diff --git a/src/HOL/Library/Infinite_Set.thy b/src/HOL/Library/Infinite_Set.thy
--- a/src/HOL/Library/Infinite_Set.thy
+++ b/src/HOL/Library/Infinite_Set.thy
@@ -1,588 +1,590 @@
 (*  Title:      HOL/Library/Infinite_Set.thy
     Author:     Stephan Merz
 *)
 
 section \<open>Infinite Sets and Related Concepts\<close>
 
 theory Infinite_Set
   imports Main
 begin
 
 subsection \<open>The set of natural numbers is infinite\<close>
 
 lemma infinite_nat_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. n \<in> S)"
   for S :: "nat set"
   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
   by (simp add: cofinite_eq_sequentially frequently_def eventually_sequentially)
 
 lemma infinite_nat_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n>m. n \<in> S)"
   for S :: "nat set"
   using frequently_cofinite[of "\<lambda>x. x \<in> S"]
   by (simp add: cofinite_eq_sequentially frequently_def eventually_at_top_dense)
 
 lemma finite_nat_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {..<k})"
   for S :: "nat set"
   using infinite_nat_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
 
 lemma finite_nat_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. S \<subseteq> {.. k})"
   for S :: "nat set"
   using infinite_nat_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
 
 lemma finite_nat_bounded: "finite S \<Longrightarrow> \<exists>k. S \<subseteq> {..<k}"
   for S :: "nat set"
   by (simp add: finite_nat_iff_bounded)
 
 
 text \<open>
   For a set of natural numbers to be infinite, it is enough to know
   that for any number larger than some \<open>k\<close>, there is some larger
   number that is an element of the set.
 \<close>
 
 lemma unbounded_k_infinite: "\<forall>m>k. \<exists>n>m. n \<in> S \<Longrightarrow> infinite (S::nat set)"
   apply (clarsimp simp add: finite_nat_set_iff_bounded)
   apply (drule_tac x="Suc (max m k)" in spec)
   using less_Suc_eq apply fastforce
   done
 
 lemma nat_not_finite: "finite (UNIV::nat set) \<Longrightarrow> R"
   by simp
 
 lemma range_inj_infinite:
   fixes f :: "nat \<Rightarrow> 'a"
   assumes "inj f"
   shows "infinite (range f)"
 proof
   assume "finite (range f)"
   from this assms have "finite (UNIV::nat set)"
     by (rule finite_imageD)
   then show False by simp
 qed
 
 
 subsection \<open>The set of integers is also infinite\<close>
 
 lemma infinite_int_iff_infinite_nat_abs: "infinite S \<longleftrightarrow> infinite ((nat \<circ> abs) ` S)"
   for S :: "int set"
 proof (unfold Not_eq_iff, rule iffI)
   assume "finite ((nat \<circ> abs) ` S)"
   then have "finite (nat ` (abs ` S))"
     by (simp add: image_image cong: image_cong)
   moreover have "inj_on nat (abs ` S)"
     by (rule inj_onI) auto
   ultimately have "finite (abs ` S)"
     by (rule finite_imageD)
   then show "finite S"
     by (rule finite_image_absD)
 qed simp
 
 proposition infinite_int_iff_unbounded_le: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> \<ge> m \<and> n \<in> S)"
   for S :: "int set"
   by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded_le o_def image_def)
     (metis abs_ge_zero nat_le_eq_zle le_nat_iff)
 
 proposition infinite_int_iff_unbounded: "infinite S \<longleftrightarrow> (\<forall>m. \<exists>n. \<bar>n\<bar> > m \<and> n \<in> S)"
   for S :: "int set"
   by (simp add: infinite_int_iff_infinite_nat_abs infinite_nat_iff_unbounded o_def image_def)
     (metis (full_types) nat_le_iff nat_mono not_le)
 
 proposition finite_int_iff_bounded: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {..<k})"
   for S :: "int set"
   using infinite_int_iff_unbounded_le[of S] by (simp add: subset_eq) (metis not_le)
 
 proposition finite_int_iff_bounded_le: "finite S \<longleftrightarrow> (\<exists>k. abs ` S \<subseteq> {.. k})"
   for S :: "int set"
   using infinite_int_iff_unbounded[of S] by (simp add: subset_eq) (metis not_le)
 
 
 subsection \<open>Infinitely Many and Almost All\<close>
 
 text \<open>
   We often need to reason about the existence of infinitely many
   (resp., all but finitely many) objects satisfying some predicate, so
   we introduce corresponding binders and their proof rules.
 \<close>
 
 lemma not_INFM [simp]: "\<not> (INFM x. P x) \<longleftrightarrow> (MOST x. \<not> P x)"
   by (rule not_frequently)
 
 lemma not_MOST [simp]: "\<not> (MOST x. P x) \<longleftrightarrow> (INFM x. \<not> P x)"
   by (rule not_eventually)
 
 lemma INFM_const [simp]: "(INFM x::'a. P) \<longleftrightarrow> P \<and> infinite (UNIV::'a set)"
   by (simp add: frequently_const_iff)
 
 lemma MOST_const [simp]: "(MOST x::'a. P) \<longleftrightarrow> P \<or> finite (UNIV::'a set)"
   by (simp add: eventually_const_iff)
 
 lemma INFM_imp_distrib: "(INFM x. P x \<longrightarrow> Q x) \<longleftrightarrow> ((MOST x. P x) \<longrightarrow> (INFM x. Q x))"
   by (rule frequently_imp_iff)
 
 lemma MOST_imp_iff: "MOST x. P x \<Longrightarrow> (MOST x. P x \<longrightarrow> Q x) \<longleftrightarrow> (MOST x. Q x)"
   by (auto intro: eventually_rev_mp eventually_mono)
 
 lemma INFM_conjI: "INFM x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> INFM x. P x \<and> Q x"
   by (rule frequently_rev_mp[of P]) (auto elim: eventually_mono)
 
 
 text \<open>Properties of quantifiers with injective functions.\<close>
 
 lemma INFM_inj: "INFM x. P (f x) \<Longrightarrow> inj f \<Longrightarrow> INFM x. P x"
   using finite_vimageI[of "{x. P x}" f] by (auto simp: frequently_cofinite)
 
 lemma MOST_inj: "MOST x. P x \<Longrightarrow> inj f \<Longrightarrow> MOST x. P (f x)"
   using finite_vimageI[of "{x. \<not> P x}" f] by (auto simp: eventually_cofinite)
 
 
 text \<open>Properties of quantifiers with singletons.\<close>
 
 lemma not_INFM_eq [simp]:
   "\<not> (INFM x. x = a)"
   "\<not> (INFM x. a = x)"
   unfolding frequently_cofinite by simp_all
 
 lemma MOST_neq [simp]:
   "MOST x. x \<noteq> a"
   "MOST x. a \<noteq> x"
   unfolding eventually_cofinite by simp_all
 
 lemma INFM_neq [simp]:
   "(INFM x::'a. x \<noteq> a) \<longleftrightarrow> infinite (UNIV::'a set)"
   "(INFM x::'a. a \<noteq> x) \<longleftrightarrow> infinite (UNIV::'a set)"
   unfolding frequently_cofinite by simp_all
 
 lemma MOST_eq [simp]:
   "(MOST x::'a. x = a) \<longleftrightarrow> finite (UNIV::'a set)"
   "(MOST x::'a. a = x) \<longleftrightarrow> finite (UNIV::'a set)"
   unfolding eventually_cofinite by simp_all
 
 lemma MOST_eq_imp:
   "MOST x. x = a \<longrightarrow> P x"
   "MOST x. a = x \<longrightarrow> P x"
   unfolding eventually_cofinite by simp_all
 
 
 text \<open>Properties of quantifiers over the naturals.\<close>
 
 lemma MOST_nat: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (auto simp add: eventually_cofinite finite_nat_iff_bounded_le subset_eq simp flip: not_le)
 
 lemma MOST_nat_le: "(\<forall>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<exists>m. \<forall>n\<ge>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (auto simp add: eventually_cofinite finite_nat_iff_bounded subset_eq simp flip: not_le)
 
 lemma INFM_nat: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (simp add: frequently_cofinite infinite_nat_iff_unbounded)
 
 lemma INFM_nat_le: "(\<exists>\<^sub>\<infinity>n. P n) \<longleftrightarrow> (\<forall>m. \<exists>n\<ge>m. P n)"
   for P :: "nat \<Rightarrow> bool"
   by (simp add: frequently_cofinite infinite_nat_iff_unbounded_le)
 
 lemma MOST_INFM: "infinite (UNIV::'a set) \<Longrightarrow> MOST x::'a. P x \<Longrightarrow> INFM x::'a. P x"
   by (simp add: eventually_frequently)
 
 lemma MOST_Suc_iff: "(MOST n. P (Suc n)) \<longleftrightarrow> (MOST n. P n)"
   by (simp add: cofinite_eq_sequentially)
 
 lemma MOST_SucI: "MOST n. P n \<Longrightarrow> MOST n. P (Suc n)"
   and MOST_SucD: "MOST n. P (Suc n) \<Longrightarrow> MOST n. P n"
   by (simp_all add: MOST_Suc_iff)
 
 lemma MOST_ge_nat: "MOST n::nat. m \<le> n"
   by (simp add: cofinite_eq_sequentially)
 
 \<comment> \<open>legacy names\<close>
 lemma Inf_many_def: "Inf_many P \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
 lemma Alm_all_def: "Alm_all P \<longleftrightarrow> \<not> (INFM x. \<not> P x)" by simp
 lemma INFM_iff_infinite: "(INFM x. P x) \<longleftrightarrow> infinite {x. P x}" by (fact frequently_cofinite)
 lemma MOST_iff_cofinite: "(MOST x. P x) \<longleftrightarrow> finite {x. \<not> P x}" by (fact eventually_cofinite)
 lemma INFM_EX: "(\<exists>\<^sub>\<infinity>x. P x) \<Longrightarrow> (\<exists>x. P x)" by (fact frequently_ex)
 lemma ALL_MOST: "\<forall>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x" by (fact always_eventually)
 lemma INFM_mono: "\<exists>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<exists>\<^sub>\<infinity>x. Q x" by (fact frequently_elim1)
 lemma MOST_mono: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_mono)
 lemma INFM_disj_distrib: "(\<exists>\<^sub>\<infinity>x. P x \<or> Q x) \<longleftrightarrow> (\<exists>\<^sub>\<infinity>x. P x) \<or> (\<exists>\<^sub>\<infinity>x. Q x)" by (fact frequently_disj_iff)
 lemma MOST_rev_mp: "\<forall>\<^sub>\<infinity>x. P x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>\<^sub>\<infinity>x. Q x" by (fact eventually_rev_mp)
 lemma MOST_conj_distrib: "(\<forall>\<^sub>\<infinity>x. P x \<and> Q x) \<longleftrightarrow> (\<forall>\<^sub>\<infinity>x. P x) \<and> (\<forall>\<^sub>\<infinity>x. Q x)" by (fact eventually_conj_iff)
 lemma MOST_conjI: "MOST x. P x \<Longrightarrow> MOST x. Q x \<Longrightarrow> MOST x. P x \<and> Q x" by (fact eventually_conj)
 lemma INFM_finite_Bex_distrib: "finite A \<Longrightarrow> (INFM y. \<exists>x\<in>A. P x y) \<longleftrightarrow> (\<exists>x\<in>A. INFM y. P x y)" by (fact frequently_bex_finite_distrib)
 lemma MOST_finite_Ball_distrib: "finite A \<Longrightarrow> (MOST y. \<forall>x\<in>A. P x y) \<longleftrightarrow> (\<forall>x\<in>A. MOST y. P x y)" by (fact eventually_ball_finite_distrib)
 lemma INFM_E: "INFM x. P x \<Longrightarrow> (\<And>x. P x \<Longrightarrow> thesis) \<Longrightarrow> thesis" by (fact frequentlyE)
 lemma MOST_I: "(\<And>x. P x) \<Longrightarrow> MOST x. P x" by (rule eventuallyI)
 lemmas MOST_iff_finiteNeg = MOST_iff_cofinite
 
 
 subsection \<open>Enumeration of an Infinite Set\<close>
 
 text \<open>The set's element type must be wellordered (e.g. the natural numbers).\<close>
 
 text \<open>
   Could be generalized to
     \<^prop>\<open>enumerate' S n = (SOME t. t \<in> s \<and> finite {s\<in>S. s < t} \<and> card {s\<in>S. s < t} = n)\<close>.
 \<close>
 
 primrec (in wellorder) enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
   where
     enumerate_0: "enumerate S 0 = (LEAST n. n \<in> S)"
   | enumerate_Suc: "enumerate S (Suc n) = enumerate (S - {LEAST n. n \<in> S}) n"
 
 lemma enumerate_Suc': "enumerate S (Suc n) = enumerate (S - {enumerate S 0}) n"
   by simp
 
 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n \<in> S"
 proof (induct n arbitrary: S)
   case 0
   then show ?case
     by (fastforce intro: LeastI dest!: infinite_imp_nonempty)
 next
   case (Suc n)
   then show ?case
     by simp (metis DiffE infinite_remove)
 qed
 
 declare enumerate_0 [simp del] enumerate_Suc [simp del]
 
 lemma enumerate_step: "infinite S \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
 proof (induction n arbitrary: S)
   case 0
   then have "enumerate S 0 \<le> enumerate S (Suc 0)"
     by (simp add: enumerate_0 Least_le enumerate_in_set)
   moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}"
     by (meson "0.prems" enumerate_in_set infinite_remove)
   then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0"
     by auto
   ultimately show ?case
     by (simp add: enumerate_Suc')
 next
   case (Suc n)
   then show ?case 
     by (simp add: enumerate_Suc')
 qed
 
 lemma enumerate_mono: "m < n \<Longrightarrow> infinite S \<Longrightarrow> enumerate S m < enumerate S n"
   by (induct m n rule: less_Suc_induct) (auto intro: enumerate_step)
 
 lemma le_enumerate:
   assumes S: "infinite S"
   shows "n \<le> enumerate S n"
   using S
 proof (induct n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "n \<le> enumerate S n" by simp
   also note enumerate_mono[of n "Suc n", OF _ \<open>infinite S\<close>]
   finally show ?case by simp
 qed
 
 lemma infinite_enumerate:
   assumes fS: "infinite S"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
   unfolding strict_mono_def
   using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
 
 lemma enumerate_Suc'':
   fixes S :: "'a::wellorder set"
   assumes "infinite S"
   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
   using assms
 proof (induct n arbitrary: S)
   case 0
   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
     by (auto simp: enumerate.simps intro: Least_le)
   then show ?case
     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
     by (intro arg_cong[where f = Least] ext) auto
 next
   case (Suc n S)
   show ?case
     using enumerate_mono[OF zero_less_Suc \<open>infinite S\<close>, of n] \<open>infinite S\<close>
     apply (subst (1 2) enumerate_Suc')
     apply (subst Suc)
      apply (use \<open>infinite S\<close> in simp)
     apply (intro arg_cong[where f = Least] ext)
     apply (auto simp flip: enumerate_Suc')
     done
 qed
 
 lemma enumerate_Ex:
   fixes S :: "nat set"
   assumes S: "infinite S"
     and s: "s \<in> S"
   shows "\<exists>n. enumerate S n = s"
   using s
 proof (induct s rule: less_induct)
   case (less s)
   show ?case
   proof (cases "\<exists>y\<in>S. y < s")
     case True
     let ?y = "Max {s'\<in>S. s' < s}"
     from True have y: "\<And>x. ?y < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
       by (subst Max_less_iff) auto
     then have y_in: "?y \<in> {s'\<in>S. s' < s}"
       by (intro Max_in) auto
     with less.hyps[of ?y] obtain n where "enumerate S n = ?y"
       by auto
     with S have "enumerate S (Suc n) = s"
       by (auto simp: y less enumerate_Suc'' intro!: Least_equality)
     then show ?thesis by auto
   next
     case False
     then have "\<forall>t\<in>S. s \<le> t" by auto
     with \<open>s \<in> S\<close> show ?thesis
       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
   qed
 qed
 
 lemma inj_enumerate:
   fixes S :: "'a::wellorder set"
   assumes S: "infinite S"
   shows "inj (enumerate S)"
   unfolding inj_on_def
 proof clarsimp
   show "\<And>x y. enumerate S x = enumerate S y \<Longrightarrow> x = y"
     by (metis neq_iff enumerate_mono[OF _ \<open>infinite S\<close>]) 
 qed
 
 text \<open>To generalise this, we'd need a condition that all initial segments were finite\<close>
 lemma bij_enumerate:
   fixes S :: "nat set"
   assumes S: "infinite S"
   shows "bij_betw (enumerate S) UNIV S"
 proof -
   have "\<forall>s \<in> S. \<exists>i. enumerate S i = s"
     using enumerate_Ex[OF S] by auto
   moreover note \<open>infinite S\<close> inj_enumerate
   ultimately show ?thesis
     unfolding bij_betw_def by (auto intro: enumerate_in_set)
 qed
 
 text \<open>A pair of weird and wonderful lemmas from HOL Light.\<close>
 lemma finite_transitivity_chain:
   assumes "finite A"
     and R: "\<And>x. \<not> R x x" "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z"
     and A: "\<And>x. x \<in> A \<Longrightarrow> \<exists>y. y \<in> A \<and> R x y"
   shows "A = {}"
   using \<open>finite A\<close> A
 proof (induct A)
   case empty
   then show ?case by simp
 next
   case (insert a A)
   have False
     using R(1)[of a] R(2)[of _ a] insert(3,4) by blast   
   thus ?case ..
 qed
 
 corollary Union_maximal_sets:
   assumes "finite \<F>"
   shows "\<Union>{T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} = \<Union>\<F>"
     (is "?lhs = ?rhs")
 proof
   show "?lhs \<subseteq> ?rhs" by force
   show "?rhs \<subseteq> ?lhs"
   proof (rule Union_subsetI)
     fix S
     assume "S \<in> \<F>"
     have "{T \<in> \<F>. S \<subseteq> T} = {}"
       if "\<not> (\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y)"
     proof -
       have \<section>: "\<And>x. x \<in> \<F> \<and> S \<subseteq> x \<Longrightarrow> \<exists>y. y \<in> \<F> \<and> S \<subseteq> y \<and> x \<subset> y"
         using that by (blast intro: dual_order.trans psubset_imp_subset)
       show ?thesis
       proof (rule finite_transitivity_chain [of _ "\<lambda>T U. S \<subseteq> T \<and> T \<subset> U"])
       qed (use assms in \<open>auto intro: \<section>\<close>)
     qed
     with \<open>S \<in> \<F>\<close> show "\<exists>y. y \<in> {T \<in> \<F>. \<forall>U\<in>\<F>. \<not> T \<subset> U} \<and> S \<subseteq> y"
       by blast
   qed
 qed
 
 subsection \<open>Properties of @{term enumerate} on finite sets\<close>
 
 lemma finite_enumerate_in_set: "\<lbrakk>finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S n \<in> S"
 proof (induction n arbitrary: S)
   case 0
   then show ?case
     by (metis all_not_in_conv card_empty enumerate.simps(1) not_less0 wellorder_Least_lemma(1))
 next
   case (Suc n)
   show ?case
     using Suc.prems Suc.IH [of "S - {LEAST n. n \<in> S}"]
     apply (simp add: enumerate.simps)
     by (metis Diff_empty Diff_insert0 Suc_lessD card.remove less_Suc_eq)
 qed
 
 lemma finite_enumerate_step: "\<lbrakk>finite S; Suc n < card S\<rbrakk> \<Longrightarrow> enumerate S n < enumerate S (Suc n)"
 proof (induction n arbitrary: S)
   case 0
   then have "enumerate S 0 \<le> enumerate S (Suc 0)"
     by (simp add: Least_le enumerate.simps(1) finite_enumerate_in_set)
   moreover have "enumerate (S - {enumerate S 0}) 0 \<in> S - {enumerate S 0}"
     by (metis 0 Suc_lessD Suc_less_eq card_Suc_Diff1 enumerate_in_set finite_enumerate_in_set)
   then have "enumerate S 0 \<noteq> enumerate (S - {enumerate S 0}) 0"
     by auto
   ultimately show ?case
     by (simp add: enumerate_Suc')
 next
   case (Suc n)
   then show ?case
     by (simp add: enumerate_Suc' finite_enumerate_in_set)
 qed
 
 lemma finite_enumerate_mono: "\<lbrakk>m < n; finite S; n < card S\<rbrakk> \<Longrightarrow> enumerate S m < enumerate S n"
   by (induct m n rule: less_Suc_induct) (auto intro: finite_enumerate_step)
 
 
 lemma finite_le_enumerate:
   assumes "finite S" "n < card S"
   shows "n \<le> enumerate S n"
   using assms
 proof (induction n)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "n \<le> enumerate S n" by simp
   also note finite_enumerate_mono[of n "Suc n", OF _ \<open>finite S\<close>]
   finally show ?case
     using Suc.prems(2) Suc_leI by blast
 qed
 
 lemma finite_enumerate:
   assumes fS: "finite S"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono_on r {..<card S} \<and> (\<forall>n<card S. r n \<in> S)"
   unfolding strict_mono_def
   using finite_enumerate_in_set[OF fS] finite_enumerate_mono[of _ _ S] fS
   by (metis lessThan_iff strict_mono_on_def)
 
 lemma finite_enumerate_Suc'':
   fixes S :: "'a::wellorder set"
   assumes "finite S" "Suc n < card S"
   shows "enumerate S (Suc n) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
   using assms
 proof (induction n arbitrary: S)
   case 0
   then have "\<forall>s \<in> S. enumerate S 0 \<le> s"
     by (auto simp: enumerate.simps intro: Least_le)
   then show ?case
     unfolding enumerate_Suc' enumerate_0[of "S - {enumerate S 0}"]
     by (metis Diff_iff dual_order.strict_iff_order singletonD singletonI)
 next
   case (Suc n S)
   then have "Suc n < card (S - {enumerate S 0})"
     using Suc.prems(2) finite_enumerate_in_set by force
   then show ?case
     apply (subst (1 2) enumerate_Suc')
     apply (simp add: Suc)
     apply (intro arg_cong[where f = Least] HOL.ext)
     using finite_enumerate_mono[OF zero_less_Suc \<open>finite S\<close>, of n] Suc.prems
     by (auto simp flip: enumerate_Suc')
 qed
 
 lemma finite_enumerate_initial_segment:
   fixes S :: "'a::wellorder set"
-  assumes "finite S" "s \<in> S" and n: "n < card (S \<inter> {..<s})"
+  assumes "finite S" and n: "n < card (S \<inter> {..<s})"
   shows "enumerate (S \<inter> {..<s}) n = enumerate S n"
   using n
 proof (induction n)
   case 0
   have "(LEAST n. n \<in> S \<and> n < s) = (LEAST n. n \<in> S)"
   proof (rule Least_equality)
     have "\<exists>t. t \<in> S \<and> t < s"
       by (metis "0" card_gt_0_iff disjoint_iff_not_equal lessThan_iff)
     then show "(LEAST n. n \<in> S) \<in> S \<and> (LEAST n. n \<in> S) < s"
       by (meson LeastI Least_le le_less_trans)
   qed (simp add: Least_le)
   then show ?case
     by (auto simp: enumerate_0)
 next
   case (Suc n)
   then have less_card: "Suc n < card S"
     by (meson assms(1) card_mono inf_sup_ord(1) leD le_less_linear order.trans)
-  obtain t where t: "t \<in> {s \<in> S. enumerate S n < s}"
+  obtain T where T: "T \<in> {s \<in> S. enumerate S n < s}"
     by (metis Infinite_Set.enumerate_step enumerate_in_set finite_enumerate_in_set finite_enumerate_step less_card mem_Collect_eq)
-  have "(LEAST x. x \<in> S \<and> x < s \<and> enumerate S n < x) = (LEAST s. s \<in> S \<and> enumerate S n < s)"
+  have "(LEAST x. x \<in> S \<and> x < s \<and> enumerate S n < x) = (LEAST x. x \<in> S \<and> enumerate S n < x)"
        (is "_ = ?r")
   proof (intro Least_equality conjI)
     show "?r \<in> S"
-      by (metis (mono_tags, lifting) LeastI mem_Collect_eq t)
-    show "?r < s"
-      using not_less_Least [of _ "\<lambda>t. t \<in> S \<and> enumerate S n < t"] Suc assms 
-      by (metis (no_types, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def linorder_cases)
+      by (metis (mono_tags, lifting) LeastI mem_Collect_eq T)
+    have "\<not> s \<le> ?r"
+      using not_less_Least [of _ "\<lambda>x. x \<in> S \<and> enumerate S n < x"] Suc assms
+      by (metis (mono_tags, lifting) Int_Collect Suc_lessD finite_Int finite_enumerate_in_set finite_enumerate_step lessThan_def less_le_trans)
+    then show "?r < s"
+      by auto
     show "enumerate S n < ?r"
-      by (metis (no_types, lifting) LeastI mem_Collect_eq t)
+      by (metis (no_types, lifting) LeastI mem_Collect_eq T)
   qed (auto simp: Least_le)
   then show ?case
     using Suc assms by (simp add: finite_enumerate_Suc'' less_card)
 qed
 
 lemma finite_enumerate_Ex:
   fixes S :: "'a::wellorder set"
   assumes S: "finite S"
     and s: "s \<in> S"
   shows "\<exists>n<card S. enumerate S n = s"
   using s S
 proof (induction s arbitrary: S rule: less_induct)
   case (less s)
   show ?case
   proof (cases "\<exists>y\<in>S. y < s")
     case True
     let ?T = "S \<inter> {..<s}"
     have "finite ?T"
       using less.prems(2) by blast
     have TS: "card ?T < card S"
       using less.prems by (blast intro: psubset_card_mono [OF \<open>finite S\<close>])
     from True have y: "\<And>x. Max ?T < x \<longleftrightarrow> (\<forall>s'\<in>S. s' < s \<longrightarrow> s' < x)"
       by (subst Max_less_iff) (auto simp: \<open>finite ?T\<close>)
     then have y_in: "Max ?T \<in> {s'\<in>S. s' < s}"
       using Max_in \<open>finite ?T\<close> by fastforce
     with less.IH[of "Max ?T" ?T] obtain n where n: "enumerate ?T n = Max ?T" "n < card ?T"
       using \<open>finite ?T\<close> by blast
     then have "Suc n < card S"
       using TS less_trans_Suc by blast
     with S n have "enumerate S (Suc n) = s"
       by (subst finite_enumerate_Suc'') (auto simp: y finite_enumerate_initial_segment less finite_enumerate_Suc'' intro!: Least_equality)
     then show ?thesis
       using \<open>Suc n < card S\<close> by blast
   next
     case False
     then have "\<forall>t\<in>S. s \<le> t" by auto
     moreover have "0 < card S"
       using card_0_eq less.prems by blast
     ultimately show ?thesis
       using \<open>s \<in> S\<close>
       by (auto intro!: exI[of _ 0] Least_equality simp: enumerate_0)
   qed
 qed
 
 lemma finite_bij_enumerate:
   fixes S :: "'a::wellorder set"
   assumes S: "finite S"
   shows "bij_betw (enumerate S) {..<card S} S"
 proof -
   have "\<And>n m. \<lbrakk>n \<noteq> m; n < card S; m < card S\<rbrakk> \<Longrightarrow> enumerate S n \<noteq> enumerate S m"
     using finite_enumerate_mono[OF _ \<open>finite S\<close>] by (auto simp: neq_iff)
   then have "inj_on (enumerate S) {..<card S}"
     by (auto simp: inj_on_def)
   moreover have "\<forall>s \<in> S. \<exists>i<card S. enumerate S i = s"
     using finite_enumerate_Ex[OF S] by auto
   moreover note \<open>finite S\<close>
   ultimately show ?thesis
     unfolding bij_betw_def by (auto intro: finite_enumerate_in_set)
 qed
 
 lemma ex_bij_betw_strict_mono_card:
   fixes M :: "'a::wellorder set"
   assumes "finite M" 
   obtains h where "bij_betw h {..<card M} M" and "strict_mono_on h {..<card M}"
 proof
   show "bij_betw (enumerate M) {..<card M} M"
     by (simp add: assms finite_bij_enumerate)
   show "strict_mono_on (enumerate M) {..<card M}"
     by (simp add: assms finite_enumerate_mono strict_mono_on_def)
 qed
 
 end