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>