diff --git a/src/HOL/Library/Ramsey.thy b/src/HOL/Library/Ramsey.thy
--- a/src/HOL/Library/Ramsey.thy
+++ b/src/HOL/Library/Ramsey.thy
@@ -1,878 +1,907 @@
 (*  Title:      HOL/Library/Ramsey.thy
     Author:     Tom Ridge. Full finite version by L C Paulson.
 *)
 
 section \<open>Ramsey's Theorem\<close>
 
 theory Ramsey
   imports Infinite_Set FuncSet
 begin
 
 subsection \<open>Preliminary definitions\<close>
 
 subsubsection \<open>The $n$-element subsets of a set $A$\<close>
 
 definition nsets :: "['a set, nat] \<Rightarrow> 'a set set" ("([_]\<^bsup>_\<^esup>)" [0,999] 999)
   where "nsets A n \<equiv> {N. N \<subseteq> A \<and> finite N \<and> card N = n}"
 
 lemma nsets_mono: "A \<subseteq> B \<Longrightarrow> nsets A n \<subseteq> nsets B n"
   by (auto simp: nsets_def)
 
 lemma nsets_2_eq: "nsets A 2 = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. {{x, y}})"
 by (auto simp: nsets_def card_2_iff)
 
+lemma nsets_doubleton_2_eq [simp]: "[{x, y}]\<^bsup>2\<^esup> = (if x=y then {} else {{x, y}})"
+  by (auto simp: nsets_2_eq)
+
+lemma doubleton_in_nsets_2 [simp]: "{x,y} \<in> [A]\<^bsup>2\<^esup> \<longleftrightarrow> x \<in> A \<and> y \<in> A \<and> x \<noteq> y"
+  by (auto simp: nsets_2_eq Set.doubleton_eq_iff)
+
 lemma binomial_eq_nsets: "n choose k = card (nsets {0..<n} k)"
   apply (simp add: binomial_def nsets_def)
   by (meson subset_eq_atLeast0_lessThan_finite)
 
 lemma nsets_eq_empty_iff: "nsets A r = {} \<longleftrightarrow> finite A \<and> card A < r"
   unfolding nsets_def
 proof (intro iffI conjI)
   assume that: "{N. N \<subseteq> A \<and> finite N \<and> card N = r} = {}"
   show "finite A"
     using infinite_arbitrarily_large that by auto
   then have "\<not> r \<le> card A"
     using that by (simp add: set_eq_iff) (metis finite_subset get_smaller_card [of A r])
   then show "card A < r"
     using not_less by blast
 next
   show "{N. N \<subseteq> A \<and> finite N \<and> card N = r} = {}"
     if "finite A \<and> card A < r"
     using that card_mono leD by auto
 qed
 
-lemma nsets_eq_empty: "n < r \<Longrightarrow> nsets {..<n} r = {}"
+lemma nsets_eq_empty: "\<lbrakk>finite A; card A < r\<rbrakk> \<Longrightarrow> nsets A r = {}"
   by (simp add: nsets_eq_empty_iff)
 
 lemma nsets_empty_iff: "nsets {} r = (if r=0 then {{}} else {})"
   by (auto simp: nsets_def)
 
 lemma nsets_singleton_iff: "nsets {a} r = (if r=0 then {{}} else if r=1 then {{a}} else {})"
   by (auto simp: nsets_def card_gt_0_iff subset_singleton_iff)
 
 lemma nsets_self [simp]: "nsets {..<m} m = {{..<m}}"
   unfolding nsets_def
   apply auto
   by (metis add.left_neutral lessThan_atLeast0 lessThan_iff subset_card_intvl_is_intvl)
 
 lemma nsets_zero [simp]: "nsets A 0 = {{}}"
   by (auto simp: nsets_def)
 
 lemma nsets_one: "nsets A (Suc 0) = (\<lambda>x. {x}) ` A"
   using card_eq_SucD by (force simp: nsets_def)
 
 lemma inj_on_nsets:
   assumes "inj_on f A"
   shows "inj_on (\<lambda>X. f ` X) ([A]\<^bsup>n\<^esup>)"
   using assms unfolding nsets_def
   by (metis (no_types, lifting) inj_on_inverseI inv_into_image_cancel mem_Collect_eq)
 
 lemma bij_betw_nsets:
   assumes "bij_betw f A B"
   shows "bij_betw (\<lambda>X. f ` X) ([A]\<^bsup>n\<^esup>) ([B]\<^bsup>n\<^esup>)"
 proof -
   have "(`) f ` [A]\<^bsup>n\<^esup> = [f ` A]\<^bsup>n\<^esup>"
     using assms
     apply (auto simp: nsets_def bij_betw_def image_iff card_image inj_on_subset)
     by (metis card_image inj_on_finite order_refl subset_image_inj)
   with assms show ?thesis
     by (auto simp: bij_betw_def inj_on_nsets)
 qed
 
+lemma nset_image_obtains:
+  assumes "X \<in> [f`A]\<^bsup>k\<^esup>" "inj_on f A"
+  obtains Y where "Y \<in> [A]\<^bsup>k\<^esup>" "X = f ` Y"
+  using assms
+  apply (clarsimp simp add: nsets_def subset_image_iff)
+  by (metis card_image finite_imageD inj_on_subset)
+
+lemma nsets_image_funcset:
+  assumes "g \<in> S \<rightarrow> T" and "inj_on g S"
+  shows "(\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
+    using assms
+    by (fastforce simp add: nsets_def card_image inj_on_subset subset_iff simp flip: image_subset_iff_funcset)
+
+lemma nsets_compose_image_funcset:
+  assumes f: "f \<in> [T]\<^bsup>k\<^esup> \<rightarrow> D" and "g \<in> S \<rightarrow> T" and "inj_on g S"
+  shows "f \<circ> (\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> D"
+proof -
+  have "(\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
+    using assms by (simp add: nsets_image_funcset)
+  then show ?thesis
+    using f by fastforce 
+qed
+
 subsubsection \<open>Partition predicates\<close>
 
 definition partn :: "'a set \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b set \<Rightarrow> bool"
   where "partn \<beta> \<alpha> \<gamma> \<delta> \<equiv> \<forall>f \<in> nsets \<beta> \<gamma>  \<rightarrow>  \<delta>. \<exists>H \<in> nsets \<beta> \<alpha>. \<exists>\<xi>\<in>\<delta>. f ` (nsets H \<gamma>) \<subseteq> {\<xi>}"
 
 definition partn_lst :: "'a set \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool"
   where "partn_lst \<beta> \<alpha> \<gamma> \<equiv> \<forall>f \<in> nsets \<beta> \<gamma>  \<rightarrow>  {..<length \<alpha>}. 
               \<exists>i < length \<alpha>. \<exists>H \<in> nsets \<beta> (\<alpha>!i). f ` (nsets H \<gamma>) \<subseteq> {i}"
 
 lemma partn_lst_greater_resource:
   fixes M::nat
   assumes M: "partn_lst {..<M} \<alpha> \<gamma>" and "M \<le> N"
   shows "partn_lst {..<N} \<alpha> \<gamma>"
 proof (clarsimp simp: partn_lst_def)
   fix f
   assume "f \<in> nsets {..<N} \<gamma> \<rightarrow> {..<length \<alpha>}"
   then have "f \<in> nsets {..<M} \<gamma> \<rightarrow> {..<length \<alpha>}"
     by (meson Pi_anti_mono \<open>M \<le> N\<close> lessThan_subset_iff nsets_mono subsetD)
   then obtain i H where i: "i < length \<alpha>" and H: "H \<in> nsets {..<M} (\<alpha> ! i)" and subi: "f ` nsets H \<gamma> \<subseteq> {i}"
     using M partn_lst_def by blast
   have "H \<in> nsets {..<N} (\<alpha> ! i)"
     using \<open>M \<le> N\<close> H by (auto simp: nsets_def subset_iff)
   then show "\<exists>i<length \<alpha>. \<exists>H\<in>nsets {..<N} (\<alpha> ! i). f ` nsets H \<gamma> \<subseteq> {i}"
     using i subi by blast
 qed
 
 lemma partn_lst_fewer_colours:
   assumes major: "partn_lst \<beta> (n#\<alpha>) \<gamma>" and "n \<ge> \<gamma>"
   shows "partn_lst \<beta> \<alpha> \<gamma>"
 proof (clarsimp simp: partn_lst_def)
   fix f :: "'a set \<Rightarrow> nat"
   assume f: "f \<in> [\<beta>]\<^bsup>\<gamma>\<^esup> \<rightarrow> {..<length \<alpha>}"
   then obtain i H where i: "i < Suc (length \<alpha>)"
       and H: "H \<in> [\<beta>]\<^bsup>((n # \<alpha>) ! i)\<^esup>"
       and hom: "\<forall>x\<in>[H]\<^bsup>\<gamma>\<^esup>. Suc (f x) = i"
     using \<open>n \<ge> \<gamma>\<close> major [unfolded partn_lst_def, rule_format, of "Suc o f"]
     by (fastforce simp: image_subset_iff nsets_eq_empty_iff)
   show "\<exists>i<length \<alpha>. \<exists>H\<in>nsets \<beta> (\<alpha> ! i). f ` [H]\<^bsup>\<gamma>\<^esup> \<subseteq> {i}"
   proof (cases i)
     case 0
     then have "[H]\<^bsup>\<gamma>\<^esup> = {}"
       using hom by blast
     then show ?thesis
       using 0 H \<open>n \<ge> \<gamma>\<close>
       by (simp add: nsets_eq_empty_iff) (simp add: nsets_def)
   next
     case (Suc i')
     then show ?thesis
       using i H hom by auto
   qed
 qed
 
 lemma partn_lst_eq_partn: "partn_lst {..<n} [m,m] 2 = partn {..<n} m 2 {..<2::nat}"
   apply (simp add: partn_lst_def partn_def numeral_2_eq_2)
   by (metis less_2_cases numeral_2_eq_2 lessThan_iff nth_Cons_0 nth_Cons_Suc)
   
  
 subsection \<open>Finite versions of Ramsey's theorem\<close>
 
 text \<open>
   To distinguish the finite and infinite ones, lower and upper case
   names are used.
 \<close>
 
 subsubsection \<open>Trivial cases\<close>
 
 text \<open>Vacuous, since we are dealing with 0-sets!\<close>
 lemma ramsey0: "\<exists>N::nat. partn_lst {..<N} [q1,q2] 0"
   by (force simp: partn_lst_def ex_in_conv less_Suc_eq nsets_eq_empty_iff)
 
 text \<open>Just the pigeon hole principle, since we are dealing with 1-sets\<close>
 lemma ramsey1: "\<exists>N::nat. partn_lst {..<N} [q0,q1] 1"
 proof -
   have "\<exists>i<Suc (Suc 0). \<exists>H\<in>nsets {..<Suc (q0 + q1)} ([q0, q1] ! i). f ` nsets H (Suc 0) \<subseteq> {i}"
     if "f \<in> nsets {..<Suc (q0 + q1)} (Suc 0) \<rightarrow> {..<Suc (Suc 0)}" for f 
   proof -
     define A where "A \<equiv> \<lambda>i. {q. q \<le> q0+q1 \<and> f {q} = i}"
     have "A 0 \<union> A 1 = {..q0 + q1}"
       using that by (auto simp: A_def PiE_iff nsets_one lessThan_Suc_atMost le_Suc_eq)
     moreover have "A 0 \<inter> A 1 = {}"
       by (auto simp: A_def)
     ultimately have "q0 + q1 \<le> card (A 0) + card (A 1)"
       by (metis card_Un_le card_atMost eq_imp_le le_SucI le_trans)
     then consider "card (A 0) \<ge> q0" | "card (A 1) \<ge> q1"
       by linarith
     then obtain i where "i < Suc (Suc 0)" "card (A i) \<ge> [q0, q1] ! i"
       by (metis One_nat_def lessI nth_Cons_0 nth_Cons_Suc zero_less_Suc)
     then obtain B where "B \<subseteq> A i" "card B = [q0, q1] ! i" "finite B"
       by (meson finite_subset get_smaller_card infinite_arbitrarily_large)
     then have "B \<in> nsets {..<Suc (q0 + q1)} ([q0, q1] ! i) \<and> f ` nsets B (Suc 0) \<subseteq> {i}"
       by (auto simp: A_def nsets_def card_1_singleton_iff)
     then show ?thesis
       using \<open>i < Suc (Suc 0)\<close> by auto
   qed
   then show ?thesis
     by (clarsimp simp: partn_lst_def) blast
 qed
 
 
 subsubsection \<open>Ramsey's theorem with two colours and arbitrary exponents (hypergraph version)\<close>
 
 proposition ramsey2_full: "\<exists>N::nat. partn_lst {..<N} [q1,q2] r"
 proof (induction r arbitrary: q1 q2)
   case 0
   then show ?case
     by (simp add: ramsey0) 
 next
   case (Suc r)
   note outer = this
   show ?case 
   proof (cases "r = 0")
     case True
     then show ?thesis
       using ramsey1 by auto
   next
     case False
     then have "r > 0"
       by simp
     show ?thesis
       using Suc.prems
     proof (induct k \<equiv> "q1 + q2" arbitrary: q1 q2)
       case 0
       show ?case 
       proof
         show "partn_lst {..<1::nat} [q1, q2] (Suc r)"
           using nsets_empty_iff subset_insert 0
           by (fastforce simp: partn_lst_def funcset_to_empty_iff nsets_eq_empty image_subset_iff)
       qed
     next
       case (Suc k)
       consider "q1 = 0 \<or> q2 = 0" | "q1 \<noteq> 0" "q2 \<noteq> 0" by auto
       then show ?case 
       proof cases
         case 1
         then have "partn_lst {..< Suc 0} [q1, q2] (Suc r)"
           unfolding partn_lst_def using \<open>r > 0\<close>
           by (fastforce simp add: nsets_empty_iff nsets_singleton_iff lessThan_Suc)
         then show ?thesis by blast
       next
         case 2
         with Suc have "k = (q1 - 1) + q2" "k = q1 + (q2 - 1)" by auto
         then obtain p1 p2::nat where  p1: "partn_lst {..<p1} [q1-1,q2] (Suc r)" and p2: "partn_lst {..<p2} [q1,q2-1] (Suc r)"
           using Suc.hyps by blast
         then obtain p::nat where p: "partn_lst {..<p} [p1,p2] r" 
           using outer Suc.prems by auto
         show ?thesis
         proof (intro exI conjI)
           have "\<exists>i<Suc (Suc 0). \<exists>H\<in>nsets {..p} ([q1,q2] ! i). f ` nsets H (Suc r) \<subseteq> {i}"
             if f: "f \<in> nsets {..p} (Suc r) \<rightarrow> {..<Suc (Suc 0)}" for f 
           proof -
             define g where "g \<equiv> \<lambda>R. f (insert p R)"
             have "f (insert p i) \<in> {..<Suc (Suc 0)}" if "i \<in> nsets {..<p} r" for i
               using that card_insert_if by (fastforce simp: nsets_def intro!: Pi_mem [OF f])
             then have g: "g \<in> nsets {..<p} r \<rightarrow> {..<Suc (Suc 0)}"
               by (force simp: g_def PiE_iff)
             then obtain i U where i: "i < Suc (Suc 0)" and gi: "g ` nsets U r \<subseteq> {i}"
               and U: "U \<in> nsets {..<p} ([p1, p2] ! i)" 
               using p by (auto simp: partn_lst_def)
             then have Usub: "U \<subseteq> {..<p}"
               by (auto simp: nsets_def)
             consider (izero) "i = 0" | (ione) "i = Suc 0"
               using i by linarith
             then show ?thesis
             proof cases
               case izero
               then have "U \<in> nsets {..<p} p1"
                 using U by simp
               then obtain u where u: "bij_betw u {..<p1} U" 
                 using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
               have u_nsets: "u ` X \<in> nsets {..p} n" if "X \<in> nsets {..<p1} n" for X n
               proof -
                 have "inj_on u X"
                   using u that bij_betw_imp_inj_on inj_on_subset by (force simp: nsets_def)
                 then show ?thesis
                   using Usub u that bij_betwE
                   by (fastforce simp add: nsets_def card_image)
               qed
               define h where "h \<equiv> \<lambda>R. f (u ` R)"
               have "h \<in> nsets {..<p1} (Suc r) \<rightarrow> {..<Suc (Suc 0)}"
                 unfolding h_def using f u_nsets by auto
               then obtain j V where j: "j <Suc (Suc 0)" and hj: "h ` nsets V (Suc r) \<subseteq> {j}"
                 and V: "V \<in> nsets {..<p1} ([q1 - Suc 0, q2] ! j)" 
                 using p1 by (auto simp: partn_lst_def)
               then have Vsub: "V \<subseteq> {..<p1}"
                 by (auto simp: nsets_def)
               have invinv_eq: "u ` inv_into {..<p1} u ` X = X" if "X \<subseteq> u ` {..<p1}" for X
                 by (simp add: image_inv_into_cancel that)
               let ?W = "insert p (u ` V)"
               consider (jzero) "j = 0" | (jone) "j = Suc 0"
                 using j by linarith
               then show ?thesis
               proof cases
                 case jzero
                 then have "V \<in> nsets {..<p1} (q1 - Suc 0)"
                   using V by simp
                 then have "u ` V \<in> nsets {..<p} (q1 - Suc 0)"
                   using u_nsets [of _ "q1 - Suc 0"] nsets_mono [OF Vsub] Usub u
                   unfolding bij_betw_def nsets_def 
                   by (fastforce elim!: subsetD)
                 then have inq1: "?W \<in> nsets {..p} q1"
                   unfolding nsets_def using \<open>q1 \<noteq> 0\<close> card_insert_if by fastforce
                 have invu_nsets: "inv_into {..<p1} u ` X \<in> nsets V r" 
                   if "X \<in> nsets (u ` V) r" for X r
                 proof -
                   have "X \<subseteq> u ` V \<and> finite X \<and> card X = r"
                     using nsets_def that by auto
                   then have [simp]: "card (inv_into {..<p1} u ` X) = card X"
                     by (meson Vsub bij_betw_def bij_betw_inv_into card_image image_mono inj_on_subset u)
                   show ?thesis
                     using that u Vsub by (fastforce simp: nsets_def bij_betw_def)
                 qed
                 have "f X = i" if X: "X \<in> nsets ?W (Suc r)" for X
                 proof (cases "p \<in> X")
                   case True
                   then have Xp: "X - {p} \<in> nsets (u ` V) r"
                     using X by (auto simp: nsets_def)
                   moreover have "u ` V \<subseteq> U"
                     using Vsub bij_betwE u by blast
                   ultimately have "X - {p} \<in> nsets U r"
                     by (meson in_mono nsets_mono)
                   then have "g (X - {p}) = i"
                     using gi by blast
                   have "f X = i"
                     using gi True \<open>X - {p} \<in> nsets U r\<close> insert_Diff
                     by (fastforce simp add: g_def image_subset_iff)
                   then show ?thesis
                     by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>) 
                 next
                   case False
                   then have Xim: "X \<in> nsets (u ` V) (Suc r)"
                     using X by (auto simp: nsets_def subset_insert)
                   then have "u ` inv_into {..<p1} u ` X = X"
                     using Vsub bij_betw_imp_inj_on u 
                     by (fastforce simp: nsets_def image_mono invinv_eq subset_trans)
                   then show ?thesis
                     using izero jzero hj Xim invu_nsets unfolding h_def
                     by (fastforce simp add: image_subset_iff)
                 qed
                 moreover have "insert p (u ` V) \<in> nsets {..p} q1"
                   by (simp add: izero inq1)
                 ultimately show ?thesis
                   by (metis izero image_subsetI insertI1 nth_Cons_0 zero_less_Suc) 
               next
                 case jone
                 then have "u ` V \<in> nsets {..p} q2"
                   using V u_nsets by auto
                 moreover have "f ` nsets (u ` V) (Suc r) \<subseteq> {j}"
                   using hj 
                   by (force simp add: h_def image_subset_iff nsets_def subset_image_inj card_image dest: finite_imageD)
                 ultimately show ?thesis
                   using jone not_less_eq by fastforce
               qed
             next 
               case ione
               then have "U \<in> nsets {..<p} p2"
                 using U by simp
               then obtain u where u: "bij_betw u {..<p2} U" 
                 using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
               have u_nsets: "u ` X \<in> nsets {..p} n" if "X \<in> nsets {..<p2} n" for X n
               proof -
                 have "inj_on u X"
                   using u that bij_betw_imp_inj_on inj_on_subset by (force simp: nsets_def)
                 then show ?thesis
                   using Usub u that bij_betwE
                   by (fastforce simp add: nsets_def card_image)
               qed
               define h where "h \<equiv> \<lambda>R. f (u ` R)"
               have "h \<in> nsets {..<p2} (Suc r) \<rightarrow> {..<Suc (Suc 0)}"
                 unfolding h_def using f u_nsets by auto
               then obtain j V where j: "j <Suc (Suc 0)" and hj: "h ` nsets V (Suc r) \<subseteq> {j}"
                 and V: "V \<in> nsets {..<p2} ([q1, q2 - Suc 0] ! j)" 
                 using p2 by (auto simp: partn_lst_def)
               then have Vsub: "V \<subseteq> {..<p2}"
                 by (auto simp: nsets_def)
               have invinv_eq: "u ` inv_into {..<p2} u ` X = X" if "X \<subseteq> u ` {..<p2}" for X
                 by (simp add: image_inv_into_cancel that)
               let ?W = "insert p (u ` V)"
               consider (jzero) "j = 0" | (jone) "j = Suc 0"
                 using j by linarith
               then show ?thesis
               proof cases
                 case jone
                 then have "V \<in> nsets {..<p2} (q2 - Suc 0)"
                   using V by simp
                 then have "u ` V \<in> nsets {..<p} (q2 - Suc 0)"
                   using u_nsets [of _ "q2 - Suc 0"] nsets_mono [OF Vsub] Usub u
                   unfolding bij_betw_def nsets_def 
                   by (fastforce elim!: subsetD)
                 then have inq1: "?W \<in> nsets {..p} q2"
                   unfolding nsets_def using \<open>q2 \<noteq> 0\<close> card_insert_if by fastforce
                 have invu_nsets: "inv_into {..<p2} u ` X \<in> nsets V r" 
                   if "X \<in> nsets (u ` V) r" for X r
                 proof -
                   have "X \<subseteq> u ` V \<and> finite X \<and> card X = r"
                     using nsets_def that by auto
                   then have [simp]: "card (inv_into {..<p2} u ` X) = card X"
                     by (meson Vsub bij_betw_def bij_betw_inv_into card_image image_mono inj_on_subset u)
                   show ?thesis
                     using that u Vsub by (fastforce simp: nsets_def bij_betw_def)
                 qed
                 have "f X = i" if X: "X \<in> nsets ?W (Suc r)" for X
                 proof (cases "p \<in> X")
                   case True
                   then have Xp: "X - {p} \<in> nsets (u ` V) r"
                     using X by (auto simp: nsets_def)
                   moreover have "u ` V \<subseteq> U"
                     using Vsub bij_betwE u by blast
                   ultimately have "X - {p} \<in> nsets U r"
                     by (meson in_mono nsets_mono)
                   then have "g (X - {p}) = i"
                     using gi by blast
                   have "f X = i"
                     using gi True \<open>X - {p} \<in> nsets U r\<close> insert_Diff
                     by (fastforce simp add: g_def image_subset_iff)
                   then show ?thesis
                     by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>) 
                 next
                   case False
                   then have Xim: "X \<in> nsets (u ` V) (Suc r)"
                     using X by (auto simp: nsets_def subset_insert)
                   then have "u ` inv_into {..<p2} u ` X = X"
                     using Vsub bij_betw_imp_inj_on u 
                     by (fastforce simp: nsets_def image_mono invinv_eq subset_trans)
                   then show ?thesis
                     using ione jone hj Xim invu_nsets unfolding h_def
                     by (fastforce simp add: image_subset_iff)
                 qed
                 moreover have "insert p (u ` V) \<in> nsets {..p} q2"
                   by (simp add: ione inq1)
                 ultimately show ?thesis
                   by (metis ione image_subsetI insertI1 lessI nth_Cons_0 nth_Cons_Suc)
               next
                 case jzero
                 then have "u ` V \<in> nsets {..p} q1"
                   using V u_nsets by auto
                 moreover have "f ` nsets (u ` V) (Suc r) \<subseteq> {j}"
                   using hj 
                   apply (clarsimp simp add: h_def image_subset_iff nsets_def)
                   by (metis Zero_not_Suc card_eq_0_iff card_image subset_image_inj)
                 ultimately show ?thesis
                   using jzero not_less_eq by fastforce
               qed
             qed
           qed
           then show "partn_lst {..<Suc p} [q1,q2] (Suc r)"
             using lessThan_Suc lessThan_Suc_atMost by (auto simp: partn_lst_def insert_commute)
         qed
       qed
     qed
   qed
 qed
 
 subsubsection \<open>Full Ramsey's theorem with multiple colours and arbitrary exponents\<close>
 
 theorem ramsey_full: "\<exists>N::nat. partn_lst {..<N} qs r"
 proof (induction k \<equiv> "length qs" arbitrary: qs)
   case 0
   then show ?case
     by (rule_tac x=" r" in exI) (simp add: partn_lst_def)
 next
   case (Suc k)
   note IH = this
   show ?case
   proof (cases k)
     case 0
     with Suc obtain q where "qs = [q]"
       by (metis length_0_conv length_Suc_conv)
     then show ?thesis
       by (rule_tac x=q in exI) (auto simp: partn_lst_def funcset_to_empty_iff)
   next
     case (Suc k')
     then obtain q1 q2 l where qs: "qs = q1#q2#l"
       by (metis Suc.hyps(2) length_Suc_conv)
     then obtain q::nat where q: "partn_lst {..<q} [q1,q2] r"
       using ramsey2_full by blast 
     then obtain p::nat where p: "partn_lst {..<p} (q#l) r"
       using IH \<open>qs = q1 # q2 # l\<close> by fastforce 
     have keq: "Suc (length l) = k"
       using IH qs by auto
     show ?thesis
     proof (intro exI conjI)
       show "partn_lst {..<p} qs r"
       proof (auto simp: partn_lst_def)
         fix f
         assume f: "f \<in> nsets {..<p} r \<rightarrow> {..<length qs}"
         define g where "g \<equiv> \<lambda>X. if f X < Suc (Suc 0) then 0 else f X - Suc 0"
         have "g \<in> nsets {..<p} r \<rightarrow> {..<k}"
           unfolding g_def using f Suc IH
           by (auto simp: Pi_def not_less)
         then obtain i U where i: "i < k" and gi: "g ` nsets U r \<subseteq> {i}"
                 and U: "U \<in> nsets {..<p} ((q#l) ! i)" 
           using p keq by (auto simp: partn_lst_def)
         show "\<exists>i<length qs. \<exists>H\<in>nsets {..<p} (qs ! i). f ` nsets H r \<subseteq> {i}"
         proof (cases "i = 0")
           case True
           then have "U \<in> nsets {..<p} q" and f01: "f ` nsets U r \<subseteq> {0, Suc 0}"
             using U gi unfolding g_def by (auto simp: image_subset_iff)
           then obtain u where u: "bij_betw u {..<q} U" 
             using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
           then have Usub: "U \<subseteq> {..<p}"
             by (smt \<open>U \<in> nsets {..<p} q\<close> mem_Collect_eq nsets_def)
           have u_nsets: "u ` X \<in> nsets {..<p} n" if "X \<in> nsets {..<q} n" for X n
           proof -
             have "inj_on u X"
               using u that bij_betw_imp_inj_on inj_on_subset              
               by (force simp: nsets_def)
             then show ?thesis
               using Usub u that bij_betwE
               by (fastforce simp add: nsets_def card_image)
           qed
           define h where "h \<equiv> \<lambda>X. f (u ` X)"
           have "f (u ` X) < Suc (Suc 0)" if "X \<in> nsets {..<q} r" for X
           proof -
             have "u ` X \<in> nsets U r"
               using u u_nsets that by (auto simp: nsets_def bij_betwE subset_eq)
             then show ?thesis
               using f01 by auto
           qed
           then have "h \<in> nsets {..<q} r \<rightarrow> {..<Suc (Suc 0)}"
             unfolding h_def by blast  
           then obtain j V where j: "j < Suc (Suc 0)" and hj: "h ` nsets V r \<subseteq> {j}"
             and V: "V \<in> nsets {..<q} ([q1,q2] ! j)" 
             using q by (auto simp: partn_lst_def)
           show ?thesis
           proof (intro exI conjI bexI)
             show "j < length qs"
               using Suc Suc.hyps(2) j by linarith
             have "nsets (u ` V) r \<subseteq> (\<lambda>x. (u ` x)) ` nsets V r"
               apply (clarsimp simp add: nsets_def image_iff)
               by (metis card_eq_0_iff card_image image_is_empty subset_image_inj)
             then have "f ` nsets (u ` V) r \<subseteq> h ` nsets V r"
               by (auto simp: h_def)
             then show "f ` nsets (u ` V) r \<subseteq> {j}"
               using hj by auto
             show "(u ` V) \<in> nsets {..<p} (qs ! j)"
               using V j less_2_cases numeral_2_eq_2 qs u_nsets by fastforce
           qed
         next
           case False
           show ?thesis
           proof (intro exI conjI bexI)
             show "Suc i < length qs"
               using Suc.hyps(2) i by auto
             show "f ` nsets U r \<subseteq> {Suc i}"
               using i gi False
               apply (auto simp: g_def image_subset_iff)
               by (metis Suc_lessD Suc_pred g_def gi image_subset_iff not_less_eq singleton_iff)
             show "U \<in> nsets {..<p} (qs ! (Suc i))"
               using False U qs by auto
           qed
         qed
       qed
     qed
   qed
 qed
 
 subsubsection \<open>Simple graph version\<close>
 
 text \<open>This is the most basic version in terms of cliques and independent
   sets, i.e. the version for graphs and 2 colours.
 \<close>
 
 definition "clique V E \<longleftrightarrow> (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<in> E)"
 definition "indep V E \<longleftrightarrow> (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<notin> E)"
 
 lemma ramsey2:
   "\<exists>r\<ge>1. \<forall>(V::'a set) (E::'a set set). finite V \<and> card V \<ge> r \<longrightarrow>
     (\<exists>R \<subseteq> V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E)"
 proof -
   obtain N where "N \<ge> Suc 0" and N: "partn_lst {..<N} [m,n] 2"
     using ramsey2_full nat_le_linear partn_lst_greater_resource by blast
   have "\<exists>R\<subseteq>V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E" 
     if "finite V" "N \<le> card V" for V :: "'a set" and E :: "'a set set"
   proof -
     from that
     obtain v where u: "inj_on v {..<N}" "v ` {..<N} \<subseteq> V"
       by (metis card_le_inj card_lessThan finite_lessThan)
     define f where "f \<equiv> \<lambda>e. if v ` e \<in> E then 0 else Suc 0"
     have f: "f \<in> nsets {..<N} 2 \<rightarrow> {..<Suc (Suc 0)}"
       by (simp add: f_def)
     then obtain i U where i: "i < 2" and gi: "f ` nsets U 2 \<subseteq> {i}"
       and U: "U \<in> nsets {..<N} ([m,n] ! i)" 
       using N numeral_2_eq_2 by (auto simp: partn_lst_def)
     show ?thesis
     proof (intro exI conjI)
       show "v ` U \<subseteq> V"
         using U u by (auto simp: image_subset_iff nsets_def)
       show "card (v ` U) = m \<and> clique (v ` U) E \<or> card (v ` U) = n \<and> indep (v ` U) E"
         using i unfolding numeral_2_eq_2
           using gi U u
           apply (simp add: image_subset_iff nsets_2_eq clique_def indep_def less_Suc_eq)
           apply (auto simp: f_def nsets_def card_image inj_on_subset split: if_split_asm)
           done
     qed
   qed
   then show ?thesis
     using \<open>Suc 0 \<le> N\<close> by auto
 qed
 
 
 subsection \<open>Preliminaries\<close>
 
 subsubsection \<open>``Axiom'' of Dependent Choice\<close>
 
 primrec choice :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a"
   where \<comment> \<open>An integer-indexed chain of choices\<close>
     choice_0: "choice P r 0 = (SOME x. P x)"
   | choice_Suc: "choice P r (Suc n) = (SOME y. P y \<and> (choice P r n, y) \<in> r)"
 
 lemma choice_n:
   assumes P0: "P x0"
     and Pstep: "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r"
   shows "P (choice P r n)"
 proof (induct n)
   case 0
   show ?case by (force intro: someI P0)
 next
   case Suc
   then show ?case by (auto intro: someI2_ex [OF Pstep])
 qed
 
 lemma dependent_choice:
   assumes trans: "trans r"
     and P0: "P x0"
     and Pstep: "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r"
   obtains f :: "nat \<Rightarrow> 'a" where "\<And>n. P (f n)" and "\<And>n m. n < m \<Longrightarrow> (f n, f m) \<in> r"
 proof
   fix n
   show "P (choice P r n)"
     by (blast intro: choice_n [OF P0 Pstep])
 next
   fix n m :: nat
   assume "n < m"
   from Pstep [OF choice_n [OF P0 Pstep]] have "(choice P r k, choice P r (Suc k)) \<in> r" for k
     by (auto intro: someI2_ex)
   then show "(choice P r n, choice P r m) \<in> r"
     by (auto intro: less_Suc_induct [OF \<open>n < m\<close>] transD [OF trans])
 qed
 
 
 subsubsection \<open>Partition functions\<close>
 
 definition part_fn :: "nat \<Rightarrow> nat \<Rightarrow> 'a set \<Rightarrow> ('a set \<Rightarrow> nat) \<Rightarrow> bool"
   \<comment> \<open>the function \<^term>\<open>f\<close> partitions the \<^term>\<open>r\<close>-subsets of the typically
       infinite set \<^term>\<open>Y\<close> into \<^term>\<open>s\<close> distinct categories.\<close>
   where "part_fn r s Y f \<longleftrightarrow> (f \<in> nsets Y r \<rightarrow> {..<s})"
 
 text \<open>For induction, we decrease the value of \<^term>\<open>r\<close> in partitions.\<close>
 lemma part_fn_Suc_imp_part_fn:
   "\<lbrakk>infinite Y; part_fn (Suc r) s Y f; y \<in> Y\<rbrakk> \<Longrightarrow> part_fn r s (Y - {y}) (\<lambda>u. f (insert y u))"
   by (simp add: part_fn_def nsets_def Pi_def subset_Diff_insert)
 
 lemma part_fn_subset: "part_fn r s YY f \<Longrightarrow> Y \<subseteq> YY \<Longrightarrow> part_fn r s Y f"
   unfolding part_fn_def nsets_def by blast
 
 
 subsection \<open>Ramsey's Theorem: Infinitary Version\<close>
 
 lemma Ramsey_induction:
   fixes s r :: nat
     and YY :: "'a set"
     and f :: "'a set \<Rightarrow> nat"
   assumes "infinite YY" "part_fn r s YY f"
   shows "\<exists>Y' t'. Y' \<subseteq> YY \<and> infinite Y' \<and> t' < s \<and> (\<forall>X. X \<subseteq> Y' \<and> finite X \<and> card X = r \<longrightarrow> f X = t')"
   using assms
 proof (induct r arbitrary: YY f)
   case 0
   then show ?case
     by (auto simp add: part_fn_def card_eq_0_iff cong: conj_cong)
 next
   case (Suc r)
   show ?case
   proof -
     from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY"
       by blast
     let ?ramr = "{((y, Y, t), (y', Y', t')). y' \<in> Y \<and> Y' \<subseteq> Y}"
     let ?propr = "\<lambda>(y, Y, t).
                  y \<in> YY \<and> y \<notin> Y \<and> Y \<subseteq> YY \<and> infinite Y \<and> t < s
                  \<and> (\<forall>X. X\<subseteq>Y \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert y) X = t)"
     from Suc.prems have infYY': "infinite (YY - {yy})" by auto
     from Suc.prems have partf': "part_fn r s (YY - {yy}) (f \<circ> insert yy)"
       by (simp add: o_def part_fn_Suc_imp_part_fn yy)
     have transr: "trans ?ramr" by (force simp add: trans_def)
     from Suc.hyps [OF infYY' partf']
     obtain Y0 and t0 where "Y0 \<subseteq> YY - {yy}" "infinite Y0" "t0 < s"
       "X \<subseteq> Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0" for X
       by blast
     with yy have propr0: "?propr(yy, Y0, t0)" by blast
     have proprstep: "\<exists>y. ?propr y \<and> (x, y) \<in> ?ramr" if x: "?propr x" for x
     proof (cases x)
       case (fields yx Yx tx)
       with x obtain yx' where yx': "yx' \<in> Yx"
         by (blast dest: infinite_imp_nonempty)
       from fields x have infYx': "infinite (Yx - {yx'})" by auto
       with fields x yx' Suc.prems have partfx': "part_fn r s (Yx - {yx'}) (f \<circ> insert yx')"
         by (simp add: o_def part_fn_Suc_imp_part_fn part_fn_subset [where YY=YY and Y=Yx])
       from Suc.hyps [OF infYx' partfx'] obtain Y' and t'
         where Y': "Y' \<subseteq> Yx - {yx'}" "infinite Y'" "t' < s"
           "X \<subseteq> Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'" for X
         by blast
       from fields x Y' yx' have "?propr (yx', Y', t') \<and> (x, (yx', Y', t')) \<in> ?ramr"
         by blast
       then show ?thesis ..
     qed
     from dependent_choice [OF transr propr0 proprstep]
     obtain g where pg: "?propr (g n)" and rg: "n < m \<Longrightarrow> (g n, g m) \<in> ?ramr" for n m :: nat
       by blast
     let ?gy = "fst \<circ> g"
     let ?gt = "snd \<circ> snd \<circ> g"
     have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
     proof (intro exI subsetI)
       fix x
       assume "x \<in> range ?gt"
       then obtain n where "x = ?gt n" ..
       with pg [of n] show "x \<in> {..<s}" by (cases "g n") auto
     qed
     from rangeg have "finite (range ?gt)"
       by (simp add: finite_nat_iff_bounded)
     then obtain s' and n' where s': "s' = ?gt n'" and infeqs': "infinite {n. ?gt n = s'}"
       by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: infinite_UNIV_nat)
     with pg [of n'] have less': "s'<s" by (cases "g n'") auto
     have inj_gy: "inj ?gy"
     proof (rule linorder_injI)
       fix m m' :: nat
       assume "m < m'"
       from rg [OF this] pg [of m] show "?gy m \<noteq> ?gy m'"
         by (cases "g m", cases "g m'") auto
     qed
     show ?thesis
     proof (intro exI conjI)
       from pg show "?gy ` {n. ?gt n = s'} \<subseteq> YY"
         by (auto simp add: Let_def split_beta)
       from infeqs' show "infinite (?gy ` {n. ?gt n = s'})"
         by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
       show "s' < s" by (rule less')
       show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} \<and> finite X \<and> card X = Suc r \<longrightarrow> f X = s'"
       proof -
         have "f X = s'"
           if X: "X \<subseteq> ?gy ` {n. ?gt n = s'}"
           and cardX: "finite X" "card X = Suc r"
           for X
         proof -
           from X obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
             by (auto simp add: subset_image_iff)
           with cardX have "AA \<noteq> {}" by auto
           then have AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
           show ?thesis
           proof (cases "g (LEAST x. x \<in> AA)")
             case (fields ya Ya ta)
             with AAleast Xeq have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
             then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
             also have "\<dots> = ta"
             proof -
               have *: "X - {ya} \<subseteq> Ya"
               proof
                 fix x assume x: "x \<in> X - {ya}"
                 then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
                   by (auto simp add: Xeq)
                 with fields x have "a' \<noteq> (LEAST x. x \<in> AA)" by auto
                 with Least_le [of "\<lambda>x. x \<in> AA", OF a'] have "(LEAST x. x \<in> AA) < a'"
                   by arith
                 from xeq fields rg [OF this] show "x \<in> Ya" by auto
               qed
               have "card (X - {ya}) = r"
                 by (simp add: cardX ya)
               with pg [of "LEAST x. x \<in> AA"] fields cardX * show ?thesis
                 by (auto simp del: insert_Diff_single)
             qed
             also from AA AAleast fields have "\<dots> = s'" by auto
             finally show ?thesis .
           qed
         qed
         then show ?thesis by blast
       qed
     qed
   qed
 qed
 
 
 theorem Ramsey:
   fixes s r :: nat
     and Z :: "'a set"
     and f :: "'a set \<Rightarrow> nat"
   shows
    "\<lbrakk>infinite Z;
       \<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = r \<longrightarrow> f X < s\<rbrakk>
     \<Longrightarrow> \<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s
             \<and> (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X = t)"
   by (blast intro: Ramsey_induction [unfolded part_fn_def nsets_def])
 
 
 corollary Ramsey2:
   fixes s :: nat
     and Z :: "'a set"
     and f :: "'a set \<Rightarrow> nat"
   assumes infZ: "infinite Z"
     and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x \<noteq> y \<longrightarrow> f {x, y} < s"
   shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>x\<in>Y. \<forall>y\<in>Y. x\<noteq>y \<longrightarrow> f {x, y} = t)"
 proof -
   from part have part2: "\<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = 2 \<longrightarrow> f X < s"
     by (fastforce simp add: eval_nat_numeral card_Suc_eq)
   obtain Y t where *:
     "Y \<subseteq> Z" "infinite Y" "t < s" "(\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = 2 \<longrightarrow> f X = t)"
     by (insert Ramsey [OF infZ part2]) auto
   then have "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
   with * show ?thesis by iprover
 qed
 
 
 subsection \<open>Disjunctive Well-Foundedness\<close>
 
 text \<open>
   An application of Ramsey's theorem to program termination. See
   @{cite "Podelski-Rybalchenko"}.
 \<close>
 
 definition disj_wf :: "('a \<times> 'a) set \<Rightarrow> bool"
   where "disj_wf r \<longleftrightarrow> (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf (T i)) \<and> r = (\<Union>i<n. T i))"
 
 definition transition_idx :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> nat set \<Rightarrow> nat"
   where "transition_idx s T A = (LEAST k. \<exists>i j. A = {i, j} \<and> i < j \<and> (s j, s i) \<in> T k)"
 
 
 lemma transition_idx_less:
   assumes "i < j" "(s j, s i) \<in> T k" "k < n"
   shows "transition_idx s T {i, j} < n"
 proof -
   from assms(1,2) have "transition_idx s T {i, j} \<le> k"
     by (simp add: transition_idx_def, blast intro: Least_le)
   with assms(3) show ?thesis by simp
 qed
 
 lemma transition_idx_in:
   assumes "i < j" "(s j, s i) \<in> T k"
   shows "(s j, s i) \<in> T (transition_idx s T {i, j})"
   using assms
   by (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR cong: conj_cong) (erule LeastI)
 
 text \<open>To be equal to the union of some well-founded relations is equivalent
   to being the subset of such a union.\<close>
 lemma disj_wf: "disj_wf r \<longleftrightarrow> (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) \<and> r \<subseteq> (\<Union>i<n. T i))"
 proof -
   have *: "\<And>T n. \<lbrakk>\<forall>i<n. wf (T i); r \<subseteq> \<Union> (T ` {..<n})\<rbrakk>
            \<Longrightarrow> (\<forall>i<n. wf (T i \<inter> r)) \<and> r = (\<Union>i<n. T i \<inter> r)"
     by (force simp add: wf_Int1)
   show ?thesis
     unfolding disj_wf_def by auto (metis "*")
 qed
 
 theorem trans_disj_wf_implies_wf:
   assumes "trans r"
     and "disj_wf r"
   shows "wf r"
 proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
   assume "\<exists>s. \<forall>i. (s (Suc i), s i) \<in> r"
   then obtain s where sSuc: "\<forall>i. (s (Suc i), s i) \<in> r" ..
   from \<open>disj_wf r\<close> obtain T and n :: nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
     by (auto simp add: disj_wf_def)
   have s_in_T: "\<exists>k. (s j, s i) \<in> T k \<and> k<n" if "i < j" for i j
   proof -
     from \<open>i < j\<close> have "(s j, s i) \<in> r"
     proof (induct rule: less_Suc_induct)
       case 1
       then show ?case by (simp add: sSuc)
     next
       case 2
       with \<open>trans r\<close> show ?case
         unfolding trans_def by blast
     qed
     then show ?thesis by (auto simp add: r)
   qed
   have "i < j \<Longrightarrow> transition_idx s T {i, j} < n" for i j
     using s_in_T transition_idx_less by blast
   then have trless: "i \<noteq> j \<Longrightarrow> transition_idx s T {i, j} < n" for i j
     by (metis doubleton_eq_iff less_linear)
   have "\<exists>K k. K \<subseteq> UNIV \<and> infinite K \<and> k < n \<and>
       (\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> transition_idx s T {i, j} = k)"
     by (rule Ramsey2) (auto intro: trless infinite_UNIV_nat)
   then obtain K and k where infK: "infinite K" and "k < n"
     and allk: "\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> transition_idx s T {i, j} = k"
     by auto
   have "(s (enumerate K (Suc m)), s (enumerate K m)) \<in> T k" for m :: nat
   proof -
     let ?j = "enumerate K (Suc m)"
     let ?i = "enumerate K m"
     have ij: "?i < ?j" by (simp add: enumerate_step infK)
     have "?j \<in> K" "?i \<in> K" by (simp_all add: enumerate_in_set infK)
     with ij have k: "k = transition_idx s T {?i, ?j}" by (simp add: allk)
     from s_in_T [OF ij] obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n" by blast
     then show "(s ?j, s ?i) \<in> T k" by (simp add: k transition_idx_in ij)
   qed
   then have "\<not> wf (T k)"
     by (meson wf_iff_no_infinite_down_chain)
   with wfT \<open>k < n\<close> show False by blast
 qed
 
 end
diff --git a/src/HOL/Num.thy b/src/HOL/Num.thy
--- a/src/HOL/Num.thy
+++ b/src/HOL/Num.thy
@@ -1,1435 +1,1438 @@
 (*  Title:      HOL/Num.thy
     Author:     Florian Haftmann
     Author:     Brian Huffman
 *)
 
 section \<open>Binary Numerals\<close>
 
 theory Num
   imports BNF_Least_Fixpoint Transfer
 begin
 
 subsection \<open>The \<open>num\<close> type\<close>
 
 datatype num = One | Bit0 num | Bit1 num
 
 text \<open>Increment function for type \<^typ>\<open>num\<close>\<close>
 
 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>\<open>num\<close> and type \<^typ>\<open>nat\<close>\<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 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
 
 lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
   by (induct x) simp_all
 
 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)"
   by (induct n) simp_all
 
 text \<open>Type \<^typ>\<open>num\<close> 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)
 
 lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
   by (induct n) (simp_all add: nat_of_num_inc)
 
 lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
   apply safe
   apply (drule arg_cong [where f=num_of_nat])
   apply (simp add: nat_of_num_inverse)
   done
 
 lemma num_induct [case_names One inc]:
   fixes P :: "num \<Rightarrow> bool"
   assumes One: "P One"
     and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
   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)
   have "P (num_of_nat (Suc n))"
   proof (induct n)
     case 0
     from One show ?case by simp
   next
     case (Suc n)
     then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
     then show "P (num_of_nat (Suc (Suc n)))" by simp
   qed
   with n show "P x"
     by (simp add: nat_of_num_inverse)
 qed
 
 text \<open>
   From now on, there are two possible models for \<^typ>\<open>num\<close>: as positive
   naturals (rule \<open>num_induct\<close>) and as digit representation (rules
   \<open>num.induct\<close>, \<open>num.cases\<close>).
 \<close>
 
 
 subsection \<open>Numeral operations\<close>
 
 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 \<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"
 
 instance
   by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
 
 end
 
 lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
   unfolding plus_num_def
   by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
 
 lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
   unfolding times_num_def
   by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
 
 lemma add_num_simps [simp, code]:
   "One + One = Bit0 One"
   "One + Bit0 n = Bit1 n"
   "One + Bit1 n = Bit0 (n + One)"
   "Bit0 m + One = Bit1 m"
   "Bit0 m + Bit0 n = Bit0 (m + n)"
   "Bit0 m + Bit1 n = Bit1 (m + n)"
   "Bit1 m + One = Bit0 (m + One)"
   "Bit1 m + Bit0 n = Bit1 (m + n)"
   "Bit1 m + Bit1 n = Bit0 (m + n + One)"
   by (simp_all add: num_eq_iff nat_of_num_add)
 
 lemma mult_num_simps [simp, code]:
   "m * One = m"
   "One * n = n"
   "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
   "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)
 
 lemma eq_num_simps:
   "One = One \<longleftrightarrow> True"
   "One = Bit0 n \<longleftrightarrow> False"
   "One = Bit1 n \<longleftrightarrow> False"
   "Bit0 m = One \<longleftrightarrow> False"
   "Bit1 m = One \<longleftrightarrow> False"
   "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
   "Bit0 m = Bit1 n \<longleftrightarrow> False"
   "Bit1 m = Bit0 n \<longleftrightarrow> False"
   "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
   by simp_all
 
 lemma le_num_simps [simp, code]:
   "One \<le> n \<longleftrightarrow> True"
   "Bit0 m \<le> One \<longleftrightarrow> False"
   "Bit1 m \<le> One \<longleftrightarrow> False"
   "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
   "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
   "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
   "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
   using nat_of_num_pos [of n] nat_of_num_pos [of m]
   by (auto simp add: less_eq_num_def less_num_def)
 
 lemma less_num_simps [simp, code]:
   "m < One \<longleftrightarrow> False"
   "One < Bit0 n \<longleftrightarrow> True"
   "One < Bit1 n \<longleftrightarrow> True"
   "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
   "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
   "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
   "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
   using nat_of_num_pos [of n] nat_of_num_pos [of m]
   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)
 
 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)
 
 lemma add_One_commute: "One + n = n + One"
   by (induct n) simp_all
 
 lemma add_inc: "x + inc y = inc (x + y)"
   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
 
 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>\<open>num_of_nat\<close> conversion.\<close>
 
 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>
 
 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
 
 lemma one_plus_BitM: "One + BitM n = Bit0 n"
   unfolding add_One_commute BitM_plus_one ..
 
 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 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)
 
 lemma sqr_conv_mult: "sqr x = x * x"
   by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
 
 lemma num_double [simp]:
   "num.Bit0 num.One * n = num.Bit0 n"
   by (simp add: num_eq_iff nat_of_num_mult)
 
 
 subsection \<open>Binary numerals\<close>
 
 text \<open>
   We embed binary representations into a generic algebraic
   structure using \<open>numeral\<close>.
 \<close>
 
 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"
 
 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"
 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
 
 declare numeral.simps [simp del]
 
 abbreviation "Numeral1 \<equiv> numeral One"
 
 declare numeral_One [code_post]
 
 end
 
 text \<open>Numeral syntax.\<close>
 
 syntax
   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
 
 ML_file \<open>Tools/numeral.ML\<close>
 
 parse_translation \<open>
   let
     fun numeral_tr [(c as Const (\<^syntax_const>\<open>_constrain\<close>, _)) $ t $ u] =
           c $ numeral_tr [t] $ u
       | numeral_tr [Const (num, _)] =
           (Numeral.mk_number_syntax o #value o Lexicon.read_num) num
       | numeral_tr ts = raise TERM ("numeral_tr", ts);
   in [(\<^syntax_const>\<open>_Numeral\<close>, K numeral_tr)] end
 \<close>
 
 typed_print_translation \<open>
   let
     fun num_tr' ctxt T [n] =
       let
         val k = Numeral.dest_num_syntax n;
         val t' =
           Syntax.const \<^syntax_const>\<open>_Numeral\<close> $
             Syntax.free (string_of_int k);
       in
         (case T of
           Type (\<^type_name>\<open>fun\<close>, [_, T']) =>
             if Printer.type_emphasis ctxt T' then
               Syntax.const \<^syntax_const>\<open>_constrain\<close> $ t' $
                 Syntax_Phases.term_of_typ ctxt T'
             else t'
         | _ => if T = dummyT then t' else raise Match)
       end;
   in
    [(\<^const_syntax>\<open>numeral\<close>, num_tr')]
   end
 \<close>
 
 
 subsection \<open>Class-specific numeral rules\<close>
 
 text \<open>\<^const>\<open>numeral\<close> is a morphism.\<close>
 
 
 subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
 
 context numeral
 begin
 
 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)
 
 lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
   by (rule numeral_add [symmetric])
 
 lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
   using numeral_add [of n One] by (simp add: numeral_One)
 
 lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
   using numeral_add [of One n] by (simp add: numeral_One)
 
 lemma one_add_one: "1 + 1 = 2"
   using numeral_add [of One One] by (simp add: numeral_One)
 
 lemmas add_numeral_special =
   numeral_plus_one one_plus_numeral one_add_one
 
 end
 
 
 subsubsection \<open>Structures with negation: class \<open>neg_numeral\<close>\<close>
 
 class neg_numeral = numeral + group_add
 begin
 
 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)"
   | "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)
 
 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 (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])
   done
 
 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 =
   add.assoc is_num_add_commute is_num_add_left_commute
   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 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)
 
 lemma dbl_simps [simp]:
   "dbl (- numeral k) = - dbl (numeral k)"
   "dbl 0 = 0"
   "dbl 1 = 2"
   "dbl (- 1) = - 2"
   "dbl (numeral k) = numeral (Bit0 k)"
   by (simp_all add: dbl_def numeral.simps minus_add)
 
 lemma dbl_inc_simps [simp]:
   "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
   "dbl_inc 0 = 1"
   "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)
 
 lemma dbl_dec_simps [simp]:
   "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
   "dbl_dec 0 = - 1"
   "dbl_dec 1 = 1"
   "dbl_dec (- 1) = - 3"
   "dbl_dec (numeral k) = numeral (BitM k)"
   by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
 
 lemma sub_num_simps [simp]:
   "sub One One = 0"
   "sub One (Bit0 l) = - numeral (BitM l)"
   "sub One (Bit1 l) = - numeral (Bit0 l)"
   "sub (Bit0 k) One = numeral (BitM k)"
   "sub (Bit1 k) One = numeral (Bit0 k)"
   "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
   "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
   "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
   "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
   by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
     numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
 
 lemma add_neg_numeral_simps:
   "numeral m + - numeral n = sub m n"
   "- 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)
 
 lemma add_neg_numeral_special:
   "1 + - numeral m = sub One m"
   "- numeral m + 1 = sub One m"
   "numeral m + - 1 = sub m One"
   "- 1 + numeral n = sub n One"
   "- 1 + - numeral n = - numeral (inc n)"
   "- numeral m + - 1 = - numeral (inc m)"
   "1 + - 1 = 0"
   "- 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)
 
 lemma diff_numeral_simps:
   "numeral m - numeral n = sub m n"
   "numeral m - - numeral n = numeral (m + n)"
   "- 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)
 
 lemma diff_numeral_special:
   "1 - numeral n = sub One n"
   "numeral m - 1 = sub m One"
   "1 - - numeral n = numeral (One + n)"
   "- numeral m - 1 = - numeral (m + One)"
   "- 1 - numeral n = - numeral (inc n)"
   "numeral m - - 1 = numeral (inc m)"
   "- 1 - - numeral n = sub n One"
   "- numeral m - - 1 = sub One m"
   "1 - 1 = 0"
   "- 1 - 1 = - 2"
   "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)
 
 end
 
 
 subsubsection \<open>Structures with multiplication: class \<open>semiring_numeral\<close>\<close>
 
 class semiring_numeral = semiring + monoid_mult
 begin
 
 subclass numeral ..
 
 lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
   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"
   by (simp add: one_add_one [symmetric] distrib_right)
 
 lemma mult_2_right: "z * 2 = z + z"
   by (simp add: one_add_one [symmetric] distrib_left)
 
 lemma left_add_twice:
   "a + (a + b) = 2 * a + b"
   by (simp add: mult_2 ac_simps)
 
 end
 
 
 subsubsection \<open>Structures with a zero: class \<open>semiring_1\<close>\<close>
 
 context semiring_1
 begin
 
 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)
 
 lemma numeral_unfold_funpow:
   "numeral k = ((+) 1 ^^ numeral k) 0"
   unfolding of_nat_def [symmetric] by simp
 
 end
 
 context
   includes lifting_syntax
 begin
 
 lemma transfer_rule_numeral:
   "((=) ===> R) numeral numeral"
     if [transfer_rule]: "R 0 0" "R 1 1"
       "(R ===> R ===> R) (+) (+)"
     for R :: "'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool"
 proof -
   have "((=) ===> R) (\<lambda>k. ((+) 1 ^^ numeral k) 0) (\<lambda>k. ((+) 1 ^^ numeral k) 0)"
     by transfer_prover
   then show ?thesis
     by (simp flip: numeral_unfold_funpow [abs_def])
 qed
 
 end
 
 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
 qed
 
 lemma nat_of_num_code [code]:
   "nat_of_num One = 1"
   "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
   "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>
 
 context semiring_char_0
 begin
 
 lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
   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])
 
 lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
   by (rule numeral_eq_iff [of One n, unfolded numeral_One])
 
 lemma numeral_neq_zero: "numeral n \<noteq> 0"
   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)
 
 lemmas eq_numeral_simps [simp] =
   numeral_eq_iff
   numeral_eq_one_iff
   one_eq_numeral_iff
   numeral_neq_zero
   zero_neq_numeral
 
 end
 
 
 subsubsection \<open>Comparisons: class \<open>linordered_nonzero_semiring\<close>\<close>
 
 context linordered_nonzero_semiring
 begin
 
 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"
     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 num.One n] by (simp add: numeral_One)
 
 lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> num.One"
   using numeral_le_iff [of n num.One] by (simp add: numeral_One)
 
 lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
 proof -
   have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
     unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
   then show ?thesis by simp
 qed
 
 lemma not_numeral_less_one: "\<not> numeral n < 1"
   using numeral_less_iff [of n num.One] by (simp add: numeral_One)
 
 lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> num.One < n"
   using numeral_less_iff [of num.One n] by (simp add: numeral_One)
 
 lemma zero_le_numeral: "0 \<le> numeral n"
   using dual_order.trans one_le_numeral zero_le_one by blast
 
 lemma zero_less_numeral: "0 < numeral n"
   using less_linear not_numeral_less_one order.strict_trans zero_less_one by blast
 
 lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
   by (simp add: not_le zero_less_numeral)
 
 lemma not_numeral_less_zero: "\<not> numeral n < 0"
   by (simp add: not_less zero_le_numeral)
 
 lemmas le_numeral_extra =
   zero_le_one not_one_le_zero
   order_refl [of 0] order_refl [of 1]
 
 lemmas less_numeral_extra =
   zero_less_one not_one_less_zero
   less_irrefl [of 0] less_irrefl [of 1]
 
 lemmas le_numeral_simps [simp] =
   numeral_le_iff
   one_le_numeral
   numeral_le_one_iff
   zero_le_numeral
   not_numeral_le_zero
 
 lemmas less_numeral_simps [simp] =
   numeral_less_iff
   one_less_numeral_iff
   not_numeral_less_one
   zero_less_numeral
   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)
 
 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
 
 text \<open>Unfold \<open>min\<close> and \<open>max\<close> on numerals.\<close>
 
 lemmas max_number_of [simp] =
   max_def [of "numeral u" "numeral v"]
   max_def [of "numeral u" "- numeral v"]
   max_def [of "- numeral u" "numeral v"]
   max_def [of "- numeral u" "- numeral v"] for u v
 
 lemmas min_number_of [simp] =
   min_def [of "numeral u" "numeral v"]
   min_def [of "numeral u" "- numeral v"]
   min_def [of "- numeral u" "numeral v"]
   min_def [of "- numeral u" "- numeral v"] for u v
 
 
 subsubsection \<open>Multiplication and negation: class \<open>ring_1\<close>\<close>
 
 context ring_1
 begin
 
 subclass neg_numeral ..
 
 lemma mult_neg_numeral_simps:
   "- numeral m * - numeral n = numeral (m * n)"
   "- numeral m * numeral n = - numeral (m * n)"
   "numeral m * - numeral n = - numeral (m * n)"
   by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)
 
 lemma mult_minus1 [simp]: "- 1 * z = - z"
   by (simp add: numeral.simps)
 
 lemma mult_minus1_right [simp]: "z * - 1 = - z"
   by (simp add: numeral.simps)
 
 end
 
 
 subsubsection \<open>Equality using \<open>iszero\<close> for rings with non-zero characteristic\<close>
 
 context ring_1
 begin
 
 definition iszero :: "'a \<Rightarrow> bool"
   where "iszero z \<longleftrightarrow> z = 0"
 
 lemma iszero_0 [simp]: "iszero 0"
   by (simp add: iszero_def)
 
 lemma not_iszero_1 [simp]: "\<not> iszero 1"
   by (simp add: iszero_def)
 
 lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
   by (simp add: numeral_One)
 
 lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
   by (simp add: iszero_def)
 
 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 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>.
 
   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:
   "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
   "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
   "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
   "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
   "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
   "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
   "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
   "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
   "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
   "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
   "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
   "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
   unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
   by simp_all
 
 end
 
 
 subsubsection \<open>Equality and negation: class \<open>ring_char_0\<close>\<close>
 
 context ring_char_0
 begin
 
 lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
   by (simp add: iszero_def)
 
 lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
   by simp
 
 lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
   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"
   by simp
 
 lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
   by simp
 
 lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
   using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
 
 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"
   using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
 
 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"
   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"
   using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
 
 lemma neg_one_neq_zero: "- 1 \<noteq> 0"
   by simp
 
 lemma zero_neq_neg_one: "0 \<noteq> - 1"
   by simp
 
 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"
   using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
 
 lemmas eq_neg_numeral_simps [simp] =
   neg_numeral_eq_iff
   numeral_neq_neg_numeral neg_numeral_neq_numeral
   one_neq_neg_numeral neg_numeral_neq_one
   zero_neq_neg_numeral neg_numeral_neq_zero
   neg_one_neq_numeral numeral_neq_neg_one
   neg_one_eq_numeral_iff numeral_eq_neg_one_iff
   neg_one_neq_zero zero_neq_neg_one
   neg_one_neq_one one_neq_neg_one
 
 end
 
 
 subsubsection \<open>Structures with negation and order: class \<open>linordered_idom\<close>\<close>
 
 context linordered_idom
 begin
 
 subclass ring_char_0 ..
 
 lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
   by (simp only: neg_le_iff_le numeral_le_iff)
 
 lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
   by (simp only: neg_less_iff_less numeral_less_iff)
 
 lemma neg_numeral_less_zero: "- numeral n < 0"
   by (simp only: neg_less_0_iff_less zero_less_numeral)
 
 lemma neg_numeral_le_zero: "- numeral n \<le> 0"
   by (simp only: neg_le_0_iff_le zero_le_numeral)
 
 lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
   by (simp only: not_less neg_numeral_le_zero)
 
 lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
   by (simp only: not_le neg_numeral_less_zero)
 
 lemma neg_numeral_less_numeral: "- numeral m < numeral n"
   using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
 
 lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
   by (simp only: less_imp_le neg_numeral_less_numeral)
 
 lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
   by (simp only: not_less neg_numeral_le_numeral)
 
 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])
 
 lemma neg_numeral_le_one: "- numeral m \<le> 1"
   by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
 
 lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
   by (simp only: not_less neg_numeral_le_one)
 
 lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
   by (simp only: not_le neg_numeral_less_one)
 
 lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
   using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
 
 lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
   using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
 
 lemma neg_one_less_numeral: "- 1 < numeral m"
   using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
 
 lemma neg_one_le_numeral: "- 1 \<le> numeral m"
   using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
 
 lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
   by (cases m) simp_all
 
 lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
   by simp
 
 lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
   by simp
 
 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"
   by (simp only: sub_def le_diff_eq) simp
 
 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"
   by (simp only: sub_def diff_le_eq) simp
 
 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] =
   neg_numeral_le_iff
   neg_numeral_le_numeral not_numeral_le_neg_numeral
   neg_numeral_le_zero not_zero_le_neg_numeral
   neg_numeral_le_one not_one_le_neg_numeral
   neg_one_le_numeral not_numeral_le_neg_one
   neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
 
 lemma le_minus_one_simps [simp]:
   "- 1 \<le> 0"
   "- 1 \<le> 1"
   "\<not> 0 \<le> - 1"
   "\<not> 1 \<le> - 1"
   by simp_all
 
 lemmas less_neg_numeral_simps [simp] =
   neg_numeral_less_iff
   neg_numeral_less_numeral not_numeral_less_neg_numeral
   neg_numeral_less_zero not_zero_less_neg_numeral
   neg_numeral_less_one not_one_less_neg_numeral
   neg_one_less_numeral not_numeral_less_neg_one
   neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
 
 lemma less_minus_one_simps [simp]:
   "- 1 < 0"
   "- 1 < 1"
   "\<not> 0 < - 1"
   "\<not> 1 < - 1"
   by (simp_all add: less_le)
 
 lemma abs_numeral [simp]: "\<bar>numeral n\<bar> = numeral n"
   by simp
 
 lemma abs_neg_numeral [simp]: "\<bar>- numeral n\<bar> = numeral n"
   by (simp only: abs_minus_cancel abs_numeral)
 
 lemma abs_neg_one [simp]: "\<bar>- 1\<bar> = 1"
   by simp
 
 end
 
 
 subsubsection \<open>Natural numbers\<close>
 
 lemma numeral_num_of_nat:
   "numeral (num_of_nat n) = n" if "n > 0"
   using that nat_of_num_numeral num_of_nat_inverse by simp
 
 lemma Suc_1 [simp]: "Suc 1 = 2"
   unfolding Suc_eq_plus1 by (rule one_add_one)
 
 lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
   unfolding Suc_eq_plus1 by (rule numeral_plus_one)
 
 definition pred_numeral :: "num \<Rightarrow> nat"
   where "pred_numeral k = numeral k - 1"
 
 declare [[code drop: pred_numeral]]
 
 lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
   by (simp add: pred_numeral_def)
 
 lemma eval_nat_numeral:
   "numeral One = Suc 0"
   "numeral (Bit0 n) = Suc (numeral (BitM n))"
   "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
   by (simp_all add: numeral.simps BitM_plus_one)
 
 lemma pred_numeral_simps [simp]:
   "pred_numeral One = 0"
   "pred_numeral (Bit0 k) = numeral (BitM k)"
   "pred_numeral (Bit1 k) = numeral (Bit0 k)"
   by (simp_all only: pred_numeral_def eval_nat_numeral diff_Suc_Suc diff_0)
 
 lemma pred_numeral_inc [simp]:
   "pred_numeral (Num.inc k) = numeral k"
   by (simp only: pred_numeral_def numeral_inc diff_add_inverse2)
 
 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
   by (simp add: eval_nat_numeral)
 
 lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
   by (simp add: eval_nat_numeral)
 
 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"
   by simp
 
 lemma numerals: "Numeral1 = (1::nat)" "2 = Suc (Suc 0)"
   by (rule numeral_One) (rule numeral_2_eq_2)
 
 lemmas numeral_nat = eval_nat_numeral BitM.simps One_nat_def
 
 text \<open>Comparisons involving \<^term>\<open>Suc\<close>.\<close>
 
 lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
   by (simp add: numeral_eq_Suc)
 
 lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
   by (simp add: numeral_eq_Suc)
 
 lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
   by (simp add: numeral_eq_Suc)
 
 lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
   by (simp add: numeral_eq_Suc)
 
 lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
   by (simp add: numeral_eq_Suc)
 
 lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
   by (simp add: numeral_eq_Suc)
 
 lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
   by (simp add: numeral_eq_Suc)
 
 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))"
   by (simp add: numeral_eq_Suc)
 
 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))"
   by (simp add: numeral_eq_Suc)
 
 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>\<open>case_nat\<close> and \<^term>\<open>rec_nat\<close>.\<close>
 
 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]:
   "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
   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))"
   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)))"
   by (simp add: numeral_eq_Suc Let_def)
 
 text \<open>Case analysis on \<^term>\<open>n < 2\<close>.\<close>
 lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
   by (auto simp add: numeral_2_eq_2)
 
+lemma less_2_cases_iff: "n < 2 \<longleftrightarrow> 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>bh: Are these rules really a good idea?\<close>
+text \<open>bh: Are these rules really a good idea? LCP: well, it already happens for 0 and 1!\<close>
 
 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
   by simp
 
 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
   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
 
 lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
 
 
 subsection \<open>Particular lemmas concerning \<^term>\<open>2\<close>\<close>
 
 context linordered_field
 begin
 
 subclass field_char_0 ..
 
 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"
   by (simp add: half_gt_zero_iff)
 
 end
 
 
 subsection \<open>Numeral equations as default simplification rules\<close>
 
 declare (in numeral) numeral_One [simp]
 declare (in numeral) numeral_plus_numeral [simp]
 declare (in numeral) add_numeral_special [simp]
 declare (in neg_numeral) add_neg_numeral_simps [simp]
 declare (in neg_numeral) add_neg_numeral_special [simp]
 declare (in neg_numeral) diff_numeral_simps [simp]
 declare (in neg_numeral) diff_numeral_special [simp]
 declare (in semiring_numeral) numeral_times_numeral [simp]
 declare (in ring_1) mult_neg_numeral_simps [simp]
 
 
 subsubsection \<open>Special Simplification for Constants\<close>
 
 text \<open>These distributive laws move literals inside sums and differences.\<close>
 
 lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
 lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
 lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
 lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v
 
 text \<open>These are actually for fields, like real\<close>
 
 lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
 lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
 lemmas zero_le_divide_iff_numeral [simp, no_atp] = zero_le_divide_iff [of "numeral w"] for w
 lemmas divide_le_0_iff_numeral [simp, no_atp] = divide_le_0_iff [of "numeral w"] for w
 
 text \<open>Replaces \<open>inverse #nn\<close> by \<open>1/#nn\<close>.  It looks
   strange, but then other simprocs simplify the quotient.\<close>
 
 lemmas inverse_eq_divide_numeral [simp] =
   inverse_eq_divide [of "numeral w"] for w
 
 lemmas inverse_eq_divide_neg_numeral [simp] =
   inverse_eq_divide [of "- numeral w"] for w
 
 text \<open>These laws simplify inequalities, moving unary minus from a term
   into the literal.\<close>
 
 lemmas equation_minus_iff_numeral [no_atp] =
   equation_minus_iff [of "numeral v"] for v
 
 lemmas minus_equation_iff_numeral [no_atp] =
   minus_equation_iff [of _ "numeral v"] for v
 
 lemmas le_minus_iff_numeral [no_atp] =
   le_minus_iff [of "numeral v"] for v
 
 lemmas minus_le_iff_numeral [no_atp] =
   minus_le_iff [of _ "numeral v"] for v
 
 lemmas less_minus_iff_numeral [no_atp] =
   less_minus_iff [of "numeral v"] for v
 
 lemmas minus_less_iff_numeral [no_atp] =
   minus_less_iff [of _ "numeral v"] for v
 
 (* FIXME maybe simproc *)
 
 text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
 
 lemmas mult_less_cancel_left_numeral [simp, no_atp] = mult_less_cancel_left [of "numeral v"] for v
 lemmas mult_less_cancel_right_numeral [simp, no_atp] = mult_less_cancel_right [of _ "numeral v"] for v
 lemmas mult_le_cancel_left_numeral [simp, no_atp] = mult_le_cancel_left [of "numeral v"] for v
 lemmas mult_le_cancel_right_numeral [simp, no_atp] = mult_le_cancel_right [of _ "numeral v"] for v
 
 text \<open>Multiplying out constant divisors in comparisons (\<open><\<close>, \<open>\<le>\<close> and \<open>=\<close>)\<close>
 
 named_theorems divide_const_simps "simplification rules to simplify comparisons involving constant divisors"
 
 lemmas le_divide_eq_numeral1 [simp,divide_const_simps] =
   pos_le_divide_eq [of "numeral w", OF zero_less_numeral]
   neg_le_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
 
 lemmas divide_le_eq_numeral1 [simp,divide_const_simps] =
   pos_divide_le_eq [of "numeral w", OF zero_less_numeral]
   neg_divide_le_eq [of "- numeral w", OF neg_numeral_less_zero] for w
 
 lemmas less_divide_eq_numeral1 [simp,divide_const_simps] =
   pos_less_divide_eq [of "numeral w", OF zero_less_numeral]
   neg_less_divide_eq [of "- numeral w", OF neg_numeral_less_zero] for w
 
 lemmas divide_less_eq_numeral1 [simp,divide_const_simps] =
   pos_divide_less_eq [of "numeral w", OF zero_less_numeral]
   neg_divide_less_eq [of "- numeral w", OF neg_numeral_less_zero] for w
 
 lemmas eq_divide_eq_numeral1 [simp,divide_const_simps] =
   eq_divide_eq [of _ _ "numeral w"]
   eq_divide_eq [of _ _ "- numeral w"] for w
 
 lemmas divide_eq_eq_numeral1 [simp,divide_const_simps] =
   divide_eq_eq [of _ "numeral w"]
   divide_eq_eq [of _ "- numeral w"] for w
 
 
 subsubsection \<open>Optional Simplification Rules Involving Constants\<close>
 
 text \<open>Simplify quotients that are compared with a literal constant.\<close>
 
 lemmas le_divide_eq_numeral [divide_const_simps] =
   le_divide_eq [of "numeral w"]
   le_divide_eq [of "- numeral w"] for w
 
 lemmas divide_le_eq_numeral [divide_const_simps] =
   divide_le_eq [of _ _ "numeral w"]
   divide_le_eq [of _ _ "- numeral w"] for w
 
 lemmas less_divide_eq_numeral [divide_const_simps] =
   less_divide_eq [of "numeral w"]
   less_divide_eq [of "- numeral w"] for w
 
 lemmas divide_less_eq_numeral [divide_const_simps] =
   divide_less_eq [of _ _ "numeral w"]
   divide_less_eq [of _ _ "- numeral w"] for w
 
 lemmas eq_divide_eq_numeral [divide_const_simps] =
   eq_divide_eq [of "numeral w"]
   eq_divide_eq [of "- numeral w"] for w
 
 lemmas divide_eq_eq_numeral [divide_const_simps] =
   divide_eq_eq [of _ _ "numeral w"]
   divide_eq_eq [of _ _ "- numeral w"] for w
 
 text \<open>Not good as automatic simprules because they cause case splits.\<close>
 
 lemmas [divide_const_simps] =
   le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
 
 
 subsection \<open>Setting up simprocs\<close>
 
 lemma mult_numeral_1: "Numeral1 * a = a"
   for a :: "'a::semiring_numeral"
   by simp
 
 lemma mult_numeral_1_right: "a * Numeral1 = a"
   for a :: "'a::semiring_numeral"
   by simp
 
 lemma divide_numeral_1: "a / Numeral1 = a"
   for a :: "'a::field"
   by simp
 
 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>
 
 lemma mult_1s_semiring_numeral:
   "Numeral1 * a = a"
   "a * Numeral1 = a"
   for a :: "'a::semiring_numeral"
   by simp_all
 
 lemma mult_1s_ring_1:
   "- Numeral1 * b = - b"
   "b * - Numeral1 = - b"
   for b :: "'a::ring_1"
   by simp_all
 
 lemmas mult_1s = mult_1s_semiring_numeral mult_1s_ring_1
 
 setup \<open>
   Reorient_Proc.add
     (fn Const (\<^const_name>\<open>numeral\<close>, _) $ _ => true
       | Const (\<^const_name>\<open>uminus\<close>, _) $ (Const (\<^const_name>\<open>numeral\<close>, _) $ _) => true
       | _ => false)
 \<close>
 
 simproc_setup reorient_numeral ("numeral w = x" | "- numeral w = y") =
   Reorient_Proc.proc
 
 
 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
 
 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
   numeral_times_numeral mult_neg_numeral_simps
   mult_zero_left mult_zero_right
   abs_numeral abs_neg_numeral
 
 text \<open>
   For making a minimal simpset, one must include these default simprules.
   Also include \<open>simp_thms\<close>.
 \<close>
 
 lemmas arith_simps =
   add_num_simps mult_num_simps sub_num_simps
   BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
   abs_zero abs_one arith_extra_simps
 
 lemmas more_arith_simps =
   neg_le_iff_le
   minus_zero left_minus right_minus
   mult_1_left mult_1_right
   mult_minus_left mult_minus_right
   minus_add_distrib minus_minus mult.assoc
 
 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>
 
 lemmas eq_numeral_extra =
   zero_neq_one one_neq_zero
 
 lemmas rel_simps =
   le_num_simps less_num_simps eq_num_simps
   le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
   less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
   eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
 
 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
   unfolding Let_def ..
 
 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
   unfolding Let_def ..
 
 declaration \<open>
 let
   fun number_of ctxt T n =
     if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, \<^sort>\<open>numeral\<close>))
     then raise CTERM ("number_of", [])
     else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
 in
   K (
     Lin_Arith.set_number_of number_of
     #> Lin_Arith.add_simps
       @{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
         arith_special numeral_One of_nat_simps uminus_numeral_One
         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})
 end
 \<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)"
   by (simp_all add: add.assoc [symmetric])
 
 lemma add_neg_numeral_left [simp]:
   "numeral v + (- numeral w + y) = (sub v w + y)"
   "- numeral v + (numeral w + y) = (sub w v + y)"
   "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
   by (simp_all add: add.assoc [symmetric])
 
 lemma mult_numeral_left_semiring_numeral:
   "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
   by (simp add: mult.assoc [symmetric])
 
 lemma mult_numeral_left_ring_1:
   "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'a::ring_1)"
   "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'a::ring_1)"
   "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'a::ring_1)"
   by (simp_all add: mult.assoc [symmetric])
 
 lemmas mult_numeral_left [simp] =
   mult_numeral_left_semiring_numeral
   mult_numeral_left_ring_1
 
 hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
 
 
 subsection \<open>Code module namespace\<close>
 
 code_identifier
   code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 subsection \<open>Printing of evaluated natural numbers as numerals\<close>
 
 lemma [code_post]:
   "Suc 0 = 1"
   "Suc 1 = 2"
   "Suc (numeral n) = numeral (Num.inc n)"
   by (simp_all add: numeral_inc)
 
 lemmas [code_post] = Num.inc.simps
 
 end