-    "\<lbrakk>\<exists>!x. P x; P a\<rbrakk> \<Longrightarrow> THE_default d P = a"
+lemma THE_default1_equality: "\<exists>!x. P x \<Longrightarrow> P a \<Longrightarrow> THE_default d P = a"
   by (simp add: the1_equality THE_default_def)
 
-lemma THE_default_none:
-    "\<not>(\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
-  by (simp add:THE_default_def)
+lemma THE_default_none: "\<not> (\<exists>!x. P x) \<Longrightarrow> THE_default d P = d"
+  by (simp add: THE_default_def)
 
 
 lemma fundef_ex1_existence:
-  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
+  assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
   assumes ex1: "\<exists>!y. G x y"
   shows "G x (f x)"
   apply (simp only: f_def)
@@ -39,7 +36,7 @@
   done
 
 lemma fundef_ex1_uniqueness:
-  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
+  assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
   assumes ex1: "\<exists>!y. G x y"
   assumes elm: "G x (h x)"
   shows "h x = f x"
@@ -50,7 +47,7 @@
   done
 
 lemma fundef_ex1_iff:
-  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
+  assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
   assumes ex1: "\<exists>!y. G x y"
   shows "(G x y) = (f x = y)"
   apply (auto simp:ex1 f_def THE_default1_equality)
@@ -59,7 +56,7 @@
   done
 
 lemma fundef_default_value:
-  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
+  assumes f_def: "f \<equiv> (\<lambda>x::'a. THE_default (d x) (\<lambda>y. G x y))"
   assumes graph: "\<And>x y. G x y \<Longrightarrow> D x"
   assumes "\<not> D x"
   shows "f x = d x"
@@ -67,21 +64,17 @@
   have "\<not>(\<exists>y. G x y)"
   proof
     assume "\<exists>y. G x y"
-    hence "D x" using graph ..
+    then have "D x" using graph ..
     with \<open>\<not> D x\<close> show False ..
   qed
-  hence "\<not>(\<exists>!y. G x y)" by blast
-
-  thus ?thesis
-    unfolding f_def
-    by (rule THE_default_none)
+  then have "\<not>(\<exists>!y. G x y)" by blast
+  then show ?thesis
+    unfolding f_def by (rule THE_default_none)
 qed
 
-definition in_rel_def[simp]:
-  "in_rel R x y == (x, y) \<in> R"
+definition in_rel_def[simp]: "in_rel R x y \<equiv> (x, y) \<in> R"
 
-lemma wf_in_rel:
-  "wf R \<Longrightarrow> wfP (in_rel R)"
+lemma wf_in_rel: "wf R \<Longrightarrow> wfP (in_rel R)"
   by (simp add: wfP_def)
 
 ML_file "Tools/Function/function_core.ML"
@@ -112,18 +105,19 @@
 subsection \<open>Measure functions\<close>
 
 inductive is_measure :: "('a \<Rightarrow> nat) \<Rightarrow> bool"
-where is_measure_trivial: "is_measure f"
+  where is_measure_trivial: "is_measure f"
 
 named_theorems measure_function "rules that guide the heuristic generation of measure functions"
 ML_file "Tools/Function/measure_functions.ML"
 
 lemma measure_size[measure_function]: "is_measure size"
-by (rule is_measure_trivial)
+  by (rule is_measure_trivial)
 
 lemma measure_fst[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (fst p))"
-by (rule is_measure_trivial)
+  by (rule is_measure_trivial)
+
 lemma measure_snd[measure_function]: "is_measure f \<Longrightarrow> is_measure (\<lambda>p. f (snd p))"
-by (rule is_measure_trivial)
+  by (rule is_measure_trivial)
 
 ML_file "Tools/Function/lexicographic_order.ML"
 
@@ -135,8 +129,7 @@
 
 subsection \<open>Congruence rules\<close>
 
-lemma let_cong [fundef_cong]:
-  "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g"
+lemma let_cong [fundef_cong]: "M = N \<Longrightarrow> (\<And>x. x = N \<Longrightarrow> f x = g x) \<Longrightarrow> Let M f = Let N g"
   unfolding Let_def by blast
 
 lemmas [fundef_cong] =
@@ -144,13 +137,11 @@
   bex_cong ball_cong imp_cong map_option_cong Option.bind_cong
 
 lemma split_cong [fundef_cong]:
-  "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q
-    \<Longrightarrow> case_prod f p = case_prod g q"
+  "(\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y) \<Longrightarrow> p = q \<Longrightarrow> case_prod f p = case_prod g q"
   by (auto simp: split_def)
 
-lemma comp_cong [fundef_cong]:
-  "f (g x) = f' (g' x') \<Longrightarrow> (f o g) x = (f' o g') x'"
-  unfolding o_apply .
+lemma comp_cong [fundef_cong]: "f (g x) = f' (g' x') \<Longrightarrow> (f \<circ> g) x = (f' \<circ> g') x'"
+  by (simp only: o_apply)
 
 
 subsection \<open>Simp rules for termination proofs\<close>
@@ -163,31 +154,25 @@
   less_imp_le_nat[termination_simp]
   le_imp_less_Suc[termination_simp]
 
-lemma size_prod_simp[termination_simp]:
-  "size_prod f g p = f (fst p) + g (snd p) + Suc 0"
-by (induct p) auto
+lemma size_prod_simp[termination_simp]: "size_prod f g p = f (fst p) + g (snd p) + Suc 0"
+  by (induct p) auto
 
 
 subsection \<open>Decomposition\<close>
 
-lemma less_by_empty:
-  "A = {} \<Longrightarrow> A \<subseteq> B"
-and  union_comp_emptyL:
-  "\<lbrakk> A O C = {}; B O C = {} \<rbrakk> \<Longrightarrow> (A \<union> B) O C = {}"
-and union_comp_emptyR:
-  "\<lbrakk> A O B = {}; A O C = {} \<rbrakk> \<Longrightarrow> A O (B \<union> C) = {}"
-and wf_no_loop:
-  "R O R = {} \<Longrightarrow> wf R"
-by (auto simp add: wf_comp_self[of R])
+lemma less_by_empty: "A = {} \<Longrightarrow> A \<subseteq> B"
+  and union_comp_emptyL: "A O C = {} \<Longrightarrow> B O C = {} \<Longrightarrow> (A \<union> B) O C = {}"
+  and union_comp_emptyR: "A O B = {} \<Longrightarrow> A O C = {} \<Longrightarrow> A O (B \<union> C) = {}"
+  and wf_no_loop: "R O R = {} \<Longrightarrow> wf R"
+  by (auto simp add: wf_comp_self [of R])
 
 
 subsection \<open>Reduction pairs\<close>
 
-definition
-  "reduction_pair P = (wf (fst P) \<and> fst P O snd P \<subseteq> fst P)"
+definition "reduction_pair P \<longleftrightarrow> wf (fst P) \<and> fst P O snd P \<subseteq> fst P"
 
 lemma reduction_pairI[intro]: "wf R \<Longrightarrow> R O S \<subseteq> R \<Longrightarrow> reduction_pair (R, S)"
-unfolding reduction_pair_def by auto
+  by (auto simp: reduction_pair_def)
 
 lemma reduction_pair_lemma:
   assumes rp: "reduction_pair P"
@@ -204,13 +189,10 @@
   ultimately show ?thesis by (rule wf_subset)
 qed
 
-definition
-  "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
+definition "rp_inv_image = (\<lambda>(R,S) f. (inv_image R f, inv_image S f))"
 
-lemma rp_inv_image_rp:
-  "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)"
-  unfolding reduction_pair_def rp_inv_image_def split_def
-  by force
+lemma rp_inv_image_rp: "reduction_pair P \<Longrightarrow> reduction_pair (rp_inv_image P f)"
+  unfolding reduction_pair_def rp_inv_image_def split_def by force
 
 
 subsection \<open>Concrete orders for SCNP termination proofs\<close>
@@ -230,70 +212,70 @@
   and pair_leqI2: "a \<le> b \<Longrightarrow> s \<le> t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_leq"
   and pair_lessI1: "a < b  \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
   and pair_lessI2: "a \<le> b \<Longrightarrow> s < t \<Longrightarrow> ((a, s), (b, t)) \<in> pair_less"
-  unfolding pair_leq_def pair_less_def by auto
+  by (auto simp: pair_leq_def pair_less_def)
 
 text \<open>Introduction rules for max\<close>
-lemma smax_emptyI:
-  "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict"
+lemma smax_emptyI: "finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> ({}, Y) \<in> max_strict"
   and smax_insertI:
-  "\<lbrakk>y \<in> Y; (x, y) \<in> pair_less; (X, Y) \<in> max_strict\<rbrakk> \<Longrightarrow> (insert x X, Y) \<in> max_strict"
-  and wmax_emptyI:
-  "finite X \<Longrightarrow> ({}, X) \<in> max_weak"
+    "y \<in> Y \<Longrightarrow> (x, y) \<in> pair_less \<Longrightarrow> (X, Y) \<in> max_strict \<Longrightarrow> (insert x X, Y) \<in> max_strict"
+  and wmax_emptyI: "finite X \<Longrightarrow> ({}, X) \<in> max_weak"
   and wmax_insertI:
-  "\<lbrakk>y \<in> YS; (x, y) \<in> pair_leq; (XS, YS) \<in> max_weak\<rbrakk> \<Longrightarrow> (insert x XS, YS) \<in> max_weak"
-unfolding max_strict_def max_weak_def by (auto elim!: max_ext.cases)
+    "y \<in> YS \<Longrightarrow> (x, y) \<in> pair_leq \<Longrightarrow> (XS, YS) \<in> max_weak \<Longrightarrow> (insert x XS, YS) \<in> max_weak"
+  by (auto simp: max_strict_def max_weak_def elim!: max_ext.cases)
 
 text \<open>Introduction rules for min\<close>
-lemma smin_emptyI:
-  "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict"
+lemma smin_emptyI: "X \<noteq> {} \<Longrightarrow> (X, {}) \<in> min_strict"
   and smin_insertI:
-  "\<lbrakk>x \<in> XS; (x, y) \<in> pair_less; (XS, YS) \<in> min_strict\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_strict"
-  and wmin_emptyI:
-  "(X, {}) \<in> min_weak"
+    "x \<in> XS \<Longrightarrow> (x, y) \<in> pair_less \<Longrightarrow> (XS, YS) \<in> min_strict \<Longrightarrow> (XS, insert y YS) \<in> min_strict"
+  and wmin_emptyI: "(X, {}) \<in> min_weak"
   and wmin_insertI:
-  "\<lbrakk>x \<in> XS; (x, y) \<in> pair_leq; (XS, YS) \<in> min_weak\<rbrakk> \<Longrightarrow> (XS, insert y YS) \<in> min_weak"
-by (auto simp: min_strict_def min_weak_def min_ext_def)
+    "x \<in> XS \<Longrightarrow> (x, y) \<in> pair_leq \<Longrightarrow> (XS, YS) \<in> min_weak \<Longrightarrow> (XS, insert y YS) \<in> min_weak"
+  by (auto simp: min_strict_def min_weak_def min_ext_def)
 
-text \<open>Reduction Pairs\<close>
+text \<open>Reduction Pairs.\<close>
 
 lemma max_ext_compat:
   assumes "R O S \<subseteq> R"
-  shows "max_ext R O (max_ext S \<union> {({},{})}) \<subseteq> max_ext R"
-using assms
-apply auto
-apply (elim max_ext.cases)
-apply rule
-apply auto[3]
-apply (drule_tac x=xa in meta_spec)
-apply simp
-apply (erule bexE)
-apply (drule_tac x=xb in meta_spec)
-by auto
+  shows "max_ext R O (max_ext S \<union> {({}, {})}) \<subseteq> max_ext R"
+  using assms
+  apply auto
+  apply (elim max_ext.cases)
+  apply rule
+     apply auto[3]
+  apply (drule_tac x=xa in meta_spec)
+  apply simp
+  apply (erule bexE)
+  apply (drule_tac x=xb in meta_spec)
+  apply auto
+  done
 
 lemma max_rpair_set: "reduction_pair (max_strict, max_weak)"
   unfolding max_strict_def max_weak_def
-apply (intro reduction_pairI max_ext_wf)
-apply simp
-apply (rule max_ext_compat)
-by (auto simp: pair_less_def pair_leq_def)
+  apply (intro reduction_pairI max_ext_wf)
+   apply simp
+  apply (rule max_ext_compat)
+  apply (auto simp: pair_less_def pair_leq_def)
+  done
 
 lemma min_ext_compat:
   assumes "R O S \<subseteq> R"
   shows "min_ext R O  (min_ext S \<union> {({},{})}) \<subseteq> min_ext R"
-using assms
-apply (auto simp: min_ext_def)
-apply (drule_tac x=ya in bspec, assumption)
-apply (erule bexE)
-apply (drule_tac x=xc in bspec)
-apply assumption
-by auto
+  using assms
+  apply (auto simp: min_ext_def)
+  apply (drule_tac x=ya in bspec, assumption)
+  apply (erule bexE)
+  apply (drule_tac x=xc in bspec)
+   apply assumption
+  apply auto
+  done
 
 lemma min_rpair_set: "reduction_pair (min_strict, min_weak)"
   unfolding min_strict_def min_weak_def
-apply (intro reduction_pairI min_ext_wf)
-apply simp
-apply (rule min_ext_compat)
-by (auto simp: pair_less_def pair_leq_def)
+  apply (intro reduction_pairI min_ext_wf)
+   apply simp
+  apply (rule min_ext_compat)
+  apply (auto simp: pair_less_def pair_leq_def)
+  done
 
 
 subsection \<open>Tool setup\<close>
diff -r 4453cfb745e5 -r f90e3926e627 src/HOL/Groups_Big.thy
--- a/src/HOL/Groups_Big.thy	Wed Aug 10 22:05:00 2016 +0200
+++ b/src/HOL/Groups_Big.thy	Wed Aug 10 22:05:36 2016 +0200
@@ -1,12 +1,14 @@
 (*  Title:      HOL/Groups_Big.thy
-    Author:     Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
-                with contributions by Jeremy Avigad
+    Author:     Tobias Nipkow
+    Author:     Lawrence C Paulson
+    Author:     Markus Wenzel
+    Author:     Jeremy Avigad
 *)
 
 section \<open>Big sum and product over finite (non-empty) sets\<close>
 
 theory Groups_Big
-imports Finite_Set Power
+  imports Finite_Set Power
 begin
 
 subsection \<open>Generic monoid operation over a set\<close>
@@ -21,60 +23,53 @@
   by (fact comp_comp_fun_commute)
 
 definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
-where
-  eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
+  where eq_fold: "F g A = Finite_Set.fold (f \<circ> g) \<^bold>1 A"
 
-lemma infinite [simp]:
-  "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
+lemma infinite [simp]: "\<not> finite A \<Longrightarrow> F g A = \<^bold>1"
   by (simp add: eq_fold)
 
-lemma empty [simp]:
-  "F g {} = \<^bold>1"
+lemma empty [simp]: "F g {} = \<^bold>1"
   by (simp add: eq_fold)
 
-lemma insert [simp]:
-  assumes "finite A" and "x \<notin> A"
-  shows "F g (insert x A) = g x \<^bold>* F g A"
-  using assms by (simp add: eq_fold)
+lemma insert [simp]: "finite A \<Longrightarrow> x \<notin> A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g A"
+  by (simp add: eq_fold)
 
 lemma remove:
   assumes "finite A" and "x \<in> A"
   shows "F g A = g x \<^bold>* F g (A - {x})"
 proof -
-  from \<open>x \<in> A\<close> obtain B where A: "A = insert x B" and "x \<notin> B"
+  from \<open>x \<in> A\<close> obtain B where B: "A = insert x B" and "x \<notin> B"
     by (auto dest: mk_disjoint_insert)
-  moreover from \<open>finite A\<close> A have "finite B" by simp
+  moreover from \<open>finite A\<close> B have "finite B" by simp
   ultimately show ?thesis by simp
 qed
 
-lemma insert_remove:
-  assumes "finite A"
-  shows "F g (insert x A) = g x \<^bold>* F g (A - {x})"
-  using assms by (cases "x \<in> A") (simp_all add: remove insert_absorb)
+lemma insert_remove: "finite A \<Longrightarrow> F g (insert x A) = g x \<^bold>* F g (A - {x})"
+  by (cases "x \<in> A") (simp_all add: remove insert_absorb)
 
-lemma neutral:
-  assumes "\<forall>x\<in>A. g x = \<^bold>1"
-  shows "F g A = \<^bold>1"
-  using assms by (induct A rule: infinite_finite_induct) simp_all
+lemma neutral: "\<forall>x\<in>A. g x = \<^bold>1 \<Longrightarrow> F g A = \<^bold>1"
+  by (induct A rule: infinite_finite_induct) simp_all
 
-lemma neutral_const [simp]:
-  "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
+lemma neutral_const [simp]: "F (\<lambda>_. \<^bold>1) A = \<^bold>1"
   by (simp add: neutral)
 
 lemma union_inter:
   assumes "finite A" and "finite B"
   shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"
   \<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
-using assms proof (induct A)
-  case empty then show ?case by simp
+  using assms
+proof (induct A)
+  case empty
+  then show ?case by simp
 next
-  case (insert x A) then show ?case
-    by (auto simp add: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
+  case (insert x A)
+  then show ?case
+    by (auto simp: insert_absorb Int_insert_left commute [of _ "g x"] assoc left_commute)
 qed
 
 corollary union_inter_neutral:
   assumes "finite A" and "finite B"
-  and I0: "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
+    and "\<forall>x \<in> A \<inter> B. g x = \<^bold>1"
   shows "F g (A \<union> B) = F g A \<^bold>* F g B"
   using assms by (simp add: union_inter [symmetric] neutral)
 
@@ -90,7 +85,8 @@
 proof -
   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
     by auto
-  with assms show ?thesis by simp (subst union_disjoint, auto)+
+  with assms show ?thesis
+    by simp (subst union_disjoint, auto)+
 qed
 
 lemma subset_diff:
@@ -116,9 +112,15 @@
 proof -
   from assms have "\<exists>a\<in>A. g a \<noteq> \<^bold>1"
   proof (induct A rule: infinite_finite_induct)
+    case infinite
+    then show ?case by simp
+  next
+    case empty
+    then show ?case by simp
+  next
     case (insert a A)
-    then show ?case by simp (rule, simp)
-  qed simp_all
+    then show ?case by fastforce
+  qed
   with that show thesis by blast
 qed
 
@@ -127,9 +129,11 @@
   shows "F g (h ` A) = F (g \<circ> h) A"
 proof (cases "finite A")
   case True
-  with assms show ?thesis by (simp add: eq_fold fold_image comp_assoc)
+  with assms show ?thesis
+    by (simp add: eq_fold fold_image comp_assoc)
 next
-  case False with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
+  case False
+  with assms have "\<not> finite (h ` A)" by (blast dest: finite_imageD)
   with False show ?thesis by simp
 qed
 
@@ -143,7 +147,7 @@
 lemma strong_cong [cong]:
   assumes "A = B" "\<And>x. x \<in> B =simp=> g x = h x"
   shows "F (\<lambda>x. g x) A = F (\<lambda>x. h x) B"
-  by (rule cong) (insert assms, simp_all add: simp_implies_def)
+  by (rule cong) (use assms in \<open>simp_all add: simp_implies_def\<close>)
 
 lemma reindex_cong:
   assumes "inj_on l B"
@@ -154,55 +158,64 @@
 
 lemma UNION_disjoint:
   assumes "finite I" and "\<forall>i\<in>I. finite (A i)"
-  and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
+    and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
   shows "F g (UNION I A) = F (\<lambda>x. F g (A x)) I"
-apply (insert assms)
-apply (induct rule: finite_induct)
-apply simp
-apply atomize
-apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
- prefer 2 apply blast
-apply (subgoal_tac "A x Int UNION Fa A = {}")
- prefer 2 apply blast
-apply (simp add: union_disjoint)
-done
+  apply (insert assms)
+  apply (induct rule: finite_induct)
+   apply simp
+  apply atomize
+  apply (subgoal_tac "\<forall>i\<in>Fa. x \<noteq> i")
+   prefer 2 apply blast
+  apply (subgoal_tac "A x \<inter> UNION Fa A = {}")
+   prefer 2 apply blast
+  apply (simp add: union_disjoint)
+  done
 
 lemma Union_disjoint:
   assumes "\<forall>A\<in>C. finite A" "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
   shows "F g (\<Union>C) = (F \<circ> F) g C"
-proof cases
-  assume "finite C"
-  from UNION_disjoint [OF this assms]
-  show ?thesis by simp
-qed (auto dest: finite_UnionD intro: infinite)
+proof (cases "finite C")
+  case True
+  from UNION_disjoint [OF this assms] show ?thesis by simp
+next
+  case False
+  then show ?thesis by (auto dest: finite_UnionD intro: infinite)
+qed
 
-lemma distrib:
-  "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
+lemma distrib: "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
   by (induct A rule: infinite_finite_induct) (simp_all add: assoc commute left_commute)
 
 lemma Sigma:
   "finite A \<Longrightarrow> \<forall>x\<in>A. finite (B x) \<Longrightarrow> F (\<lambda>x. F (g x) (B x)) A = F (case_prod g) (SIGMA x:A. B x)"
-apply (subst Sigma_def)
-apply (subst UNION_disjoint, assumption, simp)
- apply blast
-apply (rule cong)
-apply rule
-apply (simp add: fun_eq_iff)
-apply (subst UNION_disjoint, simp, simp)
- apply blast
-apply (simp add: comp_def)
-done
+  apply (subst Sigma_def)
+  apply (subst UNION_disjoint)
+     apply assumption
+    apply simp
+   apply blast
+  apply (rule cong)
+   apply rule
+  apply (simp add: fun_eq_iff)
+  apply (subst UNION_disjoint)
+     apply simp
+    apply simp
+   apply blast
+  apply (simp add: comp_def)
+  done
 
 lemma related:
   assumes Re: "R \<^bold>1 \<^bold>1"
-  and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
-  and fS: "finite S" and Rfg: "\<forall>x\<in>S. R (h x) (g x)"
+    and Rop: "\<forall>x1 y1 x2 y2. R x1 x2 \<and> R y1 y2 \<longrightarrow> R (x1 \<^bold>* y1) (x2 \<^bold>* y2)"
+    and fin: "finite S"
+    and R_h_g: "\<forall>x\<in>S. R (h x) (g x)"
   shows "R (F h S) (F g S)"
-  using fS by (rule finite_subset_induct) (insert assms, auto)
+  using fin by (rule finite_subset_induct) (use assms in auto)
 
 lemma mono_neutral_cong_left:
-  assumes "finite T" and "S \<subseteq> T" and "\<forall>i \<in> T - S. h i = \<^bold>1"
-  and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
+  assumes "finite T"
+    and "S \<subseteq> T"
+    and "\<forall>i \<in> T - S. h i = \<^bold>1"
+    and "\<And>x. x \<in> S \<Longrightarrow> g x = h x"
+  shows "F g S = F h T"
 proof-
   have eq: "T = S \<union> (T - S)" using \<open>S \<subseteq> T\<close> by blast
   have d: "S \<inter> (T - S) = {}" using \<open>S \<subseteq> T\<close> by blast
@@ -213,16 +226,14 @@
 qed
 
 lemma mono_neutral_cong_right:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = \<^bold>1; \<And>x. x \<in> S \<Longrightarrow> g x = h x \<rbrakk>
-   \<Longrightarrow> F g T = F h S"
+  "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> g x = h x) \<Longrightarrow>
+    F g T = F h S"
   by (auto intro!: mono_neutral_cong_left [symmetric])
 
-lemma mono_neutral_left:
-  "\<lbrakk> finite T; S \<subseteq> T; \<forall>i \<in> T - S. g i = \<^bold>1 \<rbrakk> \<Longrightarrow> F g S = F g T"
+lemma mono_neutral_left: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g S = F g T"
   by (blast intro: mono_neutral_cong_left)
 
-lemma mono_neutral_right:
-  "\<lbrakk> finite T;  S \<subseteq> T;  \<forall>i \<in> T - S. g i = \<^bold>1 \<rbrakk> \<Longrightarrow> F g T = F g S"
+lemma mono_neutral_right: "finite T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<forall>i \<in> T - S. g i = \<^bold>1 \<Longrightarrow> F g T = F g S"
   by (blast intro!: mono_neutral_left [symmetric])
 
 lemma reindex_bij_betw: "bij_betw h S T \<Longrightarrow> F (\<lambda>x. g (h x)) S = F g T"
@@ -256,10 +267,9 @@
 proof -
   have [simp]: "finite S \<longleftrightarrow> finite T"
     using bij_betw_finite[OF bij] fin by auto
-
   show ?thesis
-  proof cases
-    assume "finite S"
+  proof (cases "finite S")
+    case True
     with nn have "F (\<lambda>x. g (h x)) S = F (\<lambda>x. g (h x)) (S - S')"
       by (intro mono_neutral_cong_right) auto
     also have "\<dots> = F g (T - T')"
@@ -267,17 +277,20 @@
     also have "\<dots> = F g T"
       using nn \<open>finite S\<close> by (intro mono_neutral_cong_left) auto
     finally show ?thesis .
-  qed simp
+  next
+    case False
+    then show ?thesis by simp
+  qed
 qed
 
 lemma reindex_nontrivial:
   assumes "finite A"
-  and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
+    and nz: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<noteq> y \<Longrightarrow> h x = h y \<Longrightarrow> g (h x) = \<^bold>1"
   shows "F g (h ` A) = F (g \<circ> h) A"
 proof (subst reindex_bij_betw_not_neutral [symmetric])
   show "bij_betw h (A - {x \<in> A. (g \<circ> h) x = \<^bold>1}) (h ` A - h ` {x \<in> A. (g \<circ> h) x = \<^bold>1})"
     using nz by (auto intro!: inj_onI simp: bij_betw_def)
-qed (insert \<open>finite A\<close>, auto)
+qed (use \<open>finite A\<close> in auto)
 
 lemma reindex_bij_witness_not_neutral:
   assumes fin: "finite S'" "finite T'"
@@ -305,69 +318,66 @@
 lemma delta:
   assumes fS: "finite S"
   shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
-proof-
-  let ?f = "(\<lambda>k. if k=a then b k else \<^bold>1)"
-  { assume a: "a \<notin> S"
-    hence "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
-    hence ?thesis  using a by simp }
-  moreover
-  { assume a: "a \<in> S"
+proof -
+  let ?f = "(\<lambda>k. if k = a then b k else \<^bold>1)"
+  show ?thesis
+  proof (cases "a \<in> S")
+    case False
+    then have "\<forall>k\<in>S. ?f k = \<^bold>1" by simp
+    with False show ?thesis by simp
+  next
+    case True
     let ?A = "S - {a}"
     let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
+    from True have eq: "S = ?A \<union> ?B" by blast
     have dj: "?A \<inter> ?B = {}" by simp
     from fS have fAB: "finite ?A" "finite ?B" by auto
     have "F ?f S = F ?f ?A \<^bold>* F ?f ?B"
-      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]]
-      by simp
-    then have ?thesis using a by simp }
-  ultimately show ?thesis by blast
+      using union_disjoint [OF fAB dj, of ?f, unfolded eq [symmetric]] by simp
+    with True show ?thesis by simp
+  qed
 qed
 
 lemma delta':
-  assumes fS: "finite S"
+  assumes fin: "finite S"
   shows "F (\<lambda>k. if a = k then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
-  using delta [OF fS, of a b, symmetric] by (auto intro: cong)
+  using delta [OF fin, of a b, symmetric] by (auto intro: cong)
 
 lemma If_cases:
   fixes P :: "'b \<Rightarrow> bool" and g h :: "'b \<Rightarrow> 'a"
-  assumes fA: "finite A"
-  shows "F (\<lambda>x. if P x then h x else g x) A =
-    F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
+  assumes fin: "finite A"
+  shows "F (\<lambda>x. if P x then h x else g x) A = F h (A \<inter> {x. P x}) \<^bold>* F g (A \<inter> - {x. P x})"
 proof -
-  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}"
-          "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
+  have a: "A = A \<inter> {x. P x} \<union> A \<inter> -{x. P x}" "(A \<inter> {x. P x}) \<inter> (A \<inter> -{x. P x}) = {}"
     by blast+
-  from fA
-  have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
+  from fin have f: "finite (A \<inter> {x. P x})" "finite (A \<inter> -{x. P x})" by auto
   let ?g = "\<lambda>x. if P x then h x else g x"
-  from union_disjoint [OF f a(2), of ?g] a(1)
-  show ?thesis
+  from union_disjoint [OF f a(2), of ?g] a(1) show ?thesis
     by (subst (1 2) cong) simp_all
 qed
 
-lemma cartesian_product:
-   "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
-apply (rule sym)
-apply (cases "finite A")
- apply (cases "finite B")
-  apply (simp add: Sigma)
- apply (cases "A={}", simp)
- apply simp
-apply (auto intro: infinite dest: finite_cartesian_productD2)
-apply (cases "B = {}") apply (auto intro: infinite dest: finite_cartesian_productD1)
-done
+lemma cartesian_product: "F (\<lambda>x. F (g x) B) A = F (case_prod g) (A \<times> B)"
+  apply (rule sym)
+  apply (cases "finite A")
+   apply (cases "finite B")
+    apply (simp add: Sigma)
+   apply (cases "A = {}")
+    apply simp
+   apply simp
+   apply (auto intro: infinite dest: finite_cartesian_productD2)
+  apply (cases "B = {}")
+   apply (auto intro: infinite dest: finite_cartesian_productD1)
+  done
 
 lemma inter_restrict:
   assumes "finite A"
   shows "F g (A \<inter> B) = F (\<lambda>x. if x \<in> B then g x else \<^bold>1) A"
 proof -
   let ?g = "\<lambda>x. if x \<in> A \<inter> B then g x else \<^bold>1"
-  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1"
-   by simp
+  have "\<forall>i\<in>A - A \<inter> B. (if i \<in> A \<inter> B then g i else \<^bold>1) = \<^bold>1" by simp
   moreover have "A \<inter> B \<subseteq> A" by blast
-  ultimately have "F ?g (A \<inter> B) = F ?g A" using \<open>finite A\<close>
-    by (intro mono_neutral_left) auto
+  ultimately have "F ?g (A \<inter> B) = F ?g A"
+    using \<open>finite A\<close> by (intro mono_neutral_left) auto
   then show ?thesis by simp
 qed
 
@@ -377,27 +387,28 @@
 
 lemma Union_comp:
   assumes "\<forall>A \<in> B. finite A"
-    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B  \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
+    and "\<And>A1 A2 x. A1 \<in> B \<Longrightarrow> A2 \<in> B \<Longrightarrow> A1 \<noteq> A2 \<Longrightarrow> x \<in> A1 \<Longrightarrow> x \<in> A2 \<Longrightarrow> g x = \<^bold>1"
   shows "F g (\<Union>B) = (F \<circ> F) g B"
-using assms proof (induct B rule: infinite_finite_induct)
+  using assms
+proof (induct B rule: infinite_finite_induct)
   case (infinite A)
   then have "\<not> finite (\<Union>A)" by (blast dest: finite_UnionD)
   with infinite show ?case by simp
 next
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
   case (insert A B)
   then have "finite A" "finite B" "finite (\<Union>B)" "A \<notin> B"
     and "\<forall>x\<in>A \<inter> \<Union>B. g x = \<^bold>1"
-    and H: "F g (\<Union>B) = (F o F) g B" by auto
+    and H: "F g (\<Union>B) = (F \<circ> F) g B" by auto
   then have "F g (A \<union> \<Union>B) = F g A \<^bold>* F g (\<Union>B)"
     by (simp add: union_inter_neutral)
   with \<open>finite B\<close> \<open>A \<notin> B\<close> show ?case
     by (simp add: H)
 qed
 
-lemma commute:
-  "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
+lemma commute: "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
   unfolding cartesian_product
   by (rule reindex_bij_witness [where i = "\<lambda>(i, j). (j, i)" and j = "\<lambda>(i, j). (j, i)"]) auto
 
@@ -412,13 +423,11 @@
   shows "F g (A <+> B) = F (g \<circ> Inl) A \<^bold>* F (g \<circ> Inr) B"
 proof -
   have "A <+> B = Inl ` A \<union> Inr ` B" by auto
-  moreover from fin have "finite (Inl ` A :: ('b + 'c) set)" "finite (Inr ` B :: ('b + 'c) set)"
-    by auto
-  moreover have "Inl ` A \<inter> Inr ` B = ({} :: ('b + 'c) set)" by auto
-  moreover have "inj_on (Inl :: 'b \<Rightarrow> 'b + 'c) A" "inj_on (Inr :: 'c \<Rightarrow> 'b + 'c) B"
-    by (auto intro: inj_onI)
-  ultimately show ?thesis using fin
-    by (simp add: union_disjoint reindex)
+  moreover from fin have "finite (Inl ` A)" "finite (Inr ` B)" by auto
+  moreover have "Inl ` A \<inter> Inr ` B = {}" by auto
+  moreover have "inj_on Inl A" "inj_on Inr B" by (auto intro: inj_onI)
+  ultimately show ?thesis
+    using fin by (simp add: union_disjoint reindex)
 qed
 
 lemma same_carrier:
@@ -427,22 +436,22 @@
   assumes trivial: "\<And>a. a \<in> C - A \<Longrightarrow> g a = \<^bold>1" "\<And>b. b \<in> C - B \<Longrightarrow> h b = \<^bold>1"
   shows "F g A = F h B \<longleftrightarrow> F g C = F h C"
 proof -
-  from \<open>finite C\<close> subset have
-    "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
-    by (auto elim: finite_subset)
+  have "finite A" and "finite B" and "finite (C - A)" and "finite (C - B)"
+    using \<open>finite C\<close> subset by (auto elim: finite_subset)
   from subset have [simp]: "A - (C - A) = A" by auto
   from subset have [simp]: "B - (C - B) = B" by auto
   from subset have "C = A \<union> (C - A)" by auto
   then have "F g C = F g (A \<union> (C - A))" by simp
   also have "\<dots> = F g (A - (C - A)) \<^bold>* F g (C - A - A) \<^bold>* F g (A \<inter> (C - A))"
     using \<open>finite A\<close> \<open>finite (C - A)\<close> by (simp only: union_diff2)
-  finally have P: "F g C = F g A" using trivial by simp
+  finally have *: "F g C = F g A" using trivial by simp
   from subset have "C = B \<union> (C - B)" by auto
   then have "F h C = F h (B \<union> (C - B))" by simp
   also have "\<dots> = F h (B - (C - B)) \<^bold>* F h (C - B - B) \<^bold>* F h (B \<inter> (C - B))"
     using \<open>finite B\<close> \<open>finite (C - B)\<close> by (simp only: union_diff2)
-  finally have Q: "F h C = F h B" using trivial by simp
-  from P Q show ?thesis by simp
+  finally have "F h C = F h B"
+    using trivial by simp
+  with * show ?thesis by simp
 qed
 
 lemma same_carrierI:
@@ -462,8 +471,7 @@
 begin
 
 sublocale setsum: comm_monoid_set plus 0
-defines
-  setsum = setsum.F ..
+  defines setsum = setsum.F ..
 
 abbreviation Setsum ("\<Sum>_" [1000] 999)
   where "\<Sum>A \<equiv> setsum (\<lambda>x. x) A"
@@ -504,27 +512,28 @@
 in [(@{const_syntax setsum}, K setsum_tr')] end
 \<close>
 
-text \<open>TODO generalization candidates\<close>
+(* TODO generalization candidates *)
 
 lemma (in comm_monoid_add) setsum_image_gen:
-  assumes fS: "finite S"
+  assumes fin: "finite S"
   shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
-proof-
-  { fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto }
-  hence "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
+proof -
+  have "{y. y\<in> f`S \<and> f x = y} = {f x}" if "x \<in> S" for x
+    using that by auto
+  then have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
     by simp
   also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
-    by (rule setsum.commute_restrict [OF fS finite_imageI [OF fS]])
+    by (rule setsum.commute_restrict [OF fin finite_imageI [OF fin]])
   finally show ?thesis .
 qed
 
 
 subsubsection \<open>Properties in more restricted classes of structures\<close>
 
-lemma setsum_Un: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: 'a :: ab_group_add) =
-   setsum f A + setsum f B - setsum f (A Int B)"
-by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
+lemma setsum_Un:
+  "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
+  for f :: "'b \<Rightarrow> 'a::ab_group_add"
+  by (subst setsum.union_inter [symmetric]) (auto simp add: algebra_simps)
 
 lemma setsum_Un2:
   assumes "finite (A \<union> B)"
@@ -532,26 +541,30 @@
 proof -
   have "A \<union> B = A - B \<union> (B - A) \<union> A \<inter> B"
     by auto
-  with assms show ?thesis by simp (subst setsum.union_disjoint, auto)+
+  with assms show ?thesis
+    by simp (subst setsum.union_disjoint, auto)+
 qed
 
-lemma setsum_diff1: "finite A \<Longrightarrow>
-  (setsum f (A - {a}) :: ('a::ab_group_add)) =
-  (if a:A then setsum f A - f a else setsum f A)"
-by (erule finite_induct) (auto simp add: insert_Diff_if)
+lemma setsum_diff1:
+  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
+  assumes "finite A"
+  shows "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
+  using assms by induct (auto simp: insert_Diff_if)
 
 lemma setsum_diff:
-  assumes le: "finite A" "B \<subseteq> A"
-  shows "setsum f (A - B) = setsum f A - ((setsum f B)::('a::ab_group_add))"
+  fixes f :: "'b \<Rightarrow> 'a::ab_group_add"
+  assumes "finite A" "B \<subseteq> A"
+  shows "setsum f (A - B) = setsum f A - setsum f B"
 proof -
-  from le have finiteB: "finite B" using finite_subset by auto
-  show ?thesis using finiteB le
+  from assms(2,1) have "finite B" by (rule finite_subset)
+  from this \<open>B \<subseteq> A\<close>
+  show ?thesis
   proof induct
     case empty
-    thus ?case by auto
+    thus ?case by simp
   next
     case (insert x F)
-    thus ?case using le finiteB
+    with \<open>finite A\<close> \<open>finite B\<close> show ?case
       by (simp add: Diff_insert[where a=x and B=F] setsum_diff1 insert_absorb)
   qed
 qed
@@ -561,45 +574,52 @@
   shows "(\<Sum>i\<in>K. f i) \<le> (\<Sum>i\<in>K. g i)"
 proof (cases "finite K")
   case True
-  thus ?thesis using le
+  from this le show ?thesis
   proof induct
     case empty
-    thus ?case by simp
+    then show ?case by simp
   next
     case insert
-    thus ?case using add_mono by fastforce
+    then show ?case using add_mono by fastforce
   qed
 next
-  case False then show ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
 lemma (in strict_ordered_comm_monoid_add) setsum_strict_mono:
-  assumes "finite A"  "A \<noteq> {}" and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
+  assumes "finite A" "A \<noteq> {}"
+    and "\<And>x. x \<in> A \<Longrightarrow> f x < g x"
   shows "setsum f A < setsum g A"
   using assms
 proof (induct rule: finite_ne_induct)
-  case singleton thus ?case by simp
+  case singleton
+  then show ?case by simp
 next
-  case insert thus ?case by (auto simp: add_strict_mono)
+  case insert
+  then show ?case by (auto simp: add_strict_mono)
 qed
 
 lemma setsum_strict_mono_ex1:
   fixes f g :: "'i \<Rightarrow> 'a::ordered_cancel_comm_monoid_add"
-  assumes "finite A" and "\<forall>x\<in>A. f x \<le> g x" and "\<exists>a\<in>A. f a < g a"
+  assumes "finite A"
+    and "\<forall>x\<in>A. f x \<le> g x"
+    and "\<exists>a\<in>A. f a < g a"
   shows "setsum f A < setsum g A"
 proof-
-  from assms(3) obtain a where a: "a:A" "f a < g a" by blast
-  have "setsum f A = setsum f ((A-{a}) \<union> {a})"
-    by(simp add:insert_absorb[OF \<open>a:A\<close>])
-  also have "\<dots> = setsum f (A-{a}) + setsum f {a}"
+  from assms(3) obtain a where a: "a \<in> A" "f a < g a" by blast
+  have "setsum f A = setsum f ((A - {a}) \<union> {a})"
+    by(simp add: insert_absorb[OF \<open>a \<in> A\<close>])
+  also have "\<dots> = setsum f (A - {a}) + setsum f {a}"
     using \<open>finite A\<close> by(subst setsum.union_disjoint) auto
-  also have "setsum f (A-{a}) \<le> setsum g (A-{a})"
-    by(rule setsum_mono)(simp add: assms(2))
-  also have "setsum f {a} < setsum g {a}" using a by simp
-  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A-{a}) \<union> {a})"
-    using \<open>finite A\<close> by(subst setsum.union_disjoint[symmetric]) auto
-  also have "\<dots> = setsum g A" by(simp add:insert_absorb[OF \<open>a:A\<close>])
-  finally show ?thesis by (auto simp add: add_right_mono add_strict_left_mono)
+  also have "setsum f (A - {a}) \<le> setsum g (A - {a})"
+    by (rule setsum_mono) (simp add: assms(2))
+  also from a have "setsum f {a} < setsum g {a}" by simp
+  also have "setsum g (A - {a}) + setsum g {a} = setsum g((A - {a}) \<union> {a})"
+    using \<open>finite A\<close> by (subst setsum.union_disjoint[symmetric]) auto
+  also have "\<dots> = setsum g A" by (simp add: insert_absorb[OF \<open>a \<in> A\<close>])
+  finally show ?thesis
+    by (auto simp add: add_right_mono add_strict_left_mono)
 qed
 
 lemma setsum_mono_inv:
@@ -609,51 +629,67 @@
   assumes i: "i \<in> I"
   assumes I: "finite I"
   shows "f i = g i"
-proof(rule ccontr)
-  assume "f i \<noteq> g i"
+proof (rule ccontr)
+  assume "\<not> ?thesis"
   with le[OF i] have "f i < g i" by simp
-  hence "\<exists>i\<in>I. f i < g i" using i ..
-  from setsum_strict_mono_ex1[OF I _ this] le have "setsum f I < setsum g I" by blast
+  with i have "\<exists>i\<in>I. f i < g i" ..
+  from setsum_strict_mono_ex1[OF I _ this] le have "setsum f I < setsum g I"
+    by blast
   with eq show False by simp
 qed
 
-lemma setsum_negf: "(\<Sum>x\<in>A. - f x::'a::ab_group_add) = - (\<Sum>x\<in>A. f x)"
+lemma setsum_negf: "(\<Sum>x\<in>A. - f x) = - (\<Sum>x\<in>A. f x)"
+  for f :: "'b \<Rightarrow> 'a::ab_group_add"
 proof (cases "finite A")
-  case True thus ?thesis by (induct set: finite) auto
+  case True
+  then show ?thesis by (induct set: finite) auto
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
-lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x::'a::ab_group_add) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+lemma setsum_subtractf: "(\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+  for f g :: "'b \<Rightarrow>'a::ab_group_add"
   using setsum.distrib [of f "- g" A] by (simp add: setsum_negf)
 
 lemma setsum_subtractf_nat:
-  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x::nat) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
-  by (induction A rule: infinite_finite_induct) (auto simp: setsum_mono)
+  "(\<And>x. x \<in> A \<Longrightarrow> g x \<le> f x) \<Longrightarrow> (\<Sum>x\<in>A. f x - g x) = (\<Sum>x\<in>A. f x) - (\<Sum>x\<in>A. g x)"
+  for f g :: "'a \<Rightarrow> nat"
+  by (induct A rule: infinite_finite_induct) (auto simp: setsum_mono)
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg:
+context ordered_comm_monoid_add
+begin
+
+lemma setsum_nonneg:
   assumes nn: "\<forall>x\<in>A. 0 \<le> f x"
   shows "0 \<le> setsum f A"
 proof (cases "finite A")
-  case True thus ?thesis using nn
+  case True
+  then show ?thesis
+    using nn
   proof induct
-    case empty then show ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert x F)
     then have "0 + 0 \<le> f x + setsum f F" by (blast intro: add_mono)
     with insert show ?case by simp
   qed
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_nonpos:
+lemma setsum_nonpos:
   assumes np: "\<forall>x\<in>A. f x \<le> 0"
   shows "setsum f A \<le> 0"
 proof (cases "finite A")
-  case True thus ?thesis using np
+  case True
+  then show ?thesis
+    using np
   proof induct
-    case empty then show ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert x F)
     then have "f x + setsum f F \<le> 0 + 0" by (blast intro: add_mono)
@@ -663,232 +699,259 @@
   case False thus ?thesis by simp
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg_eq_0_iff:
+lemma setsum_nonneg_eq_0_iff:
   "finite A \<Longrightarrow> \<forall>x\<in>A. 0 \<le> f x \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
-  by (induct set: finite, simp) (simp add: add_nonneg_eq_0_iff setsum_nonneg)
+  by (induct set: finite) (simp_all add: add_nonneg_eq_0_iff setsum_nonneg)
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg_0:
+lemma setsum_nonneg_0:
   "finite s \<Longrightarrow> (\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0) \<Longrightarrow> (\<Sum> i \<in> s. f i) = 0 \<Longrightarrow> i \<in> s \<Longrightarrow> f i = 0"
   by (simp add: setsum_nonneg_eq_0_iff)
 
-lemma (in ordered_comm_monoid_add) setsum_nonneg_leq_bound:
+lemma setsum_nonneg_leq_bound:
   assumes "finite s" "\<And>i. i \<in> s \<Longrightarrow> f i \<ge> 0" "(\<Sum>i \<in> s. f i) = B" "i \<in> s"
   shows "f i \<le> B"
 proof -
-  have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
-    using assms by (intro add_increasing2 setsum_nonneg) auto
+  from assms have "f i \<le> f i + (\<Sum>i \<in> s - {i}. f i)"
+    by (intro add_increasing2 setsum_nonneg) auto
   also have "\<dots> = B"
     using setsum.remove[of s i f] assms by simp
   finally show ?thesis by auto
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_mono2:
-  assumes fin: "finite B" and sub: "A \<subseteq> B" and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
+lemma setsum_mono2:
+  assumes fin: "finite B"
+    and sub: "A \<subseteq> B"
+    and nn: "\<And>b. b \<in> B-A \<Longrightarrow> 0 \<le> f b"
   shows "setsum f A \<le> setsum f B"
 proof -
   have "setsum f A \<le> setsum f A + setsum f (B-A)"
     by(simp add: add_increasing2[OF setsum_nonneg] nn Ball_def)
-  also have "\<dots> = setsum f (A \<union> (B-A))" using fin finite_subset[OF sub fin]
-    by (simp add: setsum.union_disjoint del:Un_Diff_cancel)
-  also have "A \<union> (B-A) = B" using sub by blast
+  also from fin finite_subset[OF sub fin] have "\<dots> = setsum f (A \<union> (B-A))"
+    by (simp add: setsum.union_disjoint del: Un_Diff_cancel)
+  also from sub have "A \<union> (B-A) = B" by blast
   finally show ?thesis .
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_le_included:
+lemma setsum_le_included:
   assumes "finite s" "finite t"
   and "\<forall>y\<in>t. 0 \<le> g y" "(\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<le> g y)"
   shows "setsum f s \<le> setsum g t"
 proof -
   have "setsum f s \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) s"
   proof (rule setsum_mono)
-    fix y assume "y \<in> s"
+    fix y
+    assume "y \<in> s"
     with assms obtain z where z: "z \<in> t" "y = i z" "f y \<le> g z" by auto
     with assms show "f y \<le> setsum g {x \<in> t. i x = y}" (is "?A y \<le> ?B y")
       using order_trans[of "?A (i z)" "setsum g {z}" "?B (i z)", intro]
       by (auto intro!: setsum_mono2)
   qed
-  also have "... \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
+  also have "\<dots> \<le> setsum (\<lambda>y. setsum g {x. x\<in>t \<and> i x = y}) (i ` t)"
     using assms(2-4) by (auto intro!: setsum_mono2 setsum_nonneg)
-  also have "... \<le> setsum g t"
+  also have "\<dots> \<le> setsum g t"
     using assms by (auto simp: setsum_image_gen[symmetric])
   finally show ?thesis .
 qed
 
-lemma (in ordered_comm_monoid_add) setsum_mono3:
-  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
+lemma setsum_mono3: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> \<forall>x\<in>B - A. 0 \<le> f x \<Longrightarrow> setsum f A \<le> setsum f B"
   by (rule setsum_mono2) auto
 
+end
+
 lemma (in canonically_ordered_monoid_add) setsum_eq_0_iff [simp]:
   "finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
   by (intro ballI setsum_nonneg_eq_0_iff zero_le)
 
 lemma setsum_right_distrib:
-  fixes f :: "'a => ('b::semiring_0)"
-  shows "r * setsum f A = setsum (%n. r * f n) A"
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
+  shows "r * setsum f A = setsum (\<lambda>n. r * f n) A"
 proof (cases "finite A")
   case True
-  thus ?thesis
+  then show ?thesis
   proof induct
-    case empty thus ?case by simp
+    case empty
+    then show ?case by simp
   next
-    case (insert x A) thus ?case by (simp add: distrib_left)
+    case insert
+    then show ?case by (simp add: distrib_left)
   qed
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
-lemma setsum_left_distrib:
-  "setsum f A * (r::'a::semiring_0) = (\<Sum>n\<in>A. f n * r)"
+lemma setsum_left_distrib: "setsum f A * r = (\<Sum>n\<in>A. f n * r)"
+  for r :: "'a::semiring_0"
 proof (cases "finite A")
   case True
   then show ?thesis
   proof induct
-    case empty thus ?case by simp
+    case empty
+    then show ?case by simp
   next
-    case (insert x A) thus ?case by (simp add: distrib_right)
+    case insert
+    then show ?case by (simp add: distrib_right)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_divide_distrib: "setsum f A / r = (\<Sum>n\<in>A. f n / r)"
+  for r :: "'a::field"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+  proof induct
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (simp add: add_divide_distrib)
   qed
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
-lemma setsum_divide_distrib:
-  "setsum f A / (r::'a::field) = (\<Sum>n\<in>A. f n / r)"
+lemma setsum_abs[iff]: "\<bar>setsum f A\<bar> \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+  proof induct
+    case empty
+    then show ?case by simp
+  next
+    case insert
+    then show ?case by (auto intro: abs_triangle_ineq order_trans)
+  qed
+next
+  case False
+  then show ?thesis by simp
+qed
+
+lemma setsum_abs_ge_zero[iff]: "0 \<le> setsum (\<lambda>i. \<bar>f i\<bar>) A"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
+  by (simp add: setsum_nonneg)
+
+lemma abs_setsum_abs[simp]: "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
+  for f :: "'a \<Rightarrow> 'b::ordered_ab_group_add_abs"
 proof (cases "finite A")
   case True
   then show ?thesis
   proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A) thus ?case by (simp add: add_divide_distrib)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "\<bar>setsum f A\<bar> \<le> setsum (%i. \<bar>f i\<bar>) A"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
-  next
-    case (insert x A)
-    thus ?case by (auto intro: abs_triangle_ineq order_trans)
-  qed
-next
-  case False thus ?thesis by simp
-qed
-
-lemma setsum_abs_ge_zero[iff]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "0 \<le> setsum (%i. \<bar>f i\<bar>) A"
-  by (simp add: setsum_nonneg)
-
-lemma abs_setsum_abs[simp]:
-  fixes f :: "'a => ('b::ordered_ab_group_add_abs)"
-  shows "\<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar> = (\<Sum>a\<in>A. \<bar>f a\<bar>)"
-proof (cases "finite A")
-  case True
-  thus ?thesis
-  proof induct
-    case empty thus ?case by simp
+    case empty
+    then show ?case by simp
   next
     case (insert a A)
-    hence "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
-    also have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>"  using insert by simp
-    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>"
-      by (simp del: abs_of_nonneg)
-    also have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" using insert by simp
+    then have "\<bar>\<Sum>a\<in>insert a A. \<bar>f a\<bar>\<bar> = \<bar>\<bar>f a\<bar> + (\<Sum>a\<in>A. \<bar>f a\<bar>)\<bar>" by simp
+    also from insert have "\<dots> = \<bar>\<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>\<bar>" by simp
+    also have "\<dots> = \<bar>f a\<bar> + \<bar>\<Sum>a\<in>A. \<bar>f a\<bar>\<bar>" by (simp del: abs_of_nonneg)
+    also from insert have "\<dots> = (\<Sum>a\<in>insert a A. \<bar>f a\<bar>)" by simp
     finally show ?case .
   qed
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
-lemma setsum_diff1_ring: assumes "finite A" "a \<in> A"
-  shows "setsum f (A - {a}) = setsum f A - (f a::'a::ring)"
+lemma setsum_diff1_ring:
+  fixes f :: "'b \<Rightarrow> 'a::ring"
+  assumes "finite A" "a \<in> A"
+  shows "setsum f (A - {a}) = setsum f A - (f a)"
   unfolding setsum.remove [OF assms] by auto
 
 lemma setsum_product:
-  fixes f :: "'a => ('b::semiring_0)"
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
   shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
   by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
 
 lemma setsum_mult_setsum_if_inj:
-fixes f :: "'a => ('b::semiring_0)"
-shows "inj_on (%(a,b). f a * g b) (A \<times> B) ==>
-  setsum f A * setsum g B = setsum id {f a * g b|a b. a:A & b:B}"
-by(auto simp: setsum_product setsum.cartesian_product
-        intro!:  setsum.reindex_cong[symmetric])
+  fixes f :: "'a \<Rightarrow> 'b::semiring_0"
+  shows "inj_on (\<lambda>(a, b). f a * g b) (A \<times> B) \<Longrightarrow>
+    setsum f A * setsum g B = setsum id {f a * g b |a b. a \<in> A \<and> b \<in> B}"
+  by(auto simp: setsum_product setsum.cartesian_product intro!: setsum.reindex_cong[symmetric])
 
-lemma setsum_SucD: "setsum f A = Suc n ==> EX a:A. 0 < f a"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule rev_mp)
-apply (erule finite_induct, auto)
-done
+lemma setsum_SucD:
+  assumes "setsum f A = Suc n"
+  shows "\<exists>a\<in>A. 0 < f a"
+proof (cases "finite A")
+  case True
+  from this assms show ?thesis by induct auto
+next
+  case False
+  with assms show ?thesis by simp
+qed
 
-lemma setsum_eq_Suc0_iff: "finite A \<Longrightarrow>
-  setsum f A = Suc 0 \<longleftrightarrow> (EX a:A. f a = Suc 0 & (ALL b:A. a\<noteq>b \<longrightarrow> f b = 0))"
-apply(erule finite_induct)
-apply (auto simp add:add_is_1)
-done
+lemma setsum_eq_Suc0_iff:
+  assumes "finite A"
+  shows "setsum f A = Suc 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
+  using assms by induct (auto simp add:add_is_1)
 
 lemmas setsum_eq_1_iff = setsum_eq_Suc0_iff[simplified One_nat_def[symmetric]]
 
-lemma setsum_Un_nat: "finite A ==> finite B ==>
-  (setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
+lemma setsum_Un_nat:
+  "finite A \<Longrightarrow> finite B \<Longrightarrow> setsum f (A \<union> B) = setsum f A + setsum f B - setsum f (A \<inter> B)"
+  for f :: "'a \<Rightarrow> nat"
   \<comment> \<open>For the natural numbers, we have subtraction.\<close>
-by (subst setsum.union_inter [symmetric], auto simp add: algebra_simps)
+  by (subst setsum.union_inter [symmetric]) (auto simp: algebra_simps)
 
-lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
-  (if a:A then setsum f A - f a else setsum f A)"
-apply (case_tac "finite A")
- prefer 2 apply simp
-apply (erule finite_induct)
- apply (auto simp add: insert_Diff_if)
-apply (drule_tac a = a in mk_disjoint_insert, auto)
-done
+lemma setsum_diff1_nat: "setsum f (A - {a}) = (if a \<in> A then setsum f A - f a else setsum f A)"
+  for f :: "'a \<Rightarrow> nat"
+proof (cases "finite A")
+  case True
+  then show ?thesis
+    apply induct
+     apply (auto simp: insert_Diff_if)
+    apply (drule mk_disjoint_insert)
+    apply auto
+    done
+next
+  case False
+  then show ?thesis by simp
+qed
 
 lemma setsum_diff_nat:
-assumes "finite B" and "B \<subseteq> A"
-shows "(setsum f (A - B) :: nat) = (setsum f A) - (setsum f B)"
-using assms
+  fixes f :: "'a \<Rightarrow> nat"
+  assumes "finite B" and "B \<subseteq> A"
+  shows "setsum f (A - B) = setsum f A - setsum f B"
+  using assms
 proof induct
-  show "setsum f (A - {}) = (setsum f A) - (setsum f {})" by simp
+  case empty
+  then show ?case by simp
 next
-  fix F x assume finF: "finite F" and xnotinF: "x \<notin> F"
-    and xFinA: "insert x F \<subseteq> A"
-    and IH: "F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F"
-  from xnotinF xFinA have xinAF: "x \<in> (A - F)" by simp
-  from xinAF have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
+  case (insert x F)
+  note IH = \<open>F \<subseteq> A \<Longrightarrow> setsum f (A - F) = setsum f A - setsum f F\<close>
+  from \<open>x \<notin> F\<close> \<open>insert x F \<subseteq> A\<close> have "x \<in> A - F" by simp
+  then have A: "setsum f ((A - F) - {x}) = setsum f (A - F) - f x"
     by (simp add: setsum_diff1_nat)
-  from xFinA have "F \<subseteq> A" by simp
+  from \<open>insert x F \<subseteq> A\<close> have "F \<subseteq> A" by simp
   with IH have "setsum f (A - F) = setsum f A - setsum f F" by simp
   with A have B: "setsum f ((A - F) - {x}) = setsum f A - setsum f F - f x"
     by simp
-  from xnotinF have "A - insert x F = (A - F) - {x}" by auto
+  from \<open>x \<notin> F\<close> have "A - insert x F = (A - F) - {x}" by auto
   with B have C: "setsum f (A - insert x F) = setsum f A - setsum f F - f x"
     by simp
-  from finF xnotinF have "setsum f (insert x F) = setsum f F + f x" by simp
+  from \<open>finite F\<close> \<open>x \<notin> F\<close> have "setsum f (insert x F) = setsum f F + f x"
+    by simp
   with C have "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)"
     by simp
-  thus "setsum f (A - insert x F) = setsum f A - setsum f (insert x F)" by simp
+  then show ?case by simp
 qed
 
 lemma setsum_comp_morphism:
   assumes "h 0 = 0" and "\<And>x y. h (x + y) = h x + h y"
   shows "setsum (h \<circ> g) A = h (setsum g A)"
 proof (cases "finite A")
-  case False then show ?thesis by (simp add: assms)
+  case False
+  then show ?thesis by (simp add: assms)
 next
-  case True then show ?thesis by (induct A) (simp_all add: assms)
+  case True
+  then show ?thesis by (induct A) (simp_all add: assms)
 qed
 
-lemma (in comm_semiring_1) dvd_setsum:
-  "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
+lemma (in comm_semiring_1) dvd_setsum: "(\<And>a. a \<in> A \<Longrightarrow> d dvd f a) \<Longrightarrow> d dvd setsum f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in ordered_comm_monoid_add) setsum_pos:
@@ -908,17 +971,18 @@
 
 lemma setsum_cong_Suc:
   assumes "0 \<notin> A" "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)"
-  shows   "setsum f A = setsum g A"
+  shows "setsum f A = setsum g A"
 proof (rule setsum.cong)
-  fix x assume "x \<in> A"
-  with assms(1) show "f x = g x" by (cases x) (auto intro!: assms(2))
+  fix x
+  assume "x \<in> A"
+  with assms(1) show "f x = g x"
+    by (cases x) (auto intro!: assms(2))
 qed simp_all
 
 
 subsubsection \<open>Cardinality as special case of @{const setsum}\<close>
 
-lemma card_eq_setsum:
-  "card A = setsum (\<lambda>x. 1) A"
+lemma card_eq_setsum: "card A = setsum (\<lambda>x. 1) A"
 proof -
   have "plus \<circ> (\<lambda>_. Suc 0) = (\<lambda>_. Suc)"
     by (simp add: fun_eq_iff)
@@ -926,45 +990,53 @@
     by (rule arg_cong)
   then have "Finite_Set.fold (plus \<circ> (\<lambda>_. Suc 0)) 0 A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
     by (blast intro: fun_cong)
-  then show ?thesis by (simp add: card.eq_fold setsum.eq_fold)
+  then show ?thesis
+    by (simp add: card.eq_fold setsum.eq_fold)
 qed
 
-lemma setsum_constant [simp]:
-  "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
-apply (cases "finite A")
-apply (erule finite_induct)
-apply (auto simp add: algebra_simps)
-done
+lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat (card A) * y"
+proof (cases "finite A")
+  case True
+  then show ?thesis by induct (auto simp: algebra_simps)
+next
+  case False
+  then show ?thesis by simp
+qed
 
 lemma setsum_Suc: "setsum (\<lambda>x. Suc(f x)) A = setsum f A + card A"
-  using setsum.distrib[of f "\<lambda>_. 1" A]
-  by simp
+  using setsum.distrib[of f "\<lambda>_. 1" A] by simp
 
 lemma setsum_bounded_above:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{semiring_1, ordered_comm_monoid_add})"
+  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> K"
   shows "setsum f A \<le> of_nat (card A) * K"
 proof (cases "finite A")
   case True
-  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
+  then show ?thesis
+    using le setsum_mono[where K=A and g = "\<lambda>x. K"] by simp
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
 lemma setsum_bounded_above_strict:
-  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < (K::'a::{ordered_cancel_comm_monoid_add,semiring_1})"
-          "card A > 0"
+  fixes K :: "'a::{ordered_cancel_comm_monoid_add,semiring_1}"
+  assumes "\<And>i. i\<in>A \<Longrightarrow> f i < K" "card A > 0"
   shows "setsum f A < of_nat (card A) * K"
-using assms setsum_strict_mono[where A=A and g = "%x. K"]
-by (simp add: card_gt_0_iff)
+  using assms setsum_strict_mono[where A=A and g = "\<lambda>x. K"]
+  by (simp add: card_gt_0_iff)
 
 lemma setsum_bounded_below:
-  assumes le: "\<And>i. i\<in>A \<Longrightarrow> (K::'a::{semiring_1, ordered_comm_monoid_add}) \<le> f i"
+  fixes K :: "'a::{semiring_1,ordered_comm_monoid_add}"
+  assumes le: "\<And>i. i\<in>A \<Longrightarrow> K \<le> f i"
   shows "of_nat (card A) * K \<le> setsum f A"
 proof (cases "finite A")
   case True
-  thus ?thesis using le setsum_mono[where K=A and f = "%x. K"] by simp
+  then show ?thesis
+    using le setsum_mono[where K=A and f = "%x. K"] by simp
 next
-  case False thus ?thesis by simp
+  case False
+  then show ?thesis by simp
 qed
 
 lemma card_UN_disjoint:
@@ -972,24 +1044,26 @@
     and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}"
   shows "card (UNION I A) = (\<Sum>i\<in>I. card(A i))"
 proof -
-  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)" by simp
-  with assms show ?thesis by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
+  have "(\<Sum>i\<in>I. card (A i)) = (\<Sum>i\<in>I. \<Sum>x\<in>A i. 1)"
+    by simp
+  with assms show ?thesis
+    by (simp add: card_eq_setsum setsum.UNION_disjoint del: setsum_constant)
 qed
 
 lemma card_Union_disjoint:
-  "finite C ==> (ALL A:C. finite A) ==>
-   (ALL A:C. ALL B:C. A \<noteq> B --> A Int B = {})
-   ==> card (\<Union>C) = setsum card C"
-apply (frule card_UN_disjoint [of C id])
-apply simp_all
-done
+  "finite C \<Longrightarrow> \<forall>A\<in>C. finite A \<Longrightarrow> \<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {} \<Longrightarrow>
+    card (\<Union>C) = setsum card C"
+  by (frule card_UN_disjoint [of C id]) simp_all
 
 lemma setsum_multicount_gen:
   assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
-  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t" (is "?l = ?r")
+  shows "setsum (\<lambda>i. (card {j\<in>t. R i j})) s = setsum k t"
+    (is "?l = ?r")
 proof-
-  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s" by auto
-  also have "\<dots> = ?r" unfolding setsum.commute_restrict [OF assms(1-2)]
+  have "?l = setsum (\<lambda>i. setsum (\<lambda>x.1) {j\<in>t. R i j}) s"
+    by auto
+  also have "\<dots> = ?r"
+    unfolding setsum.commute_restrict [OF assms(1-2)]
     using assms(3) by auto
   finally show ?thesis .
 qed
@@ -998,17 +1072,18 @@
   assumes "finite S" "finite T" "\<forall>j\<in>T. (card {i\<in>S. R i j} = k)"
   shows "setsum (\<lambda>i. card {j\<in>T. R i j}) S = k * card T" (is "?l = ?r")
 proof-
-  have "?l = setsum (\<lambda>i. k) T" by (rule setsum_multicount_gen) (auto simp: assms)
+  have "?l = setsum (\<lambda>i. k) T"
+    by (rule setsum_multicount_gen) (auto simp: assms)
   also have "\<dots> = ?r" by (simp add: mult.commute)
   finally show ?thesis by auto
 qed
 
+
 subsubsection \<open>Cardinality of products\<close>
 
 lemma card_SigmaI [simp]:
-  "\<lbrakk> finite A; ALL a:A. finite (B a) \<rbrakk>
-  \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
-by(simp add: card_eq_setsum setsum.Sigma del:setsum_constant)
+  "finite A \<Longrightarrow> \<forall>a\<in>A. finite (B a) \<Longrightarrow> card (SIGMA x: A. B x) = (\<Sum>a\<in>A. card (B a))"
+  by (simp add: card_eq_setsum setsum.Sigma del: setsum_constant)
 
 (*
 lemma SigmaI_insert: "y \<notin> A ==>
@@ -1016,12 +1091,12 @@
   by auto
 *)
 
-lemma card_cartesian_product: "card (A \<times> B) = card(A) * card(B)"
+lemma card_cartesian_product: "card (A \<times> B) = card A * card B"
   by (cases "finite A \<and> finite B")
     (auto simp add: card_eq_0_iff dest: finite_cartesian_productD1 finite_cartesian_productD2)
 
-lemma card_cartesian_product_singleton:  "card({x} \<times> A) = card(A)"
-by (simp add: card_cartesian_product)
+lemma card_cartesian_product_singleton:  "card ({x} \<times> A) = card A"
+  by (simp add: card_cartesian_product)
 
 
 subsection \<open>Generalized product over a set\<close>
@@ -1030,12 +1105,10 @@
 begin
 
 sublocale setprod: comm_monoid_set times 1
-defines
-  setprod = setprod.F ..
+  defines setprod = setprod.F ..
 
-abbreviation
-  Setprod ("\<Prod>_" [1000] 999) where
-  "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
+abbreviation Setprod ("\<Prod>_" [1000] 999)
+  where "\<Prod>A \<equiv> setprod (\<lambda>x. x) A"
 
 end
 
@@ -1058,22 +1131,26 @@
 context comm_monoid_mult
 begin
 
-lemma setprod_dvd_setprod:
-  "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
+lemma setprod_dvd_setprod: "(\<And>a. a \<in> A \<Longrightarrow> f a dvd g a) \<Longrightarrow> setprod f A dvd setprod g A"
 proof (induct A rule: infinite_finite_induct)
-  case infinite then show ?case by (auto intro: dvdI)
+  case infinite
+  then show ?case by (auto intro: dvdI)
+next
+  case empty
+  then show ?case by (auto intro: dvdI)
 next
-  case empty then show ?case by (auto intro: dvdI)
-next
-  case (insert a A) then
-  have "f a dvd g a" and "setprod f A dvd setprod g A" by simp_all
-  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s" by (auto elim!: dvdE)
-  then have "g a * setprod g A = f a * setprod f A * (r * s)" by (simp add: ac_simps)
-  with insert.hyps show ?case by (auto intro: dvdI)
+  case (insert a A)
+  then have "f a dvd g a" and "setprod f A dvd setprod g A"
+    by simp_all
+  then obtain r s where "g a = f a * r" and "setprod g A = setprod f A * s"
+    by (auto elim!: dvdE)
+  then have "g a * setprod g A = f a * setprod f A * (r * s)"
+    by (simp add: ac_simps)
+  with insert.hyps show ?case
+    by (auto intro: dvdI)
 qed
 
-lemma setprod_dvd_setprod_subset:
-  "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
+lemma setprod_dvd_setprod_subset: "finite B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> setprod f A dvd setprod f B"
   by (auto simp add: setprod.subset_diff ac_simps intro: dvdI)
 
 end
@@ -1090,21 +1167,23 @@
 proof -
   from \<open>finite A\<close> have "setprod f (insert a (A - {a})) = f a * setprod f (A - {a})"
     by (intro setprod.insert) auto
-  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A" by blast
+  also from \<open>a \<in> A\<close> have "insert a (A - {a}) = A"
+    by blast
   finally have "setprod f A = f a * setprod f (A - {a})" .
-  with \<open>b = f a\<close> show ?thesis by simp
+  with \<open>b = f a\<close> show ?thesis
+    by simp
 qed
 
-lemma dvd_setprodI [intro]:
-  assumes "finite A" and "a \<in> A"
-  shows "f a dvd setprod f A"
-  using assms by auto
+lemma dvd_setprodI [intro]: "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> f a dvd setprod f A"
+  by auto
 
 lemma setprod_zero:
   assumes "finite A" and "\<exists>a\<in>A. f a = 0"
   shows "setprod f A = 0"
-using assms proof (induct A)
-  case empty then show ?case by simp
+  using assms
+proof (induct A)
+  case empty
+  then show ?case by simp
 next
   case (insert a A)
   then have "f a = 0 \<or> (\<exists>a\<in>A. f a = 0)" by simp
@@ -1126,71 +1205,73 @@
 end
 
 lemma setprod_zero_iff [simp]:
+  fixes f :: "'b \<Rightarrow> 'a::semidom"
   assumes "finite A"
-  shows "setprod f A = (0::'a::semidom) \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
+  shows "setprod f A = 0 \<longleftrightarrow> (\<exists>a\<in>A. f a = 0)"
   using assms by (induct A) (auto simp: no_zero_divisors)
 
 lemma (in semidom_divide) setprod_diff1:
   assumes "finite A" and "f a \<noteq> 0"
   shows "setprod f (A - {a}) = (if a \<in> A then setprod f A div f a else setprod f A)"
 proof (cases "a \<notin> A")
-  case True then show ?thesis by simp
+  case True
+  then show ?thesis by simp
 next
-  case False with assms show ?thesis
-  proof (induct A rule: finite_induct)
-    case empty then show ?case by simp
+  case False
+  with assms show ?thesis
+  proof induct
+    case empty
+    then show ?case by simp
   next
     case (insert b B)
     then show ?case
     proof (cases "a = b")
-      case True with insert show ?thesis by simp
+      case True
+      with insert show ?thesis by simp
     next
-      case False with insert have "a \<in> B" by simp
+      case False
+      with insert have "a \<in> B" by simp
       define C where "C = B - {a}"
-      with \<open>finite B\<close> \<open>a \<in> B\<close>
-        have *: "B = insert a C" "finite C" "a \<notin> C" by auto
-      with insert show ?thesis by (auto simp add: insert_commute ac_simps)
+      with \<open>finite B\<close> \<open>a \<in> B\<close> have "B = insert a C" "finite C" "a \<notin> C"
+        by auto
+      with insert show ?thesis
+        by (auto simp add: insert_commute ac_simps)
     qed
   qed
 qed
 
-lemma setsum_zero_power [simp]:
-  fixes c :: "nat \<Rightarrow> 'a::division_ring"
-  shows "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
-apply (cases "finite A")
-  by (induction A rule: finite_induct) auto
+lemma setsum_zero_power [simp]: "(\<Sum>i\<in>A. c i * 0^i) = (if finite A \<and> 0 \<in> A then c 0 else 0)"
+  for c :: "nat \<Rightarrow> 'a::division_ring"
+  by (induct A rule: infinite_finite_induct) auto
 
 lemma setsum_zero_power' [simp]:
-  fixes c :: "nat \<Rightarrow> 'a::field"
-  shows "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
-  using setsum_zero_power [of "\<lambda>i. c i / d i" A]
-  by auto
+  "(\<Sum>i\<in>A. c i * 0^i / d i) = (if finite A \<and> 0 \<in> A then c 0 / d 0 else 0)"
+  for c :: "nat \<Rightarrow> 'a::field"
+  using setsum_zero_power [of "\<lambda>i. c i / d i" A] by auto
 
 lemma (in field) setprod_inversef:
   "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = inverse (setprod f A)"
   by (induct A rule: finite_induct) simp_all
 
-lemma (in field) setprod_dividef:
-  "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
+lemma (in field) setprod_dividef: "finite A \<Longrightarrow> (\<Prod>x\<in>A. f x / g x) = setprod f A / setprod g A"
   using setprod_inversef [of A g] by (simp add: divide_inverse setprod.distrib)
 
 lemma setprod_Un:
   fixes f :: "'b \<Rightarrow> 'a :: field"
   assumes "finite A" and "finite B"
-  and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
+    and "\<forall>x\<in>A \<inter> B. f x \<noteq> 0"
   shows "setprod f (A \<union> B) = setprod f A * setprod f B / setprod f (A \<inter> B)"
 proof -
   from assms have "setprod f A * setprod f B = setprod f (A \<union> B) * setprod f (A \<inter> B)"
     by (simp add: setprod.union_inter [symmetric, of A B])
-  with assms show ?thesis by simp
+  with assms show ?thesis
+    by simp
 qed
 
-lemma (in linordered_semidom) setprod_nonneg:
-  "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
+lemma (in linordered_semidom) setprod_nonneg: "(\<forall>a\<in>A. 0 \<le> f a) \<Longrightarrow> 0 \<le> setprod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
-lemma (in linordered_semidom) setprod_pos:
-  "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
+lemma (in linordered_semidom) setprod_pos: "(\<forall>a\<in>A. 0 < f a) \<Longrightarrow> 0 < setprod f A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma (in linordered_semidom) setprod_mono:
@@ -1198,71 +1279,69 @@
   by (induct A rule: infinite_finite_induct) (auto intro!: setprod_nonneg mult_mono)
 
 lemma (in linordered_semidom) setprod_mono_strict:
-    assumes"finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
-    shows "setprod f A < setprod g A"
-using assms
-apply (induct A rule: finite_induct)
-apply (simp add: )
-apply (force intro: mult_strict_mono' setprod_nonneg)
-done
+  assumes "finite A" "\<forall>i\<in>A. 0 \<le> f i \<and> f i < g i" "A \<noteq> {}"
+  shows "setprod f A < setprod g A"
+  using assms
+proof (induct A rule: finite_induct)
+  case empty
+  then show ?case by simp
+next
+  case insert
+  then show ?case by (force intro: mult_strict_mono' setprod_nonneg)
+qed
 
-lemma (in linordered_field) abs_setprod:
-  "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
+lemma (in linordered_field) abs_setprod: "\<bar>setprod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
   by (induct A rule: infinite_finite_induct) (simp_all add: abs_mult)
 
-lemma setprod_eq_1_iff [simp]:
-  "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = (1::nat))"
+lemma setprod_eq_1_iff [simp]: "finite A \<Longrightarrow> setprod f A = 1 \<longleftrightarrow> (\<forall>a\<in>A. f a = 1)"
+  for f :: "'a \<Rightarrow> nat"
   by (induct A rule: finite_induct) simp_all
 
-lemma setprod_pos_nat_iff [simp]:
-  "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > (0::nat))"
+lemma setprod_pos_nat_iff [simp]: "finite A \<Longrightarrow> setprod f A > 0 \<longleftrightarrow> (\<forall>a\<in>A. f a > 0)"
+  for f :: "'a \<Rightarrow> nat"
   using setprod_zero_iff by (simp del: neq0_conv add: zero_less_iff_neq_zero)
 
-lemma setprod_constant:
-  "(\<Prod>x\<in> A. (y::'a::comm_monoid_mult)) = y ^ card A"
+lemma setprod_constant: "(\<Prod>x\<in> A. y) = y ^ card A"
+  for y :: "'a::comm_monoid_mult"
   by (induct A rule: infinite_finite_induct) simp_all
 
-lemma setprod_power_distrib:
-  fixes f :: "'a \<Rightarrow> 'b::comm_semiring_1"
-  shows "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
-proof (cases "finite A")
-  case True then show ?thesis
-    by (induct A rule: finite_induct) (auto simp add: power_mult_distrib)
-next
-  case False then show ?thesis
-    by simp
-qed
+lemma setprod_power_distrib: "setprod f A ^ n = setprod (\<lambda>x. (f x) ^ n) A"
+  for f :: "'a \<Rightarrow> 'b::comm_semiring_1"
+  by (induct A rule: infinite_finite_induct) (auto simp add: power_mult_distrib)
 
-lemma power_setsum:
-  "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
+lemma power_setsum: "c ^ (\<Sum>a\<in>A. f a) = (\<Prod>a\<in>A. c ^ f a)"
   by (induct A rule: infinite_finite_induct) (simp_all add: power_add)
 
 lemma setprod_gen_delta:
-  assumes fS: "finite S"
-  shows "setprod (\<lambda>k. if k=a then b k else c) S = (if a \<in> S then (b a ::'a::comm_monoid_mult) * c^ (card S - 1) else c^ card S)"
-proof-
+  fixes b :: "'b \<Rightarrow> 'a::comm_monoid_mult"
+  assumes fin: "finite S"
+  shows "setprod (\<lambda>k. if k = a then b k else c) S =
+    (if a \<in> S then b a * c ^ (card S - 1) else c ^ card S)"
+proof -
   let ?f = "(\<lambda>k. if k=a then b k else c)"
-  {assume a: "a \<notin> S"
-    hence "\<forall> k\<in> S. ?f k = c" by simp
-    hence ?thesis using a setprod_constant by simp }
-  moreover
-  {assume a: "a \<in> S"
+  show ?thesis
+  proof (cases "a \<in> S")
+    case False
+    then have "\<forall> k\<in> S. ?f k = c" by simp
+    with False show ?thesis by (simp add: setprod_constant)
+  next
+    case True
     let ?A = "S - {a}"
     let ?B = "{a}"
-    have eq: "S = ?A \<union> ?B" using a by blast
-    have dj: "?A \<inter> ?B = {}" by simp
-    from fS have fAB: "finite ?A" "finite ?B" by auto
-    have fA0:"setprod ?f ?A = setprod (\<lambda>i. c) ?A"
+    from True have eq: "S = ?A \<union> ?B" by blast
+    have disjoint: "?A \<inter> ?B = {}" by simp
+    from fin have fin': "finite ?A" "finite ?B" by auto
+    have f_A0: "setprod ?f ?A = setprod (\<lambda>i. c) ?A"
       by (rule setprod.cong) auto
-    have cA: "card ?A = card S - 1" using fS a by auto
-    have fA1: "setprod ?f ?A = c ^ card ?A"
-      unfolding fA0 by (rule setprod_constant)
+    from fin True have card_A: "card ?A = card S - 1" by auto
+    have f_A1: "setprod ?f ?A = c ^ card ?A"
+      unfolding f_A0 by (rule setprod_constant)
     have "setprod ?f ?A * setprod ?f ?B = setprod ?f S"
-      using setprod.union_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
+      using setprod.union_disjoint[OF fin' disjoint, of ?f, unfolded eq[symmetric]]
       by simp
-    then have ?thesis using a cA
-      by (simp add: fA1 field_simps cong add: setprod.cong cong del: if_weak_cong)}
-  ultimately show ?thesis by blast
+    with True card_A show ?thesis
+      by (simp add: f_A1 field_simps cong add: setprod.cong cong del: if_weak_cong)
+  qed
 qed
 
 end
diff -r 4453cfb745e5 -r f90e3926e627 src/HOL/Num.thy
--- a/src/HOL/Num.thy	Wed Aug 10 22:05:00 2016 +0200
+++ b/src/HOL/Num.thy	Wed Aug 10 22:05:36 2016 +0200
@@ -6,7 +6,7 @@
 section \<open>Binary Numerals\<close>
 
 theory Num
-imports BNF_Least_Fixpoint
+  imports BNF_Least_Fixpoint
 begin
 
 subsection \<open>The \<open>num\<close> type\<close>
@@ -15,21 +15,24 @@
 
 text \<open>Increment function for type @{typ num}\<close>
 
-primrec inc :: "num \<Rightarrow> num" where
-  "inc One = Bit0 One" |
-  "inc (Bit0 x) = Bit1 x" |
-  "inc (Bit1 x) = Bit0 (inc x)"
+primrec inc :: "num \<Rightarrow> num"
+  where
+    "inc One = Bit0 One"
+  | "inc (Bit0 x) = Bit1 x"
+  | "inc (Bit1 x) = Bit0 (inc x)"
 
 text \<open>Converting between type @{typ num} and type @{typ nat}\<close>
 
-primrec nat_of_num :: "num \<Rightarrow> nat" where
-  "nat_of_num One = Suc 0" |
-  "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |
-  "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
+primrec nat_of_num :: "num \<Rightarrow> nat"
+  where
+    "nat_of_num One = Suc 0"
+  | "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x"
+  | "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
 
-primrec num_of_nat :: "nat \<Rightarrow> num" where
-  "num_of_nat 0 = One" |
-  "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
+primrec num_of_nat :: "nat \<Rightarrow> num"
+  where
+    "num_of_nat 0 = One"
+  | "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
 
 lemma nat_of_num_pos: "0 < nat_of_num x"
   by (induct x) simp_all
@@ -40,14 +43,10 @@
 lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
   by (induct x) simp_all
 
-lemma num_of_nat_double:
-  "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
+lemma num_of_nat_double: "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
   by (induct n) simp_all
 
-text \<open>
-  Type @{typ num} is isomorphic to the strictly positive
-  natural numbers.
-\<close>
+text \<open>Type @{typ num} is isomorphic to the strictly positive natural numbers.\<close>
 
 lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
   by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
@@ -68,10 +67,11 @@
   shows "P x"
 proof -
   obtain n where n: "Suc n = nat_of_num x"
-    by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
+    by (cases "nat_of_num x") (simp_all add: nat_of_num_neq_0)
   have "P (num_of_nat (Suc n))"
   proof (induct n)
-    case 0 show ?case using One by simp
+    case 0
+    from One show ?case by simp
   next
     case (Suc n)
     then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
@@ -82,9 +82,9 @@
 qed
 
 text \<open>
-  From now on, there are two possible models for @{typ num}:
-  as positive naturals (rule \<open>num_induct\<close>)
-  and as digit representation (rules \<open>num.induct\<close>, \<open>num.cases\<close>).
+  From now on, there are two possible models for @{typ num}: as positive
+  naturals (rule \<open>num_induct\<close>) and as digit representation (rules
+  \<open>num.induct\<close>, \<open>num.cases\<close>).
 \<close>
 
 
@@ -93,17 +93,13 @@
 instantiation num :: "{plus,times,linorder}"
 begin
 
-definition [code del]:
-  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
+definition [code del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
 
-definition [code del]:
-  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
+definition [code del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
 
-definition [code del]:
-  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
+definition [code del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
 
-definition [code del]:
-  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
+definition [code del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
 
 instance
   by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
@@ -137,8 +133,7 @@
   "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
   "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
   "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
-  by (simp_all add: num_eq_iff nat_of_num_add
-    nat_of_num_mult distrib_right distrib_left)
+  by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult distrib_right distrib_left)
 
 lemma eq_num_simps:
   "One = One \<longleftrightarrow> True"
@@ -175,9 +170,9 @@
   by (auto simp add: less_eq_num_def less_num_def)
 
 lemma le_num_One_iff: "x \<le> num.One \<longleftrightarrow> x = num.One"
-by (simp add: antisym_conv)
+  by (simp add: antisym_conv)
 
-text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors\<close>
+text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors.\<close>
 
 lemma add_One: "x + One = inc x"
   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
@@ -191,22 +186,22 @@
 lemma mult_inc: "x * inc y = x * y + x"
   by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
 
-text \<open>The @{const num_of_nat} conversion\<close>
+text \<open>The @{const num_of_nat} conversion.\<close>
 
-lemma num_of_nat_One:
-  "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
+lemma num_of_nat_One: "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
   by (cases n) simp_all
 
 lemma num_of_nat_plus_distrib:
   "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
   by (induct n) (auto simp add: add_One add_One_commute add_inc)
 
-text \<open>A double-and-decrement function\<close>
+text \<open>A double-and-decrement function.\<close>
 
-primrec BitM :: "num \<Rightarrow> num" where
-  "BitM One = One" |
-  "BitM (Bit0 n) = Bit1 (BitM n)" |
-  "BitM (Bit1 n) = Bit1 (Bit0 n)"
+primrec BitM :: "num \<Rightarrow> num"
+  where
+    "BitM One = One"
+  | "BitM (Bit0 n) = Bit1 (BitM n)"
+  | "BitM (Bit1 n) = Bit1 (Bit0 n)"
 
 lemma BitM_plus_one: "BitM n + One = Bit0 n"
   by (induct n) simp_all
@@ -214,20 +209,22 @@
 lemma one_plus_BitM: "One + BitM n = Bit0 n"
   unfolding add_One_commute BitM_plus_one ..
 
-text \<open>Squaring and exponentiation\<close>
+text \<open>Squaring and exponentiation.\<close>
 
-primrec sqr :: "num \<Rightarrow> num" where
-  "sqr One = One" |
-  "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |
-  "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
+primrec sqr :: "num \<Rightarrow> num"
+  where
+    "sqr One = One"
+  | "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))"
+  | "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
 
-primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
-  "pow x One = x" |
-  "pow x (Bit0 y) = sqr (pow x y)" |
-  "pow x (Bit1 y) = sqr (pow x y) * x"
+primrec pow :: "num \<Rightarrow> num \<Rightarrow> num"
+  where
+    "pow x One = x"
+  | "pow x (Bit0 y) = sqr (pow x y)"
+  | "pow x (Bit1 y) = sqr (pow x y) * x"
 
 lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
-  by (induct x, simp_all add: algebra_simps nat_of_num_add)
+  by (induct x) (simp_all add: algebra_simps nat_of_num_add)
 
 lemma sqr_conv_mult: "sqr x = x * x"
   by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
@@ -243,32 +240,44 @@
 class numeral = one + semigroup_add
 begin
 
-primrec numeral :: "num \<Rightarrow> 'a" where
-  numeral_One: "numeral One = 1" |
-  numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |
-  numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
+primrec numeral :: "num \<Rightarrow> 'a"
+  where
+    numeral_One: "numeral One = 1"
+  | numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
+  | numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
 
 lemma numeral_code [code]:
   "numeral One = 1"
   "numeral (Bit0 n) = (let m = numeral n in m + m)"
   "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
   by (simp_all add: Let_def)
-  
+
 lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
-  apply (induct x)
-  apply simp
-  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
-  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
-  done
+proof (induct x)
+  case One
+  then show ?case by simp
+next
+  case Bit0
+  then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
+next
+  case Bit1
+  then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
+qed
 
 lemma numeral_inc: "numeral (inc x) = numeral x + 1"
 proof (induct x)
+  case One
+  then show ?case by simp
+next
+  case Bit0
+  then show ?case by simp
+next
   case (Bit1 x)
   have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
     by (simp only: one_plus_numeral_commute)
   with Bit1 show ?case
     by (simp add: add.assoc)
-qed simp_all
+qed
 
 declare numeral.simps [simp del]
 
@@ -320,9 +329,8 @@
 
 subsection \<open>Class-specific numeral rules\<close>
 
-text \<open>
-  @{const numeral} is a morphism.
-\<close>
+text \<open>@{const numeral} is a morphism.\<close>
+
 
 subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
 
@@ -331,7 +339,7 @@
 
 lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
   by (induct n rule: num_induct)
-     (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
+    (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
 
 lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
   by (rule numeral_add [symmetric])
@@ -350,44 +358,43 @@
 
 end
 
-subsubsection \<open>
-  Structures with negation: class \<open>neg_numeral\<close>
-\<close>
+
+subsubsection \<open>Structures with negation: class \<open>neg_numeral\<close>\<close>
 
 class neg_numeral = numeral + group_add
 begin
 
-lemma uminus_numeral_One:
-  "- Numeral1 = - 1"
+lemma uminus_numeral_One: "- Numeral1 = - 1"
   by (simp add: numeral_One)
 
 text \<open>Numerals form an abelian subgroup.\<close>
 
-inductive is_num :: "'a \<Rightarrow> bool" where
-  "is_num 1" |
-  "is_num x \<Longrightarrow> is_num (- x)" |
-  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
+inductive is_num :: "'a \<Rightarrow> bool"
+  where
+    "is_num 1"
+  | "is_num x \<Longrightarrow> is_num (- x)"
+  | "is_num x \<Longrightarrow> is_num y \<Longrightarrow> is_num (x + y)"
 
 lemma is_num_numeral: "is_num (numeral k)"
-  by (induct k, simp_all add: numeral.simps is_num.intros)
+  by (induct k) (simp_all add: numeral.simps is_num.intros)
 
-lemma is_num_add_commute:
-  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
+lemma is_num_add_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + y = y + x"
   apply (induct x rule: is_num.induct)
-  apply (induct y rule: is_num.induct)
-  apply simp
-  apply (rule_tac a=x in add_left_imp_eq)
-  apply (rule_tac a=x in add_right_imp_eq)
+    apply (induct y rule: is_num.induct)
+      apply simp
+     apply (rule_tac a=x in add_left_imp_eq)
+     apply (rule_tac a=x in add_right_imp_eq)
+     apply (simp add: add.assoc)
+    apply (simp add: add.assoc [symmetric])
+    apply (simp add: add.assoc)
+   apply (rule_tac a=x in add_left_imp_eq)
+   apply (rule_tac a=x in add_right_imp_eq)
+   apply (simp add: add.assoc)
   apply (simp add: add.assoc)
-  apply (simp add: add.assoc [symmetric], simp add: add.assoc)
-  apply (rule_tac a=x in add_left_imp_eq)
-  apply (rule_tac a=x in add_right_imp_eq)
-  apply (simp add: add.assoc)
-  apply (simp add: add.assoc, simp add: add.assoc [symmetric])
+  apply (simp add: add.assoc [symmetric])
   done
 
-lemma is_num_add_left_commute:
-  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
+lemma is_num_add_left_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + (y + z) = y + (x + z)"
   by (simp only: add.assoc [symmetric] is_num_add_commute)
 
 lemmas is_num_normalize =
@@ -395,12 +402,17 @@
   is_num.intros is_num_numeral
   minus_add
 
-definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
-definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
-definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
+definition dbl :: "'a \<Rightarrow> 'a"
+  where "dbl x = x + x"
+
+definition dbl_inc :: "'a \<Rightarrow> 'a"
+  where "dbl_inc x = x + x + 1"
 
-definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
-  "sub k l = numeral k - numeral l"
+definition dbl_dec :: "'a \<Rightarrow> 'a"
+  where "dbl_dec x = x + x - 1"
+
+definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a"
+  where "sub k l = numeral k - numeral l"
 
 lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
   by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
@@ -419,7 +431,8 @@
   "dbl_inc 1 = 3"
   "dbl_inc (- 1) = - 1"
   "dbl_inc (numeral k) = numeral (Bit1 k)"
-  by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
+  by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps
+      del: add_uminus_conv_diff)
 
 lemma dbl_dec_simps [simp]:
   "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
@@ -447,7 +460,7 @@
   "- numeral m + numeral n = sub n m"
   "- numeral m + - numeral n = - (numeral m + numeral n)"
   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
-    del: add_uminus_conv_diff add: diff_conv_add_uminus)
+      del: add_uminus_conv_diff add: diff_conv_add_uminus)
 
 lemma add_neg_numeral_special:
   "1 + - numeral m = sub One m"
@@ -460,7 +473,7 @@
   "- 1 + 1 = 0"
   "- 1 + - 1 = - 2"
   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
-    del: add_uminus_conv_diff add: diff_conv_add_uminus)
+      del: add_uminus_conv_diff add: diff_conv_add_uminus)
 
 lemma diff_numeral_simps:
   "numeral m - numeral n = sub m n"
@@ -468,7 +481,7 @@
   "- numeral m - numeral n = - numeral (m + n)"
   "- numeral m - - numeral n = sub n m"
   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
-    del: add_uminus_conv_diff add: diff_conv_add_uminus)
+      del: add_uminus_conv_diff add: diff_conv_add_uminus)
 
 lemma diff_numeral_special:
   "1 - numeral n = sub One n"
@@ -484,13 +497,12 @@
   "1 - - 1 = 2"
   "- 1 - - 1 = 0"
   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
-    del: add_uminus_conv_diff add: diff_conv_add_uminus)
+      del: add_uminus_conv_diff add: diff_conv_add_uminus)
 
 end
 
-subsubsection \<open>
-  Structures with multiplication: class \<open>semiring_numeral\<close>
-\<close>
+
+subsubsection \<open>Structures with multiplication: class \<open>semiring_numeral\<close>\<close>
 
 class semiring_numeral = semiring + monoid_mult
 begin
@@ -498,25 +510,22 @@
 subclass numeral ..
 
 lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
-  apply (induct n rule: num_induct)
-  apply (simp add: numeral_One)
-  apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
-  done
+  by (induct n rule: num_induct)
+    (simp_all add: numeral_One mult_inc numeral_inc numeral_add distrib_left)
 
 lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
   by (rule numeral_mult [symmetric])
 
 lemma mult_2: "2 * z = z + z"
-  unfolding one_add_one [symmetric] distrib_right by simp
+  by (simp add: one_add_one [symmetric] distrib_right)
 
 lemma mult_2_right: "z * 2 = z + z"
-  unfolding one_add_one [symmetric] distrib_left by simp
+  by (simp add: one_add_one [symmetric] distrib_left)
 
 end
 
-subsubsection \<open>
-  Structures with a zero: class \<open>semiring_1\<close>
-\<close>
+
+subsubsection \<open>Structures with a zero: class \<open>semiring_1\<close>\<close>
 
 context semiring_1
 begin
@@ -524,18 +533,17 @@
 subclass semiring_numeral ..
 
 lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
-  by (induct n,
-    simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
+  by (induct n) (simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
 
 end
 
-lemma nat_of_num_numeral [code_abbrev]:
-  "nat_of_num = numeral"
+lemma nat_of_num_numeral [code_abbrev]: "nat_of_num = numeral"
 proof
   fix n
   have "numeral n = nat_of_num n"
     by (induct n) (simp_all add: numeral.simps)
-  then show "nat_of_num n = numeral n" by simp
+  then show "nat_of_num n = numeral n"
+    by simp
 qed
 
 lemma nat_of_num_code [code]:
@@ -544,16 +552,15 @@
   "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
   by (simp_all add: Let_def)
 
-subsubsection \<open>
-  Equality: class \<open>semiring_char_0\<close>
-\<close>
+
+subsubsection \<open>Equality: class \<open>semiring_char_0\<close>\<close>
 
 context semiring_char_0
 begin
 
 lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
-  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
-    of_nat_eq_iff num_eq_iff ..
+  by (simp only: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
+    of_nat_eq_iff num_eq_iff)
 
 lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
   by (rule numeral_eq_iff [of n One, unfolded numeral_One])
@@ -562,8 +569,7 @@
   by (rule numeral_eq_iff [of One n, unfolded numeral_One])
 
 lemma numeral_neq_zero: "numeral n \<noteq> 0"
-  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
-  by (simp add: nat_of_num_pos)
+  by (simp add: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric] nat_of_num_pos)
 
 lemma zero_neq_numeral: "0 \<noteq> numeral n"
   unfolding eq_commute [of 0] by (rule numeral_neq_zero)
@@ -577,9 +583,8 @@
 
 end
 
-subsubsection \<open>
-  Comparisons: class \<open>linordered_semidom\<close>
-\<close>
+
+subsubsection \<open>Comparisons: class \<open>linordered_semidom\<close>\<close>
 
 text \<open>Could be perhaps more general than here.\<close>
 
@@ -589,15 +594,15 @@
 lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
 proof -
   have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
-    unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
+    by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
   then show ?thesis by simp
 qed
 
 lemma one_le_numeral: "1 \<le> numeral n"
-using numeral_le_iff [of One n] by (simp add: numeral_One)
+  using numeral_le_iff [of One n] by (simp add: numeral_One)
 
 lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
-using numeral_le_iff [of n One] by (simp add: numeral_One)
+  using numeral_le_iff [of n One] by (simp add: numeral_One)
 
 lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
 proof -
@@ -647,30 +652,33 @@
   not_numeral_less_zero
 
 lemma min_0_1 [simp]:
-  fixes min' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" defines "min' \<equiv> min" shows
-  "min' 0 1 = 0"
-  "min' 1 0 = 0"
-  "min' 0 (numeral x) = 0"
-  "min' (numeral x) 0 = 0"
-  "min' 1 (numeral x) = 1"
-  "min' (numeral x) 1 = 1"
-by(simp_all add: min'_def min_def le_num_One_iff)
+  fixes min' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+  defines "min' \<equiv> min"
+  shows
+    "min' 0 1 = 0"
+    "min' 1 0 = 0"
+    "min' 0 (numeral x) = 0"
+    "min' (numeral x) 0 = 0"
+    "min' 1 (numeral x) = 1"
+    "min' (numeral x) 1 = 1"
+  by (simp_all add: min'_def min_def le_num_One_iff)
 
-lemma max_0_1 [simp]: 
-  fixes max' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" defines "max' \<equiv> max" shows
-  "max' 0 1 = 1"
-  "max' 1 0 = 1"
-  "max' 0 (numeral x) = numeral x"
-  "max' (numeral x) 0 = numeral x"
-  "max' 1 (numeral x) = numeral x"
-  "max' (numeral x) 1 = numeral x"
-by(simp_all add: max'_def max_def le_num_One_iff)
+lemma max_0_1 [simp]:
+  fixes max' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+  defines "max' \<equiv> max"
+  shows
+    "max' 0 1 = 1"
+    "max' 1 0 = 1"
+    "max' 0 (numeral x) = numeral x"
+    "max' (numeral x) 0 = numeral x"
+    "max' 1 (numeral x) = numeral x"
+    "max' (numeral x) 1 = numeral x"
+  by (simp_all add: max'_def max_def le_num_One_iff)
 
 end
 
-subsubsection \<open>
-  Multiplication and negation: class \<open>ring_1\<close>
-\<close>
+
+subsubsection \<open>Multiplication and negation: class \<open>ring_1\<close>\<close>
 
 context ring_1
 begin
@@ -681,20 +689,18 @@
   "- numeral m * - numeral n = numeral (m * n)"
   "- numeral m * numeral n = - numeral (m * n)"
   "numeral m * - numeral n = - numeral (m * n)"
-  unfolding mult_minus_left mult_minus_right
-  by (simp_all only: minus_minus numeral_mult)
+  by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)
 
 lemma mult_minus1 [simp]: "- 1 * z = - z"
-  unfolding numeral.simps mult_minus_left by simp
+  by (simp add: numeral.simps)
 
 lemma mult_minus1_right [simp]: "z * - 1 = - z"
-  unfolding numeral.simps mult_minus_right by simp
+  by (simp add: numeral.simps)
 
 end
 
-subsubsection \<open>
-  Equality using \<open>iszero\<close> for rings with non-zero characteristic
-\<close>
+
+subsubsection \<open>Equality using \<open>iszero\<close> for rings with non-zero characteristic\<close>
 
 context ring_1
 begin
@@ -717,23 +723,22 @@
 lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
   by (simp add: numeral_One)
 
-lemma iszero_neg_numeral [simp]:
-  "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
-  unfolding iszero_def
-  by (rule neg_equal_0_iff_equal)
+lemma iszero_neg_numeral [simp]: "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
+  unfolding iszero_def by (rule neg_equal_0_iff_equal)
 
 lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
   unfolding iszero_def by (rule eq_iff_diff_eq_0)
 
-text \<open>The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared
-\<open>[simp]\<close> by default, because for rings of characteristic zero,
-better simp rules are possible. For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules should be added to the
-simplifier, along with a type-specific rule for deciding propositions
-of the form \<open>iszero (numeral w)\<close>.
+text \<open>
+  The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared \<open>[simp]\<close> by default,
+  because for rings of characteristic zero, better simp rules are possible.
+  For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules
+  should be added to the simplifier, along with a type-specific rule for
+  deciding propositions of the form \<open>iszero (numeral w)\<close>.
 
-bh: Maybe it would not be so bad to just declare these as simp
-rules anyway? I should test whether these rules take precedence over
-the \<open>ring_char_0\<close> rules in the simplifier.
+  bh: Maybe it would not be so bad to just declare these as simp rules anyway?
+  I should test whether these rules take precedence over the \<open>ring_char_0\<close>
+  rules in the simplifier.
 \<close>
 
 lemma eq_numeral_iff_iszero:
@@ -754,9 +759,8 @@
 
 end
 
-subsubsection \<open>
-  Equality and negation: class \<open>ring_char_0\<close>
-\<close>
+
+subsubsection \<open>Equality and negation: class \<open>ring_char_0\<close>\<close>
 
 context ring_char_0
 begin
@@ -768,17 +772,16 @@
   by simp
 
 lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
-  unfolding eq_neg_iff_add_eq_0
-  by (simp add: numeral_plus_numeral)
+  by (simp add: eq_neg_iff_add_eq_0 numeral_plus_numeral)
 
 lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
   by (rule numeral_neq_neg_numeral [symmetric])
 
 lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
-  unfolding neg_0_equal_iff_equal by simp
+  by simp
 
 lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
-  unfolding neg_equal_0_iff_equal by simp
+  by simp
 
 lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
   using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
@@ -786,36 +789,28 @@
 lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
   using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
 
-lemma neg_one_neq_numeral:
-  "- 1 \<noteq> numeral n"
+lemma neg_one_neq_numeral: "- 1 \<noteq> numeral n"
   using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
 
-lemma numeral_neq_neg_one:
-  "numeral n \<noteq> - 1"
+lemma numeral_neq_neg_one: "numeral n \<noteq> - 1"
   using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
 
-lemma neg_one_eq_numeral_iff:
-  "- 1 = - numeral n \<longleftrightarrow> n = One"
+lemma neg_one_eq_numeral_iff: "- 1 = - numeral n \<longleftrightarrow> n = One"
   using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
 
-lemma numeral_eq_neg_one_iff:
-  "- numeral n = - 1 \<longleftrightarrow> n = One"
+lemma numeral_eq_neg_one_iff: "- numeral n = - 1 \<longleftrightarrow> n = One"
   using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
 
-lemma neg_one_neq_zero:
-  "- 1 \<noteq> 0"
+lemma neg_one_neq_zero: "- 1 \<noteq> 0"
   by simp
 
-lemma zero_neq_neg_one:
-  "0 \<noteq> - 1"
+lemma zero_neq_neg_one: "0 \<noteq> - 1"
   by simp
 
-lemma neg_one_neq_one:
-  "- 1 \<noteq> 1"
+lemma neg_one_neq_one: "- 1 \<noteq> 1"
   using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
 
-lemma one_neq_neg_one:
-  "1 \<noteq> - 1"
+lemma one_neq_neg_one: "1 \<noteq> - 1"
   using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
 
 lemmas eq_neg_numeral_simps [simp] =
@@ -831,9 +826,7 @@
 end
 
 
-subsubsection \<open>
-  Structures with negation and order: class \<open>linordered_idom\<close>
-\<close>
+subsubsection \<open>Structures with negation and order: class \<open>linordered_idom\<close>\<close>
 
 context linordered_idom
 begin
@@ -869,7 +862,7 @@
 
 lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
   by (simp only: not_le neg_numeral_less_numeral)
-  
+
 lemma neg_numeral_less_one: "- numeral m < 1"
   by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
 
@@ -906,20 +899,16 @@
 lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
   by (cases m) simp_all
 
-lemma sub_non_negative:
-  "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
+lemma sub_non_negative: "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
   by (simp only: sub_def le_diff_eq) simp
 
-lemma sub_positive:
-  "sub n m > 0 \<longleftrightarrow> n > m"
+lemma sub_positive: "sub n m > 0 \<longleftrightarrow> n > m"
   by (simp only: sub_def less_diff_eq) simp
 
-lemma sub_non_positive:
-  "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
+lemma sub_non_positive: "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
   by (simp only: sub_def diff_le_eq) simp
 
-lemma sub_negative:
-  "sub n m < 0 \<longleftrightarrow> n < m"
+lemma sub_negative: "sub n m < 0 \<longleftrightarrow> n < m"
   by (simp only: sub_def diff_less_eq) simp
 
 lemmas le_neg_numeral_simps [simp] =
@@ -963,9 +952,8 @@
 
 end
 
-subsubsection \<open>
-  Natural numbers
-\<close>
+
+subsubsection \<open>Natural numbers\<close>
 
 lemma Suc_1 [simp]: "Suc 1 = 2"
   unfolding Suc_eq_plus1 by (rule one_add_one)
@@ -977,7 +965,7 @@
   where [code del]: "pred_numeral k = numeral k - 1"
 
 lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
-  unfolding pred_numeral_def by simp
+  by (simp add: pred_numeral_def)
 
 lemma eval_nat_numeral:
   "numeral One = Suc 0"
@@ -989,8 +977,7 @@
   "pred_numeral One = 0"
   "pred_numeral (Bit0 k) = numeral (BitM k)"
   "pred_numeral (Bit1 k) = numeral (Bit0 k)"
-  unfolding pred_numeral_def eval_nat_numeral
-  by (simp_all only: diff_Suc_Suc diff_0)
+  by (simp_all only: pred_numeral_def eval_nat_numeral diff_Suc_Suc diff_0)
 
 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
   by (simp add: eval_nat_numeral)
@@ -1001,12 +988,11 @@
 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
   by (simp only: numeral_One One_nat_def)
 
-lemma Suc_nat_number_of_add:
-  "Suc (numeral v + n) = numeral (v + One) + n"
+lemma Suc_nat_number_of_add: "Suc (numeral v + n) = numeral (v + One) + n"
   by simp
 
-(*Maps #n to n for n = 1, 2*)
-lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
+lemma numerals: "Numeral1 = (1::nat)" "2 = Suc (Suc 0)"
+  by (rule numeral_One) (rule numeral_2_eq_2)
 
 text \<open>Comparisons involving @{term Suc}.\<close>
 
@@ -1034,26 +1020,21 @@
 lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
   by (simp add: numeral_eq_Suc)
 
-lemma max_Suc_numeral [simp]:
-  "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
+lemma max_Suc_numeral [simp]: "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
   by (simp add: numeral_eq_Suc)
 
-lemma max_numeral_Suc [simp]:
-  "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
+lemma max_numeral_Suc [simp]: "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
   by (simp add: numeral_eq_Suc)
 
-lemma min_Suc_numeral [simp]:
-  "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
+lemma min_Suc_numeral [simp]: "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
   by (simp add: numeral_eq_Suc)
 
-lemma min_numeral_Suc [simp]:
-  "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
+lemma min_numeral_Suc [simp]: "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
   by (simp add: numeral_eq_Suc)
 
 text \<open>For @{term case_nat} and @{term rec_nat}.\<close>
 
-lemma case_nat_numeral [simp]:
-  "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
+lemma case_nat_numeral [simp]: "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
   by (simp add: numeral_eq_Suc)
 
 lemma case_nat_add_eq_if [simp]:
@@ -1061,21 +1042,18 @@
   by (simp add: numeral_eq_Suc)
 
 lemma rec_nat_numeral [simp]:
-  "rec_nat a f (numeral v) =
-    (let pv = pred_numeral v in f pv (rec_nat a f pv))"
+  "rec_nat a f (numeral v) = (let pv = pred_numeral v in f pv (rec_nat a f pv))"
   by (simp add: numeral_eq_Suc Let_def)
 
 lemma rec_nat_add_eq_if [simp]:
-  "rec_nat a f (numeral v + n) =
-    (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
+  "rec_nat a f (numeral v + n) = (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
   by (simp add: numeral_eq_Suc Let_def)
 
-text \<open>Case analysis on @{term "n < 2"}\<close>
-
+text \<open>Case analysis on @{term "n < 2"}.\<close>
 lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
   by (auto simp add: numeral_2_eq_2)
 
-text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2\<close>
+text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2.\<close>
 text \<open>bh: Are these rules really a good idea?\<close>
 
 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
@@ -1085,7 +1063,6 @@
   by simp
 
 text \<open>Can be used to eliminate long strings of Sucs, but not by default.\<close>
-
 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
   by simp
 
@@ -1099,12 +1076,10 @@
 
 subclass field_char_0 ..
 
-lemma half_gt_zero_iff:
-  "0 < a / 2 \<longleftrightarrow> 0 < a" (is "?P \<longleftrightarrow> ?Q")
+lemma half_gt_zero_iff: "0 < a / 2 \<longleftrightarrow> 0 < a"
   by (auto simp add: field_simps)
 
-lemma half_gt_zero [simp]:
-  "0 < a \<Longrightarrow> 0 < a / 2"
+lemma half_gt_zero [simp]: "0 < a \<Longrightarrow> 0 < a / 2"
   by (simp add: half_gt_zero_iff)
 
 end
@@ -1124,50 +1099,52 @@
 
 subsection \<open>Setting up simprocs\<close>
 
-lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
+lemma mult_numeral_1: "Numeral1 * a = a"
+  for a :: "'a::semiring_numeral"
   by simp
 
-lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
+lemma mult_numeral_1_right: "a * Numeral1 = a"
+  for a :: "'a::semiring_numeral"
   by simp
 
-lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
+lemma divide_numeral_1: "a / Numeral1 = a"
+  for a :: "'a::field"
   by simp
 
-lemma inverse_numeral_1:
-  "inverse Numeral1 = (Numeral1::'a::division_ring)"
+lemma inverse_numeral_1: "inverse Numeral1 = (Numeral1::'a::division_ring)"
   by simp
 
-text\<open>Theorem lists for the cancellation simprocs. The use of a binary
-numeral for 1 reduces the number of special cases.\<close>
+text \<open>
+  Theorem lists for the cancellation simprocs. The use of a binary
+  numeral for 1 reduces the number of special cases.
+\<close>
 
 lemma mult_1s:
-  fixes a :: "'a::semiring_numeral"
-    and b :: "'b::ring_1"
-  shows "Numeral1 * a = a"
-    "a * Numeral1 = a"
-    "- Numeral1 * b = - b"
-    "b * - Numeral1 = - b"
+  "Numeral1 * a = a"
+  "a * Numeral1 = a"
+  "- Numeral1 * b = - b"
+  "b * - Numeral1 = - b"
+  for a :: "'a::semiring_numeral" and b :: "'b::ring_1"
   by simp_all
 
 setup \<open>
   Reorient_Proc.add
     (fn Const (@{const_name numeral}, _) $ _ => true
-    | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
-    | _ => false)
+      | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
+      | _ => false)
 \<close>
 
-simproc_setup reorient_numeral
-  ("numeral w = x" | "- numeral w = y") = Reorient_Proc.proc
+simproc_setup reorient_numeral ("numeral w = x" | "- numeral w = y") =
+  Reorient_Proc.proc
 
 
-subsubsection \<open>Simplification of arithmetic operations on integer constants.\<close>
+subsubsection \<open>Simplification of arithmetic operations on integer constants\<close>
 
 lemmas arith_special = (* already declared simp above *)
   add_numeral_special add_neg_numeral_special
   diff_numeral_special
 
-(* rules already in simpset *)
-lemmas arith_extra_simps =
+lemmas arith_extra_simps = (* rules already in simpset *)
   numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
   minus_zero
   diff_numeral_simps diff_0 diff_0_right
@@ -1195,7 +1172,7 @@
 lemmas of_nat_simps =
   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
 
-text \<open>Simplification of relational operations\<close>
+text \<open>Simplification of relational operations.\<close>
 
 lemmas eq_numeral_extra =
   zero_neq_one one_neq_zero
@@ -1215,34 +1192,28 @@
   unfolding Let_def ..
 
 declaration \<open>
-let 
+let
   fun number_of ctxt T n =
     if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort numeral}))
     then raise CTERM ("number_of", [])
     else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
 in
   K (
-    Lin_Arith.add_simps (@{thms arith_simps} @ @{thms more_arith_simps}
-      @ @{thms rel_simps}
-      @ @{thms pred_numeral_simps}
-      @ @{thms arith_special numeral_One}
-      @ @{thms of_nat_simps})
-    #> Lin_Arith.add_simps [@{thm Suc_numeral},
-      @{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
-      @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
-      @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
-      @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
-      @{thm mult_Suc}, @{thm mult_Suc_right},
-      @{thm of_nat_numeral}]
+    Lin_Arith.add_simps
+      @{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
+        arith_special numeral_One of_nat_simps}
+    #> Lin_Arith.add_simps
+      @{thms Suc_numeral Let_numeral Let_neg_numeral Let_0 Let_1
+        le_Suc_numeral le_numeral_Suc less_Suc_numeral less_numeral_Suc
+        Suc_eq_numeral eq_numeral_Suc mult_Suc mult_Suc_right of_nat_numeral}
     #> Lin_Arith.set_number_of number_of)
 end
 \<close>
 
 
-subsubsection \<open>Simplification of arithmetic when nested to the right.\<close>
+subsubsection \<open>Simplification of arithmetic when nested to the right\<close>
 
-lemma add_numeral_left [simp]:
-  "numeral v + (numeral w + z) = (numeral(v + w) + z)"
+lemma add_numeral_left [simp]: "numeral v + (numeral w + z) = (numeral(v + w) + z)"
   by (simp_all add: add.assoc [symmetric])
 
 lemma add_neg_numeral_left [simp]:
@@ -1261,7 +1232,7 @@
 hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
 
 
-subsection \<open>code module namespace\<close>
+subsection \<open>Code module namespace\<close>
 
 code_identifier
   code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
diff -r 4453cfb745e5 -r f90e3926e627 src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Wed Aug 10 22:05:00 2016 +0200
+++ b/src/HOL/Parity.thy	Wed Aug 10 22:05:36 2016 +0200
@@ -6,7 +6,7 @@
 section \<open>Parity in rings and semirings\<close>
 
 theory Parity
-imports Nat_Transfer
+  imports Nat_Transfer
 begin
 
 subsection \<open>Ring structures with parity and \<open>even\<close>/\<open>odd\<close> predicates\<close>
@@ -21,19 +21,15 @@
 subclass semiring_numeral ..
 
 abbreviation even :: "'a \<Rightarrow> bool"
-where
-  "even a \<equiv> 2 dvd a"
+  where "even a \<equiv> 2 dvd a"
 
 abbreviation odd :: "'a \<Rightarrow> bool"
-where
-  "odd a \<equiv> \<not> 2 dvd a"
+  where "odd a \<equiv> \<not> 2 dvd a"
 
-lemma even_zero [simp]:
-  "even 0"
+lemma even_zero [simp]: "even 0"
   by (fact dvd_0_right)
 
-lemma even_plus_one_iff [simp]:
-  "even (a + 1) \<longleftrightarrow> odd a"
+lemma even_plus_one_iff [simp]: "even (a + 1) \<longleftrightarrow> odd a"
   by (auto simp add: dvd_add_right_iff intro: odd_even_add)
 
 lemma evenE [elim?]:
@@ -53,13 +49,11 @@
   with * have "a = 2 * c + 1" by simp
   with that show thesis .
 qed
- 
-lemma even_times_iff [simp]:
-  "even (a * b) \<longleftrightarrow> even a \<or> even b"
+
+lemma even_times_iff [simp]: "even (a * b) \<longleftrightarrow> even a \<or> even b"
   by (auto dest: even_multD)
 
-lemma even_numeral [simp]:
-  "even (numeral (Num.Bit0 n))"
+lemma even_numeral [simp]: "even (numeral (Num.Bit0 n))"
 proof -
   have "even (2 * numeral n)"
     unfolding even_times_iff by simp
@@ -69,8 +63,7 @@
     unfolding numeral.simps .
 qed
 
-lemma odd_numeral [simp]:
-  "odd (numeral (Num.Bit1 n))"
+lemma odd_numeral [simp]: "odd (numeral (Num.Bit1 n))"
 proof
   assume "even (numeral (num.Bit1 n))"
   then have "even (numeral n + numeral n + 1)"
@@ -79,22 +72,18 @@
     unfolding mult_2 .
   then have "2 dvd numeral n * 2 + 1"
     by (simp add: ac_simps)
-  with dvd_add_times_triv_left_iff [of 2 "numeral n" 1]
-    have "2 dvd 1"
-    by simp
+  then have "2 dvd 1"
+    using dvd_add_times_triv_left_iff [of 2 "numeral n" 1] by simp
   then show False by simp
 qed
 
-lemma even_add [simp]:
-  "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
+lemma even_add [simp]: "even (a + b) \<longleftrightarrow> (even a \<longleftrightarrow> even b)"
   by (auto simp add: dvd_add_right_iff dvd_add_left_iff odd_even_add)
 
-lemma odd_add [simp]:
-  "odd (a + b) \<longleftrightarrow> (\<not> (odd a \<longleftrightarrow> odd b))"
+lemma odd_add [simp]: "odd (a + b) \<longleftrightarrow> (\<not> (odd a \<longleftrightarrow> odd b))"
   by simp
 
-lemma even_power [simp]:
-  "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
+lemma even_power [simp]: "even (a ^ n) \<longleftrightarrow> even a \<and> n > 0"
   by (induct n) auto
 
 end
@@ -104,12 +93,10 @@
 
 subclass comm_ring_1 ..
 
-lemma even_minus [simp]:
-  "even (- a) \<longleftrightarrow> even a"
+lemma even_minus [simp]: "even (- a) \<longleftrightarrow> even a"
   by (fact dvd_minus_iff)
 
-lemma even_diff [simp]:
-  "even (a - b) \<longleftrightarrow> even (a + b)"
+lemma even_diff [simp]: "even (a - b) \<longleftrightarrow> even (a + b)"
   using even_add [of a "- b"] by simp
 
 end
@@ -117,17 +104,14 @@
 
 subsection \<open>Instances for @{typ nat} and @{typ int}\<close>
 
-lemma even_Suc_Suc_iff [simp]:
-  "2 dvd Suc (Suc n) \<longleftrightarrow> 2 dvd n"
+lemma even_Suc_Suc_iff [simp]: "2 dvd Suc (Suc n) \<longleftrightarrow> 2 dvd n"
   using dvd_add_triv_right_iff [of 2 n] by simp
 
-lemma even_Suc [simp]:
-  "2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
+lemma even_Suc [simp]: "2 dvd Suc n \<longleftrightarrow> \<not> 2 dvd n"
   by (induct n) auto
 
-lemma even_diff_nat [simp]:
-  fixes m n :: nat
-  shows "2 dvd (m - n) \<longleftrightarrow> m < n \<or> 2 dvd (m + n)"
+lemma even_diff_nat [simp]: "2 dvd (m - n) \<longleftrightarrow> m < n \<or> 2 dvd (m + n)"
+  for m n :: nat
 proof (cases "n \<le> m")
   case True
   then have "m - n + n * 2 = m + n" by (simp add: mult_2_right)
@@ -137,8 +121,8 @@
 next
   case False
   then show ?thesis by simp
-qed 
-  
+qed
+
 instance nat :: semiring_parity
 proof
   show "\<not> 2 dvd (1 :: nat)"
@@ -165,7 +149,8 @@
     then obtain r where "Suc n = 2 * r" ..
     moreover from * obtain s where "m * n = 2 * s" ..
     then have "2 * s + m = m * Suc n" by simp
-    ultimately have " 2 * s + m = 2 * (m * r)" by (simp add: algebra_simps)
+    ultimately have " 2 * s + m = 2 * (m * r)"
+      by (simp add: algebra_simps)
     then have "m = 2 * (m * r - s)" by simp
     then show "2 dvd m" ..
   qed
@@ -176,13 +161,12 @@
     by (cases n) simp_all
 qed
 
-lemma odd_pos: 
-  "odd (n :: nat) \<Longrightarrow> 0 < n"
+lemma odd_pos: "odd n \<Longrightarrow> 0 < n"
+  for n :: nat
   by (auto elim: oddE)
 
-lemma Suc_double_not_eq_double:
-  fixes m n :: nat
-  shows "Suc (2 * m) \<noteq> 2 * n"
+lemma Suc_double_not_eq_double: "Suc (2 * m) \<noteq> 2 * n"
+  for m n :: nat
 proof
   assume "Suc (2 * m) = 2 * n"
   moreover have "odd (Suc (2 * m))" and "even (2 * n)"
@@ -190,37 +174,34 @@
   ultimately show False by simp
 qed
 
-lemma double_not_eq_Suc_double:
-  fixes m n :: nat
-  shows "2 * m \<noteq> Suc (2 * n)"
+lemma double_not_eq_Suc_double: "2 * m \<noteq> Suc (2 * n)"
+  for m n :: nat
   using Suc_double_not_eq_double [of n m] by simp
 
-lemma even_diff_iff [simp]:
-  fixes k l :: int
-  shows "2 dvd (k - l) \<longleftrightarrow> 2 dvd (k + l)"
+lemma even_diff_iff [simp]: "2 dvd (k - l) \<longleftrightarrow> 2 dvd (k + l)"
+  for k l :: int
   using dvd_add_times_triv_right_iff [of 2 "k - l" l] by (simp add: mult_2_right)
 
-lemma even_abs_add_iff [simp]:
-  fixes k l :: int
-  shows "2 dvd (\<bar>k\<bar> + l) \<longleftrightarrow> 2 dvd (k + l)"
+lemma even_abs_add_iff [simp]: "2 dvd (\<bar>k\<bar> + l) \<longleftrightarrow> 2 dvd (k + l)"
+  for k l :: int
   by (cases "k \<ge> 0") (simp_all add: ac_simps)
 
-lemma even_add_abs_iff [simp]:
-  fixes k l :: int
-  shows "2 dvd (k + \<bar>l\<bar>) \<longleftrightarrow> 2 dvd (k + l)"
+lemma even_add_abs_iff [simp]: "2 dvd (k + \<bar>l\<bar>) \<longleftrightarrow> 2 dvd (k + l)"
+  for k l :: int
   using even_abs_add_iff [of l k] by (simp add: ac_simps)
 
-lemma odd_Suc_minus_one [simp]:
-  "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
+lemma odd_Suc_minus_one [simp]: "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
   by (auto elim: oddE)
 
 instance int :: ring_parity
 proof
-  show "\<not> 2 dvd (1 :: int)" by (simp add: dvd_int_unfold_dvd_nat)
+  show "\<not> 2 dvd (1 :: int)"
+    by (simp add: dvd_int_unfold_dvd_nat)
+next
   fix k l :: int
   assume "\<not> 2 dvd k"
   moreover assume "\<not> 2 dvd l"
-  ultimately have "2 dvd (nat \<bar>k\<bar> + nat \<bar>l\<bar>)" 
+  ultimately have "2 dvd (nat \<bar>k\<bar> + nat \<bar>l\<bar>)"
     by (auto simp add: dvd_int_unfold_dvd_nat intro: odd_even_add)
   then have "2 dvd (\<bar>k\<bar> + \<bar>l\<bar>)"
     by (simp add: dvd_int_unfold_dvd_nat nat_add_distrib)
@@ -237,12 +218,10 @@
   then show "\<exists>l. k = l + 1" ..
 qed
 
-lemma even_int_iff [simp]:
-  "even (int n) \<longleftrightarrow> even n"
+lemma even_int_iff [simp]: "even (int n) \<longleftrightarrow> even n"
   by (simp add: dvd_int_iff)
 
-lemma even_nat_iff:
-  "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
+lemma even_nat_iff: "0 \<le> k \<Longrightarrow> even (nat k) \<longleftrightarrow> even k"
   by (simp add: even_int_iff [symmetric])
 
 
@@ -251,58 +230,47 @@
 context ring_1
 begin
 
-lemma power_minus_even [simp]:
-  "even n \<Longrightarrow> (- a) ^ n = a ^ n"
+lemma power_minus_even [simp]: "even n \<Longrightarrow> (- a) ^ n = a ^ n"
   by (auto elim: evenE)
 
-lemma power_minus_odd [simp]:
-  "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
+lemma power_minus_odd [simp]: "odd n \<Longrightarrow> (- a) ^ n = - (a ^ n)"
   by (auto elim: oddE)
 
-lemma neg_one_even_power [simp]:
-  "even n \<Longrightarrow> (- 1) ^ n = 1"
+lemma neg_one_even_power [simp]: "even n \<Longrightarrow> (- 1) ^ n = 1"
   by simp
 
-lemma neg_one_odd_power [simp]:
-  "odd n \<Longrightarrow> (- 1) ^ n = - 1"
+lemma neg_one_odd_power [simp]: "odd n \<Longrightarrow> (- 1) ^ n = - 1"
   by simp
 
-end  
+end
 
 context linordered_idom
 begin
 
-lemma zero_le_even_power:
-  "even n \<Longrightarrow> 0 \<le> a ^ n"
+lemma zero_le_even_power: "even n \<Longrightarrow> 0 \<le> a ^ n"
   by (auto elim: evenE)
 
-lemma zero_le_odd_power:
-  "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
+lemma zero_le_odd_power: "odd n \<Longrightarrow> 0 \<le> a ^ n \<longleftrightarrow> 0 \<le> a"
   by (auto simp add: power_even_eq zero_le_mult_iff elim: oddE)
 
-lemma zero_le_power_eq:
-  "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
+lemma zero_le_power_eq: "0 \<le> a ^ n \<longleftrightarrow> even n \<or> odd n \<and> 0 \<le> a"
   by (auto simp add: zero_le_even_power zero_le_odd_power)
-  
-lemma zero_less_power_eq:
-  "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
+
+lemma zero_less_power_eq: "0 < a ^ n \<longleftrightarrow> n = 0 \<or> even n \<and> a \<noteq> 0 \<or> odd n \<and> 0 < a"
 proof -
   have [simp]: "0 = a ^ n \<longleftrightarrow> a = 0 \<and> n > 0"
     unfolding power_eq_0_iff [of a n, symmetric] by blast
   show ?thesis
-  unfolding less_le zero_le_power_eq by auto
+    unfolding less_le zero_le_power_eq by auto
 qed
 
-lemma power_less_zero_eq [simp]:
-  "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
+lemma power_less_zero_eq [simp]: "a ^ n < 0 \<longleftrightarrow> odd n \<and> a < 0"
   unfolding not_le [symmetric] zero_le_power_eq by auto
-  
-lemma power_le_zero_eq:
-  "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
-  unfolding not_less [symmetric] zero_less_power_eq by auto 
 
-lemma power_even_abs:
-  "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
+lemma power_le_zero_eq: "a ^ n \<le> 0 \<longleftrightarrow> n > 0 \<and> (odd n \<and> a \<le> 0 \<or> even n \<and> a = 0)"
+  unfolding not_less [symmetric] zero_less_power_eq by auto
+
+lemma power_even_abs: "even n \<Longrightarrow> \<bar>a\<bar> ^ n = a ^ n"
   using power_abs [of a n] by (simp add: zero_le_even_power)
 
 lemma power_mono_even:
@@ -310,30 +278,35 @@
   shows "a ^ n \<le> b ^ n"
 proof -
   have "0 \<le> \<bar>a\<bar>" by auto
-  with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close>
-  have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n" by (rule power_mono)
-  with \<open>even n\<close> show ?thesis by (simp add: power_even_abs)  
+  with \<open>\<bar>a\<bar> \<le> \<bar>b\<bar>\<close> have "\<bar>a\<bar> ^ n \<le> \<bar>b\<bar> ^ n"
+    by (rule power_mono)
+  with \<open>even n\<close> show ?thesis
+    by (simp add: power_even_abs)
 qed
 
 lemma power_mono_odd:
   assumes "odd n" and "a \<le> b"
   shows "a ^ n \<le> b ^ n"
 proof (cases "b < 0")
-  case True with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
-  hence "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
+  case True
+  with \<open>a \<le> b\<close> have "- b \<le> - a" and "0 \<le> - b" by auto
+  then have "(- b) ^ n \<le> (- a) ^ n" by (rule power_mono)
   with \<open>odd n\<close> show ?thesis by simp
 next
-  case False then have "0 \<le> b" by auto
+  case False
+  then have "0 \<le> b" by auto
   show ?thesis
   proof (cases "a < 0")
-    case True then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
+    case True
+    then have "n \<noteq> 0" and "a \<le> 0" using \<open>odd n\<close> [THEN odd_pos] by auto
     then have "a ^ n \<le> 0" unfolding power_le_zero_eq using \<open>odd n\<close> by auto
-    moreover
-    from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
+    moreover from \<open>0 \<le> b\<close> have "0 \<le> b ^ n" by auto
     ultimately show ?thesis by auto
   next
-    case False then have "0 \<le> a" by auto
-    with \<open>a \<le> b\<close> show ?thesis using power_mono by auto
+    case False
+    then have "0 \<le> a" by auto
+    with \<open>a \<le> b\<close> show ?thesis
+      using power_mono by auto
   qed
 qed
 
@@ -347,13 +320,16 @@
   by (fact zero_le_power_eq)
 
 lemma zero_less_power_eq_numeral [simp]:
-  "0 < a ^ numeral w \<longleftrightarrow> numeral w = (0 :: nat)
-    \<or> even (numeral w :: nat) \<and> a \<noteq> 0 \<or> odd (numeral w :: nat) \<and> 0 < a"
+  "0 < a ^ numeral w \<longleftrightarrow>
+    numeral w = (0 :: nat) \<or>
+    even (numeral w :: nat) \<and> a \<noteq> 0 \<or>
+    odd (numeral w :: nat) \<and> 0 < a"
   by (fact zero_less_power_eq)
 
 lemma power_le_zero_eq_numeral [simp]:
-  "a ^ numeral w \<le> 0 \<longleftrightarrow> (0 :: nat) < numeral w
-    \<and> (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
+  "a ^ numeral w \<le> 0 \<longleftrightarrow>
+    (0 :: nat) < numeral w \<and>
+    (odd (numeral w :: nat) \<and> a \<le> 0 \<or> even (numeral w :: nat) \<and> a = 0)"
   by (fact power_le_zero_eq)
 
 lemma power_less_zero_eq_numeral [simp]:
@@ -367,10 +343,8 @@
 end
 
 
-subsubsection \<open>Tools setup\<close>
+subsubsection \<open>Tool setup\<close>
 
-declare transfer_morphism_int_nat [transfer add return:
-  even_int_iff
-]
+declare transfer_morphism_int_nat [transfer add return: even_int_iff]
 
 end
diff -r 4453cfb745e5 -r f90e3926e627 src/HOL/Power.thy
--- a/src/HOL/Power.thy	Wed Aug 10 22:05:00 2016 +0200
+++ b/src/HOL/Power.thy	Wed Aug 10 22:05:36 2016 +0200
@@ -6,7 +6,7 @@
 section \<open>Exponentiation\<close>
 
 theory Power
-imports Num
+  imports Num
 begin
 
 subsection \<open>Powers for Arbitrary Monoids\<close>
@@ -15,9 +15,9 @@
 begin
 
 primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a"  (infixr "^" 80)
-where
-  power_0: "a ^ 0 = 1"
-| power_Suc: "a ^ Suc n = a * a ^ n"
+  where
+    power_0: "a ^ 0 = 1"
+  | power_Suc: "a ^ Suc n = a * a ^ n"
 
 notation (latex output)
   power ("(_\<^bsup>_\<^esup>)" [1000] 1000)
@@ -33,32 +33,25 @@
 
 subclass power .
 
-lemma power_one [simp]:
-  "1 ^ n = 1"
+lemma power_one [simp]: "1 ^ n = 1"
   by (induct n) simp_all
 
-lemma power_one_right [simp]:
-  "a ^ 1 = a"
+lemma power_one_right [simp]: "a ^ 1 = a"
   by simp
 
-lemma power_Suc0_right [simp]:
-  "a ^ Suc 0 = a"
+lemma power_Suc0_right [simp]: "a ^ Suc 0 = a"
   by simp
 
-lemma power_commutes:
-  "a ^ 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 ^ Suc n = a ^ n * a"
+lemma power_Suc2: "a ^ Suc n = a ^ n * a"
   by (simp add: power_commutes)
 
-lemma power_add:
-  "a ^ (m + n) = a ^ m * a ^ n"
+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"
+lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"
   by (induct n) (simp_all add: power_add)
 
 lemma power2_eq_square: "a\<^sup>2 = a * a"
@@ -67,51 +60,49 @@
 lemma power3_eq_cube: "a ^ 3 = a * a * a"
   by (simp add: numeral_3_eq_3 mult.assoc)
 
-lemma power_even_eq:
-  "a ^ (2 * n) = (a ^ n)\<^sup>2"
+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"
+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)"
-  unfolding numeral_Bit0 power_add Let_def ..
+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)"
-  unfolding numeral_Bit1 One_nat_def add_Suc_right add_0_right
-  unfolding power_Suc power_add Let_def mult.assoc ..
+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 funpow_times_power:
-  "(times x ^^ f x) = times (x ^ f x)"
+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)
+  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)
+  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"
-    unfolding Suc power_Suc2
-    by (simp add: ac_simps)
+    by (simp only: Suc power_Suc2) (simp add: ac_simps)
   finally show ?case .
-qed simp
+qed
 
-lemma power_minus_mult:
-  "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
+lemma power_minus_mult: "0 < n \<Longrightarrow> a ^ (n - 1) * a = a ^ n"
   by (simp add: power_commutes split: nat_diff_split)
 
 end
@@ -119,29 +110,25 @@
 context comm_monoid_mult
 begin
 
-lemma power_mult_distrib [field_simps]:
-  "(a * b) ^ n = (a ^ n) * (b ^ n)"
+lemma power_mult_distrib [field_simps]: "(a * b) ^ n = (a ^ n) * (b ^ n)"
   by (induct n) (simp_all add: ac_simps)
 
 end
 
-text\<open>Extract constant factors from powers\<close>
+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]:
-  fixes a :: "'a :: monoid_mult"
-  shows "a^numeral m * a^numeral n = a^numeral (m + n)"
+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]:
-  fixes a :: "'a :: monoid_mult"
-  shows "a^numeral m * (a^numeral n * b) = a^numeral (m + n) * b"
+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]:
-  fixes a :: "'a :: monoid_mult"
-  shows"(a^numeral m)^numeral n = a^numeral (m * n)"
+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
@@ -151,8 +138,9 @@
   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)
+  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])
@@ -162,16 +150,13 @@
 context semiring_1
 begin
 
-lemma of_nat_power [simp]:
-  "of_nat (m ^ n) = of_nat m ^ n"
+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"
+lemma zero_power: "0 < n \<Longrightarrow> 0 ^ n = 0"
   by (cases n) simp_all
 
-lemma power_zero_numeral [simp]:
-  "0 ^ numeral k = 0"
+lemma power_zero_numeral [simp]: "0 ^ numeral k = 0"
   by (simp add: numeral_eq_Suc)
 
 lemma zero_power2: "0\<^sup>2 = 0" (* delete? *)
@@ -180,13 +165,11 @@
 lemma one_power2: "1\<^sup>2 = 1" (* delete? *)
   by (rule power_one)
 
-lemma power_0_Suc [simp]:
-  "0 ^ Suc n = 0"
+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)"
+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
@@ -194,34 +177,32 @@
 context comm_semiring_1
 begin
 
-text \<open>The divides relation\<close>
+text \<open>The divides relation.\<close>
 
 lemma le_imp_power_dvd:
-  assumes "m \<le> n" shows "a ^ m dvd a ^ n"
+  assumes "m \<le> n"
+  shows "a ^ m dvd a ^ n"
 proof
-  have "a ^ n = a ^ (m + (n - m))"
-    using \<open>m \<le> n\<close> by simp
-  also have "\<dots> = a ^ m * a ^ (n - m)"
-    by (rule power_add)
+  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"
+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"
+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"
+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]:
-  assumes "n > (0::nat) \<or> x = 1"
+  fixes n :: nat
+  assumes "n > 0 \<or> x = 1"
   shows "x dvd (x ^ n)"
-using assms proof
+  using assms
+proof
   assume "0 < n"
   then have "x ^ n = x ^ Suc (n - 1)" by simp
   then show "x dvd (x ^ n)" by simp
@@ -237,16 +218,13 @@
 
 subclass power .
 
-lemma power_eq_0_iff [simp]:
-  "a ^ n = 0 \<longleftrightarrow> a = 0 \<and> n > 0"
+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"
+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"
+lemma zero_eq_power2 [simp]: "a\<^sup>2 = 0 \<longleftrightarrow> a = 0"
   unfolding power2_eq_square by simp
 
 end
@@ -254,45 +232,42 @@
 context ring_1
 begin
 
-lemma power_minus:
-  "(- a) ^ n = (- 1) ^ n * a ^ n"
+lemma power_minus: "(- a) ^ n = (- 1) ^ n * a ^ n"
 proof (induct n)
-  case 0 show ?case by simp
+  case 0
+  show ?case by simp
 next
-  case (Suc n) then show ?case
+  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)"
+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))"
+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"
+lemma power2_minus [simp]: "(- a)\<^sup>2 = a\<^sup>2"
   by (fact power_minus_Bit0)
 
-lemma power_minus1_even [simp]:
-  "(- 1) ^ (2*n) = 1"
+lemma power_minus1_even [simp]: "(- 1) ^ (2*n) = 1"
 proof (induct n)
-  case 0 show ?case by simp
+  case 0
+  show ?case by simp
 next
-  case (Suc n) then show ?case by (simp add: power_add power2_eq_square)
+  case (Suc n)
+  then show ?case by (simp add: power_add power2_eq_square)
 qed
 
-lemma power_minus1_odd:
-  "(- 1) ^ Suc (2*n) = -1"
+lemma power_minus1_odd: "(- 1) ^ Suc (2*n) = -1"
   by simp
 
-lemma power_minus_even [simp]:
-  "(-a) ^ (2*n) = a ^ (2*n)"
+lemma power_minus_even [simp]: "(-a) ^ (2*n) = a ^ (2*n)"
   by (simp add: power_minus [of a])
 
 end
@@ -300,8 +275,7 @@
 context ring_1_no_zero_divisors
 begin
 
-lemma power2_eq_1_iff:
-  "a\<^sup>2 = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
+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
@@ -317,13 +291,10 @@
 context algebraic_semidom
 begin
 
-lemma div_power:
-  assumes "b dvd a"
-  shows "(a div b) ^ n = a ^ n div b ^ n"
-  using assms by (induct n) (simp_all add: div_mult_div_if_dvd dvd_power_same)
+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"
+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)
 
 end
@@ -331,12 +302,10 @@
 context normalization_semidom
 begin
 
-lemma normalize_power:
-  "normalize (a ^ n) = normalize a ^ n"
+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"
+lemma unit_factor_power: "unit_factor (a ^ n) = unit_factor a ^ n"
   by (induct n) (simp_all add: unit_factor_mult)
 
 end
@@ -344,19 +313,19 @@
 context division_ring
 begin
 
-text\<open>Perhaps these should be simprules.\<close>
-lemma power_inverse [field_simps, divide_simps]:
-  "inverse a ^ n = inverse (a ^ n)"
+text \<open>Perhaps these should be simprules.\<close>
+lemma power_inverse [field_simps, divide_simps]: "inverse a ^ n = inverse (a ^ n)"
 proof (cases "a = 0")
-  case True then show ?thesis by (simp add: power_0_left)
+  case True
+  then show ?thesis by (simp add: power_0_left)
 next
-  case False then have "inverse (a ^ n) = inverse a ^ n"
+  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, divide_simps]:
-  "(1 / a) ^ n = 1 / a ^ n"
+lemma power_one_over [field_simps, divide_simps]: "(1 / a) ^ n = 1 / a ^ n"
   using power_inverse [of a] by (simp add: divide_inverse)
 
 end
@@ -365,12 +334,11 @@
 begin
 
 lemma power_diff:
-  assumes nz: "a \<noteq> 0"
+  assumes "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 power_not_zero)
+  by (induct m n rule: diff_induct) (simp_all add: assms power_not_zero)
 
-lemma power_divide [field_simps, divide_simps]:
-  "(a / b) ^ n = a ^ n / b ^ n"
+lemma power_divide [field_simps, divide_simps]: "(a / b) ^ n = a ^ n / b ^ n"
   by (induct n) simp_all
 
 end
@@ -381,22 +349,19 @@
 context linordered_semidom
 begin
 
-lemma zero_less_power [simp]:
-  "0 < a \<Longrightarrow> 0 < a ^ n"
+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"
+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"
+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: "\<lbrakk>0 \<le> a; a \<le> 1\<rbrakk> \<Longrightarrow> a ^ n \<le> 1"
+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:
@@ -405,19 +370,16 @@
 proof -
   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 \<open>0 \<le> a\<close> one_le_power order_less_imp_le
-        zero_le_one order_refl)
+  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"
+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"
+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:
@@ -431,123 +393,122 @@
   show ?case
   proof (cases n)
     case 0
-    with Suc.prems Suc.hyps have "a * a ^ m \<le> 1" by simp
+    with Suc have "a * a ^ m \<le> 1" by simp
     with gt1 show ?thesis
-      by (force simp only: power_gt1_lemma
-          not_less [symmetric])
+      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])
+      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"
+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"
+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 "0<a"} 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)
+text \<open>
+  Can relax the first premise to @{term "0<a"} 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 add: mult_strict_mono le_less_trans [of 0 a b])
+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])
 
 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 add: mult_strict_left_mono)
+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"
+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
+  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
+  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 [rule_format]:
-  "n \<le> N \<Longrightarrow> 0 \<le> a \<Longrightarrow> a \<le> 1 \<longrightarrow> a ^ N \<le> a ^ n"
+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
+  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
+  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_Suc_less_one:
-  "0 < a \<Longrightarrow> a < 1 \<Longrightarrow> a ^ Suc n < 1"
+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 [rule_format]:
-  "n \<le> N \<Longrightarrow> 1 \<le> a \<Longrightarrow> a ^ n \<le> a ^ N"
+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
+  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
+  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 add: mult_strict_left_mono less_trans [OF zero_less_one])
+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 [rule_format]:
-  "n < N \<Longrightarrow> 1 < a \<longrightarrow> a ^ n < a ^ N"
+lemma power_strict_increasing: "n < N \<Longrightarrow> 1 < a \<Longrightarrow> a ^ n < a ^ N"
 proof (induct N)
-  case 0 then show ?case by simp
+  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
+  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"
+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_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 ynonneg: "0 \<le> b"
+    and "0 \<le> b"
   shows "a \<le> b"
 proof (rule ccontr)
-  assume "~ a \<le> b"
+  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 power_strict_mono)
-  from le and this show False
+    by (simp only: assms(2) power_strict_mono)
+  with le show False
     by (simp add: linorder_not_less [symmetric])
 qed
 
@@ -556,38 +517,31 @@
   assumes nonneg: "0 \<le> b"
   shows "a < b"
 proof (rule contrapos_pp [OF less])
-  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 "\<not> a ^ n < b ^ n" by (simp only: linorder_not_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_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"
+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"
+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"
+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"
+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"
+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:
-  shows "0 \<le> a \<Longrightarrow> a \<le> 1 \<Longrightarrow> a ^ Suc n \<le> a"
+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
 
 end
@@ -595,16 +549,13 @@
 context linordered_ring_strict
 begin
 
-lemma sum_squares_eq_zero_iff:
-  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
+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"
+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"
+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
@@ -620,28 +571,26 @@
 
 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
+  case 0
+  show ?case by simp
 next
-  case (Suc n) show ?case by (auto simp add: Suc zero_less_mult_iff)
+  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 zero_le_power2 [simp]:
-  "0 \<le> a\<^sup>2"
+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"
+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"
+lemma power2_less_0 [simp]: "\<not> a\<^sup>2 < 0"
   by (force simp add: power2_eq_square mult_less_0_iff)
 
-lemma power2_less_eq_zero_iff [simp]:
-  "a\<^sup>2 \<le> 0 \<longleftrightarrow> a = 0"
+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"
@@ -650,8 +599,7 @@
 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"
+lemma odd_power_less_zero: "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
 proof (induct n)
   case 0
   then show ?case by simp
@@ -659,160 +607,152 @@
   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)
-  thus ?case
+  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"
+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])
+  by (force simp add: linorder_not_less [symmetric])
 
-lemma zero_le_even_power'[simp]:
-  "0 \<le> a ^ (2*n)"
+lemma zero_le_even_power'[simp]: "0 \<le> a ^ (2*n)"
 proof (induct n)
   case 0
-    show ?case by simp
+  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)
-    thus ?case
-      by (simp add: Suc zero_le_mult_iff)
+  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"
+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"
+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"
+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"
+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"
+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"
+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 "\<bar>x\<bar> \<le> \<bar>y\<bar>"
-  then have "\<bar>x\<bar>\<^sup>2 \<le> \<bar>y\<bar>\<^sup>2" by (rule power_mono, simp)
-  then show "x\<^sup>2 \<le> y\<^sup>2" by simp
+  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 "x\<^sup>2 \<le> y\<^sup>2"
-  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>"
+  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
+  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)
+  using  abs_square_eq_1 [of x] abs_square_le_1 [of x] by (auto simp add: le_less)
 
 end
 
 
 subsection \<open>Miscellaneous rules\<close>
 
-lemma (in linordered_semidom) self_le_power:
-  "1 \<le> a \<Longrightarrow> 0 < n \<Longrightarrow> a \<le> a ^ n"
+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)))"
+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"
+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)
 
-lemma (in comm_ring_1) power2_diff:
-  "(x - y)\<^sup>2 = x\<^sup>2 + y\<^sup>2 - 2 * x * y"
+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 (in comm_ring_1) power2_commute:
-  "(x - y)\<^sup>2 = (y - x)\<^sup>2"
+lemma power2_commute: "(x - y)\<^sup>2 = (y - x)\<^sup>2"
   by (simp add: algebra_simps power2_eq_square)
 
-lemma (in comm_ring_1) 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 (in comm_ring_1) minus_one_mult_self [simp]:
-  "(- 1) ^ n * (- 1) ^ n = 1"
+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 (in comm_ring_1) left_minus_one_mult_self [simp]:
-  "(- 1) ^ n * ((- 1) ^ n * a) = a"
+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
+  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"
+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::nat) \<or> n = 0"
+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"
+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"
+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 @{term "i=0"},
-@{term "m=1"} and @{term "n=0"}.\<close>
+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:
-  assumes nonneg: "0 < (i::nat)"
+  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
+  case True
+  with less power_one [where 'a = nat] show ?thesis by simp
 next
-  case False with nonneg have "1 < i" by auto
+  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::nat) < i \<Longrightarrow> m \<le> n"
-  apply (rule power_le_imp_le_exp, assumption)
-  apply (erule dvd_imp_le, simp)
+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 power2_nat_le_eq_le:
-  fixes m n :: nat
-  shows "m\<^sup>2 \<le> n\<^sup>2 \<longleftrightarrow> m \<le> n"
+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:
@@ -820,18 +760,20 @@
   assumes "m\<^sup>2 \<le> n"
   shows "m \<le> n"
 proof (cases m)
-  case 0 then show ?thesis by simp
+  case 0
+  then show ?thesis by simp
 next
   case (Suc k)
   show ?thesis
   proof (rule ccontr)
-    assume "\<not> m \<le> n"
+    assume "\<not> ?thesis"
     then have "n < m" by simp
     with assms Suc show False
       by (simp add: power2_eq_square)
   qed
 qed
 
+
 subsubsection \<open>Cardinality of the Powerset\<close>
 
 lemma card_UNIV_bool [simp]: "card (UNIV :: bool set) = 2"
@@ -840,16 +782,17 @@
 lemma card_Pow: "finite A \<Longrightarrow> card (Pow A) = 2 ^ card A"
 proof (induct rule: finite_induct)
   case empty
-    show ?case by auto
+  show ?case by auto
 next
   case (insert x A)
   then have "inj_on (insert x) (Pow A)"
     unfolding inj_on_def by (blast elim!: equalityE)
   then have "card (Pow A) + card (insert x ` Pow A) = 2 * 2 ^ card A"
     by (simp add: mult_2 card_image Pow_insert insert.hyps)
-  then show ?case using insert
+  with insert show ?case
     apply (simp add: Pow_insert)
-    apply (subst card_Un_disjoint, auto)
+    apply (subst card_Un_disjoint)
+       apply auto
     done
 qed