diff --git a/src/HOL/Analysis/Brouwer_Fixpoint.thy b/src/HOL/Analysis/Brouwer_Fixpoint.thy
--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy
@@ -1,2524 +1,2524 @@
 (*  Author:     John Harrison
     Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light) and LCP
 *)
 
 (* At the moment this is just Brouwer's fixpoint theorem. The proof is from  *)
 (* Kuhn: "some combinatorial lemmas in topology", IBM J. v4. (1960) p. 518   *)
 (* See "http://www.research.ibm.com/journal/rd/045/ibmrd0405K.pdf".          *)
 (*                                                                           *)
 (* The script below is quite messy, but at least we avoid formalizing any    *)
 (* topological machinery; we don't even use barycentric subdivision; this is *)
 (* the big advantage of Kuhn's proof over the usual Sperner's lemma one.     *)
 (*                                                                           *)
 (*              (c) Copyright, John Harrison 1998-2008                       *)
 
 section \<open>Brouwer's Fixed Point Theorem\<close>
 
 theory Brouwer_Fixpoint
   imports Homeomorphism Derivative
 begin
 
 subsection \<open>Retractions\<close>
 
 lemma retract_of_contractible:
   assumes "contractible T" "S retract_of T"
   shows "contractible S"
   using assms
   apply (clarsimp simp add: retract_of_def contractible_def retraction_def homotopic_with image_subset_iff_funcset)
   apply (rule_tac x="r a" in exI)
   apply (rule_tac x="r \<circ> h" in exI)
   apply (intro conjI continuous_intros continuous_on_compose)
       apply (erule continuous_on_subset | force)+
   done
 
 lemma retract_of_path_connected:
     "\<lbrakk>path_connected T; S retract_of T\<rbrakk> \<Longrightarrow> path_connected S"
   by (metis path_connected_continuous_image retract_of_def retraction)
 
 lemma retract_of_simply_connected:
-  "\<lbrakk>simply_connected T; S retract_of T\<rbrakk> \<Longrightarrow> simply_connected S"
-  apply (simp add: retract_of_def retraction_def Pi_iff, clarify)
-  apply (rule simply_connected_retraction_gen)
-       apply (force elim!: continuous_on_subset)+
-  done
+  assumes T: "simply_connected T" and "S retract_of T"
+  shows "simply_connected S"
+proof -
+  obtain r where r: "retraction T S r"
+    using assms by (metis retract_of_def) 
+  have "S \<subseteq> T"
+    by (meson \<open>retraction T S r\<close> retraction)
+  then have "(\<lambda>a. a) \<in> S \<rightarrow> T"
+    by blast
+  then show ?thesis
+    using simply_connected_retraction_gen [OF T]
+    by (metis (no_types) r retraction retraction_refl)
+qed
 
 lemma retract_of_homotopically_trivial:
   assumes ts: "T retract_of S"
       and hom: "\<And>f g. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S;
                        continuous_on U g; g \<in> U \<rightarrow> S\<rbrakk>
                        \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g"
       and "continuous_on U f" "f \<in> U \<rightarrow> T"
       and "continuous_on U g" "g \<in> U \<rightarrow> T"
     shows "homotopic_with_canon (\<lambda>x. True) U T f g"
 proof -
   obtain r where "r \<in> S \<rightarrow> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
     using ts by (auto simp: retract_of_def retraction)
   then obtain k where "Retracts S r T k"
     unfolding Retracts_def using continuous_on_id by blast
   then show ?thesis
-    apply (rule Retracts.homotopically_trivial_retraction_gen)
-    using assms
-    apply (force simp: hom)+
-    done
+    by (rule Retracts.homotopically_trivial_retraction_gen) (use assms hom in force)+
 qed
 
 lemma retract_of_homotopically_trivial_null:
   assumes ts: "T retract_of S"
       and hom: "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S\<rbrakk>
                      \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>x. True) U S f (\<lambda>x. c)"
       and "continuous_on U f" "f \<in> U \<rightarrow> T"
   obtains c where "homotopic_with_canon (\<lambda>x. True) U T f (\<lambda>x. c)"
 proof -
   obtain r where "r \<in> S \<rightarrow> S" "continuous_on S r" "\<forall>x\<in>S. r (r x) = r x" "T = r ` S"
     using ts by (auto simp: retract_of_def retraction)
   then obtain k where "Retracts S r T k"
     unfolding Retracts_def by fastforce
   then show ?thesis
-    apply (rule Retracts.homotopically_trivial_retraction_null_gen)
-    apply (rule TrueI refl assms that | assumption)+
-    done
+  proof (rule Retracts.homotopically_trivial_retraction_null_gen)
+    show "\<And>f. \<lbrakk>continuous_on U f; f \<in> U \<rightarrow> S\<rbrakk>
+         \<Longrightarrow> \<exists>c. homotopic_with_canon (\<lambda>a. True) U S f (\<lambda>x. c)"
+      using hom by blast
+  qed (use assms that in auto)
 qed
 
 lemma retraction_openin_vimage_iff:
   "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U"
   if "retraction S T r" and "U \<subseteq> T"
   by (simp add: retraction_openin_vimage_iff that)
 
 lemma retract_of_locally_compact:
     fixes S :: "'a :: {heine_borel,real_normed_vector} set"
     shows  "\<lbrakk> locally compact S; T retract_of S\<rbrakk> \<Longrightarrow> locally compact T"
   by (metis locally_compact_closedin closedin_retract)
 
 lemma homotopic_into_retract:
    "\<lbrakk>f \<in> S \<rightarrow> T; g \<in> S \<rightarrow> T; T retract_of U; homotopic_with_canon (\<lambda>x. True) S U f g\<rbrakk>
         \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g"
   apply (subst (asm) homotopic_with_def)
   apply (simp add: homotopic_with retract_of_def retraction_def Pi_iff, clarify)
   apply (rule_tac x="r \<circ> h" in exI)
   by (smt (verit, ccfv_SIG) comp_def continuous_on_compose continuous_on_subset image_subset_iff)
 
 lemma retract_of_locally_connected:
   assumes "locally connected T" "S retract_of T"
   shows "locally connected S"
   using assms
-  by (metis Abstract_Topology_2.retraction_openin_vimage_iff idempotent_imp_retraction locally_connected_quotient_image retract_ofE)
+  by (metis retraction_openin_vimage_iff idempotent_imp_retraction locally_connected_quotient_image retract_ofE)
 
 lemma retract_of_locally_path_connected:
   assumes "locally path_connected T" "S retract_of T"
   shows "locally path_connected S"
   using assms
-  by (metis Abstract_Topology_2.retraction_openin_vimage_iff idempotent_imp_retraction locally_path_connected_quotient_image retract_ofE)
+  by (metis retraction_openin_vimage_iff idempotent_imp_retraction locally_path_connected_quotient_image retract_ofE)
 
 text \<open>A few simple lemmas about deformation retracts\<close>
 
 lemma deformation_retract_imp_homotopy_eqv:
   fixes S :: "'a::euclidean_space set"
   assumes "homotopic_with_canon (\<lambda>x. True) S S id r" and r: "retraction S T r"
   shows "S homotopy_eqv T"
 proof -
   have "homotopic_with_canon (\<lambda>x. True) S S (id \<circ> r) id"
     by (simp add: assms(1) homotopic_with_symD)
   moreover have "homotopic_with_canon (\<lambda>x. True) T T (r \<circ> id) id"
     using r unfolding retraction_def
     by (metis eq_id_iff homotopic_with_id2 topspace_euclidean_subtopology)
   ultimately
   show ?thesis
     unfolding homotopy_equivalent_space_def
     by (smt (verit, del_insts) continuous_map_id continuous_map_subtopology_eu id_def r retraction retraction_comp subset_refl) 
 qed
 
 lemma deformation_retract:
   fixes S :: "'a::euclidean_space set"
     shows "(\<exists>r. homotopic_with_canon (\<lambda>x. True) S S id r \<and> retraction S T r) \<longleftrightarrow>
            T retract_of S \<and> (\<exists>f. homotopic_with_canon (\<lambda>x. True) S S id f \<and> f \<in> S \<rightarrow> T)"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (auto simp: retract_of_def retraction_def)
 next
-  assume ?rhs
-  then show ?lhs
-    apply (clarsimp simp add: retract_of_def retraction_def)
-    apply (rule_tac x=r in exI, simp)
-     apply (rule homotopic_with_trans, assumption)
-     apply (rule_tac f = "r \<circ> f" and g="r \<circ> id" in homotopic_with_eq)
-        apply (rule_tac Y=S in homotopic_with_compose_continuous_left)
+  assume R: ?rhs
+  have "\<And>r f. \<lbrakk>T \<subseteq> S; continuous_on S r; homotopic_with_canon (\<lambda>x. True) S S id f;
+            f \<in> S \<rightarrow> T; r \<in> S \<rightarrow> T; \<forall>x\<in>T. r x = x\<rbrakk>
+           \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) S S f r"
+    apply (rule_tac f = "r \<circ> f" and g="r \<circ> id" in homotopic_with_eq)
+       apply (rule_tac Y=S in homotopic_with_compose_continuous_left)
          apply (auto simp: homotopic_with_sym Pi_iff)
     done
+  with R homotopic_with_trans show ?lhs
+    unfolding retract_of_def retraction_def by blast
 qed
 
 lemma deformation_retract_of_contractible_sing:
   fixes S :: "'a::euclidean_space set"
   assumes "contractible S" "a \<in> S"
   obtains r where "homotopic_with_canon (\<lambda>x. True) S S id r" "retraction S {a} r"
 proof -
   have "{a} retract_of S"
     by (simp add: \<open>a \<in> S\<close>)
   moreover have "homotopic_with_canon (\<lambda>x. True) S S id (\<lambda>x. a)"
       using assms
       by (auto simp: contractible_def homotopic_into_contractible image_subset_iff)
   moreover have "(\<lambda>x. a) \<in> S \<rightarrow> {a}"
     by (simp add: image_subsetI)
   ultimately show ?thesis
     by (metis that deformation_retract)
 qed
 
 
 lemma continuous_on_compact_surface_projection_aux:
   fixes S :: "'a::t2_space set"
   assumes "compact S" "S \<subseteq> T" "image q T \<subseteq> S"
       and contp: "continuous_on T p"
       and "\<And>x. x \<in> S \<Longrightarrow> q x = x"
       and [simp]: "\<And>x. x \<in> T \<Longrightarrow> q(p x) = q x"
       and "\<And>x. x \<in> T \<Longrightarrow> p(q x) = p x"
     shows "continuous_on T q"
 proof -
   have *: "image p T = image p S"
     using assms by auto (metis imageI subset_iff)
   have contp': "continuous_on S p"
     by (rule continuous_on_subset [OF contp \<open>S \<subseteq> T\<close>])
   have "continuous_on (p ` T) q"
     by (simp add: "*" assms(1) assms(2) assms(5) continuous_on_inv contp' rev_subsetD)
   then have "continuous_on T (q \<circ> p)"
     by (rule continuous_on_compose [OF contp])
   then show ?thesis
     by (rule continuous_on_eq [of _ "q \<circ> p"]) (simp add: o_def)
 qed
 
 lemma continuous_on_compact_surface_projection:
   fixes S :: "'a::real_normed_vector set"
   assumes "compact S"
       and S: "S \<subseteq> V - {0}" and "cone V"
       and iff: "\<And>x k. x \<in> V - {0} \<Longrightarrow> 0 < k \<and> (k *\<^sub>R x) \<in> S \<longleftrightarrow> d x = k"
   shows "continuous_on (V - {0}) (\<lambda>x. d x *\<^sub>R x)"
 proof (rule continuous_on_compact_surface_projection_aux [OF \<open>compact S\<close> S])
   show "(\<lambda>x. d x *\<^sub>R x) ` (V - {0}) \<subseteq> S"
     using iff by auto
   show "continuous_on (V - {0}) (\<lambda>x. inverse(norm x) *\<^sub>R x)"
     by (intro continuous_intros) force
   show "\<And>x. x \<in> S \<Longrightarrow> d x *\<^sub>R x = x"
     by (metis S zero_less_one local.iff scaleR_one subset_eq)
   show "d (x /\<^sub>R norm x) *\<^sub>R (x /\<^sub>R norm x) = d x *\<^sub>R x" if "x \<in> V - {0}" for x
     using iff [of "inverse(norm x) *\<^sub>R x" "norm x * d x", symmetric] iff that \<open>cone V\<close>
     by (simp add: field_simps cone_def zero_less_mult_iff)
   show "d x *\<^sub>R x /\<^sub>R norm (d x *\<^sub>R x) = x /\<^sub>R norm x" if "x \<in> V - {0}" for x
   proof -
     have "0 < d x"
       using local.iff that by blast
     then show ?thesis
       by simp
   qed
 qed
 
 subsection \<open>Kuhn Simplices\<close>
 
 lemma bij_betw_singleton_eq:
   assumes f: "bij_betw f A B" and g: "bij_betw g A B" and a: "a \<in> A"
   assumes eq: "(\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x = g x)"
   shows "f a = g a"
 proof -
   have "f ` (A - {a}) = g ` (A - {a})"
     by (intro image_cong) (simp_all add: eq)
   then have "B - {f a} = B - {g a}"
     using f g a  by (auto simp: bij_betw_def inj_on_image_set_diff set_eq_iff)
   moreover have "f a \<in> B" "g a \<in> B"
     using f g a by (auto simp: bij_betw_def)
   ultimately show ?thesis
     by auto
 qed
 
 lemmas swap_apply1 = swap_apply(1)
 lemmas swap_apply2 = swap_apply(2)
 
 lemma pointwise_minimal_pointwise_maximal:
   fixes s :: "(nat \<Rightarrow> nat) set"
   assumes "finite s"
     and "s \<noteq> {}"
     and "\<forall>x\<in>s. \<forall>y\<in>s. x \<le> y \<or> y \<le> x"
   shows "\<exists>a\<in>s. \<forall>x\<in>s. a \<le> x"
     and "\<exists>a\<in>s. \<forall>x\<in>s. x \<le> a"
   using assms
 proof (induct s rule: finite_ne_induct)
   case (insert b s)
   assume *: "\<forall>x\<in>insert b s. \<forall>y\<in>insert b s. x \<le> y \<or> y \<le> x"
   then obtain u l where "l \<in> s" "\<forall>b\<in>s. l \<le> b" "u \<in> s" "\<forall>b\<in>s. b \<le> u"
     using insert by auto
   with * show "\<exists>a\<in>insert b s. \<forall>x\<in>insert b s. a \<le> x" "\<exists>a\<in>insert b s. \<forall>x\<in>insert b s. x \<le> a"
     by (metis insert_iff order.trans)+
 qed auto
 
 lemma kuhn_labelling_lemma:
   fixes P Q :: "'a::euclidean_space \<Rightarrow> bool"
   assumes "\<forall>x. P x \<longrightarrow> P (f x)"
     and "\<forall>x. P x \<longrightarrow> (\<forall>i\<in>Basis. Q i \<longrightarrow> 0 \<le> x\<bullet>i \<and> x\<bullet>i \<le> 1)"
   shows "\<exists>l. (\<forall>x.\<forall>i\<in>Basis. l x i \<le> (1::nat)) \<and>
              (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 0) \<longrightarrow> (l x i = 0)) \<and>
              (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (x\<bullet>i = 1) \<longrightarrow> (l x i = 1)) \<and>
              (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x\<bullet>i \<le> f x\<bullet>i) \<and>
              (\<forall>x.\<forall>i\<in>Basis. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f x\<bullet>i \<le> x\<bullet>i)"
 proof -
   { fix x i
     let ?R = "\<lambda>y. (P x \<and> Q i \<and> x \<bullet> i = 0 \<longrightarrow> y = (0::nat)) \<and>
         (P x \<and> Q i \<and> x \<bullet> i = 1 \<longrightarrow> y = 1) \<and>
         (P x \<and> Q i \<and> y = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i) \<and>
         (P x \<and> Q i \<and> y = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i)"
     { assume "P x" "Q i" "i \<in> Basis" with assms have "0 \<le> f x \<bullet> i \<and> f x \<bullet> i \<le> 1" by auto }
     then have "i \<in> Basis \<Longrightarrow> ?R 0 \<or> ?R 1" by auto }
   then show ?thesis
     unfolding all_conj_distrib[symmetric] Ball_def (* FIXME: shouldn't this work by metis? *)
     by (subst choice_iff[symmetric])+ blast
 qed
 
 
 subsubsection \<open>The key "counting" observation, somewhat abstracted\<close>
 
 lemma kuhn_counting_lemma:
   fixes bnd compo compo' face S F
   defines "nF s == card {f\<in>F. face f s \<and> compo' f}"
   assumes [simp, intro]: "finite F" \<comment> \<open>faces\<close> and [simp, intro]: "finite S" \<comment> \<open>simplices\<close>
     and "\<And>f. f \<in> F \<Longrightarrow> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 1"
     and "\<And>f. f \<in> F \<Longrightarrow> \<not> bnd f \<Longrightarrow> card {s\<in>S. face f s} = 2"
     and "\<And>s. s \<in> S \<Longrightarrow> compo s \<Longrightarrow> nF s = 1"
     and "\<And>s. s \<in> S \<Longrightarrow> \<not> compo s \<Longrightarrow> nF s = 0 \<or> nF s = 2"
     and "odd (card {f\<in>F. compo' f \<and> bnd f})"
   shows "odd (card {s\<in>S. compo s})"
 proof -
   have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + (\<Sum>s | s \<in> S \<and> compo s. nF s) = (\<Sum>s\<in>S. nF s)"
     by (subst sum.union_disjoint[symmetric]) (auto intro!: sum.cong)
   also have "\<dots> = (\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> bnd f}. face f s}) +
                   (\<Sum>s\<in>S. card {f \<in> {f\<in>F. compo' f \<and> \<not> bnd f}. face f s})"
     unfolding sum.distrib[symmetric]
     by (subst card_Un_disjoint[symmetric])
        (auto simp: nF_def intro!: sum.cong arg_cong[where f=card])
   also have "\<dots> = 1 * card {f\<in>F. compo' f \<and> bnd f} + 2 * card {f\<in>F. compo' f \<and> \<not> bnd f}"
     using assms(4,5) by (fastforce intro!: arg_cong2[where f="(+)"] sum_multicount)
   finally have "odd ((\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) + card {s\<in>S. compo s})"
     using assms(6,8) by simp
   moreover have "(\<Sum>s | s \<in> S \<and> \<not> compo s. nF s) =
     (\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 0. nF s) + (\<Sum>s | s \<in> S \<and> \<not> compo s \<and> nF s = 2. nF s)"
     using assms(7) by (subst sum.union_disjoint[symmetric]) (fastforce intro!: sum.cong)+
   ultimately show ?thesis
     by auto
 qed
 
 subsubsection \<open>The odd/even result for faces of complete vertices, generalized\<close>
 
 lemma kuhn_complete_lemma:
   assumes [simp]: "finite simplices"
     and face: "\<And>f s. face f s \<longleftrightarrow> (\<exists>a\<in>s. f = s - {a})"
     and card_s[simp]:  "\<And>s. s \<in> simplices \<Longrightarrow> card s = n + 2"
     and rl_bd: "\<And>s. s \<in> simplices \<Longrightarrow> rl ` s \<subseteq> {..Suc n}"
     and bnd: "\<And>f s. s \<in> simplices \<Longrightarrow> face f s \<Longrightarrow> bnd f \<Longrightarrow> card {s\<in>simplices. face f s} = 1"
     and nbnd: "\<And>f s. s \<in> simplices \<Longrightarrow> face f s \<Longrightarrow> \<not> bnd f \<Longrightarrow> card {s\<in>simplices. face f s} = 2"
     and odd_card: "odd (card {f. (\<exists>s\<in>simplices. face f s) \<and> rl ` f = {..n} \<and> bnd f})"
   shows "odd (card {s\<in>simplices. (rl ` s = {..Suc n})})"
 proof (rule kuhn_counting_lemma)
   have finite_s[simp]: "\<And>s. s \<in> simplices \<Longrightarrow> finite s"
     by (metis add_is_0 zero_neq_numeral card.infinite assms(3))
 
   let ?F = "{f. \<exists>s\<in>simplices. face f s}"
   have F_eq: "?F = (\<Union>s\<in>simplices. \<Union>a\<in>s. {s - {a}})"
     by (auto simp: face)
   show "finite ?F"
     using \<open>finite simplices\<close> unfolding F_eq by auto
 
   show "card {s \<in> simplices. face f s} = 1" if "f \<in> ?F" "bnd f" for f
     using bnd that by auto
 
   show "card {s \<in> simplices. face f s} = 2" if "f \<in> ?F" "\<not> bnd f" for f
     using nbnd that by auto
 
   show "odd (card {f \<in> {f. \<exists>s\<in>simplices. face f s}. rl ` f = {..n} \<and> bnd f})"
     using odd_card by simp
 
   fix s assume s[simp]: "s \<in> simplices"
   let ?S = "{f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {..n}}"
   have "?S = (\<lambda>a. s - {a}) ` {a\<in>s. rl ` (s - {a}) = {..n}}"
     using s by (fastforce simp: face)
   then have card_S: "card ?S = card {a\<in>s. rl ` (s - {a}) = {..n}}"
     by (auto intro!: card_image inj_onI)
 
   { assume rl: "rl ` s = {..Suc n}"
     then have inj_rl: "inj_on rl s"
       by (intro eq_card_imp_inj_on) auto
     moreover obtain a where "rl a = Suc n" "a \<in> s"
       by (metis atMost_iff image_iff le_Suc_eq rl)
     ultimately have n: "{..n} = rl ` (s - {a})"
       by (auto simp: inj_on_image_set_diff rl)
     have "{a\<in>s. rl ` (s - {a}) = {..n}} = {a}"
       using inj_rl \<open>a \<in> s\<close> by (auto simp: n inj_on_image_eq_iff[OF inj_rl])
     then show "card ?S = 1"
       unfolding card_S by simp }
 
   { assume rl: "rl ` s \<noteq> {..Suc n}"
     show "card ?S = 0 \<or> card ?S = 2"
     proof cases
       assume *: "{..n} \<subseteq> rl ` s"
       with rl rl_bd[OF s] have rl_s: "rl ` s = {..n}"
         by (auto simp: atMost_Suc subset_insert_iff split: if_split_asm)
       then have "\<not> inj_on rl s"
         by (intro pigeonhole) simp
       then obtain a b where ab: "a \<in> s" "b \<in> s" "rl a = rl b" "a \<noteq> b"
         by (auto simp: inj_on_def)
       then have eq: "rl ` (s - {a}) = rl ` s"
         by auto
       with ab have inj: "inj_on rl (s - {a})"
         by (intro eq_card_imp_inj_on) (auto simp: rl_s card_Diff_singleton_if)
 
       { fix x assume "x \<in> s" "x \<notin> {a, b}"
         then have "rl ` s - {rl x} = rl ` ((s - {a}) - {x})"
           by (auto simp: eq inj_on_image_set_diff[OF inj])
         also have "\<dots> = rl ` (s - {x})"
           using ab \<open>x \<notin> {a, b}\<close> by auto
         also assume "\<dots> = rl ` s"
         finally have False
           using \<open>x\<in>s\<close> by auto }
       moreover
       { fix x assume "x \<in> {a, b}" with ab have "x \<in> s \<and> rl ` (s - {x}) = rl ` s"
           by (simp add: set_eq_iff image_iff Bex_def) metis }
       ultimately have "{a\<in>s. rl ` (s - {a}) = {..n}} = {a, b}"
         unfolding rl_s[symmetric] by fastforce
       with \<open>a \<noteq> b\<close> show "card ?S = 0 \<or> card ?S = 2"
         unfolding card_S by simp
     next
       assume "\<not> {..n} \<subseteq> rl ` s"
       then have "\<And>x. rl ` (s - {x}) \<noteq> {..n}"
         by auto
       then show "card ?S = 0 \<or> card ?S = 2"
         unfolding card_S by simp
     qed }
 qed fact
 
 locale kuhn_simplex =
   fixes p n and base upd and S :: "(nat \<Rightarrow> nat) set"
   assumes base: "base \<in> {..< n} \<rightarrow> {..< p}"
   assumes base_out: "\<And>i. n \<le> i \<Longrightarrow> base i = p"
   assumes upd: "bij_betw upd {..< n} {..< n}"
   assumes s_pre: "S = (\<lambda>i j. if j \<in> upd`{..< i} then Suc (base j) else base j) ` {.. n}"
 begin
 
 definition "enum i j = (if j \<in> upd`{..< i} then Suc (base j) else base j)"
 
 lemma s_eq: "S = enum ` {.. n}"
   unfolding s_pre enum_def[abs_def] ..
 
 lemma upd_space: "i < n \<Longrightarrow> upd i < n"
   using upd by (auto dest!: bij_betwE)
 
 lemma s_space: "S \<subseteq> {..< n} \<rightarrow> {.. p}"
 proof -
   { fix i assume "i \<le> n" then have "enum i \<in> {..< n} \<rightarrow> {.. p}"
     proof (induct i)
       case 0 then show ?case
         using base by (auto simp: Pi_iff less_imp_le enum_def)
     next
       case (Suc i) with base show ?case
         by (auto simp: Pi_iff Suc_le_eq less_imp_le enum_def intro: upd_space)
     qed }
   then show ?thesis
     by (auto simp: s_eq)
 qed
 
 lemma inj_upd: "inj_on upd {..< n}"
   using upd by (simp add: bij_betw_def)
 
 lemma inj_enum: "inj_on enum {.. n}"
 proof -
   { fix x y :: nat assume "x \<noteq> y" "x \<le> n" "y \<le> n"
     with upd have "upd ` {..< x} \<noteq> upd ` {..< y}"
       by (subst inj_on_image_eq_iff[where C="{..< n}"]) (auto simp: bij_betw_def)
     then have "enum x \<noteq> enum y"
       by (auto simp: enum_def fun_eq_iff) }
   then show ?thesis
     by (auto simp: inj_on_def)
 qed
 
 lemma enum_0: "enum 0 = base"
   by (simp add: enum_def[abs_def])
 
 lemma base_in_s: "base \<in> S"
   unfolding s_eq by (subst enum_0[symmetric]) auto
 
 lemma enum_in: "i \<le> n \<Longrightarrow> enum i \<in> S"
   unfolding s_eq by auto
 
 lemma one_step:
   assumes a: "a \<in> S" "j < n"
   assumes *: "\<And>a'. a' \<in> S \<Longrightarrow> a' \<noteq> a \<Longrightarrow> a' j = p'"
   shows "a j \<noteq> p'"
 proof
   assume "a j = p'"
   with * a have "\<And>a'. a' \<in> S \<Longrightarrow> a' j = p'"
     by auto
   then have "\<And>i. i \<le> n \<Longrightarrow> enum i j = p'"
     unfolding s_eq by auto
   from this[of 0] this[of n] have "j \<notin> upd ` {..< n}"
     by (auto simp: enum_def fun_eq_iff split: if_split_asm)
   with upd \<open>j < n\<close> show False
     by (auto simp: bij_betw_def)
 qed
 
 lemma upd_inj: "i < n \<Longrightarrow> j < n \<Longrightarrow> upd i = upd j \<longleftrightarrow> i = j"
   using upd by (auto simp: bij_betw_def inj_on_eq_iff)
 
 lemma upd_surj: "upd ` {..< n} = {..< n}"
   using upd by (auto simp: bij_betw_def)
 
 lemma in_upd_image: "A \<subseteq> {..< n} \<Longrightarrow> i < n \<Longrightarrow> upd i \<in> upd ` A \<longleftrightarrow> i \<in> A"
   using inj_on_image_mem_iff[of upd "{..< n}"] upd
   by (auto simp: bij_betw_def)
 
 lemma enum_inj: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i = enum j \<longleftrightarrow> i = j"
   using inj_enum by (auto simp: inj_on_eq_iff)
 
 lemma in_enum_image: "A \<subseteq> {.. n} \<Longrightarrow> i \<le> n \<Longrightarrow> enum i \<in> enum ` A \<longleftrightarrow> i \<in> A"
   using inj_on_image_mem_iff[OF inj_enum] by auto
 
 lemma enum_mono: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i \<le> enum j \<longleftrightarrow> i \<le> j"
   by (auto simp: enum_def le_fun_def in_upd_image Ball_def[symmetric])
 
 lemma enum_strict_mono: "i \<le> n \<Longrightarrow> j \<le> n \<Longrightarrow> enum i < enum j \<longleftrightarrow> i < j"
   using enum_mono[of i j] enum_inj[of i j] by (auto simp: le_less)
 
 lemma chain: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> a \<le> b \<or> b \<le> a"
   by (auto simp: s_eq enum_mono)
 
 lemma less: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> a i < b i \<Longrightarrow> a < b"
   using chain[of a b] by (auto simp: less_fun_def le_fun_def not_le[symmetric])
 
 lemma enum_0_bot: "a \<in> S \<Longrightarrow> a = enum 0 \<longleftrightarrow> (\<forall>a'\<in>S. a \<le> a')"
   unfolding s_eq by (auto simp: enum_mono Ball_def)
 
 lemma enum_n_top: "a \<in> S \<Longrightarrow> a = enum n \<longleftrightarrow> (\<forall>a'\<in>S. a' \<le> a)"
   unfolding s_eq by (auto simp: enum_mono Ball_def)
 
 lemma enum_Suc: "i < n \<Longrightarrow> enum (Suc i) = (enum i)(upd i := Suc (enum i (upd i)))"
   by (auto simp: fun_eq_iff enum_def upd_inj)
 
 lemma enum_eq_p: "i \<le> n \<Longrightarrow> n \<le> j \<Longrightarrow> enum i j = p"
   by (induct i) (auto simp: enum_Suc enum_0 base_out upd_space not_less[symmetric])
 
 lemma out_eq_p: "a \<in> S \<Longrightarrow> n \<le> j \<Longrightarrow> a j = p"
   unfolding s_eq by (auto simp: enum_eq_p)
 
 lemma s_le_p: "a \<in> S \<Longrightarrow> a j \<le> p"
   using out_eq_p[of a j] s_space by (cases "j < n") auto
 
 lemma le_Suc_base: "a \<in> S \<Longrightarrow> a j \<le> Suc (base j)"
   unfolding s_eq by (auto simp: enum_def)
 
 lemma base_le: "a \<in> S \<Longrightarrow> base j \<le> a j"
   unfolding s_eq by (auto simp: enum_def)
 
 lemma enum_le_p: "i \<le> n \<Longrightarrow> j < n \<Longrightarrow> enum i j \<le> p"
   using enum_in[of i] s_space by auto
 
 lemma enum_less: "a \<in> S \<Longrightarrow> i < n \<Longrightarrow> enum i < a \<longleftrightarrow> enum (Suc i) \<le> a"
   unfolding s_eq by (auto simp: enum_strict_mono enum_mono)
 
 lemma ksimplex_0:
   "n = 0 \<Longrightarrow> S = {(\<lambda>x. p)}"
   using s_eq enum_def base_out by auto
 
 lemma replace_0:
   assumes "j < n" "a \<in> S" and p: "\<forall>x\<in>S - {a}. x j = 0" and "x \<in> S"
   shows "x \<le> a"
 proof cases
   assume "x \<noteq> a"
   have "a j \<noteq> 0"
     using assms by (intro one_step[where a=a]) auto
   with less[OF \<open>x\<in>S\<close> \<open>a\<in>S\<close>, of j] p[rule_format, of x] \<open>x \<in> S\<close> \<open>x \<noteq> a\<close>
   show ?thesis
     by auto
 qed simp
 
 lemma replace_1:
   assumes "j < n" "a \<in> S" and p: "\<forall>x\<in>S - {a}. x j = p" and "x \<in> S"
   shows "a \<le> x"
 proof cases
   assume "x \<noteq> a"
   have "a j \<noteq> p"
     using assms by (intro one_step[where a=a]) auto
   with enum_le_p[of _ j] \<open>j < n\<close> \<open>a\<in>S\<close>
   have "a j < p"
     by (auto simp: less_le s_eq)
   with less[OF \<open>a\<in>S\<close> \<open>x\<in>S\<close>, of j] p[rule_format, of x] \<open>x \<in> S\<close> \<open>x \<noteq> a\<close>
   show ?thesis
     by auto
 qed simp
 
 end
 
 locale kuhn_simplex_pair = s: kuhn_simplex p n b_s u_s s + t: kuhn_simplex p n b_t u_t t
   for p n b_s u_s s b_t u_t t
 begin
 
 lemma enum_eq:
   assumes l: "i \<le> l" "l \<le> j" and "j + d \<le> n"
   assumes eq: "s.enum ` {i .. j} = t.enum ` {i + d .. j + d}"
   shows "s.enum l = t.enum (l + d)"
 using l proof (induct l rule: dec_induct)
   case base
   then have s: "s.enum i \<in> t.enum ` {i + d .. j + d}" and t: "t.enum (i + d) \<in> s.enum ` {i .. j}"
     using eq by auto
   from t \<open>i \<le> j\<close> \<open>j + d \<le> n\<close> have "s.enum i \<le> t.enum (i + d)"
     by (auto simp: s.enum_mono)
   moreover from s \<open>i \<le> j\<close> \<open>j + d \<le> n\<close> have "t.enum (i + d) \<le> s.enum i"
     by (auto simp: t.enum_mono)
   ultimately show ?case
     by auto
 next
   case (step l)
   moreover from step.prems \<open>j + d \<le> n\<close> have
       "s.enum l < s.enum (Suc l)"
       "t.enum (l + d) < t.enum (Suc l + d)"
     by (simp_all add: s.enum_strict_mono t.enum_strict_mono)
   moreover have
       "s.enum (Suc l) \<in> t.enum ` {i + d .. j + d}"
       "t.enum (Suc l + d) \<in> s.enum ` {i .. j}"
     using step \<open>j + d \<le> n\<close> eq by (auto simp: s.enum_inj t.enum_inj)
   ultimately have "s.enum (Suc l) = t.enum (Suc (l + d))"
     using \<open>j + d \<le> n\<close>
     by (intro antisym s.enum_less[THEN iffD1] t.enum_less[THEN iffD1])
        (auto intro!: s.enum_in t.enum_in)
   then show ?case by simp
 qed
 
 lemma ksimplex_eq_bot:
   assumes a: "a \<in> s" "\<And>a'. a' \<in> s \<Longrightarrow> a \<le> a'"
   assumes b: "b \<in> t" "\<And>b'. b' \<in> t \<Longrightarrow> b \<le> b'"
   assumes eq: "s - {a} = t - {b}"
   shows "s = t"
 proof cases
   assume "n = 0" with s.ksimplex_0 t.ksimplex_0 show ?thesis by simp
 next
   assume "n \<noteq> 0"
   have "s.enum 0 = (s.enum (Suc 0)) (u_s 0 := s.enum (Suc 0) (u_s 0) - 1)"
        "t.enum 0 = (t.enum (Suc 0)) (u_t 0 := t.enum (Suc 0) (u_t 0) - 1)"
     using \<open>n \<noteq> 0\<close> by (simp_all add: s.enum_Suc t.enum_Suc)
   moreover have e0: "a = s.enum 0" "b = t.enum 0"
     using a b by (simp_all add: s.enum_0_bot t.enum_0_bot)
   moreover
   { fix j assume "0 < j" "j \<le> n"
     moreover have "s - {a} = s.enum ` {Suc 0 .. n}" "t - {b} = t.enum ` {Suc 0 .. n}"
       unfolding s.s_eq t.s_eq e0 by (auto simp: s.enum_inj t.enum_inj)
     ultimately have "s.enum j = t.enum j"
       using enum_eq[of "1" j n 0] eq by auto }
   note enum_eq = this
   then have "s.enum (Suc 0) = t.enum (Suc 0)"
     using \<open>n \<noteq> 0\<close> by auto
   moreover
   { fix j assume "Suc j < n"
     with enum_eq[of "Suc j"] enum_eq[of "Suc (Suc j)"]
     have "u_s (Suc j) = u_t (Suc j)"
       using s.enum_Suc[of "Suc j"] t.enum_Suc[of "Suc j"]
       by (auto simp: fun_eq_iff split: if_split_asm) }
   then have "\<And>j. 0 < j \<Longrightarrow> j < n \<Longrightarrow> u_s j = u_t j"
     by (auto simp: gr0_conv_Suc)
   with \<open>n \<noteq> 0\<close> have "u_t 0 = u_s 0"
     by (intro bij_betw_singleton_eq[OF t.upd s.upd, of 0]) auto
   ultimately have "a = b"
     by simp
   with assms show "s = t"
     by auto
 qed
 
 lemma ksimplex_eq_top:
   assumes a: "a \<in> s" "\<And>a'. a' \<in> s \<Longrightarrow> a' \<le> a"
   assumes b: "b \<in> t" "\<And>b'. b' \<in> t \<Longrightarrow> b' \<le> b"
   assumes eq: "s - {a} = t - {b}"
   shows "s = t"
 proof (cases n)
   assume "n = 0" with s.ksimplex_0 t.ksimplex_0 show ?thesis by simp
 next
   case (Suc n')
   have "s.enum n = (s.enum n') (u_s n' := Suc (s.enum n' (u_s n')))"
        "t.enum n = (t.enum n') (u_t n' := Suc (t.enum n' (u_t n')))"
     using Suc by (simp_all add: s.enum_Suc t.enum_Suc)
   moreover have en: "a = s.enum n" "b = t.enum n"
     using a b by (simp_all add: s.enum_n_top t.enum_n_top)
   moreover
   { fix j assume "j < n"
     moreover have "s - {a} = s.enum ` {0 .. n'}" "t - {b} = t.enum ` {0 .. n'}"
       unfolding s.s_eq t.s_eq en by (auto simp: s.enum_inj t.enum_inj Suc)
     ultimately have "s.enum j = t.enum j"
       using enum_eq[of "0" j n' 0] eq Suc by auto }
   note enum_eq = this
   then have "s.enum n' = t.enum n'"
     using Suc by auto
   moreover
   { fix j assume "j < n'"
     with enum_eq[of j] enum_eq[of "Suc j"]
     have "u_s j = u_t j"
       using s.enum_Suc[of j] t.enum_Suc[of j]
       by (auto simp: Suc fun_eq_iff split: if_split_asm) }
   then have "\<And>j. j < n' \<Longrightarrow> u_s j = u_t j"
     by (auto simp: gr0_conv_Suc)
   then have "u_t n' = u_s n'"
     by (intro bij_betw_singleton_eq[OF t.upd s.upd, of n']) (auto simp: Suc)
   ultimately have "a = b"
     by simp
   with assms show "s = t"
     by auto
 qed
 
 end
 
 inductive ksimplex for p n :: nat where
   ksimplex: "kuhn_simplex p n base upd s \<Longrightarrow> ksimplex p n s"
 
 lemma finite_ksimplexes: "finite {s. ksimplex p n s}"
 proof (rule finite_subset)
   { fix a s assume "ksimplex p n s" "a \<in> s"
     then obtain b u where "kuhn_simplex p n b u s" by (auto elim: ksimplex.cases)
     then interpret kuhn_simplex p n b u s .
     from s_space \<open>a \<in> s\<close> out_eq_p[OF \<open>a \<in> s\<close>]
     have "a \<in> (\<lambda>f x. if n \<le> x then p else f x) ` ({..< n} \<rightarrow>\<^sub>E {.. p})"
       by (auto simp: image_iff subset_eq Pi_iff split: if_split_asm
                intro!: bexI[of _ "restrict a {..< n}"]) }
   then show "{s. ksimplex p n s} \<subseteq> Pow ((\<lambda>f x. if n \<le> x then p else f x) ` ({..< n} \<rightarrow>\<^sub>E {.. p}))"
     by auto
 qed (simp add: finite_PiE)
 
 lemma ksimplex_card:
   assumes "ksimplex p n s" shows "card s = Suc n"
 using assms proof cases
   case (ksimplex u b)
   then interpret kuhn_simplex p n u b s .
   show ?thesis
     by (simp add: card_image s_eq inj_enum)
 qed
 
 lemma simplex_top_face:
   assumes "0 < p" "\<forall>x\<in>s'. x n = p"
   shows "ksimplex p n s' \<longleftrightarrow> (\<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> s' = s - {a})"
   using assms
 proof safe
   fix s a assume "ksimplex p (Suc n) s" and a: "a \<in> s" and na: "\<forall>x\<in>s - {a}. x n = p"
   then show "ksimplex p n (s - {a})"
   proof cases
     case (ksimplex base upd)
     then interpret kuhn_simplex p "Suc n" base upd "s" .
 
     have "a n < p"
       using one_step[of a n p] na \<open>a\<in>s\<close> s_space by (auto simp: less_le)
     then have "a = enum 0"
       using \<open>a \<in> s\<close> na by (subst enum_0_bot) (auto simp: le_less intro!: less[of a _ n])
     then have s_eq: "s - {a} = enum ` Suc ` {.. n}"
       using s_eq by (simp add: atMost_Suc_eq_insert_0 insert_ident in_enum_image subset_eq)
     then have "enum 1 \<in> s - {a}"
       by auto
     then have "upd 0 = n"
       using \<open>a n < p\<close> \<open>a = enum 0\<close> na[rule_format, of "enum 1"]
       by (auto simp: fun_eq_iff enum_Suc split: if_split_asm)
     then have "bij_betw upd (Suc ` {..< n}) {..< n}"
       using upd
       by (subst notIn_Un_bij_betw3[where b=0])
          (auto simp: lessThan_Suc[symmetric] lessThan_Suc_eq_insert_0)
     then have "bij_betw (upd\<circ>Suc) {..<n} {..<n}"
       by (rule bij_betw_trans[rotated]) (auto simp: bij_betw_def)
 
     have "a n = p - 1"
       using enum_Suc[of 0] na[rule_format, OF \<open>enum 1 \<in> s - {a}\<close>] \<open>a = enum 0\<close> by (auto simp: \<open>upd 0 = n\<close>)
 
     show ?thesis
     proof (rule ksimplex.intros, standard)
       show "bij_betw (upd\<circ>Suc) {..< n} {..< n}" by fact
       show "base(n := p) \<in> {..<n} \<rightarrow> {..<p}" "\<And>i. n\<le>i \<Longrightarrow> (base(n := p)) i = p"
         using base base_out by (auto simp: Pi_iff)
 
       have "\<And>i. Suc ` {..< i} = {..< Suc i} - {0}"
         by (auto simp: image_iff Ball_def) arith
       then have upd_Suc: "\<And>i. i \<le> n \<Longrightarrow> (upd\<circ>Suc) ` {..< i} = upd ` {..< Suc i} - {n}"
         using \<open>upd 0 = n\<close> upd_inj by (auto simp add: image_iff less_Suc_eq_0_disj)
       have n_in_upd: "\<And>i. n \<in> upd ` {..< Suc i}"
         using \<open>upd 0 = n\<close> by auto
 
       define f' where "f' i j =
         (if j \<in> (upd\<circ>Suc)`{..< i} then Suc ((base(n := p)) j) else (base(n := p)) j)" for i j
       { fix x i
         assume i [arith]: "i \<le> n"
         with upd_Suc have "(upd \<circ> Suc) ` {..<i} = upd ` {..<Suc i} - {n}" .
         with \<open>a n < p\<close> \<open>a = enum 0\<close> \<open>upd 0 = n\<close> \<open>a n = p - 1\<close>
         have "enum (Suc i) x = f' i x"
           by (auto simp add: f'_def enum_def)  }
       then show "s - {a} = f' ` {.. n}"
         unfolding s_eq image_comp by (intro image_cong) auto
     qed
   qed
 next
   assume "ksimplex p n s'" and *: "\<forall>x\<in>s'. x n = p"
   then show "\<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> s' = s - {a}"
   proof cases
     case (ksimplex base upd)
     then interpret kuhn_simplex p n base upd s' .
     define b where "b = base (n := p - 1)"
     define u where "u i = (case i of 0 \<Rightarrow> n | Suc i \<Rightarrow> upd i)" for i
 
     have "ksimplex p (Suc n) (s' \<union> {b})"
     proof (rule ksimplex.intros, standard)
       show "b \<in> {..<Suc n} \<rightarrow> {..<p}"
         using base \<open>0 < p\<close> unfolding lessThan_Suc b_def by (auto simp: PiE_iff)
       show "\<And>i. Suc n \<le> i \<Longrightarrow> b i = p"
         using base_out by (auto simp: b_def)
 
       have "bij_betw u (Suc ` {..< n} \<union> {0}) ({..<n} \<union> {u 0})"
         using upd
         by (intro notIn_Un_bij_betw) (auto simp: u_def bij_betw_def image_comp comp_def inj_on_def)
       then show "bij_betw u {..<Suc n} {..<Suc n}"
         by (simp add: u_def lessThan_Suc[symmetric] lessThan_Suc_eq_insert_0)
 
       define f' where "f' i j = (if j \<in> u`{..< i} then Suc (b j) else b j)" for i j
 
       have u_eq: "\<And>i. i \<le> n \<Longrightarrow> u ` {..< Suc i} = upd ` {..< i} \<union> { n }"
         by (auto simp: u_def image_iff upd_inj Ball_def split: nat.split) arith
 
       { fix x have "x \<le> n \<Longrightarrow> n \<notin> upd ` {..<x}"
           using upd_space by (simp add: image_iff neq_iff) }
       note n_not_upd = this
 
       have *: "f' ` {.. Suc n} = f' ` (Suc ` {.. n} \<union> {0})"
         unfolding atMost_Suc_eq_insert_0 by simp
       also have "\<dots> = (f' \<circ> Suc) ` {.. n} \<union> {b}"
         by (auto simp: f'_def)
       also have "(f' \<circ> Suc) ` {.. n} = s'"
         using \<open>0 < p\<close> base_out[of n]
         unfolding s_eq enum_def[abs_def] f'_def[abs_def] upd_space
         by (intro image_cong) (simp_all add: u_eq b_def fun_eq_iff n_not_upd)
       finally show "s' \<union> {b} = f' ` {.. Suc n}" ..
     qed
     moreover have "b \<notin> s'"
       using * \<open>0 < p\<close> by (auto simp: b_def)
     ultimately show ?thesis by auto
   qed
 qed
 
 lemma ksimplex_replace_0:
   assumes s: "ksimplex p n s" and a: "a \<in> s"
   assumes j: "j < n" and p: "\<forall>x\<in>s - {a}. x j = 0"
   shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
   using s
 proof cases
   case (ksimplex b_s u_s)
 
   { fix t b assume "ksimplex p n t"
     then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
       by (auto elim: ksimplex.cases)
     interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
       by intro_locales fact+
 
     assume b: "b \<in> t" "t - {b} = s - {a}"
     with a j p s.replace_0[of _ a] t.replace_0[of _ b] have "s = t"
       by (intro ksimplex_eq_top[of a b]) auto }
   then have "{s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = {s}"
     using s \<open>a \<in> s\<close> by auto
   then show ?thesis
     by simp
 qed
 
 lemma ksimplex_replace_1:
   assumes s: "ksimplex p n s" and a: "a \<in> s"
   assumes j: "j < n" and p: "\<forall>x\<in>s - {a}. x j = p"
   shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 1"
   using s
 proof cases
   case (ksimplex b_s u_s)
 
   { fix t b assume "ksimplex p n t"
     then obtain b_t u_t where "kuhn_simplex p n b_t u_t t"
       by (auto elim: ksimplex.cases)
     interpret kuhn_simplex_pair p n b_s u_s s b_t u_t t
       by intro_locales fact+
 
     assume b: "b \<in> t" "t - {b} = s - {a}"
     with a j p s.replace_1[of _ a] t.replace_1[of _ b] have "s = t"
       by (intro ksimplex_eq_bot[of a b]) auto }
   then have "{s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = {s}"
     using s \<open>a \<in> s\<close> by auto
   then show ?thesis
     by simp
 qed
 
 lemma ksimplex_replace_2:
   assumes s: "ksimplex p n s" and "a \<in> s" and "n \<noteq> 0"
     and lb: "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> 0"
     and ub: "\<forall>j<n. \<exists>x\<in>s - {a}. x j \<noteq> p"
   shows "card {s'. ksimplex p n s' \<and> (\<exists>b\<in>s'. s' - {b} = s - {a})} = 2"
   using s
 proof cases
   case (ksimplex base upd)
   then interpret kuhn_simplex p n base upd s .
 
   from \<open>a \<in> s\<close> obtain i where "i \<le> n" "a = enum i"
     unfolding s_eq by auto
 
   from \<open>i \<le> n\<close> have "i = 0 \<or> i = n \<or> (0 < i \<and> i < n)"
     by linarith
   then have "\<exists>!s'. s' \<noteq> s \<and> ksimplex p n s' \<and> (\<exists>b\<in>s'. s - {a} = s'- {b})"
   proof (elim disjE conjE)
     assume "i = 0"
     define rot where [abs_def]: "rot i = (if i + 1 = n then 0 else i + 1)" for i
     let ?upd = "upd \<circ> rot"
 
     have rot: "bij_betw rot {..< n} {..< n}"
       by (auto simp: bij_betw_def inj_on_def image_iff Ball_def rot_def)
          arith+
     from rot upd have "bij_betw ?upd {..<n} {..<n}"
       by (rule bij_betw_trans)
 
     define f' where [abs_def]: "f' i j =
       (if j \<in> ?upd`{..< i} then Suc (enum (Suc 0) j) else enum (Suc 0) j)" for i j
 
     interpret b: kuhn_simplex p n "enum (Suc 0)" "upd \<circ> rot" "f' ` {.. n}"
     proof
       from \<open>a = enum i\<close> ub \<open>n \<noteq> 0\<close> \<open>i = 0\<close>
       obtain i' where "i' \<le> n" "enum i' \<noteq> enum 0" "enum i' (upd 0) \<noteq> p"
         unfolding s_eq by (auto intro: upd_space simp: enum_inj)
       then have "enum 1 \<le> enum i'" "enum i' (upd 0) < p"
         using enum_le_p[of i' "upd 0"] by (auto simp: enum_inj enum_mono upd_space)
       then have "enum 1 (upd 0) < p"
         by (auto simp: le_fun_def intro: le_less_trans)
       then show "enum (Suc 0) \<in> {..<n} \<rightarrow> {..<p}"
         using base \<open>n \<noteq> 0\<close> by (auto simp: enum_0 enum_Suc PiE_iff extensional_def upd_space)
 
       { fix i assume "n \<le> i" then show "enum (Suc 0) i = p"
         using \<open>n \<noteq> 0\<close> by (auto simp: enum_eq_p) }
       show "bij_betw ?upd {..<n} {..<n}" by fact
     qed (simp add: f'_def)
     have ks_f': "ksimplex p n (f' ` {.. n})"
       by rule unfold_locales
 
     have b_enum: "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
     with b.inj_enum have inj_f': "inj_on f' {.. n}" by simp
 
     have f'_eq_enum: "f' j = enum (Suc j)" if "j < n" for j
     proof -
       from that have "rot ` {..< j} = {0 <..< Suc j}"
         by (auto simp: rot_def image_Suc_lessThan cong: image_cong_simp)
       with that \<open>n \<noteq> 0\<close> show ?thesis
         by (simp only: f'_def enum_def fun_eq_iff image_comp [symmetric])
           (auto simp add: upd_inj)
     qed
     then have "enum ` Suc ` {..< n} = f' ` {..< n}"
       by (force simp: enum_inj)
     also have "Suc ` {..< n} = {.. n} - {0}"
       by (auto simp: image_iff Ball_def) arith
     also have "{..< n} = {.. n} - {n}"
       by auto
     finally have eq: "s - {a} = f' ` {.. n} - {f' n}"
       unfolding s_eq \<open>a = enum i\<close> \<open>i = 0\<close>
       by (simp add: inj_on_image_set_diff[OF inj_enum] inj_on_image_set_diff[OF inj_f'])
 
     have "enum 0 < f' 0"
       using \<open>n \<noteq> 0\<close> by (simp add: enum_strict_mono f'_eq_enum)
     also have "\<dots> < f' n"
       using \<open>n \<noteq> 0\<close> b.enum_strict_mono[of 0 n] unfolding b_enum by simp
     finally have "a \<noteq> f' n"
       using \<open>a = enum i\<close> \<open>i = 0\<close> by auto
 
     { fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
       obtain b u where "kuhn_simplex p n b u t"
         using \<open>ksimplex p n t\<close> by (auto elim: ksimplex.cases)
       then interpret t: kuhn_simplex p n b u t .
 
       { fix x assume "x \<in> s" "x \<noteq> a"
          then have "x (upd 0) = enum (Suc 0) (upd 0)"
            by (auto simp: \<open>a = enum i\<close> \<open>i = 0\<close> s_eq enum_def enum_inj) }
       then have eq_upd0: "\<forall>x\<in>t-{c}. x (upd 0) = enum (Suc 0) (upd 0)"
         unfolding eq_sma[symmetric] by auto
       then have "c (upd 0) \<noteq> enum (Suc 0) (upd 0)"
         using \<open>n \<noteq> 0\<close> by (intro t.one_step[OF \<open>c\<in>t\<close> ]) (auto simp: upd_space)
       then have "c (upd 0) < enum (Suc 0) (upd 0) \<or> c (upd 0) > enum (Suc 0) (upd 0)"
         by auto
       then have "t = s \<or> t = f' ` {..n}"
       proof (elim disjE conjE)
         assume *: "c (upd 0) < enum (Suc 0) (upd 0)"
         interpret st: kuhn_simplex_pair p n base upd s b u t ..
         { fix x assume "x \<in> t" with * \<open>c\<in>t\<close> eq_upd0[rule_format, of x] have "c \<le> x"
             by (auto simp: le_less intro!: t.less[of _ _ "upd 0"]) }
         note top = this
         have "s = t"
           using \<open>a = enum i\<close> \<open>i = 0\<close> \<open>c \<in> t\<close>
           by (intro st.ksimplex_eq_bot[OF _ _ _ _ eq_sma])
              (auto simp: s_eq enum_mono t.s_eq t.enum_mono top)
         then show ?thesis by simp
       next
         assume *: "c (upd 0) > enum (Suc 0) (upd 0)"
         interpret st: kuhn_simplex_pair p n "enum (Suc 0)" "upd \<circ> rot" "f' ` {.. n}" b u t ..
         have eq: "f' ` {..n} - {f' n} = t - {c}"
           using eq_sma eq by simp
         { fix x assume "x \<in> t" with * \<open>c\<in>t\<close> eq_upd0[rule_format, of x] have "x \<le> c"
             by (auto simp: le_less intro!: t.less[of _ _ "upd 0"]) }
         note top = this
         have "f' ` {..n} = t"
           using \<open>a = enum i\<close> \<open>i = 0\<close> \<open>c \<in> t\<close>
           by (intro st.ksimplex_eq_top[OF _ _ _ _ eq])
              (auto simp: b.s_eq b.enum_mono t.s_eq t.enum_mono b_enum[symmetric] top)
         then show ?thesis by simp
       qed }
     with ks_f' eq \<open>a \<noteq> f' n\<close> \<open>n \<noteq> 0\<close> show ?thesis
       apply (intro ex1I[of _ "f' ` {.. n}"])
-      apply auto []
+       apply auto []
       apply metis
       done
   next
     assume "i = n"
     from \<open>n \<noteq> 0\<close> obtain n' where n': "n = Suc n'"
       by (cases n) auto
 
     define rot where "rot i = (case i of 0 \<Rightarrow> n' | Suc i \<Rightarrow> i)" for i
     let ?upd = "upd \<circ> rot"
 
     have rot: "bij_betw rot {..< n} {..< n}"
       by (auto simp: bij_betw_def inj_on_def image_iff Bex_def rot_def n' split: nat.splits)
          arith
     from rot upd have "bij_betw ?upd {..<n} {..<n}"
       by (rule bij_betw_trans)
 
     define b where "b = base (upd n' := base (upd n') - 1)"
     define f' where [abs_def]: "f' i j = (if j \<in> ?upd`{..< i} then Suc (b j) else b j)" for i j
 
     interpret b: kuhn_simplex p n b "upd \<circ> rot" "f' ` {.. n}"
     proof
       { fix i assume "n \<le> i" then show "b i = p"
           using base_out[of i] upd_space[of n'] by (auto simp: b_def n') }
       show "b \<in> {..<n} \<rightarrow> {..<p}"
         using base \<open>n \<noteq> 0\<close> upd_space[of n']
         by (auto simp: b_def PiE_def Pi_iff Ball_def upd_space extensional_def n')
 
       show "bij_betw ?upd {..<n} {..<n}" by fact
     qed (simp add: f'_def)
     have f': "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
     have ks_f': "ksimplex p n (b.enum ` {.. n})"
       unfolding f' by rule unfold_locales
 
     have "0 < n"
       using \<open>n \<noteq> 0\<close> by auto
 
     { from \<open>a = enum i\<close> \<open>n \<noteq> 0\<close> \<open>i = n\<close> lb upd_space[of n']
       obtain i' where "i' \<le> n" "enum i' \<noteq> enum n" "0 < enum i' (upd n')"
         unfolding s_eq by (auto simp: enum_inj n')
       moreover have "enum i' (upd n') = base (upd n')"
         unfolding enum_def using \<open>i' \<le> n\<close> \<open>enum i' \<noteq> enum n\<close> by (auto simp: n' upd_inj enum_inj)
       ultimately have "0 < base (upd n')"
         by auto }
     then have benum1: "b.enum (Suc 0) = base"
       unfolding b.enum_Suc[OF \<open>0<n\<close>] b.enum_0 by (auto simp: b_def rot_def)
 
     have [simp]: "\<And>j. Suc j < n \<Longrightarrow> rot ` {..< Suc j} = {n'} \<union> {..< j}"
       by (auto simp: rot_def image_iff Ball_def split: nat.splits)
     have rot_simps: "\<And>j. rot (Suc j) = j" "rot 0 = n'"
       by (simp_all add: rot_def)
 
     { fix j assume j: "Suc j \<le> n" then have "b.enum (Suc j) = enum j"
         by (induct j) (auto simp: benum1 enum_0 b.enum_Suc enum_Suc rot_simps) }
     note b_enum_eq_enum = this
     then have "enum ` {..< n} = b.enum ` Suc ` {..< n}"
       by (auto simp: image_comp intro!: image_cong)
     also have "Suc ` {..< n} = {.. n} - {0}"
       by (auto simp: image_iff Ball_def) arith
     also have "{..< n} = {.. n} - {n}"
       by auto
     finally have eq: "s - {a} = b.enum ` {.. n} - {b.enum 0}"
       unfolding s_eq \<open>a = enum i\<close> \<open>i = n\<close>
       using inj_on_image_set_diff[OF inj_enum Diff_subset, of "{n}"]
             inj_on_image_set_diff[OF b.inj_enum Diff_subset, of "{0}"]
       by (simp add: comp_def)
 
     have "b.enum 0 \<le> b.enum n"
       by (simp add: b.enum_mono)
     also have "b.enum n < enum n"
       using \<open>n \<noteq> 0\<close> by (simp add: enum_strict_mono b_enum_eq_enum n')
     finally have "a \<noteq> b.enum 0"
       using \<open>a = enum i\<close> \<open>i = n\<close> by auto
 
     { fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
       obtain b' u where "kuhn_simplex p n b' u t"
         using \<open>ksimplex p n t\<close> by (auto elim: ksimplex.cases)
       then interpret t: kuhn_simplex p n b' u t .
 
       { fix x assume "x \<in> s" "x \<noteq> a"
          then have "x (upd n') = enum n' (upd n')"
            by (auto simp: \<open>a = enum i\<close> n' \<open>i = n\<close> s_eq enum_def enum_inj in_upd_image) }
       then have eq_upd0: "\<forall>x\<in>t-{c}. x (upd n') = enum n' (upd n')"
         unfolding eq_sma[symmetric] by auto
       then have "c (upd n') \<noteq> enum n' (upd n')"
         using \<open>n \<noteq> 0\<close> by (intro t.one_step[OF \<open>c\<in>t\<close> ]) (auto simp: n' upd_space[unfolded n'])
       then have "c (upd n') < enum n' (upd n') \<or> c (upd n') > enum n' (upd n')"
         by auto
       then have "t = s \<or> t = b.enum ` {..n}"
       proof (elim disjE conjE)
         assume *: "c (upd n') > enum n' (upd n')"
         interpret st: kuhn_simplex_pair p n base upd s b' u t ..
         { fix x assume "x \<in> t" with * \<open>c\<in>t\<close> eq_upd0[rule_format, of x] have "x \<le> c"
             by (auto simp: le_less intro!: t.less[of _ _ "upd n'"]) }
         note top = this
         have "s = t"
           using \<open>a = enum i\<close> \<open>i = n\<close> \<open>c \<in> t\<close>
           by (intro st.ksimplex_eq_top[OF _ _ _ _ eq_sma])
              (auto simp: s_eq enum_mono t.s_eq t.enum_mono top)
         then show ?thesis by simp
       next
         assume *: "c (upd n') < enum n' (upd n')"
         interpret st: kuhn_simplex_pair p n b "upd \<circ> rot" "f' ` {.. n}" b' u t ..
         have eq: "f' ` {..n} - {b.enum 0} = t - {c}"
           using eq_sma eq f' by simp
         { fix x assume "x \<in> t" with * \<open>c\<in>t\<close> eq_upd0[rule_format, of x] have "c \<le> x"
             by (auto simp: le_less intro!: t.less[of _ _ "upd n'"]) }
         note bot = this
         have "f' ` {..n} = t"
           using \<open>a = enum i\<close> \<open>i = n\<close> \<open>c \<in> t\<close>
           by (intro st.ksimplex_eq_bot[OF _ _ _ _ eq])
              (auto simp: b.s_eq b.enum_mono t.s_eq t.enum_mono bot)
         with f' show ?thesis by simp
       qed }
     with ks_f' eq \<open>a \<noteq> b.enum 0\<close> \<open>n \<noteq> 0\<close> show ?thesis
       apply (intro ex1I[of _ "b.enum ` {.. n}"])
       apply fastforce
       apply metis
       done
   next
     assume i: "0 < i" "i < n"
     define i' where "i' = i - 1"
     with i have "Suc i' < n"
       by simp
     with i have Suc_i': "Suc i' = i"
       by (simp add: i'_def)
 
     let ?upd = "Fun.swap i' i upd"
     from i upd have "bij_betw ?upd {..< n} {..< n}"
       by (subst bij_betw_swap_iff) (auto simp: i'_def)
 
     define f' where [abs_def]: "f' i j = (if j \<in> ?upd`{..< i} then Suc (base j) else base j)"
       for i j
     interpret b: kuhn_simplex p n base ?upd "f' ` {.. n}"
     proof
       show "base \<in> {..<n} \<rightarrow> {..<p}" by (rule base)
       { fix i assume "n \<le> i" then show "base i = p" by (rule base_out) }
       show "bij_betw ?upd {..<n} {..<n}" by fact
     qed (simp add: f'_def)
     have f': "b.enum = f'" unfolding f'_def b.enum_def[abs_def] ..
     have ks_f': "ksimplex p n (b.enum ` {.. n})"
       unfolding f' by rule unfold_locales
 
     have "{i} \<subseteq> {..n}"
       using i by auto
     { fix j assume "j \<le> n"
       with i Suc_i' have "enum j = b.enum j \<longleftrightarrow> j \<noteq> i"
         unfolding fun_eq_iff enum_def b.enum_def image_comp [symmetric]
         apply (cases \<open>i = j\<close>)
          apply (metis imageI in_upd_image lessI lessThan_iff lessThan_subset_iff order_less_le transpose_apply_first)
         by (metis lessThan_iff linorder_not_less not_less_eq_eq order_less_le transpose_image_eq)
       }
     note enum_eq_benum = this
     then have "enum ` ({.. n} - {i}) = b.enum ` ({.. n} - {i})"
       by (intro image_cong) auto
     then have eq: "s - {a} = b.enum ` {.. n} - {b.enum i}"
       unfolding s_eq \<open>a = enum i\<close>
       using inj_on_image_set_diff[OF inj_enum Diff_subset \<open>{i} \<subseteq> {..n}\<close>]
             inj_on_image_set_diff[OF b.inj_enum Diff_subset \<open>{i} \<subseteq> {..n}\<close>]
       by (simp add: comp_def)
 
     have "a \<noteq> b.enum i"
       using \<open>a = enum i\<close> enum_eq_benum i by auto
 
     { fix t c assume "ksimplex p n t" "c \<in> t" and eq_sma: "s - {a} = t - {c}"
       obtain b' u where "kuhn_simplex p n b' u t"
         using \<open>ksimplex p n t\<close> by (auto elim: ksimplex.cases)
       then interpret t: kuhn_simplex p n b' u t .
       have "enum i' \<in> s - {a}" "enum (i + 1) \<in> s - {a}"
         using \<open>a = enum i\<close> i enum_in by (auto simp: enum_inj i'_def)
       then obtain l k where
         l: "t.enum l = enum i'" "l \<le> n" "t.enum l \<noteq> c" and
         k: "t.enum k = enum (i + 1)" "k \<le> n" "t.enum k \<noteq> c"
         unfolding eq_sma by (auto simp: t.s_eq)
       with i have "t.enum l < t.enum k"
         by (simp add: enum_strict_mono i'_def)
       with \<open>l \<le> n\<close> \<open>k \<le> n\<close> have "l < k"
         by (simp add: t.enum_strict_mono)
       { assume "Suc l = k"
         have "enum (Suc (Suc i')) = t.enum (Suc l)"
           using i by (simp add: k \<open>Suc l = k\<close> i'_def)
         then have False
           using \<open>l < k\<close> \<open>k \<le> n\<close> \<open>Suc i' < n\<close>
           by (auto simp: t.enum_Suc enum_Suc l upd_inj fun_eq_iff split: if_split_asm)
              (metis Suc_lessD n_not_Suc_n upd_inj) }
       with \<open>l < k\<close> have "Suc l < k"
         by arith
       have c_eq: "c = t.enum (Suc l)"
       proof (rule ccontr)
         assume "c \<noteq> t.enum (Suc l)"
         then have "t.enum (Suc l) \<in> s - {a}"
           using \<open>l < k\<close> \<open>k \<le> n\<close> by (simp add: t.s_eq eq_sma)
         then obtain j where "t.enum (Suc l) = enum j" "j \<le> n" "enum j \<noteq> enum i"
           unfolding s_eq \<open>a = enum i\<close> by auto
         with i have "t.enum (Suc l) \<le> t.enum l \<or> t.enum k \<le> t.enum (Suc l)"
           by (auto simp: i'_def enum_mono enum_inj l k)
         with \<open>Suc l < k\<close> \<open>k \<le> n\<close> show False
           by (simp add: t.enum_mono)
       qed
 
       { have "t.enum (Suc (Suc l)) \<in> s - {a}"
           unfolding eq_sma c_eq t.s_eq using \<open>Suc l < k\<close> \<open>k \<le> n\<close> by (auto simp: t.enum_inj)
         then obtain j where eq: "t.enum (Suc (Suc l)) = enum j" and "j \<le> n" "j \<noteq> i"
           by (auto simp: s_eq \<open>a = enum i\<close>)
         moreover have "enum i' < t.enum (Suc (Suc l))"
           unfolding l(1)[symmetric] using \<open>Suc l < k\<close> \<open>k \<le> n\<close> by (auto simp: t.enum_strict_mono)
         ultimately have "i' < j"
           using i by (simp add: enum_strict_mono i'_def)
         with \<open>j \<noteq> i\<close> \<open>j \<le> n\<close> have "t.enum k \<le> t.enum (Suc (Suc l))"
           unfolding i'_def by (simp add: enum_mono k eq)
         then have "k \<le> Suc (Suc l)"
           using \<open>k \<le> n\<close> \<open>Suc l < k\<close> by (simp add: t.enum_mono) }
       with \<open>Suc l < k\<close> have "Suc (Suc l) = k" by simp
       then have "enum (Suc (Suc i')) = t.enum (Suc (Suc l))"
         using i by (simp add: k i'_def)
       also have "\<dots> = (enum i') (u l := Suc (enum i' (u l)), u (Suc l) := Suc (enum i' (u (Suc l))))"
         using \<open>Suc l < k\<close> \<open>k \<le> n\<close> by (simp add: t.enum_Suc l t.upd_inj)
       finally have "(u l = upd i' \<and> u (Suc l) = upd (Suc i')) \<or>
         (u l = upd (Suc i') \<and> u (Suc l) = upd i')"
         using \<open>Suc i' < n\<close> by (auto simp: enum_Suc fun_eq_iff split: if_split_asm)
 
       then have "t = s \<or> t = b.enum ` {..n}"
       proof (elim disjE conjE)
         assume u: "u l = upd i'"
         have "c = t.enum (Suc l)" unfolding c_eq ..
         also have "t.enum (Suc l) = enum (Suc i')"
           using u \<open>l < k\<close> \<open>k \<le> n\<close> \<open>Suc i' < n\<close> by (simp add: enum_Suc t.enum_Suc l)
         also have "\<dots> = a"
           using \<open>a = enum i\<close> i by (simp add: i'_def)
         finally show ?thesis
           using eq_sma \<open>a \<in> s\<close> \<open>c \<in> t\<close> by auto
       next
         assume u: "u l = upd (Suc i')"
         define B where "B = b.enum ` {..n}"
         have "b.enum i' = enum i'"
           using enum_eq_benum[of i'] i by (auto simp: i'_def gr0_conv_Suc)
         have "c = t.enum (Suc l)" unfolding c_eq ..
         also have "t.enum (Suc l) = b.enum (Suc i')"
           using u \<open>l < k\<close> \<open>k \<le> n\<close> \<open>Suc i' < n\<close>
           by (simp_all add: enum_Suc t.enum_Suc l b.enum_Suc \<open>b.enum i' = enum i'\<close>)
              (simp add: Suc_i')
         also have "\<dots> = b.enum i"
           using i by (simp add: i'_def)
         finally have "c = b.enum i" .
         then have "t - {c} = B - {c}" "c \<in> B"
           unfolding eq_sma[symmetric] eq B_def using i by auto
         with \<open>c \<in> t\<close> have "t = B"
           by auto
         then show ?thesis
           by (simp add: B_def)
       qed }
     with ks_f' eq \<open>a \<noteq> b.enum i\<close> \<open>n \<noteq> 0\<close> \<open>i \<le> n\<close> show ?thesis
       apply (intro ex1I[of _ "b.enum ` {.. n}"])
       apply auto []
       apply metis
       done
   qed
   then show ?thesis
     using s \<open>a \<in> s\<close> by (simp add: card_2_iff' Ex1_def) metis
 qed
 
 text \<open>Hence another step towards concreteness.\<close>
 
 lemma kuhn_simplex_lemma:
   assumes "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> rl ` s \<subseteq> {.. Suc n}"
     and "odd (card {f. \<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> (f = s - {a}) \<and>
       rl ` f = {..n} \<and> ((\<exists>j\<le>n. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>f. x j = p))})"
   shows "odd (card {s. ksimplex p (Suc n) s \<and> rl ` s = {..Suc n}})"
 proof (rule kuhn_complete_lemma[OF finite_ksimplexes refl, unfolded mem_Collect_eq,
     where bnd="\<lambda>f. (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = p)"],
     safe del: notI)
 
   have *: "\<And>x y. x = y \<Longrightarrow> odd (card x) \<Longrightarrow> odd (card y)"
     by auto
   show "odd (card {f. (\<exists>s\<in>{s. ksimplex p (Suc n) s}. \<exists>a\<in>s. f = s - {a}) \<and>
     rl ` f = {..n} \<and> ((\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<in>{..n}. \<forall>x\<in>f. x j = p))})"
     apply (rule *[OF _ assms(2)])
     apply (auto simp: atLeast0AtMost)
     done
 
 next
 
   fix s assume s: "ksimplex p (Suc n) s"
   then show "card s = n + 2"
     by (simp add: ksimplex_card)
 
   fix a assume a: "a \<in> s" then show "rl a \<le> Suc n"
     using assms(1) s by (auto simp: subset_eq)
 
   let ?S = "{t. ksimplex p (Suc n) t \<and> (\<exists>b\<in>t. s - {a} = t - {b})}"
   { fix j assume j: "j \<le> n" "\<forall>x\<in>s - {a}. x j = 0"
     with s a show "card ?S = 1"
       using ksimplex_replace_0[of p "n + 1" s a j]
       by (subst eq_commute) simp }
 
   { fix j assume j: "j \<le> n" "\<forall>x\<in>s - {a}. x j = p"
     with s a show "card ?S = 1"
       using ksimplex_replace_1[of p "n + 1" s a j]
       by (subst eq_commute) simp }
 
   { assume "card ?S \<noteq> 2" "\<not> (\<exists>j\<in>{..n}. \<forall>x\<in>s - {a}. x j = p)"
     with s a show "\<exists>j\<in>{..n}. \<forall>x\<in>s - {a}. x j = 0"
       using ksimplex_replace_2[of p "n + 1" s a]
       by (subst (asm) eq_commute) auto }
 qed
 
 subsubsection \<open>Reduced labelling\<close>
 
 definition reduced :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> nat" where "reduced n x = (LEAST k. k = n \<or> x k \<noteq> 0)"
 
 lemma reduced_labelling:
   shows "reduced n x \<le> n"
     and "\<forall>i<reduced n x. x i = 0"
     and "reduced n x = n \<or> x (reduced n x) \<noteq> 0"
 proof -
   show "reduced n x \<le> n"
     unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) auto
   show "\<forall>i<reduced n x. x i = 0"
     unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) fastforce+
   show "reduced n x = n \<or> x (reduced n x) \<noteq> 0"
     unfolding reduced_def by (rule LeastI2_wellorder[where a=n]) fastforce+
 qed
 
 lemma reduced_labelling_unique:
   "r \<le> n \<Longrightarrow> \<forall>i<r. x i = 0 \<Longrightarrow> r = n \<or> x r \<noteq> 0 \<Longrightarrow> reduced n x = r"
   by (metis linorder_less_linear linorder_not_le reduced_labelling)
 
 lemma reduced_labelling_zero: "j < n \<Longrightarrow> x j = 0 \<Longrightarrow> reduced n x \<noteq> j"
   using reduced_labelling[of n x] by auto
 
 lemma reduce_labelling_zero[simp]: "reduced 0 x = 0"
   by (rule reduced_labelling_unique) auto
 
 lemma reduced_labelling_nonzero: "j < n \<Longrightarrow> x j \<noteq> 0 \<Longrightarrow> reduced n x \<le> j"
   using reduced_labelling[of n x] by (elim allE[where x=j]) auto
 
 lemma reduced_labelling_Suc: "reduced (Suc n) x \<noteq> Suc n \<Longrightarrow> reduced (Suc n) x = reduced n x"
   using reduced_labelling[of "Suc n" x]
   by (intro reduced_labelling_unique[symmetric]) auto
 
 lemma complete_face_top:
   assumes "\<forall>x\<in>f. \<forall>j\<le>n. x j = 0 \<longrightarrow> lab x j = 0"
     and "\<forall>x\<in>f. \<forall>j\<le>n. x j = p \<longrightarrow> lab x j = 1"
     and eq: "(reduced (Suc n) \<circ> lab) ` f = {..n}"
   shows "((\<exists>j\<le>n. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>f. x j = p)) \<longleftrightarrow> (\<forall>x\<in>f. x n = p)"
 proof (safe del: disjCI)
   fix x j assume j: "j \<le> n" "\<forall>x\<in>f. x j = 0"
   { fix x assume "x \<in> f" with assms j have "reduced (Suc n) (lab x) \<noteq> j"
       by (intro reduced_labelling_zero) auto }
   moreover have "j \<in> (reduced (Suc n) \<circ> lab) ` f"
     using j eq by auto
   ultimately show "x n = p"
     by force
 next
   fix x j assume j: "j \<le> n" "\<forall>x\<in>f. x j = p" and x: "x \<in> f"
   have "j = n"
   proof (rule ccontr)
     assume "\<not> ?thesis"
     { fix x assume "x \<in> f"
       with assms j have "reduced (Suc n) (lab x) \<le> j"
         by (intro reduced_labelling_nonzero) auto
       then have "reduced (Suc n) (lab x) \<noteq> n"
         using \<open>j \<noteq> n\<close> \<open>j \<le> n\<close> by simp }
     moreover
     have "n \<in> (reduced (Suc n) \<circ> lab) ` f"
       using eq by auto
     ultimately show False
       by force
   qed
   moreover have "j \<in> (reduced (Suc n) \<circ> lab) ` f"
     using j eq by auto
   ultimately show "x n = p"
     using j x by auto
 qed auto
 
 text \<open>Hence we get just about the nice induction.\<close>
 
 lemma kuhn_induction:
   assumes "0 < p"
     and lab_0: "\<forall>x. \<forall>j\<le>n. (\<forall>j. x j \<le> p) \<and> x j = 0 \<longrightarrow> lab x j = 0"
     and lab_1: "\<forall>x. \<forall>j\<le>n. (\<forall>j. x j \<le> p) \<and> x j = p \<longrightarrow> lab x j = 1"
     and odd: "odd (card {s. ksimplex p n s \<and> (reduced n\<circ>lab) ` s = {..n}})"
   shows "odd (card {s. ksimplex p (Suc n) s \<and> (reduced (Suc n)\<circ>lab) ` s = {..Suc n}})"
 proof -
   let ?rl = "reduced (Suc n) \<circ> lab" and ?ext = "\<lambda>f v. \<exists>j\<le>n. \<forall>x\<in>f. x j = v"
   let ?ext = "\<lambda>s. (\<exists>j\<le>n. \<forall>x\<in>s. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>s. x j = p)"
   have "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> ?rl ` s \<subseteq> {..Suc n}"
     by (simp add: reduced_labelling subset_eq)
   moreover
   have "{s. ksimplex p n s \<and> (reduced n \<circ> lab) ` s = {..n}} =
         {f. \<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> f = s - {a} \<and> ?rl ` f = {..n} \<and> ?ext f}"
   proof (intro set_eqI, safe del: disjCI equalityI disjE)
     fix s assume s: "ksimplex p n s" and rl: "(reduced n \<circ> lab) ` s = {..n}"
     from s obtain u b where "kuhn_simplex p n u b s" by (auto elim: ksimplex.cases)
     then interpret kuhn_simplex p n u b s .
     have all_eq_p: "\<forall>x\<in>s. x n = p"
       by (auto simp: out_eq_p)
     moreover
     { fix x assume "x \<in> s"
       with lab_1[rule_format, of n x] all_eq_p s_le_p[of x] 
       have "?rl x \<le> n"
         by (auto intro!: reduced_labelling_nonzero)
       then have "?rl x = reduced n (lab x)"
         by (auto intro!: reduced_labelling_Suc) }
     then have "?rl ` s = {..n}"
       using rl by (simp cong: image_cong)
     moreover
     obtain t a where "ksimplex p (Suc n) t" "a \<in> t" "s = t - {a}"
       using s unfolding simplex_top_face[OF \<open>0 < p\<close> all_eq_p] by auto
     ultimately
     show "\<exists>t a. ksimplex p (Suc n) t \<and> a \<in> t \<and> s = t - {a} \<and> ?rl ` s = {..n} \<and> ?ext s"
       by auto
   next
     fix x s a assume s: "ksimplex p (Suc n) s" and rl: "?rl ` (s - {a}) = {.. n}"
       and a: "a \<in> s" and "?ext (s - {a})"
     from s obtain u b where "kuhn_simplex p (Suc n) u b s" by (auto elim: ksimplex.cases)
     then interpret kuhn_simplex p "Suc n" u b s .
     have all_eq_p: "\<forall>x\<in>s. x (Suc n) = p"
       by (auto simp: out_eq_p)
 
     { fix x assume "x \<in> s - {a}"
       then have "?rl x \<in> ?rl ` (s - {a})"
         by auto
       then have "?rl x \<le> n"
         unfolding rl by auto
       then have "?rl x = reduced n (lab x)"
         by (auto intro!: reduced_labelling_Suc) }
     then show rl': "(reduced n\<circ>lab) ` (s - {a}) = {..n}"
       unfolding rl[symmetric] by (intro image_cong) auto
 
     from \<open>?ext (s - {a})\<close>
     have all_eq_p: "\<forall>x\<in>s - {a}. x n = p"
     proof (elim disjE exE conjE)
       fix j assume "j \<le> n" "\<forall>x\<in>s - {a}. x j = 0"
       with lab_0[rule_format, of j] all_eq_p s_le_p
       have "\<And>x. x \<in> s - {a} \<Longrightarrow> reduced (Suc n) (lab x) \<noteq> j"
         by (intro reduced_labelling_zero) auto
       moreover have "j \<in> ?rl ` (s - {a})"
         using \<open>j \<le> n\<close> unfolding rl by auto
       ultimately show ?thesis
         by force
     next
       fix j assume "j \<le> n" and eq_p: "\<forall>x\<in>s - {a}. x j = p"
       show ?thesis
       proof cases
         assume "j = n" with eq_p show ?thesis by simp
       next
         assume "j \<noteq> n"
         { fix x assume x: "x \<in> s - {a}"
           have "reduced n (lab x) \<le> j"
           proof (rule reduced_labelling_nonzero)
             show "lab x j \<noteq> 0"
               using lab_1[rule_format, of j x] x s_le_p[of x] eq_p \<open>j \<le> n\<close> by auto
             show "j < n"
               using \<open>j \<le> n\<close> \<open>j \<noteq> n\<close> by simp
           qed
           then have "reduced n (lab x) \<noteq> n"
             using \<open>j \<le> n\<close> \<open>j \<noteq> n\<close> by simp }
         moreover have "n \<in> (reduced n\<circ>lab) ` (s - {a})"
           unfolding rl' by auto
         ultimately show ?thesis
           by force
       qed
     qed
     show "ksimplex p n (s - {a})"
       unfolding simplex_top_face[OF \<open>0 < p\<close> all_eq_p] using s a by auto
   qed
   ultimately show ?thesis
     using assms by (intro kuhn_simplex_lemma) auto
 qed
 
 text \<open>And so we get the final combinatorial result.\<close>
 
 lemma ksimplex_0: "ksimplex p 0 s \<longleftrightarrow> s = {(\<lambda>x. p)}"
 proof
   assume "ksimplex p 0 s" then show "s = {(\<lambda>x. p)}"
     by (blast dest: kuhn_simplex.ksimplex_0 elim: ksimplex.cases)
 next
   assume s: "s = {(\<lambda>x. p)}"
   show "ksimplex p 0 s"
   proof (intro ksimplex, unfold_locales)
     show "(\<lambda>_. p) \<in> {..<0::nat} \<rightarrow> {..<p}" by auto
     show "bij_betw id {..<0} {..<0}"
       by simp
   qed (auto simp: s)
 qed
 
 lemma kuhn_combinatorial:
   assumes "0 < p"
     and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n \<and> x j = 0 \<longrightarrow> lab x j = 0"
     and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n  \<and> x j = p \<longrightarrow> lab x j = 1"
   shows "odd (card {s. ksimplex p n s \<and> (reduced n\<circ>lab) ` s = {..n}})"
     (is "odd (card (?M n))")
   using assms
 proof (induct n)
   case 0 then show ?case
     by (simp add: ksimplex_0 cong: conj_cong)
 next
   case (Suc n)
   then have "odd (card (?M n))"
     by force
   with Suc show ?case
     using kuhn_induction[of p n] by (auto simp: comp_def)
 qed
 
 lemma kuhn_lemma:
   fixes n p :: nat
   assumes "0 < p"
     and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. label x i = (0::nat) \<or> label x i = 1)"
     and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow> label x i = 0)"
     and "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow> label x i = 1)"
   obtains q where "\<forall>i<n. q i < p"
     and "\<forall>i<n. \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and> (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and> label r i \<noteq> label s i"
 proof -
   let ?rl = "reduced n \<circ> label"
   let ?A = "{s. ksimplex p n s \<and> ?rl ` s = {..n}}"
   have "odd (card ?A)"
     using assms by (intro kuhn_combinatorial[of p n label]) auto
   then have "?A \<noteq> {}"
     by (rule odd_card_imp_not_empty)
   then obtain s b u where "kuhn_simplex p n b u s" and rl: "?rl ` s = {..n}"
     by (auto elim: ksimplex.cases)
   interpret kuhn_simplex p n b u s by fact
 
   show ?thesis
   proof (intro that[of b] allI impI)
     fix i
     assume "i < n"
     then show "b i < p"
       using base by auto
   next
     fix i
     assume "i < n"
     then have "i \<in> {.. n}" "Suc i \<in> {.. n}"
       by auto
     then obtain u v where u: "u \<in> s" "Suc i = ?rl u" and v: "v \<in> s" "i = ?rl v"
       unfolding rl[symmetric] by blast
 
     have "label u i \<noteq> label v i"
       using reduced_labelling [of n "label u"] reduced_labelling [of n "label v"]
         u(2)[symmetric] v(2)[symmetric] \<open>i < n\<close>
       by auto
     moreover
     have "b j \<le> u j" "u j \<le> b j + 1" "b j \<le> v j" "v j \<le> b j + 1" if "j < n" for j
       using that base_le[OF \<open>u\<in>s\<close>] le_Suc_base[OF \<open>u\<in>s\<close>] base_le[OF \<open>v\<in>s\<close>] le_Suc_base[OF \<open>v\<in>s\<close>]
       by auto
     ultimately show "\<exists>r s. (\<forall>j<n. b j \<le> r j \<and> r j \<le> b j + 1) \<and>
         (\<forall>j<n. b j \<le> s j \<and> s j \<le> b j + 1) \<and> label r i \<noteq> label s i"
       by blast
   qed
 qed
 
 subsubsection \<open>Main result for the unit cube\<close>
 
 lemma kuhn_labelling_lemma':
   assumes "(\<forall>x::nat\<Rightarrow>real. P x \<longrightarrow> P (f x))"
     and "\<forall>x. P x \<longrightarrow> (\<forall>i::nat. Q i \<longrightarrow> 0 \<le> x i \<and> x i \<le> 1)"
   shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
              (\<forall>x i. P x \<and> Q i \<and> x i = 0 \<longrightarrow> l x i = 0) \<and>
              (\<forall>x i. P x \<and> Q i \<and> x i = 1 \<longrightarrow> l x i = 1) \<and>
              (\<forall>x i. P x \<and> Q i \<and> l x i = 0 \<longrightarrow> x i \<le> f x i) \<and>
              (\<forall>x i. P x \<and> Q i \<and> l x i = 1 \<longrightarrow> f x i \<le> x i)"
   unfolding all_conj_distrib [symmetric] 
   apply (subst choice_iff[symmetric])+
-  by (metis assms bot_nat_0.extremum nle_le zero_neq_one)
+  by (metis assms choice_iff bot_nat_0.extremum nle_le zero_neq_one)
 
 subsection \<open>Brouwer's fixed point theorem\<close>
 
 text \<open>We start proving Brouwer's fixed point theorem for the unit cube = \<open>cbox 0 One\<close>.\<close>
 
 lemma brouwer_cube:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "continuous_on (cbox 0 One) f"
     and "f ` cbox 0 One \<subseteq> cbox 0 One"
   shows "\<exists>x\<in>cbox 0 One. f x = x"
 proof (rule ccontr)
   define n where "n = DIM('a)"
   have n: "1 \<le> n" "0 < n" "n \<noteq> 0"
     unfolding n_def by (auto simp: Suc_le_eq)
   assume "\<not> ?thesis"
   then have *: "\<not> (\<exists>x\<in>cbox 0 One. f x - x = 0)"
     by auto
   obtain d where
       d: "d > 0" "\<And>x. x \<in> cbox 0 One \<Longrightarrow> d \<le> norm (f x - x)"
-    apply (rule brouwer_compactness_lemma[OF compact_cbox _ *])
-    apply (rule continuous_intros assms)+
-    apply blast
-    done
+    using brouwer_compactness_lemma[OF compact_cbox _ *] assms
+    by (metis (no_types, lifting) continuous_on_cong continuous_on_diff continuous_on_id)
   have *: "\<forall>x. x \<in> cbox 0 One \<longrightarrow> f x \<in> cbox 0 One"
     "\<forall>x. x \<in> (cbox 0 One::'a set) \<longrightarrow> (\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
     using assms(2)[unfolded image_subset_iff Ball_def]
     unfolding cbox_def
     by auto
   obtain label :: "'a \<Rightarrow> 'a \<Rightarrow> nat" where label [rule_format]:
     "\<forall>x. \<forall>i\<in>Basis. label x i \<le> 1"
     "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> x \<bullet> i = 0 \<longrightarrow> label x i = 0"
     "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> x \<bullet> i = 1 \<longrightarrow> label x i = 1"
     "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> label x i = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i"
     "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> label x i = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i"
     using kuhn_labelling_lemma[OF *] by auto
   note label = this [rule_format]
   have lem1: "\<forall>x\<in>cbox 0 One. \<forall>y\<in>cbox 0 One. \<forall>i\<in>Basis. label x i \<noteq> label y i \<longrightarrow>
     \<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
   proof safe
     fix x y :: 'a
     assume x: "x \<in> cbox 0 One" and y: "y \<in> cbox 0 One"
     fix i
     assume i: "label x i \<noteq> label y i" "i \<in> Basis"
     have *: "\<And>x y fx fy :: real. x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy \<Longrightarrow>
       \<bar>fx - x\<bar> \<le> \<bar>fy - fx\<bar> + \<bar>y - x\<bar>" by auto
     have "\<bar>(f x - x) \<bullet> i\<bar> \<le> \<bar>(f y - f x)\<bullet>i\<bar> + \<bar>(y - x)\<bullet>i\<bar>"
     proof (cases "label x i = 0")
       case True
       then have fxy: "\<not> f y \<bullet> i \<le> y \<bullet> i \<Longrightarrow> f x \<bullet> i \<le> x \<bullet> i"
         by (metis True i label(1) label(5) le_antisym less_one not_le_imp_less y)
       show ?thesis
       unfolding inner_simps         
       by (rule *) (auto simp: True i label x y fxy)
     next
       case False
       then show ?thesis
         using label [OF \<open>i \<in> Basis\<close>] i(1) x y
-        apply (auto simp: inner_diff_left le_Suc_eq)
-        by (metis "*")
+        by (smt (verit, ccfv_threshold) inner_diff_left less_one order_le_less)
     qed
     also have "\<dots> \<le> norm (f y - f x) + norm (y - x)"
       by (simp add: add_mono i(2) norm_bound_Basis_le)
     finally show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
       unfolding inner_simps .
   qed
   have "\<exists>e>0. \<forall>x\<in>cbox 0 One. \<forall>y\<in>cbox 0 One. \<forall>z\<in>cbox 0 One. \<forall>i\<in>Basis.
     norm (x - z) < e \<longrightarrow> norm (y - z) < e \<longrightarrow> label x i \<noteq> label y i \<longrightarrow>
       \<bar>(f(z) - z)\<bullet>i\<bar> < d / (real n)"
   proof -
     have d': "d / real n / 8 > 0"
       using d(1) by (simp add: n_def)
     have *: "uniformly_continuous_on (cbox 0 One) f"
       by (rule compact_uniformly_continuous[OF assms(1) compact_cbox])
     obtain e where e:
         "e > 0"
         "\<And>x x'. x \<in> cbox 0 One \<Longrightarrow>
           x' \<in> cbox 0 One \<Longrightarrow>
           norm (x' - x) < e \<Longrightarrow>
           norm (f x' - f x) < d / real n / 8"
       using *[unfolded uniformly_continuous_on_def,rule_format,OF d']
       unfolding dist_norm
       by blast
     show ?thesis
     proof (intro exI conjI ballI impI)
       show "0 < min (e / 2) (d / real n / 8)"
         using d' e by auto
       fix x y z i
       assume as:
         "x \<in> cbox 0 One" "y \<in> cbox 0 One" "z \<in> cbox 0 One"
         "norm (x - z) < min (e / 2) (d / real n / 8)"
         "norm (y - z) < min (e / 2) (d / real n / 8)"
         "label x i \<noteq> label y i"
       assume i: "i \<in> Basis"
       have *: "\<And>z fz x fx n1 n2 n3 n4 d4 d :: real. \<bar>fx - x\<bar> \<le> n1 + n2 \<Longrightarrow>
         \<bar>fx - fz\<bar> \<le> n3 \<Longrightarrow> \<bar>x - z\<bar> \<le> n4 \<Longrightarrow>
         n1 < d4 \<Longrightarrow> n2 < 2 * d4 \<Longrightarrow> n3 < d4 \<Longrightarrow> n4 < d4 \<Longrightarrow>
         (8 * d4 = d) \<Longrightarrow> \<bar>fz - z\<bar> < d"
         by auto
       show "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
         unfolding inner_simps
       proof (rule *)
         show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y -f x) + norm (y - x)"
           using as(1) as(2) as(6) i lem1 by blast
         show "norm (f x - f z) < d / real n / 8"
           using d' e as by auto
         show "\<bar>f x \<bullet> i - f z \<bullet> i\<bar> \<le> norm (f x - f z)" "\<bar>x \<bullet> i - z \<bullet> i\<bar> \<le> norm (x - z)"
           unfolding inner_diff_left[symmetric]
           by (rule Basis_le_norm[OF i])+
         have tria: "norm (y - x) \<le> norm (y - z) + norm (x - z)"
           using dist_triangle[of y x z, unfolded dist_norm]
           unfolding norm_minus_commute
           by auto
         also have "\<dots> < e / 2 + e / 2"
           using as(4) as(5) by auto
         finally show "norm (f y - f x) < d / real n / 8"
           using as(1) as(2) e(2) by auto
         have "norm (y - z) + norm (x - z) < d / real n / 8 + d / real n / 8"
           using as(4) as(5) by auto
         with tria show "norm (y - x) < 2 * (d / real n / 8)"
           by auto
       qed (use as in auto)
     qed
   qed
   then
   obtain e where e:
     "e > 0"
     "\<And>x y z i. x \<in> cbox 0 One \<Longrightarrow>
       y \<in> cbox 0 One \<Longrightarrow>
       z \<in> cbox 0 One \<Longrightarrow>
       i \<in> Basis \<Longrightarrow>
       norm (x - z) < e \<and> norm (y - z) < e \<and> label x i \<noteq> label y i \<Longrightarrow>
       \<bar>(f z - z) \<bullet> i\<bar> < d / real n"
     by blast
   obtain p :: nat where p: "1 + real n / e \<le> real p"
     using real_arch_simple ..
   have "1 + real n / e > 0"
     using e(1) n by (simp add: add_pos_pos)
   then have "p > 0"
     using p by auto
 
   obtain b :: "nat \<Rightarrow> 'a" where b: "bij_betw b {..< n} Basis"
     by atomize_elim (auto simp: n_def intro!: finite_same_card_bij)
   define b' where "b' = inv_into {..< n} b"
   then have b': "bij_betw b' Basis {..< n}"
     using bij_betw_inv_into[OF b] by auto
   then have b'_Basis: "\<And>i. i \<in> Basis \<Longrightarrow> b' i \<in> {..< n}"
     unfolding bij_betw_def by (auto simp: set_eq_iff)
   have bb'[simp]:"\<And>i. i \<in> Basis \<Longrightarrow> b (b' i) = i"
     unfolding b'_def
     using b
     by (auto simp: f_inv_into_f bij_betw_def)
   have b'b[simp]:"\<And>i. i < n \<Longrightarrow> b' (b i) = i"
     unfolding b'_def
     using b
     by (auto simp: inv_into_f_eq bij_betw_def)
   have *: "\<And>x :: nat. x = 0 \<or> x = 1 \<longleftrightarrow> x \<le> 1"
     by auto
   have b'': "\<And>j. j < n \<Longrightarrow> b j \<in> Basis"
     using b unfolding bij_betw_def by auto
   have q1: "0 < p" "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow>
     (\<forall>i<n. (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0 \<or>
            (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
     unfolding *
     using \<open>p > 0\<close> \<open>n > 0\<close>
     using label(1)[OF b'']
     by auto
   { fix x :: "nat \<Rightarrow> nat" and i assume "\<forall>i<n. x i \<le> p" "i < n" "x i = p \<or> x i = 0"
     then have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> (cbox 0 One::'a set)"
       using b'_Basis
       by (auto simp: cbox_def bij_betw_def zero_le_divide_iff divide_le_eq_1) }
   note cube = this
   have q2: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow>
       (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
     unfolding o_def using cube \<open>p > 0\<close> by (intro allI impI label(2)) (auto simp: b'')
   have q3: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = p \<longrightarrow>
       (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 1)"
     using cube \<open>p > 0\<close> unfolding o_def by (intro allI impI label(3)) (auto simp: b'')
   obtain q where q:
       "\<forall>i<n. q i < p"
       "\<forall>i<n.
          \<exists>r s. (\<forall>j<n. q j \<le> r j \<and> r j \<le> q j + 1) \<and>
                (\<forall>j<n. q j \<le> s j \<and> s j \<le> q j + 1) \<and>
                (label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) i \<noteq>
                (label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) i"
     by (rule kuhn_lemma[OF q1 q2 q3])
   define z :: 'a where "z = (\<Sum>i\<in>Basis. (real (q (b' i)) / real p) *\<^sub>R i)"
   have "\<exists>i\<in>Basis. d / real n \<le> \<bar>(f z - z)\<bullet>i\<bar>"
   proof (rule ccontr)
     have "\<forall>i\<in>Basis. q (b' i) \<in> {0..p}"
       using q(1) b'
       by (auto intro: less_imp_le simp: bij_betw_def)
     then have "z \<in> cbox 0 One"
       unfolding z_def cbox_def
       using b'_Basis
       by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
     then have d_fz_z: "d \<le> norm (f z - z)"
       by (rule d)
     assume "\<not> ?thesis"
     then have as: "\<forall>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar> < d / real n"
       using \<open>n > 0\<close>
       by (auto simp: not_le inner_diff)
     have "norm (f z - z) \<le> (\<Sum>i\<in>Basis. \<bar>f z \<bullet> i - z \<bullet> i\<bar>)"
       unfolding inner_diff_left[symmetric]
       by (rule norm_le_l1)
     also have "\<dots> < (\<Sum>(i::'a) \<in> Basis. d / real n)"
       by (meson as finite_Basis nonempty_Basis sum_strict_mono)
     also have "\<dots> = d"
       using DIM_positive[where 'a='a] by (auto simp: n_def)
     finally show False
       using d_fz_z by auto
   qed
   then obtain i where i: "i \<in> Basis" "d / real n \<le> \<bar>(f z - z) \<bullet> i\<bar>" ..
   have *: "b' i < n"
     using i and b'[unfolded bij_betw_def]
     by auto
   obtain r s where rs:
     "\<And>j. j < n \<Longrightarrow> q j \<le> r j \<and> r j \<le> q j + 1"
     "\<And>j. j < n \<Longrightarrow> q j \<le> s j \<and> s j \<le> q j + 1"
     "(label (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i) \<noteq>
       (label (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i) \<circ> b) (b' i)"
     using q(2)[rule_format,OF *] by blast
   have b'_im: "\<And>i. i \<in> Basis \<Longrightarrow>  b' i < n"
     using b' unfolding bij_betw_def by auto
   define r' ::'a where "r' = (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i)"
   have "\<And>i. i \<in> Basis \<Longrightarrow> r (b' i) \<le> p"
     using b'_im q(1) rs(1) by fastforce
   then have "r' \<in> cbox 0 One"
     unfolding r'_def cbox_def
     using b'_Basis
     by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
   define s' :: 'a where "s' = (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)"
   have "\<And>i. i \<in> Basis \<Longrightarrow> s (b' i) \<le> p"
     using b'_im q(1) rs(2) by fastforce
   then have "s' \<in> cbox 0 One"
     unfolding s'_def cbox_def
     using b'_Basis by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
   have "z \<in> cbox 0 One"
     unfolding z_def cbox_def
     using b'_Basis q(1)[rule_format,OF b'_im] \<open>p > 0\<close>
     by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1 less_imp_le)
   {
     have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       by (rule sum_mono) (use rs(1)[OF b'_im] in force)
     also have "\<dots> < e * real p"
       using p \<open>e > 0\<close> \<open>p > 0\<close>
       by (auto simp: field_simps n_def)
     finally have "(\<Sum>i\<in>Basis. \<bar>real (r (b' i)) - real (q (b' i))\<bar>) < e * real p" .
   }
   moreover
   {
     have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) \<le> (\<Sum>(i::'a)\<in>Basis. 1)"
       by (rule sum_mono) (use rs(2)[OF b'_im] in force)
     also have "\<dots> < e * real p"
       using p \<open>e > 0\<close> \<open>p > 0\<close>
       by (auto simp: field_simps n_def)
     finally have "(\<Sum>i\<in>Basis. \<bar>real (s (b' i)) - real (q (b' i))\<bar>) < e * real p" .
   }
   ultimately
   have "norm (r' - z) < e" and "norm (s' - z) < e"
     unfolding r'_def s'_def z_def
     using \<open>p > 0\<close>
     apply (rule_tac[!] le_less_trans[OF norm_le_l1])
     apply (auto simp: field_simps sum_divide_distrib[symmetric] inner_diff_left)
     done
   then have "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
     using rs(3) i
     unfolding r'_def[symmetric] s'_def[symmetric] o_def bb'
     by (intro e(2)[OF \<open>r'\<in>cbox 0 One\<close> \<open>s'\<in>cbox 0 One\<close> \<open>z\<in>cbox 0 One\<close>]) auto
   then show False
     using i by auto
 qed
 
 text \<open>Next step is to prove it for nonempty interiors.\<close>
 
 lemma brouwer_weak:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "compact S"
     and "convex S"
     and "interior S \<noteq> {}"
     and "continuous_on S f"
     and "f \<in> S \<rightarrow> S"
   obtains x where "x \<in> S" and "f x = x"
 proof -
   let ?U = "cbox 0 One :: 'a set"
   have "\<Sum>Basis /\<^sub>R 2 \<in> interior ?U"
   proof (rule interiorI)
     let ?I = "(\<Inter>i\<in>Basis. {x::'a. 0 < x \<bullet> i} \<inter> {x. x \<bullet> i < 1})"
     show "open ?I"
       by (intro open_INT finite_Basis ballI open_Int, auto intro: open_Collect_less simp: continuous_on_inner)
     show "\<Sum>Basis /\<^sub>R 2 \<in> ?I"
       by simp
     show "?I \<subseteq> cbox 0 One"
       unfolding cbox_def by force
   qed
   then have *: "interior ?U \<noteq> {}" by fast
   have *: "?U homeomorphic S"
     using homeomorphic_convex_compact[OF convex_box(1) compact_cbox * assms(2,1,3)] .
   have "\<forall>f. continuous_on ?U f \<and> f \<in> ?U \<rightarrow> ?U \<longrightarrow> (\<exists>x\<in>?U. f x = x)"
     using brouwer_cube by auto
   then show ?thesis
     unfolding homeomorphic_fixpoint_property[OF *]
     using assms
     by (auto intro: that)
 qed
 
 text \<open>Then the particular case for closed balls.\<close>
 
 lemma brouwer_ball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "e > 0"
     and "continuous_on (cball a e) f"
     and "f \<in> cball a e \<rightarrow> cball a e"
   obtains x where "x \<in> cball a e" and "f x = x"
   using brouwer_weak[OF compact_cball convex_cball, of a e f]
   unfolding interior_cball ball_eq_empty
   using assms by auto
 
 text \<open>And finally we prove Brouwer's fixed point theorem in its general version.\<close>
 
 theorem brouwer:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes S: "compact S" "convex S" "S \<noteq> {}"
     and contf: "continuous_on S f"
     and fim: "f \<in> S \<rightarrow> S"
   obtains x where "x \<in> S" and "f x = x"
 proof -
   have "\<exists>e>0. S \<subseteq> cball 0 e"
     using compact_imp_bounded[OF \<open>compact S\<close>]  unfolding bounded_pos
     by auto
   then obtain e where e: "e > 0" "S \<subseteq> cball 0 e"
     by blast
   have "\<exists>x\<in> cball 0 e. (f \<circ> closest_point S) x = x"
   proof (rule_tac brouwer_ball[OF e(1)])
     show "continuous_on (cball 0 e) (f \<circ> closest_point S)"
       by (meson assms closest_point_in_set compact_eq_bounded_closed contf continuous_on_closest_point
           continuous_on_compose continuous_on_subset image_subsetI)
     show "f \<circ> closest_point S \<in> cball 0 e \<rightarrow> cball 0 e"
       by (smt (verit) Pi_iff assms(1) assms(3) closest_point_in_set comp_apply compact_eq_bounded_closed e(2) fim subset_eq)
   qed (use assms in auto)
   then obtain x where x: "x \<in> cball 0 e" "(f \<circ> closest_point S) x = x" ..
   with S have "x \<in> S"
     by (metis PiE closest_point_in_set comp_apply compact_imp_closed fim)
   then have *: "closest_point S x = x"
     by (rule closest_point_self)
   show thesis
   proof
     show "closest_point S x \<in> S"
       by (simp add: "*" \<open>x \<in> S\<close>)
     show "f (closest_point S x) = closest_point S x"
       using "*" x by auto
   qed
 qed
 
 subsection \<open>Applications\<close>
 
 text \<open>So we get the no-retraction theorem.\<close>
 
 corollary no_retraction_cball:
   fixes a :: "'a::euclidean_space"
   assumes "e > 0"
   shows "\<not> (frontier (cball a e) retract_of (cball a e))"
 proof
   assume *: "frontier (cball a e) retract_of (cball a e)"
   have **: "\<And>xa. a - (2 *\<^sub>R a - xa) = - (a - xa)"
     using scaleR_left_distrib[of 1 1 a] by auto
   obtain x where x: "x \<in> {x. norm (a - x) = e}" "2 *\<^sub>R a - x = x"
   proof (rule retract_fixpoint_property[OF *, of "\<lambda>x. scaleR 2 a - x"])
     show "continuous_on (frontier (cball a e)) ((-) (2 *\<^sub>R a))"
       by (intro continuous_intros)
     show "(-) (2 *\<^sub>R a) \<in> frontier (cball a e) \<rightarrow> frontier (cball a e)"
       by clarsimp (metis "**" dist_norm norm_minus_cancel)
   qed (auto simp: dist_norm intro: brouwer_ball[OF assms])
   then have "scaleR 2 a = scaleR 1 x + scaleR 1 x"
     by (auto simp: algebra_simps)
   then have "a = x"
     unfolding scaleR_left_distrib[symmetric] by auto
   then show False
     using x assms by auto
 qed
 
 corollary contractible_sphere:
   fixes a :: "'a::euclidean_space"
   shows "contractible(sphere a r) \<longleftrightarrow> r \<le> 0"
 proof (cases "0 < r")
   case True
   then show ?thesis
     unfolding contractible_def nullhomotopic_from_sphere_extension
     using no_retraction_cball [OF True, of a]
     by (auto simp: retract_of_def retraction_def)
 next
   case False
   then show ?thesis
     unfolding contractible_def nullhomotopic_from_sphere_extension
     using less_eq_real_def by auto
 qed
 
 corollary connected_sphere_eq:
   fixes a :: "'a :: euclidean_space"
   shows "connected(sphere a r) \<longleftrightarrow> 2 \<le> DIM('a) \<or> r \<le> 0"
     (is "?lhs = ?rhs")
 proof (cases r "0::real" rule: linorder_cases)
   case less
   then show ?thesis by auto
 next
   case equal
   then show ?thesis by auto
 next
   case greater
   show ?thesis
   proof
     assume L: ?lhs
     have "False" if 1: "DIM('a) = 1"
     proof -
       obtain x y where xy: "sphere a r = {x,y}" "x \<noteq> y"
         using sphere_1D_doubleton [OF 1 greater]
         by (metis dist_self greater insertI1 less_add_same_cancel1 mem_sphere mult_2 not_le zero_le_dist)
       then have "finite (sphere a r)"
         by auto
       with L \<open>r > 0\<close> xy show "False"
         using connected_finite_iff_sing by auto
     qed
     with greater show ?rhs
       by (metis DIM_ge_Suc0 One_nat_def Suc_1 le_antisym not_less_eq_eq)
   next
     assume ?rhs
     then show ?lhs
       using connected_sphere greater by auto
   qed
 qed
 
 corollary path_connected_sphere_eq:
   fixes a :: "'a :: euclidean_space"
   shows "path_connected(sphere a r) \<longleftrightarrow> 2 \<le> DIM('a) \<or> r \<le> 0"
          (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     using connected_sphere_eq path_connected_imp_connected by blast
 next
   assume R: ?rhs
   then show ?lhs
     by (auto simp: contractible_imp_path_connected contractible_sphere path_connected_sphere)
 qed
 
 proposition frontier_subset_retraction:
   fixes S :: "'a::euclidean_space set"
   assumes "bounded S" and fros: "frontier S \<subseteq> T"
       and contf: "continuous_on (closure S) f"
       and fim: "f \<in> S \<rightarrow> T"
       and fid: "\<And>x. x \<in> T \<Longrightarrow> f x = x"
     shows "S \<subseteq> T"
 proof (rule ccontr)
   assume "\<not> S \<subseteq> T"
   then obtain a where "a \<in> S" "a \<notin> T" by blast
   define g where "g \<equiv> \<lambda>z. if z \<in> closure S then f z else z"
   have "continuous_on (closure S \<union> closure(-S)) g"
-    unfolding g_def
-    apply (rule continuous_on_cases)
-    using fros fid frontier_closures by (auto simp: contf)
+    unfolding g_def using fros fid frontier_closures
+    by (intro continuous_on_cases) (auto simp: contf)
   moreover have "closure S \<union> closure(- S) = UNIV"
     using closure_Un by fastforce
   ultimately have contg: "continuous_on UNIV g" by metis
   obtain B where "0 < B" and B: "closure S \<subseteq> ball a B"
     using \<open>bounded S\<close> bounded_subset_ballD by blast
   have notga: "g x \<noteq> a" for x
     unfolding g_def using fros fim \<open>a \<notin> T\<close>
     by (metis PiE Un_iff \<open>a \<in> S\<close> closure_Un_frontier fid subsetD)
   define h where "h \<equiv> (\<lambda>y. a + (B / norm(y - a)) *\<^sub>R (y - a)) \<circ> g"
   have "\<not> (frontier (cball a B) retract_of (cball a B))"
     by (metis no_retraction_cball \<open>0 < B\<close>)
   then have "\<And>k. \<not> retraction (cball a B) (frontier (cball a B)) k"
     by (simp add: retract_of_def)
   moreover have "retraction (cball a B) (frontier (cball a B)) h"
     unfolding retraction_def
   proof (intro conjI ballI)
     show "frontier (cball a B) \<subseteq> cball a B"
       by force
     show "continuous_on (cball a B) h"
       unfolding h_def
       by (intro continuous_intros) (use contg continuous_on_subset notga in auto)
     show "h \<in> cball a B \<rightarrow> frontier (cball a B)"
       using \<open>0 < B\<close> by (auto simp: h_def notga dist_norm)
     show "\<And>x. x \<in> frontier (cball a B) \<Longrightarrow> h x = x"
       using notga \<open>0 < B\<close>
       apply (simp add: g_def h_def field_simps)
       by (metis B dist_commute dist_norm mem_ball order_less_irrefl subset_eq)
   qed
   ultimately show False by simp
 qed
 
 subsubsection \<open>Punctured affine hulls, etc\<close>
 
 lemma rel_frontier_deformation_retract_of_punctured_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "convex T" "bounded S"
       and arelS: "a \<in> rel_interior S"
       and relS: "rel_frontier S \<subseteq> T"
       and affS: "T \<subseteq> affine hull S"
   obtains r where "homotopic_with_canon (\<lambda>x. True) (T - {a}) (T - {a}) id r"
                   "retraction (T - {a}) (rel_frontier S) r"
 proof -
   have "\<exists>d. 0 < d \<and> (a + d *\<^sub>R l) \<in> rel_frontier S \<and>
             (\<forall>e. 0 \<le> e \<and> e < d \<longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S)"
-       if "(a + l) \<in> affine hull S" "l \<noteq> 0" for l
-    apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> arelS])
-    apply (rule that)+
-    by metis
+    if "(a + l) \<in> affine hull S" "l \<noteq> 0" for l
+    using ray_to_rel_frontier [OF \<open>bounded S\<close> arelS] that by metis
   then obtain dd
     where dd1: "\<And>l. \<lbrakk>(a + l) \<in> affine hull S; l \<noteq> 0\<rbrakk> \<Longrightarrow> 0 < dd l \<and> (a + dd l *\<^sub>R l) \<in> rel_frontier S"
       and dd2: "\<And>l e. \<lbrakk>(a + l) \<in> affine hull S; e < dd l; 0 \<le> e; l \<noteq> 0\<rbrakk>
                       \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
     by metis+
   have aaffS: "a \<in> affine hull S"
     by (meson arelS subsetD hull_inc rel_interior_subset)
   have "((\<lambda>z. z - a) ` (affine hull S - {a})) = ((\<lambda>z. z - a) ` (affine hull S)) - {0}"
     by auto
   moreover have "continuous_on (((\<lambda>z. z - a) ` (affine hull S)) - {0}) (\<lambda>x. dd x *\<^sub>R x)"
   proof (rule continuous_on_compact_surface_projection)
     show "compact (rel_frontier ((\<lambda>z. z - a) ` S))"
       by (simp add: \<open>bounded S\<close> bounded_translation_minus compact_rel_frontier_bounded)
     have releq: "rel_frontier ((\<lambda>z. z - a) ` S) = (\<lambda>z. z - a) ` rel_frontier S"
       using rel_frontier_translation [of "-a"] add.commute by simp
     also have "\<dots> \<subseteq> (\<lambda>z. z - a) ` (affine hull S) - {0}"
       using rel_frontier_affine_hull arelS rel_frontier_def by fastforce
     finally show "rel_frontier ((\<lambda>z. z - a) ` S) \<subseteq> (\<lambda>z. z - a) ` (affine hull S) - {0}" .
     show "cone ((\<lambda>z. z - a) ` (affine hull S))"
       by (rule subspace_imp_cone)
          (use aaffS in \<open>simp add: subspace_affine image_comp o_def affine_translation_aux [of a]\<close>)
     show "(0 < k \<and> k *\<^sub>R x \<in> rel_frontier ((\<lambda>z. z - a) ` S)) \<longleftrightarrow> (dd x = k)"
          if x: "x \<in> (\<lambda>z. z - a) ` (affine hull S) - {0}" for k x
     proof
       show "dd x = k \<Longrightarrow> 0 < k \<and> k *\<^sub>R x \<in> rel_frontier ((\<lambda>z. z - a) ` S)"
       using dd1 [of x] that image_iff by (fastforce simp add: releq)
     next
       assume k: "0 < k \<and> k *\<^sub>R x \<in> rel_frontier ((\<lambda>z. z - a) ` S)"
       have False if "dd x < k"
       proof -
         have "k \<noteq> 0" "a + k *\<^sub>R x \<in> closure S"
           using k closure_translation [of "-a"]
           by (auto simp: rel_frontier_def cong: image_cong_simp)
         then have segsub: "open_segment a (a + k *\<^sub>R x) \<subseteq> rel_interior S"
           by (metis rel_interior_closure_convex_segment [OF \<open>convex S\<close> arelS])
         have "x \<noteq> 0" and xaffS: "a + x \<in> affine hull S"
           using x by auto
         then have "0 < dd x" and inS: "a + dd x *\<^sub>R x \<in> rel_frontier S"
           using dd1 by auto
         moreover have "a + dd x *\<^sub>R x \<in> open_segment a (a + k *\<^sub>R x)"
-          using k \<open>x \<noteq> 0\<close> \<open>0 < dd x\<close>
-          apply (simp add: in_segment)
-          apply (rule_tac x = "dd x / k" in exI)
-          apply (simp add: that vector_add_divide_simps algebra_simps)
-          done
+          unfolding in_segment
+        proof (intro conjI exI)
+          show "a + dd x *\<^sub>R x = (1 - dd x / k) *\<^sub>R a + (dd x / k) *\<^sub>R (a + k *\<^sub>R x)"
+            using k by (simp add: that algebra_simps)
+        qed (use \<open>x \<noteq> 0\<close> \<open>0 < dd x\<close> that in auto)
         ultimately show ?thesis
           using segsub by (auto simp: rel_frontier_def)
       qed
       moreover have False if "k < dd x"
         using x k that rel_frontier_def
         by (fastforce simp: algebra_simps releq dest!: dd2)
       ultimately show "dd x = k"
         by fastforce
     qed
   qed
   ultimately have *: "continuous_on ((\<lambda>z. z - a) ` (affine hull S - {a})) (\<lambda>x. dd x *\<^sub>R x)"
     by auto
   have "continuous_on (affine hull S - {a}) ((\<lambda>x. a + dd x *\<^sub>R x) \<circ> (\<lambda>z. z - a))"
     by (intro * continuous_intros continuous_on_compose)
   with affS have contdd: "continuous_on (T - {a}) ((\<lambda>x. a + dd x *\<^sub>R x) \<circ> (\<lambda>z. z - a))"
     by (blast intro: continuous_on_subset)
   show ?thesis
   proof
-    show "homotopic_with_canon (\<lambda>x. True) (T - {a}) (T - {a}) id (\<lambda>x. a + dd (x - a) *\<^sub>R (x - a))"
+    show "homotopic_with_canon (\<lambda>x. True) (T - {a}) (T - {a}) id (\<lambda>x. a + dd (x-a) *\<^sub>R (x-a))"
     proof (rule homotopic_with_linear)
       show "continuous_on (T - {a}) id"
         by (intro continuous_intros continuous_on_compose)
-      show "continuous_on (T - {a}) (\<lambda>x. a + dd (x - a) *\<^sub>R (x - a))"
+      show "continuous_on (T - {a}) (\<lambda>x. a + dd (x-a) *\<^sub>R (x-a))"
         using contdd by (simp add: o_def)
-      show "closed_segment (id x) (a + dd (x - a) *\<^sub>R (x - a)) \<subseteq> T - {a}"
+      show "closed_segment (id x) (a + dd (x-a) *\<^sub>R (x-a)) \<subseteq> T - {a}"
            if "x \<in> T - {a}" for x
       proof (clarsimp simp: in_segment, intro conjI)
         fix u::real assume u: "0 \<le> u" "u \<le> 1"
-        have "a + dd (x - a) *\<^sub>R (x - a) \<in> T"
+        have "a + dd (x-a) *\<^sub>R (x-a) \<in> T"
           by (metis DiffD1 DiffD2 add.commute add.right_neutral affS dd1 diff_add_cancel relS singletonI subsetCE that)
-        then show "(1 - u) *\<^sub>R x + u *\<^sub>R (a + dd (x - a) *\<^sub>R (x - a)) \<in> T"
+        then show "(1 - u) *\<^sub>R x + u *\<^sub>R (a + dd (x-a) *\<^sub>R (x-a)) \<in> T"
           using convexD [OF \<open>convex T\<close>] that u by simp
-        have iff: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + d *\<^sub>R (x - a)) = a \<longleftrightarrow>
-                  (1 - u + u * d) *\<^sub>R (x - a) = 0" for d
+        have iff: "(1 - u) *\<^sub>R x + u *\<^sub>R (a + d *\<^sub>R (x-a)) = a \<longleftrightarrow>
+                  (1 - u + u * d) *\<^sub>R (x-a) = 0" for d
           by (auto simp: algebra_simps)
         have "x \<in> T" "x \<noteq> a" using that by auto
-        then have axa: "a + (x - a) \<in> affine hull S"
+        then have axa: "a + (x-a) \<in> affine hull S"
            by (metis (no_types) add.commute affS diff_add_cancel rev_subsetD)
-        then have "\<not> dd (x - a) \<le> 0 \<and> a + dd (x - a) *\<^sub>R (x - a) \<in> rel_frontier S"
+        then have "\<not> dd (x-a) \<le> 0 \<and> a + dd (x-a) *\<^sub>R (x-a) \<in> rel_frontier S"
           using \<open>x \<noteq> a\<close> dd1 by fastforce
-        with \<open>x \<noteq> a\<close> show "(1 - u) *\<^sub>R x + u *\<^sub>R (a + dd (x - a) *\<^sub>R (x - a)) \<noteq> a"
+        with \<open>x \<noteq> a\<close> show "(1 - u) *\<^sub>R x + u *\<^sub>R (a + dd (x-a) *\<^sub>R (x-a)) \<noteq> a"
           using less_eq_real_def mult_le_0_iff not_less u by (fastforce simp: iff)
       qed
     qed
-    show "retraction (T - {a}) (rel_frontier S) (\<lambda>x. a + dd (x - a) *\<^sub>R (x - a))"
+    show "retraction (T - {a}) (rel_frontier S) (\<lambda>x. a + dd (x-a) *\<^sub>R (x-a))"
     proof (simp add: retraction_def, intro conjI ballI)
       show "rel_frontier S \<subseteq> T - {a}"
         using arelS relS rel_frontier_def by fastforce
-      show "continuous_on (T - {a}) (\<lambda>x. a + dd (x - a) *\<^sub>R (x - a))"
+      show "continuous_on (T - {a}) (\<lambda>x. a + dd (x-a) *\<^sub>R (x-a))"
         using contdd by (simp add: o_def)
-      show "(\<lambda>x. a + dd (x - a) *\<^sub>R (x - a)) \<in> (T - {a}) \<rightarrow> rel_frontier S"
-        apply (auto simp: rel_frontier_def)
-        apply (metis Diff_subset add.commute affS dd1 diff_add_cancel eq_iff_diff_eq_0 rel_frontier_def subset_iff)
-        by (metis DiffE add.commute affS dd1 diff_add_cancel eq_iff_diff_eq_0 rel_frontier_def rev_subsetD)
-      show "a + dd (x - a) *\<^sub>R (x - a) = x" if x: "x \<in> rel_frontier S" for x
+      show "(\<lambda>x. a + dd (x-a) *\<^sub>R (x-a)) \<in> (T - {a}) \<rightarrow> rel_frontier S"
+        unfolding Pi_iff using affS dd1 subset_eq by force
+      show "a + dd (x-a) *\<^sub>R (x-a) = x" if x: "x \<in> rel_frontier S" for x
       proof -
         have "x \<noteq> a"
           using that arelS by (auto simp: rel_frontier_def)
-        have False if "dd (x - a) < 1"
+        have False if "dd (x-a) < 1"
         proof -
           have "x \<in> closure S"
             using x by (auto simp: rel_frontier_def)
           then have segsub: "open_segment a x \<subseteq> rel_interior S"
             by (metis rel_interior_closure_convex_segment [OF \<open>convex S\<close> arelS])
           have  xaffS: "x \<in> affine hull S"
             using affS relS x by auto
-          then have "0 < dd (x - a)" and inS: "a + dd (x - a) *\<^sub>R (x - a) \<in> rel_frontier S"
+          then have "0 < dd (x-a)" and inS: "a + dd (x-a) *\<^sub>R (x-a) \<in> rel_frontier S"
             using dd1 by (auto simp: \<open>x \<noteq> a\<close>)
-          moreover have "a + dd (x - a) *\<^sub>R (x - a) \<in> open_segment a x"
-            using  \<open>x \<noteq> a\<close> \<open>0 < dd (x - a)\<close>
-            apply (simp add: in_segment)
-            apply (rule_tac x = "dd (x - a)" in exI)
-            apply (simp add: algebra_simps that)
-            done
+          moreover have "a + dd (x-a) *\<^sub>R (x-a) \<in> open_segment a x"
+            unfolding in_segment
+          proof (intro exI conjI)
+            show "a + dd (x-a) *\<^sub>R (x-a) = (1 - dd (x-a)) *\<^sub>R a + (dd (x-a)) *\<^sub>R x"
+              by (simp add: algebra_simps)
+          qed (use  \<open>x \<noteq> a\<close> \<open>0 < dd (x-a)\<close> that in auto)
           ultimately show ?thesis
             using segsub by (auto simp: rel_frontier_def)
         qed
-        moreover have False if "1 < dd (x - a)"
+        moreover have False if "1 < dd (x-a)"
           using x that dd2 [of "x - a" 1] \<open>x \<noteq> a\<close> closure_affine_hull
           by (auto simp: rel_frontier_def)
-        ultimately have "dd (x - a) = 1" \<comment> \<open>similar to another proof above\<close>
+        ultimately have "dd (x-a) = 1" \<comment> \<open>similar to another proof above\<close>
           by fastforce
         with that show ?thesis
           by (simp add: rel_frontier_def)
       qed
     qed
   qed
 qed
 
 corollary rel_frontier_retract_of_punctured_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "bounded S" "convex S" "a \<in> rel_interior S"
     shows "rel_frontier S retract_of (affine hull S - {a})"
   by (meson assms convex_affine_hull dual_order.refl rel_frontier_affine_hull 
       rel_frontier_deformation_retract_of_punctured_convex retract_of_def)
 
 corollary rel_boundary_retract_of_punctured_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "compact S" "convex S" "a \<in> rel_interior S"
     shows "(S - rel_interior S) retract_of (affine hull S - {a})"
 by (metis assms closure_closed compact_eq_bounded_closed rel_frontier_def
           rel_frontier_retract_of_punctured_affine_hull)
 
 lemma homotopy_eqv_rel_frontier_punctured_convex:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "bounded S" "a \<in> rel_interior S" "convex T" "rel_frontier S \<subseteq> T" "T \<subseteq> affine hull S"
   shows "(rel_frontier S) homotopy_eqv (T - {a})"
   by (meson assms deformation_retract_imp_homotopy_eqv homotopy_equivalent_space_sym 
       rel_frontier_deformation_retract_of_punctured_convex[of S T])
 
 lemma homotopy_eqv_rel_frontier_punctured_affine_hull:
   fixes S :: "'a::euclidean_space set"
   assumes "convex S" "bounded S" "a \<in> rel_interior S"
     shows "(rel_frontier S) homotopy_eqv (affine hull S - {a})"
   by (simp add: assms homotopy_eqv_rel_frontier_punctured_convex rel_frontier_affine_hull)
 
 lemma path_connected_sphere_gen:
   assumes "convex S" "bounded S" "aff_dim S \<noteq> 1"
   shows "path_connected(rel_frontier S)"
 proof -
   have "convex (closure S)" 
     using assms by auto
   then show ?thesis
     by (metis Diff_empty aff_dim_affine_hull assms convex_affine_hull convex_imp_path_connected equals0I 
        path_connected_punctured_convex rel_frontier_def rel_frontier_retract_of_punctured_affine_hull retract_of_path_connected)
 qed
 
 lemma connected_sphere_gen:
   assumes "convex S" "bounded S" "aff_dim S \<noteq> 1"
   shows "connected(rel_frontier S)"
   by (simp add: assms path_connected_imp_connected path_connected_sphere_gen)
 
 subsubsection\<open>Borsuk-style characterization of separation\<close>
 
 lemma continuous_on_Borsuk_map:
-   "a \<notin> S \<Longrightarrow>  continuous_on S (\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a))"
+   "a \<notin> S \<Longrightarrow>  continuous_on S (\<lambda>x. inverse(norm (x-a)) *\<^sub>R (x-a))"
 by (rule continuous_intros | force)+
 
 lemma Borsuk_map_into_sphere:
-   "(\<lambda>x. inverse(norm (x - a)) *\<^sub>R (x - a)) \<in> S \<rightarrow> sphere 0 1 \<longleftrightarrow> (a \<notin> S)"
+   "(\<lambda>x. inverse(norm (x-a)) *\<^sub>R (x-a)) \<in> S \<rightarrow> sphere 0 1 \<longleftrightarrow> (a \<notin> S)"
 proof -
-  have "\<And>x. \<lbrakk>a \<notin> S; x \<in> S\<rbrakk> \<Longrightarrow> inverse (norm (x - a)) * norm (x - a) = 1"
+  have "\<And>x. \<lbrakk>a \<notin> S; x \<in> S\<rbrakk> \<Longrightarrow> inverse (norm (x-a)) * norm (x-a) = 1"
     by (metis left_inverse norm_eq_zero right_minus_eq)
   then show ?thesis
     by force
 qed
 
 lemma Borsuk_maps_homotopic_in_path_component:
   assumes "path_component (- S) a b"
     shows "homotopic_with_canon (\<lambda>x. True) S (sphere 0 1)
-                   (\<lambda>x. inverse(norm(x - a)) *\<^sub>R (x - a))
+                   (\<lambda>x. inverse(norm(x-a)) *\<^sub>R (x-a))
                    (\<lambda>x. inverse(norm(x - b)) *\<^sub>R (x - b))"
 proof -
-  obtain g where "path g" "path_image g \<subseteq> -S" "pathstart g = a" "pathfinish g = b"
+  obtain g where g: "path g" "path_image g \<subseteq> -S" "pathstart g = a" "pathfinish g = b"
     using assms by (auto simp: path_component_def)
-  then show ?thesis
-    apply (simp add: path_def path_image_def pathstart_def pathfinish_def homotopic_with_def)
-    apply (rule_tac x = "\<lambda>z. inverse(norm(snd z - (g \<circ> fst)z)) *\<^sub>R (snd z - (g \<circ> fst)z)" in exI)
-    apply (rule conjI continuous_intros | erule continuous_on_subset | fastforce simp: divide_simps sphere_def)+
-    done
+  define h where "h \<equiv> \<lambda>z. (snd z - (g \<circ> fst) z) /\<^sub>R norm (snd z - (g \<circ> fst) z)"
+  have "continuous_on ({0..1} \<times> S) h"
+    unfolding h_def using g by (intro continuous_intros) (auto simp: path_defs)
+  moreover
+  have "h ` ({0..1} \<times> S) \<subseteq> sphere 0 1"
+    unfolding h_def using g  by (auto simp: divide_simps path_defs)
+  ultimately show ?thesis
+    using g by (auto simp: h_def path_defs homotopic_with_def)
 qed
 
 lemma non_extensible_Borsuk_map:
   fixes a :: "'a :: euclidean_space"
   assumes "compact S" and cin: "C \<in> components(- S)" and boc: "bounded C" and "a \<in> C"
     shows "\<not> (\<exists>g. continuous_on (S \<union> C) g \<and>
                   g \<in> (S \<union> C) \<rightarrow> sphere 0 1 \<and>
-                  (\<forall>x \<in> S. g x = inverse(norm(x - a)) *\<^sub>R (x - a)))"
+                  (\<forall>x \<in> S. g x = inverse(norm(x-a)) *\<^sub>R (x-a)))"
 proof -
   have "closed S" using assms by (simp add: compact_imp_closed)
   have "C \<subseteq> -S"
     using assms by (simp add: in_components_subset)
   with \<open>a \<in> C\<close> have "a \<notin> S" by blast
   then have ceq: "C = connected_component_set (- S) a"
     by (metis \<open>a \<in> C\<close> cin components_iff connected_component_eq)
   then have "bounded (S \<union> connected_component_set (- S) a)"
     using \<open>compact S\<close> boc compact_imp_bounded by auto
   with bounded_subset_ballD obtain r where "0 < r" and r: "(S \<union> connected_component_set (- S) a) \<subseteq> ball a r"
     by blast
   { fix g
     assume "continuous_on (S \<union> C) g"
             "g \<in> (S \<union> C) \<rightarrow> sphere 0 1"
-       and [simp]: "\<And>x. x \<in> S \<Longrightarrow> g x = (x - a) /\<^sub>R norm (x - a)"
+       and [simp]: "\<And>x. x \<in> S \<Longrightarrow> g x = (x-a) /\<^sub>R norm (x-a)"
     then have norm_g1[simp]: "\<And>x. x \<in> S \<union> C \<Longrightarrow> norm (g x) = 1"
       by force
     have cb_eq: "cball a r = (S \<union> connected_component_set (- S) a) \<union>
                       (cball a r - connected_component_set (- S) a)"
       using ball_subset_cball [of a r] r by auto
     have cont1: "continuous_on (S \<union> connected_component_set (- S) a)
                      (\<lambda>x. a + r *\<^sub>R g x)"
       using \<open>continuous_on (S \<union> C) g\<close> ceq
       by (intro continuous_intros) blast
     have cont2: "continuous_on (cball a r - connected_component_set (- S) a)
-            (\<lambda>x. a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
+            (\<lambda>x. a + r *\<^sub>R ((x-a) /\<^sub>R norm (x-a)))"
       by (rule continuous_intros | force simp: \<open>a \<notin> S\<close>)+
     have 1: "continuous_on (cball a r)
              (\<lambda>x. if connected_component (- S) a x
                   then a + r *\<^sub>R g x
-                  else a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
+                  else a + r *\<^sub>R ((x-a) /\<^sub>R norm (x-a)))"
       apply (subst cb_eq)
       apply (rule continuous_on_cases [OF _ _ cont1 cont2])
       using \<open>closed S\<close> ceq cin 
       by (force simp: closed_Diff open_Compl closed_Un_complement_component open_connected_component)+
     have 2: "(\<lambda>x. a + r *\<^sub>R g x) ` (cball a r \<inter> connected_component_set (- S) a)
              \<subseteq> sphere a r "
       using \<open>0 < r\<close> by (force simp: dist_norm ceq)
     have "retraction (cball a r) (sphere a r)
             (\<lambda>x. if x \<in> connected_component_set (- S) a
                  then a + r *\<^sub>R g x
-                 else a + r *\<^sub>R ((x - a) /\<^sub>R norm (x - a)))"
+                 else a + r *\<^sub>R ((x-a) /\<^sub>R norm (x-a)))"
       using  \<open>0 < r\<close> \<open>a \<notin> S\<close> \<open>a \<in> C\<close> r
       by (auto simp: norm_minus_commute retraction_def Pi_iff ceq dist_norm abs_if 
           mult_less_0_iff divide_simps 1 2)
     then have False
       using no_retraction_cball
              [OF \<open>0 < r\<close>, of a, unfolded retract_of_def, simplified, rule_format,
               of "\<lambda>x. if x \<in> connected_component_set (- S) a
                       then a + r *\<^sub>R g x
-                      else a + r *\<^sub>R inverse(norm(x - a)) *\<^sub>R (x - a)"]
+                      else a + r *\<^sub>R inverse(norm(x-a)) *\<^sub>R (x-a)"]
       by blast
   }
   then show ?thesis
     by blast
 qed
 
 subsubsection \<open>Proving surjectivity via Brouwer fixpoint theorem\<close>
 
 lemma brouwer_surjective:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
-  assumes "compact T"
-    and "convex T"
-    and "T \<noteq> {}"
-    and "continuous_on T f"
+  assumes T: "compact T" "convex T" "T \<noteq> {}"
+    and f: "continuous_on T f"
     and "\<And>x y. \<lbrakk>x\<in>S; y\<in>T\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> T"
     and "x \<in> S"
   shows "\<exists>y\<in>T. f y = x"
 proof -
   have *: "\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
     by (auto simp add: algebra_simps)
   show ?thesis
     unfolding *
-    apply (rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
-    apply (intro continuous_intros)
-    using assms
-    apply auto
-    done
+  proof (rule brouwer[OF T])
+    show "continuous_on T (\<lambda>y. x + (y - f y))"
+      by (intro continuous_intros f)
+  qed (use assms in auto)
 qed
 
 lemma brouwer_surjective_cball:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "continuous_on (cball a e) f"
     and "e > 0"
     and "x \<in> S"
     and "\<And>x y. \<lbrakk>x\<in>S; y\<in>cball a e\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> cball a e"
   shows "\<exists>y\<in>cball a e. f y = x"
   by (smt (verit, best) assms brouwer_surjective cball_eq_empty compact_cball convex_cball)
 
 subsubsection \<open>Inverse function theorem\<close>
 
 text \<open>See Sussmann: "Multidifferential calculus", Theorem 2.1.1\<close>
 
 lemma sussmann_open_mapping:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
   assumes "open S"
     and contf: "continuous_on S f"
     and "x \<in> S"
     and derf: "(f has_derivative f') (at x)"
     and "bounded_linear g'" "f' \<circ> g' = id"
     and "T \<subseteq> S"
     and x: "x \<in> interior T"
   shows "f x \<in> interior (f ` T)"
 proof -
   interpret f': bounded_linear f'
     using assms unfolding has_derivative_def by auto
   interpret g': bounded_linear g'
     using assms by auto
   obtain B where B: "0 < B" "\<forall>x. norm (g' x) \<le> norm x * B"
     using bounded_linear.pos_bounded[OF assms(5)] by blast
   hence *: "1 / (2 * B) > 0" by auto
   obtain e0 where e0:
       "0 < e0"
       "\<forall>y. norm (y - x) < e0 \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> 1 / (2 * B) * norm (y - x)"
     using derf unfolding has_derivative_at_alt
     using * by blast
   obtain e1 where e1: "0 < e1" "cball x e1 \<subseteq> T"
     using mem_interior_cball x by blast
   have *: "0 < e0 / B" "0 < e1 / B" using e0 e1 B by auto
   obtain e where e: "0 < e" "e < e0 / B" "e < e1 / B"
     using field_lbound_gt_zero[OF *] by blast
   have lem: "\<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z" if "z\<in>cball (f x) (e / 2)" for z
   proof (rule brouwer_surjective_cball)
     have z: "z \<in> S" if as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))" for y z
     proof-
       have "dist x z = norm (g' (f x) - g' y)"
         unfolding as(2) and dist_norm by auto
       also have "\<dots> \<le> norm (f x - y) * B"
         by (metis B(2) g'.diff)
       also have "\<dots> \<le> e * B"
         by (metis B(1) dist_norm mem_cball mult_le_cancel_right_pos that(1))
       also have "\<dots> \<le> e1"
         using B(1) e(3) pos_less_divide_eq by fastforce
       finally have "z \<in> cball x e1"
         by force
       then show "z \<in> S"
         using e1 assms(7) by auto
     qed
     show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
       unfolding g'.diff
     proof (rule continuous_on_compose2 [OF _ _ order_refl, of _ _ f])
       show "continuous_on ((\<lambda>y. x + (g' y - g' (f x))) ` cball (f x) e) f"
         by (rule continuous_on_subset[OF contf]) (use z in blast)
       show "continuous_on (cball (f x) e) (\<lambda>y. x + (g' y - g' (f x)))"
         by (intro continuous_intros linear_continuous_on[OF \<open>bounded_linear g'\<close>])
     qed
   next
     fix y z
     assume y: "y \<in> cball (f x) (e / 2)" and z: "z \<in> cball (f x) e"
     have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
       using B by auto
     also have "\<dots> \<le> e * B"
       by (metis B(1) z dist_norm mem_cball norm_minus_commute mult_le_cancel_right_pos)
     also have "\<dots> < e0"
       using B(1) e(2) pos_less_divide_eq by blast
     finally have *: "norm (x + g' (z - f x) - x) < e0"
       by auto
     have **: "f x + f' (x + g' (z - f x) - x) = z"
       using assms(6)[unfolded o_def id_def,THEN cong]
       by auto
     have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le>
           norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
       using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
       by (auto simp add: algebra_simps)
     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
       using e0(2)[rule_format, OF *]
       by (simp only: algebra_simps **) auto
     also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
       using y by (auto simp: dist_norm)
     also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
       using * B by (auto simp add: field_simps)
     also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2"
       by auto
     also have "\<dots> \<le> e/2 + e/2"
       using B(1) \<open>norm (z - f x) * B \<le> e * B\<close> by auto
     finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
       by (auto simp: dist_norm)
   qed (use e that in auto) 
   show ?thesis
     unfolding mem_interior
   proof (intro exI conjI subsetI)
     fix y
     assume "y \<in> ball (f x) (e / 2)"
     then have *: "y \<in> cball (f x) (e / 2)"
       by auto
     obtain z where z: "z \<in> cball (f x) e" "f (x + g' (z - f x)) = y"
       using lem * by blast
     then have "norm (g' (z - f x)) \<le> norm (z - f x) * B"
       using B
       by (auto simp add: field_simps)
     also have "\<dots> \<le> e * B"
       by (metis B(1) dist_norm mem_cball norm_minus_commute mult_le_cancel_right_pos z(1))
     also have "\<dots> \<le> e1"
       using e B unfolding less_divide_eq by auto
     finally have "x + g'(z - f x) \<in> T"
       by (metis add_diff_cancel diff_diff_add dist_norm e1(2) mem_cball norm_minus_commute subset_eq)
     then show "y \<in> f ` T"
       using z by auto
   qed (use e in auto)
 qed
 
 text \<open>Hence the following eccentric variant of the inverse function theorem.
   This has no continuity assumptions, but we do need the inverse function.
   We could put \<open>f' \<circ> g = I\<close> but this happens to fit with the minimal linear
   algebra theory I've set up so far.\<close>
 
 lemma has_derivative_inverse_strong:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes S: "open S" "x \<in> S"
     and contf: "continuous_on S f"
     and gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     and derf: "(f has_derivative f') (at x)"
     and id: "f' \<circ> g' = id"
   shows "(g has_derivative g') (at (f x))"
 proof -
   have linf: "bounded_linear f'"
     using derf unfolding has_derivative_def by auto
   then have ling: "bounded_linear g'"
     unfolding linear_conv_bounded_linear[symmetric]
     using id right_inverse_linear by blast
   moreover have "g' \<circ> f' = id"
     using id linear_inverse_left linear_linear linf ling by blast
   moreover have *: "\<And>T. \<lbrakk>T \<subseteq> S; x \<in> interior T\<rbrakk> \<Longrightarrow> f x \<in> interior (f ` T)"
     using S derf contf id ling sussmann_open_mapping by blast
   have "continuous (at (f x)) g"
     unfolding continuous_at Lim_at
   proof (intro strip)
     fix e :: real
     assume "e > 0"
     then have "f x \<in> interior (f ` (ball x e \<inter> S))"
       by (simp add: "*" S interior_open)
     then obtain d where d: "0 < d" "ball (f x) d \<subseteq> f ` (ball x e \<inter> S)"
       unfolding mem_interior by blast
     show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
     proof (intro exI allI impI conjI)
       fix y
       assume "0 < dist y (f x) \<and> dist y (f x) < d"
       then have "g y \<in> g ` f ` (ball x e \<inter> S)"
         by (metis d(2) dist_commute mem_ball rev_image_eqI subset_iff)
       then show "dist (g y) (g (f x)) < e"
         using \<open>x \<in> S\<close> by (simp add: gf dist_commute image_iff)
     qed (use d in auto)
   qed
   moreover have "f x \<in> interior (f ` S)"
     using "*" S interior_eq by blast
   moreover have "f (g y) = y" if "y \<in> interior (f ` S)" for y
     by (metis gf imageE interiorE subsetD that)
   ultimately show ?thesis using assms
     by (metis has_derivative_inverse_basic_x open_interior)
 qed
 
 text \<open>A rewrite based on the other domain.\<close>
 
 lemma has_derivative_inverse_strong_x:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "open S"
     and "g y \<in> S"
     and "continuous_on S f"
     and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     and "(f has_derivative f') (at (g y))"
     and "f' \<circ> g' = id"
     and f: "f (g y) = y"
   shows "(g has_derivative g') (at y)"
   using has_derivative_inverse_strong[OF assms(1-6)] by (simp add: f)
 
 text \<open>On a region.\<close>
 
 theorem has_derivative_inverse_on:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
   assumes "open S"
     and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f'(x)) (at x)"
     and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
     and "f' x \<circ> g' x = id"
     and "x \<in> S"
   shows "(g has_derivative g'(x)) (at (f x))"
   by (meson assms continuous_on_eq_continuous_at has_derivative_continuous has_derivative_inverse_strong)
 
 end
diff --git a/src/HOL/Library/Groups_Big_Fun.thy b/src/HOL/Library/Groups_Big_Fun.thy
--- a/src/HOL/Library/Groups_Big_Fun.thy
+++ b/src/HOL/Library/Groups_Big_Fun.thy
@@ -1,326 +1,290 @@
 (* Author: Florian Haftmann, TU Muenchen *)
 
 section \<open>Big sum and product over function bodies\<close>
 
 theory Groups_Big_Fun
 imports
   Main
 begin
 
 subsection \<open>Abstract product\<close>
 
 locale comm_monoid_fun = comm_monoid
 begin
 
 definition G :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a"
 where
   expand_set: "G g = comm_monoid_set.F f \<^bold>1 g {a. g a \<noteq> \<^bold>1}"
 
 interpretation F: comm_monoid_set f "\<^bold>1"
   ..
 
 lemma expand_superset:
   assumes "finite A" and "{a. g a \<noteq> \<^bold>1} \<subseteq> A"
   shows "G g = F.F g A"
-  apply (simp add: expand_set)
-  apply (rule F.same_carrierI [of A])
-  apply (simp_all add: assms)
-  done
+  using F.mono_neutral_right assms expand_set by fastforce
 
 lemma conditionalize:
   assumes "finite A"
   shows "F.F g A = G (\<lambda>a. if a \<in> A then g a else \<^bold>1)"
   using assms
-  apply (simp add: expand_set)
-  apply (rule F.same_carrierI [of A])
-  apply auto
-  done
+  by (smt (verit, ccfv_threshold) Diff_iff F.mono_neutral_cong_right expand_set mem_Collect_eq subsetI)
+
 
 lemma neutral [simp]:
   "G (\<lambda>a. \<^bold>1) = \<^bold>1"
   by (simp add: expand_set)
 
 lemma update [simp]:
   assumes "finite {a. g a \<noteq> \<^bold>1}"
   assumes "g a = \<^bold>1"
   shows "G (g(a := b)) = b \<^bold>* G g"
 proof (cases "b = \<^bold>1")
   case True with \<open>g a = \<^bold>1\<close> show ?thesis
     by (simp add: expand_set) (rule F.cong, auto)
 next
   case False
   moreover have "{a'. a' \<noteq> a \<longrightarrow> g a' \<noteq> \<^bold>1} = insert a {a. g a \<noteq> \<^bold>1}"
     by auto
   moreover from \<open>g a = \<^bold>1\<close> have "a \<notin> {a. g a \<noteq> \<^bold>1}"
     by simp
   moreover have "F.F (\<lambda>a'. if a' = a then b else g a') {a. g a \<noteq> \<^bold>1} = F.F g {a. g a \<noteq> \<^bold>1}"
     by (rule F.cong) (auto simp add: \<open>g a = \<^bold>1\<close>)
   ultimately show ?thesis using \<open>finite {a. g a \<noteq> \<^bold>1}\<close> by (simp add: expand_set)
 qed
 
 lemma infinite [simp]:
   "\<not> finite {a. g a \<noteq> \<^bold>1} \<Longrightarrow> G g = \<^bold>1"
   by (simp add: expand_set)
 
 lemma cong [cong]:
   assumes "\<And>a. g a = h a"
   shows "G g = G h"
   using assms by (simp add: expand_set)
 
 lemma not_neutral_obtains_not_neutral:
   assumes "G g \<noteq> \<^bold>1"
   obtains a where "g a \<noteq> \<^bold>1"
   using assms by (auto elim: F.not_neutral_contains_not_neutral simp add: expand_set)
 
 lemma reindex_cong:
   assumes "bij l"
   assumes "g \<circ> l = h"
   shows "G g = G h"
 proof -
   from assms have unfold: "h = g \<circ> l" by simp
   from \<open>bij l\<close> have "inj l" by (rule bij_is_inj)
   then have "inj_on l {a. h a \<noteq> \<^bold>1}" by (rule subset_inj_on) simp
   moreover from \<open>bij l\<close> have "{a. g a \<noteq> \<^bold>1} = l ` {a. h a \<noteq> \<^bold>1}"
     by (auto simp add: image_Collect unfold elim: bij_pointE)
   moreover have "\<And>x. x \<in> {a. h a \<noteq> \<^bold>1} \<Longrightarrow> g (l x) = h x"
     by (simp add: unfold)
   ultimately have "F.F g {a. g a \<noteq> \<^bold>1} = F.F h {a. h a \<noteq> \<^bold>1}"
     by (rule F.reindex_cong)
   then show ?thesis by (simp add: expand_set)
 qed
 
 lemma distrib:
   assumes "finite {a. g a \<noteq> \<^bold>1}" and "finite {a. h a \<noteq> \<^bold>1}"
   shows "G (\<lambda>a. g a \<^bold>* h a) = G g \<^bold>* G h"
 proof -
   from assms have "finite ({a. g a \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1})" by simp
   moreover have "{a. g a \<^bold>* h a \<noteq> \<^bold>1} \<subseteq> {a. g a \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1}"
     by auto (drule sym, simp)
   ultimately show ?thesis
     using assms
     by (simp add: expand_superset [of "{a. g a \<noteq> \<^bold>1} \<union> {a. h a \<noteq> \<^bold>1}"] F.distrib)
 qed
 
 lemma swap:
   assumes "finite C"
   assumes subset: "{a. \<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
   shows "G (\<lambda>a. G (g a)) = G (\<lambda>b. G (\<lambda>a. g a b))"
 proof -
   from \<open>finite C\<close> subset
     have "finite ({a. \<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1})"
     by (rule rev_finite_subset)
   then have fins:
     "finite {b. \<exists>a. g a b \<noteq> \<^bold>1}" "finite {a. \<exists>b. g a b \<noteq> \<^bold>1}"
     by (auto simp add: finite_cartesian_product_iff)
   have subsets: "\<And>a. {b. g a b \<noteq> \<^bold>1} \<subseteq> {b. \<exists>a. g a b \<noteq> \<^bold>1}"
     "\<And>b. {a. g a b \<noteq> \<^bold>1} \<subseteq> {a. \<exists>b. g a b \<noteq> \<^bold>1}"
     "{a. F.F (g a) {b. \<exists>a. g a b \<noteq> \<^bold>1} \<noteq> \<^bold>1} \<subseteq> {a. \<exists>b. g a b \<noteq> \<^bold>1}"
     "{a. F.F (\<lambda>aa. g aa a) {a. \<exists>b. g a b \<noteq> \<^bold>1} \<noteq> \<^bold>1} \<subseteq> {b. \<exists>a. g a b \<noteq> \<^bold>1}"
     by (auto elim: F.not_neutral_contains_not_neutral)
   from F.swap have
     "F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> \<^bold>1}) {a. \<exists>b. g a b \<noteq> \<^bold>1} =
       F.F (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> \<^bold>1}) {b. \<exists>a. g a b \<noteq> \<^bold>1}" .
   with subsets fins have "G (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> \<^bold>1}) =
     G (\<lambda>b. F.F (\<lambda>a. g a b) {a. \<exists>b. g a b \<noteq> \<^bold>1})"
     by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> \<^bold>1}"]
       expand_superset [of "{a. \<exists>b. g a b \<noteq> \<^bold>1}"])
   with subsets fins show ?thesis
     by (auto simp add: expand_superset [of "{b. \<exists>a. g a b \<noteq> \<^bold>1}"]
       expand_superset [of "{a. \<exists>b. g a b \<noteq> \<^bold>1}"])
 qed
 
 lemma cartesian_product:
   assumes "finite C"
   assumes subset: "{a. \<exists>b. g a b \<noteq> \<^bold>1} \<times> {b. \<exists>a. g a b \<noteq> \<^bold>1} \<subseteq> C" (is "?A \<times> ?B \<subseteq> C")
   shows "G (\<lambda>a. G (g a)) = G (\<lambda>(a, b). g a b)"
 proof -
   from subset \<open>finite C\<close> have fin_prod: "finite (?A \<times> ?B)"
     by (rule finite_subset)
   from fin_prod have "finite ?A" and "finite ?B"
     by (auto simp add: finite_cartesian_product_iff)
   have *: "G (\<lambda>a. G (g a)) =
     (F.F (\<lambda>a. F.F (g a) {b. \<exists>a. g a b \<noteq> \<^bold>1}) {a. \<exists>b. g a b \<noteq> \<^bold>1})"
-    apply (subst expand_superset [of "?B"])
-    apply (rule \<open>finite ?B\<close>)
-    apply auto
-    apply (subst expand_superset [of "?A"])
-    apply (rule \<open>finite ?A\<close>)
-    apply auto
-    apply (erule F.not_neutral_contains_not_neutral)
-    apply auto
-    done
-  have "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> \<^bold>1} \<subseteq> ?A \<times> ?B"
+    using \<open>finite ?A\<close> \<open>finite ?B\<close> expand_superset
+    by (smt (verit, del_insts) Collect_mono local.cong not_neutral_obtains_not_neutral)
+  have **: "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> \<^bold>1} \<subseteq> ?A \<times> ?B"
     by auto
-  with subset have **: "{p. (case p of (a, b) \<Rightarrow> g a b) \<noteq> \<^bold>1} \<subseteq> C"
-    by blast
   show ?thesis
-    apply (simp add: *)
-    apply (simp add: F.cartesian_product)
-    apply (subst expand_superset [of C])
-    apply (rule \<open>finite C\<close>)
-    apply (simp_all add: **)
-    apply (rule F.same_carrierI [of C])
-    apply (rule \<open>finite C\<close>)
-    apply (simp_all add: subset)
-    apply auto
-    done
+    using \<open>finite C\<close> expand_superset
+    using "*" ** F.cartesian_product fin_prod by force
 qed
 
 lemma cartesian_product2:
   assumes fin: "finite D"
   assumes subset: "{(a, b). \<exists>c. g a b c \<noteq> \<^bold>1} \<times> {c. \<exists>a b. g a b c \<noteq> \<^bold>1} \<subseteq> D" (is "?AB \<times> ?C \<subseteq> D")
   shows "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>(a, b, c). g a b c)"
 proof -
   have bij: "bij (\<lambda>(a, b, c). ((a, b), c))"
     by (auto intro!: bijI injI simp add: image_def)
   have "{p. \<exists>c. g (fst p) (snd p) c \<noteq> \<^bold>1} \<times> {c. \<exists>p. g (fst p) (snd p) c \<noteq> \<^bold>1} \<subseteq> D"
     by auto (insert subset, blast)
   with fin have "G (\<lambda>p. G (g (fst p) (snd p))) = G (\<lambda>(p, c). g (fst p) (snd p) c)"
     by (rule cartesian_product)
   then have "G (\<lambda>(a, b). G (g a b)) = G (\<lambda>((a, b), c). g a b c)"
     by (auto simp add: split_def)
   also have "G (\<lambda>((a, b), c). g a b c) = G (\<lambda>(a, b, c). g a b c)"
     using bij by (rule reindex_cong [of "\<lambda>(a, b, c). ((a, b), c)"]) (simp add: fun_eq_iff)
   finally show ?thesis .
 qed
 
 lemma delta [simp]:
   "G (\<lambda>b. if b = a then g b else \<^bold>1) = g a"
 proof -
   have "{b. (if b = a then g b else \<^bold>1) \<noteq> \<^bold>1} \<subseteq> {a}" by auto
   then show ?thesis by (simp add: expand_superset [of "{a}"])
 qed
 
 lemma delta' [simp]:
   "G (\<lambda>b. if a = b then g b else \<^bold>1) = g a"
 proof -
   have "(\<lambda>b. if a = b then g b else \<^bold>1) = (\<lambda>b. if b = a then g b else \<^bold>1)"
     by (simp add: fun_eq_iff)
   then have "G (\<lambda>b. if a = b then g b else \<^bold>1) = G (\<lambda>b. if b = a then g b else \<^bold>1)"
     by (simp cong del: cong)
   then show ?thesis by simp
 qed
 
 end
 
 
 subsection \<open>Concrete sum\<close>
 
 context comm_monoid_add
 begin
 
 sublocale Sum_any: comm_monoid_fun plus 0
   rewrites "comm_monoid_set.F plus 0 = sum"
   defines Sum_any = Sum_any.G
 proof -
   show "comm_monoid_fun plus 0" ..
   then interpret Sum_any: comm_monoid_fun plus 0 .
   from sum_def show "comm_monoid_set.F plus 0 = sum" by (auto intro: sym)
 qed
 
 end
 
 syntax (ASCII)
   "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"    ("(3SUM _. _)" [0, 10] 10)
 syntax
   "_Sum_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"    ("(3\<Sum>_. _)" [0, 10] 10)
 translations
   "\<Sum>a. b" \<rightleftharpoons> "CONST Sum_any (\<lambda>a. b)"
 
 lemma Sum_any_left_distrib:
   fixes r :: "'a :: semiring_0"
   assumes "finite {a. g a \<noteq> 0}"
   shows "Sum_any g * r = (\<Sum>n. g n * r)"
-proof -
-  note assms
-  moreover have "{a. g a * r \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
-  ultimately show ?thesis
-    by (simp add: sum_distrib_right Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
-qed  
+  by (metis (mono_tags, lifting) Collect_mono Sum_any.expand_superset assms mult_zero_left sum_distrib_right)
 
 lemma Sum_any_right_distrib:
   fixes r :: "'a :: semiring_0"
   assumes "finite {a. g a \<noteq> 0}"
   shows "r * Sum_any g = (\<Sum>n. r * g n)"
-proof -
-  note assms
-  moreover have "{a. r * g a \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
-  ultimately show ?thesis
-    by (simp add: sum_distrib_left Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
-qed
+  by (metis (mono_tags, lifting) Collect_mono Sum_any.expand_superset assms mult_zero_right sum_distrib_left)
 
 lemma Sum_any_product:
   fixes f g :: "'b \<Rightarrow> 'a::semiring_0"
   assumes "finite {a. f a \<noteq> 0}" and "finite {b. g b \<noteq> 0}"
   shows "Sum_any f * Sum_any g = (\<Sum>a. \<Sum>b. f a * g b)"
 proof -
-  have subset_f: "{a. (\<Sum>b. f a * g b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}"
-    by rule (simp, rule, auto)
-  moreover have subset_g: "\<And>a. {b. f a * g b \<noteq> 0} \<subseteq> {b. g b \<noteq> 0}"
-    by rule (simp, rule, auto)
-  ultimately show ?thesis using assms
-    by (auto simp add: Sum_any.expand_set [of f] Sum_any.expand_set [of g]
-      Sum_any.expand_superset [of "{a. f a \<noteq> 0}"] Sum_any.expand_superset [of "{b. g b \<noteq> 0}"]
-      sum_product)
+  have "\<forall>a. (\<Sum>b. a * g b) = a * Sum_any g"
+    by (simp add: Sum_any_right_distrib assms(2))
+  then show ?thesis
+    by (simp add: Sum_any_left_distrib assms(1))
 qed
 
 lemma Sum_any_eq_zero_iff [simp]: 
   fixes f :: "'a \<Rightarrow> nat"
   assumes "finite {a. f a \<noteq> 0}"
   shows "Sum_any f = 0 \<longleftrightarrow> f = (\<lambda>_. 0)"
   using assms by (simp add: Sum_any.expand_set fun_eq_iff)
 
 
 subsection \<open>Concrete product\<close>
 
 context comm_monoid_mult
 begin
 
 sublocale Prod_any: comm_monoid_fun times 1
   rewrites "comm_monoid_set.F times 1 = prod"
   defines Prod_any = Prod_any.G
 proof -
   show "comm_monoid_fun times 1" ..
   then interpret Prod_any: comm_monoid_fun times 1 .
   from prod_def show "comm_monoid_set.F times 1 = prod" by (auto intro: sym)
 qed
 
 end
 
 syntax (ASCII)
   "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _. _)" [0, 10] 10)
 syntax
   "_Prod_any" :: "pttrn \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3\<Prod>_. _)" [0, 10] 10)
 translations
   "\<Prod>a. b" == "CONST Prod_any (\<lambda>a. b)"
 
 lemma Prod_any_zero:
   fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
   assumes "finite {a. f a \<noteq> 1}"
   assumes "f a = 0"
   shows "(\<Prod>a. f a) = 0"
 proof -
   from \<open>f a = 0\<close> have "f a \<noteq> 1" by simp
   with \<open>f a = 0\<close> have "\<exists>a. f a \<noteq> 1 \<and> f a = 0" by blast
   with \<open>finite {a. f a \<noteq> 1}\<close> show ?thesis
     by (simp add: Prod_any.expand_set prod_zero)
 qed
 
 lemma Prod_any_not_zero:
   fixes f :: "'b \<Rightarrow> 'a :: comm_semiring_1"
   assumes "finite {a. f a \<noteq> 1}"
   assumes "(\<Prod>a. f a) \<noteq> 0"
   shows "f a \<noteq> 0"
   using assms Prod_any_zero [of f] by blast
 
 lemma power_Sum_any:
   assumes "finite {a. f a \<noteq> 0}"
   shows "c ^ (\<Sum>a. f a) = (\<Prod>a. c ^ f a)"
 proof -
   have "{a. c ^ f a \<noteq> 1} \<subseteq> {a. f a \<noteq> 0}"
     by (auto intro: ccontr)
   with assms show ?thesis
     by (simp add: Sum_any.expand_set Prod_any.expand_superset power_sum)
 qed
 
 end
diff --git a/src/HOL/Library/Multiset.thy b/src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy
+++ b/src/HOL/Library/Multiset.thy
@@ -1,4782 +1,4677 @@
 (*  Title:      HOL/Library/Multiset.thy
     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
     Author:     Andrei Popescu, TU Muenchen
     Author:     Jasmin Blanchette, Inria, LORIA, MPII
     Author:     Dmitriy Traytel, TU Muenchen
     Author:     Mathias Fleury, MPII
     Author:     Martin Desharnais, MPI-INF Saarbruecken
 *)
 
 section \<open>(Finite) Multisets\<close>
 
 theory Multiset
   imports Cancellation
 begin
 
 subsection \<open>The type of multisets\<close>
 
 typedef 'a multiset = \<open>{f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}\<close>
   morphisms count Abs_multiset
 proof
   show \<open>(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}\<close>
     by simp
 qed
 
 setup_lifting type_definition_multiset
 
 lemma count_Abs_multiset:
   \<open>count (Abs_multiset f) = f\<close> if \<open>finite {x. f x > 0}\<close>
   by (rule Abs_multiset_inverse) (simp add: that)
 
 lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
   by (simp only: count_inject [symmetric] fun_eq_iff)
 
 lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
   using multiset_eq_iff by auto
 
 text \<open>Preservation of the representing set \<^term>\<open>multiset\<close>.\<close>
 
 lemma diff_preserves_multiset:
   \<open>finite {x. 0 < M x - N x}\<close> if \<open>finite {x. 0 < M x}\<close> for M N :: \<open>'a \<Rightarrow> nat\<close>
   using that by (rule rev_finite_subset) auto
 
 lemma filter_preserves_multiset:
   \<open>finite {x. 0 < (if P x then M x else 0)}\<close> if \<open>finite {x. 0 < M x}\<close> for M N :: \<open>'a \<Rightarrow> nat\<close>
   using that by (rule rev_finite_subset) auto
 
 lemmas in_multiset = diff_preserves_multiset filter_preserves_multiset
 
 
 subsection \<open>Representing multisets\<close>
 
 text \<open>Multiset enumeration\<close>
 
 instantiation multiset :: (type) cancel_comm_monoid_add
 begin
 
 lift_definition zero_multiset :: \<open>'a multiset\<close>
   is \<open>\<lambda>a. 0\<close>
   by simp
 
 abbreviation empty_mset :: \<open>'a multiset\<close> (\<open>{#}\<close>)
   where \<open>empty_mset \<equiv> 0\<close>
 
 lift_definition plus_multiset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>M N a. M a + N a\<close>
   by simp
 
 lift_definition minus_multiset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>M N a. M a - N a\<close>
   by (rule diff_preserves_multiset)
 
 instance
   by (standard; transfer) (simp_all add: fun_eq_iff)
 
 end
 
 context
 begin
 
 qualified definition is_empty :: "'a multiset \<Rightarrow> bool" where
   [code_abbrev]: "is_empty A \<longleftrightarrow> A = {#}"
 
 end
 
 lemma add_mset_in_multiset:
   \<open>finite {x. 0 < (if x = a then Suc (M x) else M x)}\<close>
   if \<open>finite {x. 0 < M x}\<close>
   using that by (simp add: flip: insert_Collect)
 
 lift_definition add_mset :: "'a \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is
   "\<lambda>a M b. if b = a then Suc (M b) else M b"
 by (rule add_mset_in_multiset)
 
 syntax
   "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
 translations
   "{#x, xs#}" == "CONST add_mset x {#xs#}"
   "{#x#}" == "CONST add_mset x {#}"
 
 lemma count_empty [simp]: "count {#} a = 0"
   by (simp add: zero_multiset.rep_eq)
 
 lemma count_add_mset [simp]:
   "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
   by (simp add: add_mset.rep_eq)
 
 lemma count_single: "count {#b#} a = (if b = a then 1 else 0)"
   by simp
 
 lemma
   add_mset_not_empty [simp]: \<open>add_mset a A \<noteq> {#}\<close> and
   empty_not_add_mset [simp]: "{#} \<noteq> add_mset a A"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_add_mset_same_iff [simp]:
   "add_mset a A = add_mset a B \<longleftrightarrow> A = B"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_commute:
   "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
   by (auto simp: multiset_eq_iff)
 
 
 subsection \<open>Basic operations\<close>
 
 subsubsection \<open>Conversion to set and membership\<close>
 
 definition set_mset :: \<open>'a multiset \<Rightarrow> 'a set\<close>
   where \<open>set_mset M = {x. count M x > 0}\<close>
 
 abbreviation member_mset :: \<open>'a \<Rightarrow> 'a multiset \<Rightarrow> bool\<close>
   where \<open>member_mset a M \<equiv> a \<in> set_mset M\<close>
 
 notation
   member_mset  (\<open>'(\<in>#')\<close>) and
   member_mset  (\<open>(_/ \<in># _)\<close> [50, 51] 50)
 
 notation  (ASCII)
   member_mset  (\<open>'(:#')\<close>) and
   member_mset  (\<open>(_/ :# _)\<close> [50, 51] 50)
 
 abbreviation not_member_mset :: \<open>'a \<Rightarrow> 'a multiset \<Rightarrow> bool\<close>
   where \<open>not_member_mset a M \<equiv> a \<notin> set_mset M\<close>
 
 notation
   not_member_mset  (\<open>'(\<notin>#')\<close>) and
   not_member_mset  (\<open>(_/ \<notin># _)\<close> [50, 51] 50)
 
 notation  (ASCII)
   not_member_mset  (\<open>'(~:#')\<close>) and
   not_member_mset  (\<open>(_/ ~:# _)\<close> [50, 51] 50)
 
 context
 begin
 
 qualified abbreviation Ball :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Ball M \<equiv> Set.Ball (set_mset M)"
 
 qualified abbreviation Bex :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
   where "Bex M \<equiv> Set.Bex (set_mset M)"
 
 end
 
 syntax
   "_MBall"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>_\<in>#_./ _)" [0, 0, 10] 10)
   "_MBex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>_\<in>#_./ _)" [0, 0, 10] 10)
 
 syntax  (ASCII)
   "_MBall"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>_:#_./ _)" [0, 0, 10] 10)
   "_MBex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>_:#_./ _)" [0, 0, 10] 10)
 
 translations
   "\<forall>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Ball A (\<lambda>x. P)"
   "\<exists>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Bex A (\<lambda>x. P)"
 
 print_translation \<open>
  [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Multiset.Ball\<close> \<^syntax_const>\<open>_MBall\<close>,
   Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Multiset.Bex\<close> \<^syntax_const>\<open>_MBex\<close>]
 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
 
 lemma count_eq_zero_iff:
   "count M x = 0 \<longleftrightarrow> x \<notin># M"
   by (auto simp add: set_mset_def)
 
 lemma not_in_iff:
   "x \<notin># M \<longleftrightarrow> count M x = 0"
   by (auto simp add: count_eq_zero_iff)
 
 lemma count_greater_zero_iff [simp]:
   "count M x > 0 \<longleftrightarrow> x \<in># M"
   by (auto simp add: set_mset_def)
 
 lemma count_inI:
   assumes "count M x = 0 \<Longrightarrow> False"
   shows "x \<in># M"
 proof (rule ccontr)
   assume "x \<notin># M"
   with assms show False by (simp add: not_in_iff)
 qed
 
 lemma in_countE:
   assumes "x \<in># M"
   obtains n where "count M x = Suc n"
 proof -
   from assms have "count M x > 0" by simp
   then obtain n where "count M x = Suc n"
     using gr0_conv_Suc by blast
   with that show thesis .
 qed
 
 lemma count_greater_eq_Suc_zero_iff [simp]:
   "count M x \<ge> Suc 0 \<longleftrightarrow> x \<in># M"
   by (simp add: Suc_le_eq)
 
 lemma count_greater_eq_one_iff [simp]:
   "count M x \<ge> 1 \<longleftrightarrow> x \<in># M"
   by simp
 
 lemma set_mset_empty [simp]:
   "set_mset {#} = {}"
   by (simp add: set_mset_def)
 
 lemma set_mset_single:
   "set_mset {#b#} = {b}"
   by (simp add: set_mset_def)
 
 lemma set_mset_eq_empty_iff [simp]:
   "set_mset M = {} \<longleftrightarrow> M = {#}"
   by (auto simp add: multiset_eq_iff count_eq_zero_iff)
 
 lemma finite_set_mset [iff]:
   "finite (set_mset M)"
   using count [of M] by simp
 
 lemma set_mset_add_mset_insert [simp]: \<open>set_mset (add_mset a A) = insert a (set_mset A)\<close>
   by (auto simp flip: count_greater_eq_Suc_zero_iff split: if_splits)
 
 lemma multiset_nonemptyE [elim]:
   assumes "A \<noteq> {#}"
   obtains x where "x \<in># A"
 proof -
   have "\<exists>x. x \<in># A" by (rule ccontr) (insert assms, auto)
   with that show ?thesis by blast
 qed
 
 lemma count_gt_imp_in_mset: "count M x > n \<Longrightarrow> x \<in># M"
   using count_greater_zero_iff by fastforce
 
 
 subsubsection \<open>Union\<close>
 
 lemma count_union [simp]:
   "count (M + N) a = count M a + count N a"
   by (simp add: plus_multiset.rep_eq)
 
 lemma set_mset_union [simp]:
   "set_mset (M + N) = set_mset M \<union> set_mset N"
   by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_union) simp
 
 lemma union_mset_add_mset_left [simp]:
   "add_mset a A + B = add_mset a (A + B)"
   by (auto simp: multiset_eq_iff)
 
 lemma union_mset_add_mset_right [simp]:
   "A + add_mset a B = add_mset a (A + B)"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_add_single: \<open>add_mset a A = A + {#a#}\<close>
   by (subst union_mset_add_mset_right, subst add.comm_neutral) standard
 
 
 subsubsection \<open>Difference\<close>
 
 instance multiset :: (type) comm_monoid_diff
   by standard (transfer; simp add: fun_eq_iff)
 
 lemma count_diff [simp]:
   "count (M - N) a = count M a - count N a"
   by (simp add: minus_multiset.rep_eq)
 
 lemma add_mset_diff_bothsides:
   \<open>add_mset a M - add_mset a A = M - A\<close>
   by (auto simp: multiset_eq_iff)
 
 lemma in_diff_count:
   "a \<in># M - N \<longleftrightarrow> count N a < count M a"
   by (simp add: set_mset_def)
 
 lemma count_in_diffI:
   assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False"
   shows "x \<in># M - N"
 proof (rule ccontr)
   assume "x \<notin># M - N"
   then have "count N x = (count N x - count M x) + count M x"
     by (simp add: in_diff_count not_less)
   with assms show False by auto
 qed
 
 lemma in_diff_countE:
   assumes "x \<in># M - N"
   obtains n where "count M x = Suc n + count N x"
 proof -
   from assms have "count M x - count N x > 0" by (simp add: in_diff_count)
   then have "count M x > count N x" by simp
   then obtain n where "count M x = Suc n + count N x"
     using less_iff_Suc_add by auto
   with that show thesis .
 qed
 
 lemma in_diffD:
   assumes "a \<in># M - N"
   shows "a \<in># M"
 proof -
   have "0 \<le> count N a" by simp
   also from assms have "count N a < count M a"
     by (simp add: in_diff_count)
   finally show ?thesis by simp
 qed
 
 lemma set_mset_diff:
   "set_mset (M - N) = {a. count N a < count M a}"
   by (simp add: set_mset_def)
 
 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
   by rule (fact Groups.diff_zero, fact Groups.zero_diff)
 
 lemma diff_cancel: "A - A = {#}"
   by (fact Groups.diff_cancel)
 
 lemma diff_union_cancelR: "M + N - N = (M::'a multiset)"
   by (fact add_diff_cancel_right')
 
 lemma diff_union_cancelL: "N + M - N = (M::'a multiset)"
   by (fact add_diff_cancel_left')
 
 lemma diff_right_commute:
   fixes M N Q :: "'a multiset"
   shows "M - N - Q = M - Q - N"
   by (fact diff_right_commute)
 
 lemma diff_add:
   fixes M N Q :: "'a multiset"
   shows "M - (N + Q) = M - N - Q"
   by (rule sym) (fact diff_diff_add)
 
 lemma insert_DiffM [simp]: "x \<in># M \<Longrightarrow> add_mset x (M - {#x#}) = M"
   by (clarsimp simp: multiset_eq_iff)
 
 lemma insert_DiffM2: "x \<in># M \<Longrightarrow> (M - {#x#}) + {#x#} = M"
   by simp
 
 lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> add_mset b (M - {#a#}) = add_mset b M - {#a#}"
   by (auto simp add: multiset_eq_iff)
 
 lemma diff_add_mset_swap [simp]: "b \<notin># A \<Longrightarrow> add_mset b M - A = add_mset b (M - A)"
   by (auto simp add: multiset_eq_iff simp: not_in_iff)
 
 lemma diff_union_swap2 [simp]: "y \<in># M \<Longrightarrow> add_mset x M - {#y#} = add_mset x (M - {#y#})"
   by (metis add_mset_diff_bothsides diff_union_swap diff_zero insert_DiffM)
 
 lemma diff_diff_add_mset [simp]: "(M::'a multiset) - N - P = M - (N + P)"
   by (rule diff_diff_add)
 
 lemma diff_union_single_conv:
   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
   by (simp add: multiset_eq_iff Suc_le_eq)
 
 lemma mset_add [elim?]:
   assumes "a \<in># A"
   obtains B where "A = add_mset a B"
 proof -
   from assms have "A = add_mset a (A - {#a#})"
     by simp
   with that show thesis .
 qed
 
 lemma union_iff:
   "a \<in># A + B \<longleftrightarrow> a \<in># A \<or> a \<in># B"
   by auto
 
 lemma count_minus_inter_lt_count_minus_inter_iff:
   "count (M2 - M1) y < count (M1 - M2) y \<longleftrightarrow> y \<in># M1 - M2"
   by (meson count_greater_zero_iff gr_implies_not_zero in_diff_count leI order.strict_trans2
       order_less_asym)
 
 lemma minus_inter_eq_minus_inter_iff:
   "(M1 - M2) = (M2 - M1) \<longleftrightarrow> set_mset (M1 - M2) = set_mset (M2 - M1)"
   by (metis add.commute count_diff count_eq_zero_iff diff_add_zero in_diff_countE multiset_eq_iff)
 
 
 subsubsection \<open>Min and Max\<close>
 
 abbreviation Min_mset :: "'a::linorder multiset \<Rightarrow> 'a" where
 "Min_mset m \<equiv> Min (set_mset m)"
 
 abbreviation Max_mset :: "'a::linorder multiset \<Rightarrow> 'a" where
 "Max_mset m \<equiv> Max (set_mset m)"
 
 lemma
   Min_in_mset: "M \<noteq> {#} \<Longrightarrow> Min_mset M \<in># M" and
   Max_in_mset: "M \<noteq> {#} \<Longrightarrow> Max_mset M \<in># M"
   by simp+
 
 
 subsubsection \<open>Equality of multisets\<close>
 
 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
   by (auto simp add: multiset_eq_iff)
 
 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
   by (auto simp add: multiset_eq_iff)
 
 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
   by (auto simp add: multiset_eq_iff)
 
 lemma multi_self_add_other_not_self [simp]: "M = add_mset x M \<longleftrightarrow> False"
   by (auto simp add: multiset_eq_iff)
 
 lemma add_mset_remove_trivial [simp]: \<open>add_mset x M - {#x#} = M\<close>
   by (auto simp: multiset_eq_iff)
 
 lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
   by (auto simp add: multiset_eq_iff not_in_iff)
 
 lemma diff_single_eq_union: "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = add_mset x N"
   by auto
 
 lemma union_single_eq_diff: "add_mset x M = N \<Longrightarrow> M = N - {#x#}"
   unfolding add_mset_add_single[of _ M] by (fact add_implies_diff)
 
 lemma union_single_eq_member: "add_mset x M = N \<Longrightarrow> x \<in># N"
   by auto
 
 lemma add_mset_remove_trivial_If:
   "add_mset a (N - {#a#}) = (if a \<in># N then N else add_mset a N)"
   by (simp add: diff_single_trivial)
 
 lemma add_mset_remove_trivial_eq: \<open>N = add_mset a (N - {#a#}) \<longleftrightarrow> a \<in># N\<close>
   by (auto simp: add_mset_remove_trivial_If)
 
 lemma union_is_single:
   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#}"
   (is "?lhs = ?rhs")
 proof
   show ?lhs if ?rhs using that by auto
   show ?rhs if ?lhs
     by (metis Multiset.diff_cancel add.commute add_diff_cancel_left' diff_add_zero diff_single_trivial insert_DiffM that)
 qed
 
 lemma single_is_union: "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
 
 lemma add_eq_conv_diff:
   "add_mset a M = add_mset b N \<longleftrightarrow> M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#})"
   (is "?lhs \<longleftrightarrow> ?rhs")
 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
 proof
   show ?lhs if ?rhs
     using that
     by (auto simp add: add_mset_commute[of a b])
   show ?rhs if ?lhs
   proof (cases "a = b")
     case True with \<open>?lhs\<close> show ?thesis by simp
   next
     case False
     from \<open>?lhs\<close> have "a \<in># add_mset b N" by (rule union_single_eq_member)
     with False have "a \<in># N" by auto
     moreover from \<open>?lhs\<close> have "M = add_mset b N - {#a#}" by (rule union_single_eq_diff)
     moreover note False
     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"])
   qed
 qed
 
 lemma add_mset_eq_single [iff]: "add_mset b M = {#a#} \<longleftrightarrow> b = a \<and> M = {#}"
   by (auto simp: add_eq_conv_diff)
 
 lemma single_eq_add_mset [iff]: "{#a#} = add_mset b M \<longleftrightarrow> b = a \<and> M = {#}"
   by (auto simp: add_eq_conv_diff)
 
 lemma insert_noteq_member:
   assumes BC: "add_mset b B = add_mset c C"
    and bnotc: "b \<noteq> c"
   shows "c \<in># B"
 proof -
   have "c \<in># add_mset c C" by simp
   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
   then have "c \<in># add_mset b B" using BC by simp
   then show "c \<in># B" using nc by simp
 qed
 
 lemma add_eq_conv_ex:
   "(add_mset a M = add_mset b N) =
     (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
   by (auto simp add: add_eq_conv_diff)
 
 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = add_mset x A"
   by (rule exI [where x = "M - {#x#}"]) simp
 
 lemma multiset_add_sub_el_shuffle:
   assumes "c \<in># B"
     and "b \<noteq> c"
   shows "add_mset b (B - {#c#}) = add_mset b B - {#c#}"
 proof -
   from \<open>c \<in># B\<close> obtain A where B: "B = add_mset c A"
     by (blast dest: multi_member_split)
   have "add_mset b A = add_mset c (add_mset b A) - {#c#}" by simp
   then have "add_mset b A = add_mset b (add_mset c A) - {#c#}"
     by (simp add: \<open>b \<noteq> c\<close>)
   then show ?thesis using B by simp
 qed
 
 lemma add_mset_eq_singleton_iff[iff]:
   "add_mset x M = {#y#} \<longleftrightarrow> M = {#} \<and> x = y"
   by auto
 
 
 subsubsection \<open>Pointwise ordering induced by count\<close>
 
 definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<subseteq>#" 50)
   where "A \<subseteq># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
 
 definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50)
   where "A \<subset># B \<longleftrightarrow> A \<subseteq># B \<and> A \<noteq> B"
 
 abbreviation (input) supseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<supseteq>#" 50)
   where "supseteq_mset A B \<equiv> B \<subseteq># A"
 
 abbreviation (input) supset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<supset>#" 50)
   where "supset_mset A B \<equiv> B \<subset># A"
 
 notation (input)
   subseteq_mset  (infix "\<le>#" 50) and
   supseteq_mset  (infix "\<ge>#" 50)
 
 notation (ASCII)
   subseteq_mset  (infix "<=#" 50) and
   subset_mset  (infix "<#" 50) and
   supseteq_mset  (infix ">=#" 50) and
   supset_mset  (infix ">#" 50)
 
 global_interpretation subset_mset: ordering \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order.trans order.antisym)
 
 interpretation subset_mset: ordered_ab_semigroup_add_imp_le \<open>(+)\<close> \<open>(-)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 interpretation subset_mset: ordered_ab_semigroup_monoid_add_imp_le "(+)" 0 "(-)" "(\<subseteq>#)" "(\<subset>#)"
   by standard
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 lemma mset_subset_eqI:
   "(\<And>a. count A a \<le> count B a) \<Longrightarrow> A \<subseteq># B"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eq_count:
   "A \<subseteq># B \<Longrightarrow> count A a \<le> count B a"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eq_exists_conv: "(A::'a multiset) \<subseteq># B \<longleftrightarrow> (\<exists>C. B = A + C)"
   unfolding subseteq_mset_def
-  apply (rule iffI)
-   apply (rule exI [where x = "B - A"])
-   apply (auto intro: multiset_eq_iff [THEN iffD2])
-  done
+  by (metis add_diff_cancel_left' count_diff count_union le_Suc_ex le_add_same_cancel1 multiset_eq_iff zero_le)
 
 interpretation subset_mset: ordered_cancel_comm_monoid_diff "(+)" 0 "(\<subseteq>#)" "(\<subset>#)" "(-)"
   by standard (simp, fact mset_subset_eq_exists_conv)
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 declare subset_mset.add_diff_assoc[simp] subset_mset.add_diff_assoc2[simp]
 
 lemma mset_subset_eq_mono_add_right_cancel: "(A::'a multiset) + C \<subseteq># B + C \<longleftrightarrow> A \<subseteq># B"
    by (fact subset_mset.add_le_cancel_right)
 
 lemma mset_subset_eq_mono_add_left_cancel: "C + (A::'a multiset) \<subseteq># C + B \<longleftrightarrow> A \<subseteq># B"
    by (fact subset_mset.add_le_cancel_left)
 
 lemma mset_subset_eq_mono_add: "(A::'a multiset) \<subseteq># B \<Longrightarrow> C \<subseteq># D \<Longrightarrow> A + C \<subseteq># B + D"
    by (fact subset_mset.add_mono)
 
 lemma mset_subset_eq_add_left: "(A::'a multiset) \<subseteq># A + B"
    by simp
 
 lemma mset_subset_eq_add_right: "B \<subseteq># (A::'a multiset) + B"
    by simp
 
 lemma single_subset_iff [simp]:
   "{#a#} \<subseteq># M \<longleftrightarrow> a \<in># M"
   by (auto simp add: subseteq_mset_def Suc_le_eq)
 
 lemma mset_subset_eq_single: "a \<in># B \<Longrightarrow> {#a#} \<subseteq># B"
   by simp
 
 lemma mset_subset_eq_add_mset_cancel: \<open>add_mset a A \<subseteq># add_mset a B \<longleftrightarrow> A \<subseteq># B\<close>
   unfolding add_mset_add_single[of _ A] add_mset_add_single[of _ B]
   by (rule mset_subset_eq_mono_add_right_cancel)
 
 lemma multiset_diff_union_assoc:
   fixes A B C D :: "'a multiset"
   shows "C \<subseteq># B \<Longrightarrow> A + B - C = A + (B - C)"
   by (fact subset_mset.diff_add_assoc)
 
 lemma mset_subset_eq_multiset_union_diff_commute:
   fixes A B C D :: "'a multiset"
   shows "B \<subseteq># A \<Longrightarrow> A - B + C = A + C - B"
   by (fact subset_mset.add_diff_assoc2)
 
 lemma diff_subset_eq_self[simp]:
   "(M::'a multiset) - N \<subseteq># M"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eqD:
   assumes "A \<subseteq># B" and "x \<in># A"
   shows "x \<in># B"
 proof -
   from \<open>x \<in># A\<close> have "count A x > 0" by simp
   also from \<open>A \<subseteq># B\<close> have "count A x \<le> count B x"
     by (simp add: subseteq_mset_def)
   finally show ?thesis by simp
 qed
 
 lemma mset_subsetD:
   "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
   by (auto intro: mset_subset_eqD [of A])
 
 lemma set_mset_mono:
   "A \<subseteq># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
   by (metis mset_subset_eqD subsetI)
 
 lemma mset_subset_eq_insertD:
-  "add_mset x A \<subseteq># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
-apply (rule conjI)
- apply (simp add: mset_subset_eqD)
- apply (clarsimp simp: subset_mset_def subseteq_mset_def)
- apply safe
-  apply (erule_tac x = a in allE)
-  apply (auto split: if_split_asm)
-done
+  assumes "add_mset x A \<subseteq># B"
+  shows "x \<in># B \<and> A \<subset># B"
+proof 
+  show "x \<in># B"
+    using assms by (simp add: mset_subset_eqD)
+  have "A \<subseteq># add_mset x A"
+    by (metis (no_types) add_mset_add_single mset_subset_eq_add_left)
+  then have "A \<subset># add_mset x A"
+    by (meson multi_self_add_other_not_self subset_mset.le_imp_less_or_eq)
+  then show "A \<subset># B"
+    using assms subset_mset.strict_trans2 by blast
+qed
 
 lemma mset_subset_insertD:
   "add_mset x A \<subset># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
   by (rule mset_subset_eq_insertD) simp
 
 lemma mset_subset_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False"
   by (simp only: subset_mset.not_less_zero)
 
 lemma empty_subset_add_mset[simp]: "{#} \<subset># add_mset x M"
   by (auto intro: subset_mset.gr_zeroI)
 
 lemma empty_le: "{#} \<subseteq># A"
   by (fact subset_mset.zero_le)
 
 lemma insert_subset_eq_iff:
   "add_mset a A \<subseteq># B \<longleftrightarrow> a \<in># B \<and> A \<subseteq># B - {#a#}"
-  using le_diff_conv2 [of "Suc 0" "count B a" "count A a"]
-  apply (auto simp add: subseteq_mset_def not_in_iff Suc_le_eq)
-  apply (rule ccontr)
-  apply (auto simp add: not_in_iff)
-  done
+  using mset_subset_eq_insertD subset_mset.le_diff_conv2 by fastforce
 
 lemma insert_union_subset_iff:
   "add_mset a A \<subset># B \<longleftrightarrow> a \<in># B \<and> A \<subset># B - {#a#}"
   by (auto simp add: insert_subset_eq_iff subset_mset_def)
 
 lemma subset_eq_diff_conv:
   "A - C \<subseteq># B \<longleftrightarrow> A \<subseteq># B + C"
   by (simp add: subseteq_mset_def le_diff_conv)
 
 lemma multi_psub_of_add_self [simp]: "A \<subset># add_mset x A"
   by (auto simp: subset_mset_def subseteq_mset_def)
 
 lemma multi_psub_self: "A \<subset># A = False"
   by simp
 
 lemma mset_subset_add_mset [simp]: "add_mset x N \<subset># add_mset x M \<longleftrightarrow> N \<subset># M"
   unfolding add_mset_add_single[of _ N] add_mset_add_single[of _ M]
   by (fact subset_mset.add_less_cancel_right)
 
 lemma mset_subset_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
   by (auto simp: subset_mset_def elim: mset_add)
 
 lemma Diff_eq_empty_iff_mset: "A - B = {#} \<longleftrightarrow> A \<subseteq># B"
   by (auto simp: multiset_eq_iff subseteq_mset_def)
 
 lemma add_mset_subseteq_single_iff[iff]: "add_mset a M \<subseteq># {#b#} \<longleftrightarrow> M = {#} \<and> a = b"
 proof
   assume A: "add_mset a M \<subseteq># {#b#}"
   then have \<open>a = b\<close>
     by (auto dest: mset_subset_eq_insertD)
   then show "M={#} \<and> a=b"
     using A by (simp add: mset_subset_eq_add_mset_cancel)
 qed simp
 
 lemma nonempty_subseteq_mset_eq_single: "M \<noteq> {#} \<Longrightarrow> M \<subseteq># {#x#} \<Longrightarrow> M = {#x#}"
   by (cases M) (metis single_is_union subset_mset.less_eqE)
 
 lemma nonempty_subseteq_mset_iff_single: "(M \<noteq> {#} \<and> M \<subseteq># {#x#} \<and> P) \<longleftrightarrow> M = {#x#} \<and> P"
   by (cases M) (metis empty_not_add_mset nonempty_subseteq_mset_eq_single subset_mset.order_refl)
 
 
 subsubsection \<open>Intersection and bounded union\<close>
 
 definition inter_mset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>  (infixl \<open>\<inter>#\<close> 70)
   where \<open>A \<inter># B = A - (A - B)\<close>
 
 lemma count_inter_mset [simp]:
   \<open>count (A \<inter># B) x = min (count A x) (count B x)\<close>
   by (simp add: inter_mset_def)
 
 (*global_interpretation subset_mset: semilattice_order \<open>(\<inter>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (simp_all add: multiset_eq_iff subseteq_mset_def subset_mset_def min_def)*)
 
 interpretation subset_mset: semilattice_inf \<open>(\<inter>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (simp_all add: multiset_eq_iff subseteq_mset_def)
   \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 definition union_mset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>  (infixl \<open>\<union>#\<close> 70)
   where \<open>A \<union># B = A + (B - A)\<close>
 
 lemma count_union_mset [simp]:
   \<open>count (A \<union># B) x = max (count A x) (count B x)\<close>
   by (simp add: union_mset_def)
 
 global_interpretation subset_mset: semilattice_neutr_order \<open>(\<union>#)\<close> \<open>{#}\<close> \<open>(\<supseteq>#)\<close> \<open>(\<supset>#)\<close>
-  apply standard
-      apply (simp_all add: multiset_eq_iff subseteq_mset_def subset_mset_def max_def)
-   apply (auto simp add: le_antisym dest: sym)
-   apply (metis nat_le_linear)+
-  done
+proof 
+  show "\<And>a b. (b \<subseteq># a) = (a = a \<union># b)"
+    by (simp add: Diff_eq_empty_iff_mset union_mset_def)
+  show "\<And>a b. (b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
+    by (metis Diff_eq_empty_iff_mset add_cancel_left_right subset_mset_def union_mset_def)
+qed (auto simp: multiset_eqI union_mset_def)
 
 interpretation subset_mset: semilattice_sup \<open>(\<union>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
 proof -
   have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
     by arith
   show "class.semilattice_sup (\<union>#) (\<subseteq>#) (\<subset>#)"
     by standard (auto simp add: union_mset_def subseteq_mset_def)
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 interpretation subset_mset: bounded_lattice_bot "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)"
   "(\<union>#)" "{#}"
   by standard auto
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 
 subsubsection \<open>Additional intersection facts\<close>
 
 lemma set_mset_inter [simp]:
   "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
   by (simp only: set_mset_def) auto
 
 lemma diff_intersect_left_idem [simp]:
   "M - M \<inter># N = M - N"
   by (simp add: multiset_eq_iff min_def)
 
 lemma diff_intersect_right_idem [simp]:
   "M - N \<inter># M = M - N"
   by (simp add: multiset_eq_iff min_def)
 
 lemma multiset_inter_single[simp]: "a \<noteq> b \<Longrightarrow> {#a#} \<inter># {#b#} = {#}"
   by (rule multiset_eqI) auto
 
 lemma multiset_union_diff_commute:
   assumes "B \<inter># C = {#}"
   shows "A + B - C = A - C + B"
 proof (rule multiset_eqI)
   fix x
   from assms have "min (count B x) (count C x) = 0"
     by (auto simp add: multiset_eq_iff)
   then have "count B x = 0 \<or> count C x = 0"
     unfolding min_def by (auto split: if_splits)
   then show "count (A + B - C) x = count (A - C + B) x"
     by auto
 qed
 
 lemma disjunct_not_in:
-  "A \<inter># B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)" (is "?P \<longleftrightarrow> ?Q")
-proof
-  assume ?P
-  show ?Q
-  proof
-    fix a
-    from \<open>?P\<close> have "min (count A a) (count B a) = 0"
-      by (simp add: multiset_eq_iff)
-    then have "count A a = 0 \<or> count B a = 0"
-      by (cases "count A a \<le> count B a") (simp_all add: min_def)
-    then show "a \<notin># A \<or> a \<notin># B"
-      by (simp add: not_in_iff)
-  qed
-next
-  assume ?Q
-  show ?P
-  proof (rule multiset_eqI)
-    fix a
-    from \<open>?Q\<close> have "count A a = 0 \<or> count B a = 0"
-      by (auto simp add: not_in_iff)
-    then show "count (A \<inter># B) a = count {#} a"
-      by auto
-  qed
-qed
+  "A \<inter># B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)"
+  by (metis disjoint_iff set_mset_eq_empty_iff set_mset_inter)
 
 lemma inter_mset_empty_distrib_right: "A \<inter># (B + C) = {#} \<longleftrightarrow> A \<inter># B = {#} \<and> A \<inter># C = {#}"
   by (meson disjunct_not_in union_iff)
 
 lemma inter_mset_empty_distrib_left: "(A + B) \<inter># C = {#} \<longleftrightarrow> A \<inter># C = {#} \<and> B \<inter># C = {#}"
   by (meson disjunct_not_in union_iff)
 
 lemma add_mset_inter_add_mset [simp]:
   "add_mset a A \<inter># add_mset a B = add_mset a (A \<inter># B)"
   by (rule multiset_eqI) simp
 
 lemma add_mset_disjoint [simp]:
   "add_mset a A \<inter># B = {#} \<longleftrightarrow> a \<notin># B \<and> A \<inter># B = {#}"
   "{#} = add_mset a A \<inter># B \<longleftrightarrow> a \<notin># B \<and> {#} = A \<inter># B"
   by (auto simp: disjunct_not_in)
 
 lemma disjoint_add_mset [simp]:
   "B \<inter># add_mset a A = {#} \<longleftrightarrow> a \<notin># B \<and> B \<inter># A = {#}"
   "{#} = A \<inter># add_mset b B \<longleftrightarrow> b \<notin># A \<and> {#} = A \<inter># B"
   by (auto simp: disjunct_not_in)
 
 lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = M \<inter># N"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma inter_add_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = add_mset x (M \<inter># (N - {#x#}))"
   by (auto simp add: multiset_eq_iff elim: mset_add)
 
 lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = N \<inter># M"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma inter_add_right2: "x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = add_mset x ((N - {#x#}) \<inter># M)"
   by (auto simp add: multiset_eq_iff elim: mset_add)
 
 lemma disjunct_set_mset_diff:
   assumes "M \<inter># N = {#}"
   shows "set_mset (M - N) = set_mset M"
 proof (rule set_eqI)
   fix a
   from assms have "a \<notin># M \<or> a \<notin># N"
     by (simp add: disjunct_not_in)
   then show "a \<in># M - N \<longleftrightarrow> a \<in># M"
     by (auto dest: in_diffD) (simp add: in_diff_count not_in_iff)
 qed
 
 lemma at_most_one_mset_mset_diff:
   assumes "a \<notin># M - {#a#}"
   shows "set_mset (M - {#a#}) = set_mset M - {a}"
   using assms by (auto simp add: not_in_iff in_diff_count set_eq_iff)
 
 lemma more_than_one_mset_mset_diff:
   assumes "a \<in># M - {#a#}"
   shows "set_mset (M - {#a#}) = set_mset M"
 proof (rule set_eqI)
   fix b
   have "Suc 0 < count M b \<Longrightarrow> count M b > 0" by arith
   then show "b \<in># M - {#a#} \<longleftrightarrow> b \<in># M"
     using assms by (auto simp add: in_diff_count)
 qed
 
 lemma inter_iff:
   "a \<in># A \<inter># B \<longleftrightarrow> a \<in># A \<and> a \<in># B"
   by simp
 
 lemma inter_union_distrib_left:
   "A \<inter># B + C = (A + C) \<inter># (B + C)"
   by (simp add: multiset_eq_iff min_add_distrib_left)
 
 lemma inter_union_distrib_right:
   "C + A \<inter># B = (C + A) \<inter># (C + B)"
   using inter_union_distrib_left [of A B C] by (simp add: ac_simps)
 
 lemma inter_subset_eq_union:
   "A \<inter># B \<subseteq># A + B"
   by (auto simp add: subseteq_mset_def)
 
 
 subsubsection \<open>Additional bounded union facts\<close>
 
 lemma set_mset_sup [simp]:
   \<open>set_mset (A \<union># B) = set_mset A \<union> set_mset B\<close>
   by (simp only: set_mset_def) (auto simp add: less_max_iff_disj)
 
 lemma sup_union_left1 [simp]: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># N)"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma sup_union_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># (N - {#x#}))"
   by (simp add: multiset_eq_iff)
 
 lemma sup_union_right1 [simp]: "\<not> x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x (N \<union># M)"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma sup_union_right2: "x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x ((N - {#x#}) \<union># M)"
   by (simp add: multiset_eq_iff)
 
 lemma sup_union_distrib_left:
   "A \<union># B + C = (A + C) \<union># (B + C)"
   by (simp add: multiset_eq_iff max_add_distrib_left)
 
 lemma union_sup_distrib_right:
   "C + A \<union># B = (C + A) \<union># (C + B)"
   using sup_union_distrib_left [of A B C] by (simp add: ac_simps)
 
 lemma union_diff_inter_eq_sup:
   "A + B - A \<inter># B = A \<union># B"
   by (auto simp add: multiset_eq_iff)
 
 lemma union_diff_sup_eq_inter:
   "A + B - A \<union># B = A \<inter># B"
   by (auto simp add: multiset_eq_iff)
 
 lemma add_mset_union:
   \<open>add_mset a A \<union># add_mset a B = add_mset a (A \<union># B)\<close>
   by (auto simp: multiset_eq_iff max_def)
 
 
 subsection \<open>Replicate and repeat operations\<close>
 
 definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
   "replicate_mset n x = (add_mset x ^^ n) {#}"
 
 lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
   unfolding replicate_mset_def by simp
 
 lemma replicate_mset_Suc [simp]: "replicate_mset (Suc n) x = add_mset x (replicate_mset n x)"
   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
 
 lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
   unfolding replicate_mset_def by (induct n) auto
 
 lift_definition repeat_mset :: \<open>nat \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>n M a. n * M a\<close> by simp
 
 lemma count_repeat_mset [simp]: "count (repeat_mset i A) a = i * count A a"
   by transfer rule
 
 lemma repeat_mset_0 [simp]:
   \<open>repeat_mset 0 M = {#}\<close>
   by transfer simp
 
 lemma repeat_mset_Suc [simp]:
   \<open>repeat_mset (Suc n) M = M + repeat_mset n M\<close>
   by transfer simp
 
 lemma repeat_mset_right [simp]: "repeat_mset a (repeat_mset b A) = repeat_mset (a * b) A"
   by (auto simp: multiset_eq_iff left_diff_distrib')
 
 lemma left_diff_repeat_mset_distrib': \<open>repeat_mset (i - j) u = repeat_mset i u - repeat_mset j u\<close>
   by (auto simp: multiset_eq_iff left_diff_distrib')
 
 lemma left_add_mult_distrib_mset:
   "repeat_mset i u + (repeat_mset j u + k) = repeat_mset (i+j) u + k"
   by (auto simp: multiset_eq_iff add_mult_distrib)
 
 lemma repeat_mset_distrib:
   "repeat_mset (m + n) A = repeat_mset m A + repeat_mset n A"
   by (auto simp: multiset_eq_iff Nat.add_mult_distrib)
 
 lemma repeat_mset_distrib2[simp]:
   "repeat_mset n (A + B) = repeat_mset n A + repeat_mset n B"
   by (auto simp: multiset_eq_iff add_mult_distrib2)
 
 lemma repeat_mset_replicate_mset[simp]:
   "repeat_mset n {#a#} = replicate_mset n a"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_distrib_add_mset[simp]:
   "repeat_mset n (add_mset a A) = replicate_mset n a + repeat_mset n A"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_empty[simp]: "repeat_mset n {#} = {#}"
   by transfer simp
 
 
 subsubsection \<open>Simprocs\<close>
 
 lemma repeat_mset_iterate_add: \<open>repeat_mset n M = iterate_add n M\<close>
   unfolding iterate_add_def by (induction n) auto
 
 lemma mset_subseteq_add_iff1:
   "j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (repeat_mset (i-j) u + m \<subseteq># n)"
   by (auto simp add: subseteq_mset_def nat_le_add_iff1)
 
 lemma mset_subseteq_add_iff2:
   "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (m \<subseteq># repeat_mset (j-i) u + n)"
   by (auto simp add: subseteq_mset_def nat_le_add_iff2)
 
 lemma mset_subset_add_iff1:
   "j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<subset># repeat_mset j u + n) = (repeat_mset (i-j) u + m \<subset># n)"
   unfolding subset_mset_def repeat_mset_iterate_add
   by (simp add: iterate_add_eq_add_iff1 mset_subseteq_add_iff1[unfolded repeat_mset_iterate_add])
 
 lemma mset_subset_add_iff2:
   "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<subset># repeat_mset j u + n) = (m \<subset># repeat_mset (j-i) u + n)"
   unfolding subset_mset_def repeat_mset_iterate_add
   by (simp add: iterate_add_eq_add_iff2 mset_subseteq_add_iff2[unfolded repeat_mset_iterate_add])
 
 ML_file \<open>multiset_simprocs.ML\<close>
 
 lemma add_mset_replicate_mset_safe[cancelation_simproc_pre]: \<open>NO_MATCH {#} M \<Longrightarrow> add_mset a M = {#a#} + M\<close>
   by simp
 
 declare repeat_mset_iterate_add[cancelation_simproc_pre]
 
 declare iterate_add_distrib[cancelation_simproc_pre]
 declare repeat_mset_iterate_add[symmetric, cancelation_simproc_post]
 
 declare add_mset_not_empty[cancelation_simproc_eq_elim]
     empty_not_add_mset[cancelation_simproc_eq_elim]
     subset_mset.le_zero_eq[cancelation_simproc_eq_elim]
     empty_not_add_mset[cancelation_simproc_eq_elim]
     add_mset_not_empty[cancelation_simproc_eq_elim]
     subset_mset.le_zero_eq[cancelation_simproc_eq_elim]
     le_zero_eq[cancelation_simproc_eq_elim]
 
 simproc_setup mseteq_cancel
   ("(l::'a multiset) + m = n" | "(l::'a multiset) = m + n" |
    "add_mset a m = n" | "m = add_mset a n" |
    "replicate_mset p a = n" | "m = replicate_mset p a" |
    "repeat_mset p m = n" | "m = repeat_mset p m") =
   \<open>K Cancel_Simprocs.eq_cancel\<close>
 
 simproc_setup msetsubset_cancel
   ("(l::'a multiset) + m \<subset># n" | "(l::'a multiset) \<subset># m + n" |
    "add_mset a m \<subset># n" | "m \<subset># add_mset a n" |
    "replicate_mset p r \<subset># n" | "m \<subset># replicate_mset p r" |
    "repeat_mset p m \<subset># n" | "m \<subset># repeat_mset p m") =
   \<open>K Multiset_Simprocs.subset_cancel_msets\<close>
 
 simproc_setup msetsubset_eq_cancel
   ("(l::'a multiset) + m \<subseteq># n" | "(l::'a multiset) \<subseteq># m + n" |
    "add_mset a m \<subseteq># n" | "m \<subseteq># add_mset a n" |
    "replicate_mset p r \<subseteq># n" | "m \<subseteq># replicate_mset p r" |
    "repeat_mset p m \<subseteq># n" | "m \<subseteq># repeat_mset p m") =
   \<open>K Multiset_Simprocs.subseteq_cancel_msets\<close>
 
 simproc_setup msetdiff_cancel
   ("((l::'a multiset) + m) - n" | "(l::'a multiset) - (m + n)" |
    "add_mset a m - n" | "m - add_mset a n" |
    "replicate_mset p r - n" | "m - replicate_mset p r" |
    "repeat_mset p m - n" | "m - repeat_mset p m") =
   \<open>K Cancel_Simprocs.diff_cancel\<close>
 
 
 subsubsection \<open>Conditionally complete lattice\<close>
 
 instantiation multiset :: (type) Inf
 begin
 
 lift_definition Inf_multiset :: "'a multiset set \<Rightarrow> 'a multiset" is
   "\<lambda>A i. if A = {} then 0 else Inf ((\<lambda>f. f i) ` A)"
 proof -
   fix A :: "('a \<Rightarrow> nat) set"
   assume *: "\<And>f. f \<in> A \<Longrightarrow> finite {x. 0 < f x}"
   show \<open>finite {i. 0 < (if A = {} then 0 else INF f\<in>A. f i)}\<close>
   proof (cases "A = {}")
     case False
     then obtain f where "f \<in> A" by blast
     hence "{i. Inf ((\<lambda>f. f i) ` A) > 0} \<subseteq> {i. f i > 0}"
       by (auto intro: less_le_trans[OF _ cInf_lower])
     moreover from \<open>f \<in> A\<close> * have "finite \<dots>" by simp
     ultimately have "finite {i. Inf ((\<lambda>f. f i) ` A) > 0}" by (rule finite_subset)
     with False show ?thesis by simp
   qed simp_all
 qed
 
 instance ..
 
 end
 
 lemma Inf_multiset_empty: "Inf {} = {#}"
   by transfer simp_all
 
 lemma count_Inf_multiset_nonempty: "A \<noteq> {} \<Longrightarrow> count (Inf A) x = Inf ((\<lambda>X. count X x) ` A)"
   by transfer simp_all
 
 
 instantiation multiset :: (type) Sup
 begin
 
 definition Sup_multiset :: "'a multiset set \<Rightarrow> 'a multiset" where
   "Sup_multiset A = (if A \<noteq> {} \<and> subset_mset.bdd_above A then
            Abs_multiset (\<lambda>i. Sup ((\<lambda>X. count X i) ` A)) else {#})"
 
 lemma Sup_multiset_empty: "Sup {} = {#}"
   by (simp add: Sup_multiset_def)
 
 lemma Sup_multiset_unbounded: "\<not> subset_mset.bdd_above A \<Longrightarrow> Sup A = {#}"
   by (simp add: Sup_multiset_def)
 
 instance ..
 
 end
 
 lemma bdd_above_multiset_imp_bdd_above_count:
   assumes "subset_mset.bdd_above (A :: 'a multiset set)"
   shows   "bdd_above ((\<lambda>X. count X x) ` A)"
 proof -
   from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y"
     by (meson subset_mset.bdd_above.E)
   hence "count X x \<le> count Y x" if "X \<in> A" for X
     using that by (auto intro: mset_subset_eq_count)
   thus ?thesis by (intro bdd_aboveI[of _ "count Y x"]) auto
 qed
 
 lemma bdd_above_multiset_imp_finite_support:
   assumes "A \<noteq> {}" "subset_mset.bdd_above (A :: 'a multiset set)"
   shows   "finite (\<Union>X\<in>A. {x. count X x > 0})"
 proof -
   from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y"
     by (meson subset_mset.bdd_above.E)
   hence "count X x \<le> count Y x" if "X \<in> A" for X x
     using that by (auto intro: mset_subset_eq_count)
   hence "(\<Union>X\<in>A. {x. count X x > 0}) \<subseteq> {x. count Y x > 0}"
     by safe (erule less_le_trans)
   moreover have "finite \<dots>" by simp
   ultimately show ?thesis by (rule finite_subset)
 qed
 
 lemma Sup_multiset_in_multiset:
   \<open>finite {i. 0 < (SUP M\<in>A. count M i)}\<close>
   if \<open>A \<noteq> {}\<close> \<open>subset_mset.bdd_above A\<close>
 proof -
   have "{i. Sup ((\<lambda>X. count X i) ` A) > 0} \<subseteq> (\<Union>X\<in>A. {i. 0 < count X i})"
   proof safe
     fix i assume pos: "(SUP X\<in>A. count X i) > 0"
     show "i \<in> (\<Union>X\<in>A. {i. 0 < count X i})"
     proof (rule ccontr)
       assume "i \<notin> (\<Union>X\<in>A. {i. 0 < count X i})"
       hence "\<forall>X\<in>A. count X i \<le> 0" by (auto simp: count_eq_zero_iff)
       with that have "(SUP X\<in>A. count X i) \<le> 0"
         by (intro cSup_least bdd_above_multiset_imp_bdd_above_count) auto
       with pos show False by simp
     qed
   qed
   moreover from that have "finite \<dots>"
     by (rule bdd_above_multiset_imp_finite_support)
   ultimately show "finite {i. Sup ((\<lambda>X. count X i) ` A) > 0}"
     by (rule finite_subset)
 qed
 
 lemma count_Sup_multiset_nonempty:
   \<open>count (Sup A) x = (SUP X\<in>A. count X x)\<close>
   if \<open>A \<noteq> {}\<close> \<open>subset_mset.bdd_above A\<close>
   using that by (simp add: Sup_multiset_def Sup_multiset_in_multiset count_Abs_multiset)
 
 interpretation subset_mset: conditionally_complete_lattice Inf Sup "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)" "(\<union>#)"
 proof
   fix X :: "'a multiset" and A
   assume "X \<in> A"
   show "Inf A \<subseteq># X"
-  proof (rule mset_subset_eqI)
-    fix x
-    from \<open>X \<in> A\<close> have "A \<noteq> {}" by auto
-    hence "count (Inf A) x = (INF X\<in>A. count X x)"
-      by (simp add: count_Inf_multiset_nonempty)
-    also from \<open>X \<in> A\<close> have "\<dots> \<le> count X x"
-      by (intro cInf_lower) simp_all
-    finally show "count (Inf A) x \<le> count X x" .
-  qed
+    by (metis \<open>X \<in> A\<close> count_Inf_multiset_nonempty empty_iff image_eqI mset_subset_eqI wellorder_Inf_le1)
 next
   fix X :: "'a multiset" and A
   assume nonempty: "A \<noteq> {}" and le: "\<And>Y. Y \<in> A \<Longrightarrow> X \<subseteq># Y"
   show "X \<subseteq># Inf A"
   proof (rule mset_subset_eqI)
     fix x
     from nonempty have "count X x \<le> (INF X\<in>A. count X x)"
       by (intro cInf_greatest) (auto intro: mset_subset_eq_count le)
     also from nonempty have "\<dots> = count (Inf A) x" by (simp add: count_Inf_multiset_nonempty)
     finally show "count X x \<le> count (Inf A) x" .
   qed
 next
   fix X :: "'a multiset" and A
   assume X: "X \<in> A" and bdd: "subset_mset.bdd_above A"
   show "X \<subseteq># Sup A"
   proof (rule mset_subset_eqI)
     fix x
     from X have "A \<noteq> {}" by auto
     have "count X x \<le> (SUP X\<in>A. count X x)"
       by (intro cSUP_upper X bdd_above_multiset_imp_bdd_above_count bdd)
     also from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd]
       have "(SUP X\<in>A. count X x) = count (Sup A) x" by simp
     finally show "count X x \<le> count (Sup A) x" .
   qed
 next
   fix X :: "'a multiset" and A
   assume nonempty: "A \<noteq> {}" and ge: "\<And>Y. Y \<in> A \<Longrightarrow> Y \<subseteq># X"
   from ge have bdd: "subset_mset.bdd_above A"
     by blast
   show "Sup A \<subseteq># X"
   proof (rule mset_subset_eqI)
     fix x
     from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd]
       have "count (Sup A) x = (SUP X\<in>A. count X x)" .
     also from nonempty have "\<dots> \<le> count X x"
       by (intro cSup_least) (auto intro: mset_subset_eq_count ge)
     finally show "count (Sup A) x \<le> count X x" .
   qed
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 lemma set_mset_Inf:
   assumes "A \<noteq> {}"
   shows   "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)"
 proof safe
   fix x X assume "x \<in># Inf A" "X \<in> A"
   hence nonempty: "A \<noteq> {}" by (auto simp: Inf_multiset_empty)
   from \<open>x \<in># Inf A\<close> have "{#x#} \<subseteq># Inf A" by auto
   also from \<open>X \<in> A\<close> have "\<dots> \<subseteq># X" by (rule subset_mset.cInf_lower) simp_all
   finally show "x \<in># X" by simp
 next
   fix x assume x: "x \<in> (\<Inter>X\<in>A. set_mset X)"
   hence "{#x#} \<subseteq># X" if "X \<in> A" for X using that by auto
   from assms and this have "{#x#} \<subseteq># Inf A" by (rule subset_mset.cInf_greatest)
   thus "x \<in># Inf A" by simp
 qed
 
 lemma in_Inf_multiset_iff:
   assumes "A \<noteq> {}"
   shows   "x \<in># Inf A \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)"
 proof -
   from assms have "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)" by (rule set_mset_Inf)
   also have "x \<in> \<dots> \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)" by simp
   finally show ?thesis .
 qed
 
 lemma in_Inf_multisetD: "x \<in># Inf A \<Longrightarrow> X \<in> A \<Longrightarrow> x \<in># X"
   by (subst (asm) in_Inf_multiset_iff) auto
 
 lemma set_mset_Sup:
   assumes "subset_mset.bdd_above A"
   shows   "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)"
 proof safe
   fix x assume "x \<in># Sup A"
   hence nonempty: "A \<noteq> {}" by (auto simp: Sup_multiset_empty)
   show "x \<in> (\<Union>X\<in>A. set_mset X)"
   proof (rule ccontr)
     assume x: "x \<notin> (\<Union>X\<in>A. set_mset X)"
     have "count X x \<le> count (Sup A) x" if "X \<in> A" for X x
       using that by (intro mset_subset_eq_count subset_mset.cSup_upper assms)
     with x have "X \<subseteq># Sup A - {#x#}" if "X \<in> A" for X
       using that by (auto simp: subseteq_mset_def algebra_simps not_in_iff)
     hence "Sup A \<subseteq># Sup A - {#x#}" by (intro subset_mset.cSup_least nonempty)
     with \<open>x \<in># Sup A\<close> show False
-      by (auto simp: subseteq_mset_def simp flip: count_greater_zero_iff
-               dest!: spec[of _ x])
+      using mset_subset_diff_self by fastforce
   qed
 next
   fix x X assume "x \<in> set_mset X" "X \<in> A"
   hence "{#x#} \<subseteq># X" by auto
   also have "X \<subseteq># Sup A" by (intro subset_mset.cSup_upper \<open>X \<in> A\<close> assms)
   finally show "x \<in> set_mset (Sup A)" by simp
 qed
 
 lemma in_Sup_multiset_iff:
   assumes "subset_mset.bdd_above A"
   shows   "x \<in># Sup A \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)"
-proof -
-  from assms have "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)" by (rule set_mset_Sup)
-  also have "x \<in> \<dots> \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)" by simp
-  finally show ?thesis .
-qed
+  by (simp add: assms set_mset_Sup)
 
 lemma in_Sup_multisetD:
   assumes "x \<in># Sup A"
   shows   "\<exists>X\<in>A. x \<in># X"
-proof -
-  have "subset_mset.bdd_above A"
-    by (rule ccontr) (insert assms, simp_all add: Sup_multiset_unbounded)
-  with assms show ?thesis by (simp add: in_Sup_multiset_iff)
-qed
+  using Sup_multiset_unbounded assms in_Sup_multiset_iff by fastforce
 
 interpretation subset_mset: distrib_lattice "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)" "(\<union>#)"
 proof
   fix A B C :: "'a multiset"
   show "A \<union># (B \<inter># C) = A \<union># B \<inter># (A \<union># C)"
     by (intro multiset_eqI) simp_all
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 
 subsubsection \<open>Filter (with comprehension syntax)\<close>
 
 text \<open>Multiset comprehension\<close>
 
 lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
 is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
 by (rule filter_preserves_multiset)
 
 syntax (ASCII)
   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{#_ :# _./ _#})")
 syntax
   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{#_ \<in># _./ _#})")
 translations
   "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"
 
 lemma count_filter_mset [simp]:
   "count (filter_mset P M) a = (if P a then count M a else 0)"
   by (simp add: filter_mset.rep_eq)
 
 lemma set_mset_filter [simp]:
   "set_mset (filter_mset P M) = {a \<in> set_mset M. P a}"
   by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_filter_mset) simp
 
 lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}"
   by (rule multiset_eqI) simp
 
 lemma filter_single_mset: "filter_mset P {#x#} = (if P x then {#x#} else {#})"
   by (rule multiset_eqI) simp
 
 lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
   by (rule multiset_eqI) simp
 
 lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
   by (rule multiset_eqI) simp
 
 lemma filter_inter_mset [simp]: "filter_mset P (M \<inter># N) = filter_mset P M \<inter># filter_mset P N"
   by (rule multiset_eqI) simp
 
 lemma filter_sup_mset[simp]: "filter_mset P (A \<union># B) = filter_mset P A \<union># filter_mset P B"
   by (rule multiset_eqI) simp
 
 lemma filter_mset_add_mset [simp]:
    "filter_mset P (add_mset x A) =
      (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
    by (auto simp: multiset_eq_iff)
 
 lemma multiset_filter_subset[simp]: "filter_mset f M \<subseteq># M"
   by (simp add: mset_subset_eqI)
 
 lemma multiset_filter_mono:
   assumes "A \<subseteq># B"
   shows "filter_mset f A \<subseteq># filter_mset f B"
-proof -
-  from assms[unfolded mset_subset_eq_exists_conv]
-  obtain C where B: "B = A + C" by auto
-  show ?thesis unfolding B by auto
-qed
+  by (metis assms filter_sup_mset subset_mset.order_iff)
 
 lemma filter_mset_eq_conv:
   "filter_mset P M = N \<longleftrightarrow> N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a)" (is "?P \<longleftrightarrow> ?Q")
 proof
   assume ?P then show ?Q by auto (simp add: multiset_eq_iff in_diff_count)
 next
   assume ?Q
   then obtain Q where M: "M = N + Q"
     by (auto simp add: mset_subset_eq_exists_conv)
   then have MN: "M - N = Q" by simp
   show ?P
   proof (rule multiset_eqI)
     fix a
     from \<open>?Q\<close> MN have *: "\<not> P a \<Longrightarrow> a \<notin># N" "P a \<Longrightarrow> a \<notin># Q"
       by auto
     show "count (filter_mset P M) a = count N a"
     proof (cases "a \<in># M")
       case True
       with * show ?thesis
         by (simp add: not_in_iff M)
     next
       case False then have "count M a = 0"
         by (simp add: not_in_iff)
       with M show ?thesis by simp
     qed
   qed
 qed
 
 lemma filter_filter_mset: "filter_mset P (filter_mset Q M) = {#x \<in># M. Q x \<and> P x#}"
   by (auto simp: multiset_eq_iff)
 
 lemma
   filter_mset_True[simp]: "{#y \<in># M. True#} = M" and
   filter_mset_False[simp]: "{#y \<in># M. False#} = {#}"
   by (auto simp: multiset_eq_iff)
 
 lemma filter_mset_cong0:
   assumes "\<And>x. x \<in># M \<Longrightarrow> f x \<longleftrightarrow> g x"
   shows "filter_mset f M = filter_mset g M"
 proof (rule subset_mset.antisym; unfold subseteq_mset_def; rule allI)
   fix x
   show "count (filter_mset f M) x \<le> count (filter_mset g M) x"
     using assms by (cases "x \<in># M") (simp_all add: not_in_iff)
 next
   fix x
   show "count (filter_mset g M) x \<le> count (filter_mset f M) x"
     using assms by (cases "x \<in># M") (simp_all add: not_in_iff)
 qed
 
 lemma filter_mset_cong:
   assumes "M = M'" and "\<And>x. x \<in># M' \<Longrightarrow> f x \<longleftrightarrow> g x"
   shows "filter_mset f M = filter_mset g M'"
   unfolding \<open>M = M'\<close>
   using assms by (auto intro: filter_mset_cong0)
 
 lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
   by (induct D) (simp add: multiset_eqI)
 
 
 subsubsection \<open>Size\<close>
 
 definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"
 
 lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
   by (auto simp: wcount_def add_mult_distrib)
 
 lemma wcount_add_mset:
   "wcount f (add_mset x M) a = (if x = a then Suc (f a) else 0) + wcount f M a"
   unfolding add_mset_add_single[of _ M] wcount_union by (auto simp: wcount_def)
 
 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
   "size_multiset f M = sum (wcount f M) (set_mset M)"
 
 lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]
 
 instantiation multiset :: (type) size
 begin
 
 definition size_multiset where
   size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
 instance ..
 
 end
 
 lemmas size_multiset_overloaded_eq =
   size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]
 
 lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
-by (simp add: size_multiset_def)
+  by (simp add: size_multiset_def)
 
 lemma size_empty [simp]: "size {#} = 0"
-by (simp add: size_multiset_overloaded_def)
+  by (simp add: size_multiset_overloaded_def)
 
 lemma size_multiset_single : "size_multiset f {#b#} = Suc (f b)"
-by (simp add: size_multiset_eq)
+  by (simp add: size_multiset_eq)
 
 lemma size_single: "size {#b#} = 1"
-by (simp add: size_multiset_overloaded_def size_multiset_single)
+  by (simp add: size_multiset_overloaded_def size_multiset_single)
 
 lemma sum_wcount_Int:
   "finite A \<Longrightarrow> sum (wcount f N) (A \<inter> set_mset N) = sum (wcount f N) A"
   by (induct rule: finite_induct)
     (simp_all add: Int_insert_left wcount_def count_eq_zero_iff)
 
 lemma size_multiset_union [simp]:
   "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
-apply (simp add: size_multiset_def sum_Un_nat sum.distrib sum_wcount_Int wcount_union)
-apply (subst Int_commute)
-apply (simp add: sum_wcount_Int)
-done
+  apply (simp add: size_multiset_def sum_Un_nat sum.distrib sum_wcount_Int wcount_union)
+  by (metis add_implies_diff finite_set_mset inf.commute sum_wcount_Int)
 
 lemma size_multiset_add_mset [simp]:
   "size_multiset f (add_mset a M) = Suc (f a) + size_multiset f M"
-  unfolding add_mset_add_single[of _ M] size_multiset_union by (auto simp: size_multiset_single)
+  by (metis add.commute add_mset_add_single size_multiset_single size_multiset_union)
 
 lemma size_add_mset [simp]: "size (add_mset a A) = Suc (size A)"
-by (simp add: size_multiset_overloaded_def wcount_add_mset)
+  by (simp add: size_multiset_overloaded_def wcount_add_mset)
 
 lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
-by (auto simp add: size_multiset_overloaded_def)
+  by (auto simp add: size_multiset_overloaded_def)
 
 lemma size_multiset_eq_0_iff_empty [iff]:
   "size_multiset f M = 0 \<longleftrightarrow> M = {#}"
   by (auto simp add: size_multiset_eq count_eq_zero_iff)
 
 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
 by (auto simp add: size_multiset_overloaded_def)
 
 lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
-by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
+  by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
 
 lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
-apply (unfold size_multiset_overloaded_eq)
-apply (drule sum_SucD)
-apply auto
-done
+  using all_not_in_conv by fastforce
 
 lemma size_eq_Suc_imp_eq_union:
   assumes "size M = Suc n"
   shows "\<exists>a N. M = add_mset a N"
-proof -
-  from assms obtain a where "a \<in># M"
-    by (erule size_eq_Suc_imp_elem [THEN exE])
-  then have "M = add_mset a (M - {#a#})" by simp
-  then show ?thesis by blast
-qed
+  by (metis assms insert_DiffM size_eq_Suc_imp_elem)
 
 lemma size_mset_mono:
   fixes A B :: "'a multiset"
   assumes "A \<subseteq># B"
   shows "size A \<le> size B"
 proof -
   from assms[unfolded mset_subset_eq_exists_conv]
   obtain C where B: "B = A + C" by auto
   show ?thesis unfolding B by (induct C) auto
 qed
 
 lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
-by (rule size_mset_mono[OF multiset_filter_subset])
+  by (rule size_mset_mono[OF multiset_filter_subset])
 
 lemma size_Diff_submset:
   "M \<subseteq># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
 by (metis add_diff_cancel_left' size_union mset_subset_eq_exists_conv)
 
 lemma size_lt_imp_ex_count_lt: "size M < size N \<Longrightarrow> \<exists>x \<in># N. count M x < count N x"
   by (metis count_eq_zero_iff leD not_le_imp_less not_less_zero size_mset_mono subseteq_mset_def)
 
 
 subsection \<open>Induction and case splits\<close>
 
 theorem multiset_induct [case_names empty add, induct type: multiset]:
   assumes empty: "P {#}"
   assumes add: "\<And>x M. P M \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case 0 thus "P M" by (simp add: empty)
 next
   case (Suc k)
   obtain N x where "M = add_mset x N"
     using \<open>Suc k = size M\<close> [symmetric]
     using size_eq_Suc_imp_eq_union by fast
   with Suc add show "P M" by simp
 qed
 
 lemma multiset_induct_min[case_names empty add]:
   fixes M :: "'a::linorder multiset"
   assumes
     empty: "P {#}" and
     add: "\<And>x M. P M \<Longrightarrow> (\<forall>y \<in># M. y \<ge> x) \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case (Suc k)
   note ih = this(1) and Sk_eq_sz_M = this(2)
 
   let ?y = "Min_mset M"
   let ?N = "M - {#?y#}"
 
   have M: "M = add_mset ?y ?N"
     by (metis Min_in Sk_eq_sz_M finite_set_mset insert_DiffM lessI not_less_zero
       set_mset_eq_empty_iff size_empty)
   show ?case
     by (subst M, rule add, rule ih, metis M Sk_eq_sz_M nat.inject size_add_mset,
       meson Min_le finite_set_mset in_diffD)
 qed (simp add: empty)
 
 lemma multiset_induct_max[case_names empty add]:
   fixes M :: "'a::linorder multiset"
   assumes
     empty: "P {#}" and
     add: "\<And>x M. P M \<Longrightarrow> (\<forall>y \<in># M. y \<le> x) \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case (Suc k)
   note ih = this(1) and Sk_eq_sz_M = this(2)
 
   let ?y = "Max_mset M"
   let ?N = "M - {#?y#}"
 
   have M: "M = add_mset ?y ?N"
     by (metis Max_in Sk_eq_sz_M finite_set_mset insert_DiffM lessI not_less_zero
       set_mset_eq_empty_iff size_empty)
   show ?case
     by (subst M, rule add, rule ih, metis M Sk_eq_sz_M nat.inject size_add_mset,
       meson Max_ge finite_set_mset in_diffD)
 qed (simp add: empty)
 
 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A"
-by (induct M) auto
+  by (induct M) auto
 
 lemma multiset_cases [cases type]:
-  obtains (empty) "M = {#}"
-    | (add) x N where "M = add_mset x N"
+  obtains (empty) "M = {#}" | (add) x N where "M = add_mset x N"
   by (induct M) simp_all
 
 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
-by (cases "B = {#}") (auto dest: multi_member_split)
+  by (cases "B = {#}") (auto dest: multi_member_split)
 
 lemma union_filter_mset_complement[simp]:
   "\<forall>x. P x = (\<not> Q x) \<Longrightarrow> filter_mset P M + filter_mset Q M = M"
-by (subst multiset_eq_iff) auto
+  by (subst multiset_eq_iff) auto
 
 lemma multiset_partition: "M = {#x \<in># M. P x#} + {#x \<in># M. \<not> P x#}"
-by simp
+  by simp
 
 lemma mset_subset_size: "A \<subset># B \<Longrightarrow> size A < size B"
 proof (induct A arbitrary: B)
   case empty
   then show ?case
     using nonempty_has_size by auto
 next
   case (add x A)
   have "add_mset x A \<subseteq># B"
     by (meson add.prems subset_mset_def)
   then show ?case
-    by (metis (no_types) add.prems add.right_neutral add_diff_cancel_left' leD nat_neq_iff
-      size_Diff_submset size_eq_0_iff_empty size_mset_mono subset_mset.le_iff_add subset_mset_def)
+    using add.prems subset_mset.less_eqE by fastforce
 qed
 
 lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
   by (cases M) auto
 
 lemma set_mset_subset_singletonD:
   assumes "set_mset A \<subseteq> {x}"
   shows   "A = replicate_mset (size A) x"
   using assms by (induction A) auto
 
 
 subsubsection \<open>Strong induction and subset induction for multisets\<close>
 
 text \<open>Well-foundedness of strict subset relation\<close>
 
 lemma wf_subset_mset_rel: "wf {(M, N :: 'a multiset). M \<subset># N}"
-apply (rule wf_measure [THEN wf_subset, where f1=size])
-apply (clarsimp simp: measure_def inv_image_def mset_subset_size)
-done
+  using mset_subset_size wfP_def wfP_if_convertible_to_nat by blast
 
 lemma wfP_subset_mset[simp]: "wfP (\<subset>#)"
   by (rule wf_subset_mset_rel[to_pred])
 
 lemma full_multiset_induct [case_names less]:
-assumes ih: "\<And>B. \<forall>(A::'a multiset). A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
-shows "P B"
-apply (rule wf_subset_mset_rel [THEN wf_induct])
-apply (rule ih, auto)
-done
+  assumes ih: "\<And>B. \<forall>(A::'a multiset). A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
+  shows "P B"
+  apply (rule wf_subset_mset_rel [THEN wf_induct])
+  apply (rule ih, auto)
+  done
 
 lemma multi_subset_induct [consumes 2, case_names empty add]:
   assumes "F \<subseteq># A"
     and empty: "P {#}"
     and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (add_mset a F)"
   shows "P F"
 proof -
   from \<open>F \<subseteq># A\<close>
   show ?thesis
   proof (induct F)
     show "P {#}" by fact
   next
     fix x F
     assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "add_mset x F \<subseteq># A"
     show "P (add_mset x F)"
     proof (rule insert)
       from i show "x \<in># A" by (auto dest: mset_subset_eq_insertD)
       from i have "F \<subseteq># A" by (auto dest: mset_subset_eq_insertD)
       with P show "P F" .
     qed
   qed
 qed
 
 
 subsection \<open>Least and greatest elements\<close>
 
 context begin
 
 qualified lemma
   assumes
     "M \<noteq> {#}" and
     "transp_on (set_mset M) R" and
     "totalp_on (set_mset M) R"
   shows
     bex_least_element: "(\<exists>l \<in># M. \<forall>x \<in># M. x \<noteq> l \<longrightarrow> R l x)" and
     bex_greatest_element: "(\<exists>g \<in># M. \<forall>x \<in># M. x \<noteq> g \<longrightarrow> R x g)"
   using assms
   by (auto intro: Finite_Set.bex_least_element Finite_Set.bex_greatest_element)
 
 end
 
 
 subsection \<open>The fold combinator\<close>
 
 definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
 where
   "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
 
 lemma fold_mset_empty [simp]: "fold_mset f s {#} = s"
   by (simp add: fold_mset_def)
 
 lemma fold_mset_single [simp]: "fold_mset f s {#x#} = f x s"
   by (simp add: fold_mset_def)
 
 context comp_fun_commute
 begin
 
 lemma fold_mset_add_mset [simp]: "fold_mset f s (add_mset x M) = f x (fold_mset f s M)"
 proof -
   interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
     by (fact comp_fun_commute_funpow)
   interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (add_mset x M) y"
     by (fact comp_fun_commute_funpow)
   show ?thesis
   proof (cases "x \<in> set_mset M")
     case False
     then have *: "count (add_mset x M) x = 1"
       by (simp add: not_in_iff)
     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (add_mset x M) y) s (set_mset M) =
       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)"
       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_on_funpow)
     with False * show ?thesis
       by (simp add: fold_mset_def del: count_add_mset)
   next
     case True
     define N where "N = set_mset M - {x}"
     from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto
     then have "Finite_Set.fold (\<lambda>y. f y ^^ count (add_mset x M) y) s N =
       Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
       by (auto intro!: Finite_Set.fold_cong comp_fun_commute_on_funpow)
     with * show ?thesis by (simp add: fold_mset_def del: count_add_mset) simp
   qed
 qed
 
 lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M"
   by (induct M) (simp_all add: fun_left_comm)
 
 lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
   by (induct M) (simp_all add: fold_mset_fun_left_comm)
 
 lemma fold_mset_fusion:
   assumes "comp_fun_commute g"
     and *: "\<And>x y. h (g x y) = f x (h y)"
   shows "h (fold_mset g w A) = fold_mset f (h w) A"
 proof -
   interpret comp_fun_commute g by (fact assms)
   from * show ?thesis by (induct A) auto
 qed
 
 end
 
 lemma union_fold_mset_add_mset: "A + B = fold_mset add_mset A B"
 proof -
   interpret comp_fun_commute add_mset
     by standard auto
   show ?thesis
     by (induction B) auto
 qed
 
 text \<open>
   A note on code generation: When defining some function containing a
   subterm \<^term>\<open>fold_mset F\<close>, code generation is not automatic. When
   interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
   would be code thms for \<^const>\<open>fold_mset\<close> become thms like
   \<^term>\<open>fold_mset F z {#} = z\<close> where \<open>F\<close> is not a pattern but
   contains defined symbols, i.e.\ is not a code thm. Hence a separate
   constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below.
 \<close>
 
 
 subsection \<open>Image\<close>
 
 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
   "image_mset f = fold_mset (add_mset \<circ> f) {#}"
 
 lemma comp_fun_commute_mset_image: "comp_fun_commute (add_mset \<circ> f)"
   by unfold_locales (simp add: fun_eq_iff)
 
 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
   by (simp add: image_mset_def)
 
 lemma image_mset_single: "image_mset f {#x#} = {#f x#}"
   by (simp add: comp_fun_commute.fold_mset_add_mset comp_fun_commute_mset_image image_mset_def)
 
 lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
 proof -
   interpret comp_fun_commute "add_mset \<circ> f"
     by (fact comp_fun_commute_mset_image)
   show ?thesis by (induct N) (simp_all add: image_mset_def)
 qed
 
 corollary image_mset_add_mset [simp]:
   "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
   unfolding image_mset_union add_mset_add_single[of a M] by (simp add: image_mset_single)
 
 lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)"
   by (induct M) simp_all
 
 lemma size_image_mset [simp]: "size (image_mset f M) = size M"
   by (induct M) simp_all
 
 lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
   by (cases M) auto
 
 lemma image_mset_If:
   "image_mset (\<lambda>x. if P x then f x else g x) A =
      image_mset f (filter_mset P A) + image_mset g (filter_mset (\<lambda>x. \<not>P x) A)"
   by (induction A) auto
 
 lemma image_mset_Diff:
   assumes "B \<subseteq># A"
   shows   "image_mset f (A - B) = image_mset f A - image_mset f B"
 proof -
   have "image_mset f (A - B + B) = image_mset f (A - B) + image_mset f B"
     by simp
   also from assms have "A - B + B = A"
     by (simp add: subset_mset.diff_add)
   finally show ?thesis by simp
 qed
 
 lemma count_image_mset:
   \<open>count (image_mset f A) x = (\<Sum>y\<in>f -` {x} \<inter> set_mset A. count A y)\<close>
 proof (induction A)
   case empty
   then show ?case by simp
 next
   case (add x A)
   moreover have *: "(if x = y then Suc n else n) = n + (if x = y then 1 else 0)" for n y
     by simp
   ultimately show ?case
     by (auto simp: sum.distrib intro!: sum.mono_neutral_left)
 qed
 
 lemma count_image_mset':
   \<open>count (image_mset f X) y = (\<Sum>x | x \<in># X \<and> y = f x. count X x)\<close>
   by (auto simp add: count_image_mset simp flip: singleton_conv2 simp add: Collect_conj_eq ac_simps)
 
 lemma image_mset_subseteq_mono: "A \<subseteq># B \<Longrightarrow> image_mset f A \<subseteq># image_mset f B"
   by (metis image_mset_union subset_mset.le_iff_add)
 
 lemma image_mset_subset_mono: "M \<subset># N \<Longrightarrow> image_mset f M \<subset># image_mset f N"
   by (metis (no_types) Diff_eq_empty_iff_mset image_mset_Diff image_mset_is_empty_iff
     image_mset_subseteq_mono subset_mset.less_le_not_le)
 
 syntax (ASCII)
   "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ :# _#})")
 syntax
   "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ \<in># _#})")
 translations
   "{#e. x \<in># M#}" \<rightleftharpoons> "CONST image_mset (\<lambda>x. e) M"
 
 syntax (ASCII)
   "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ :# _./ _#})")
 syntax
   "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ \<in># _./ _#})")
 translations
   "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"
 
 text \<open>
   This allows to write not just filters like \<^term>\<open>{#x\<in>#M. x<c#}\<close>
   but also images like \<^term>\<open>{#x+x. x\<in>#M #}\<close> and @{term [source]
   "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
   \<^term>\<open>{#x+x|x\<in>#M. x<c#}\<close>.
 \<close>
 
 lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
   by simp
 
 functor image_mset: image_mset
 proof -
   fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
   proof
     fix A
     show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
       by (induct A) simp_all
   qed
   show "image_mset id = id"
   proof
     fix A
     show "image_mset id A = id A"
       by (induct A) simp_all
   qed
 qed
 
 declare
   image_mset.id [simp]
   image_mset.identity [simp]
 
 lemma image_mset_id[simp]: "image_mset id x = x"
   unfolding id_def by auto
 
 lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
   by (induct M) auto
 
 lemma image_mset_cong_pair:
   "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
   by (metis image_mset_cong split_cong)
 
 lemma image_mset_const_eq:
   "{#c. a \<in># M#} = replicate_mset (size M) c"
   by (induct M) simp_all
 
 lemma image_mset_filter_mset_swap:
   "image_mset f (filter_mset (\<lambda>x. P (f x)) M) = filter_mset P (image_mset f M)"
   by (induction M rule: multiset_induct) simp_all
 
 lemma image_mset_eq_plusD:
   "image_mset f A = B + C \<Longrightarrow> \<exists>B' C'. A = B' + C' \<and> B = image_mset f B' \<and> C = image_mset f C'"
 proof (induction A arbitrary: B C)
   case empty
   thus ?case by simp
 next
   case (add x A)
   show ?case
   proof (cases "f x \<in># B")
     case True
     with add.prems have "image_mset f A = (B - {#f x#}) + C"
       by (metis add_mset_remove_trivial image_mset_add_mset mset_subset_eq_single
           subset_mset.add_diff_assoc2)
     thus ?thesis
       using add.IH add.prems by force
   next
     case False
     with add.prems have "image_mset f A = B + (C - {#f x#})"
       by (metis diff_single_eq_union diff_union_single_conv image_mset_add_mset union_iff
           union_single_eq_member)
     then show ?thesis
       using add.IH add.prems by force
   qed
 qed
 
 lemma image_mset_eq_image_mset_plusD:
   assumes "image_mset f A = image_mset f B + C" and inj_f: "inj_on f (set_mset A \<union> set_mset B)"
   shows "\<exists>C'. A = B + C' \<and> C = image_mset f C'"
   using assms
 proof (induction A arbitrary: B C)
   case empty
   thus ?case by simp
 next
   case (add x A)
   show ?case
   proof (cases "x \<in># B")
     case True
     with add.prems have "image_mset f A = image_mset f (B - {#x#}) + C"
-      by (smt (verit, del_insts) add.left_commute add_cancel_right_left diff_union_cancelL
-          diff_union_single_conv image_mset_union union_mset_add_mset_left
-          union_mset_add_mset_right)
+      by (smt (verit) add_mset_add_mset_same_iff image_mset_add_mset insert_DiffM union_mset_add_mset_left)
     with add.IH have "\<exists>M3'. A = B - {#x#} + M3' \<and> image_mset f M3' = C"
       by (smt (verit, del_insts) True Un_insert_left Un_insert_right add.prems(2) inj_on_insert
           insert_DiffM set_mset_add_mset_insert)
     with True show ?thesis
       by auto
   next
     case False
     with add.prems(2) have "f x \<notin># image_mset f B"
       by auto
     with add.prems(1) have "image_mset f A = image_mset f B + (C - {#f x#})"
       by (metis (no_types, lifting) diff_union_single_conv image_eqI image_mset_Diff
           image_mset_single mset_subset_eq_single set_image_mset union_iff union_single_eq_diff
           union_single_eq_member)
     with add.prems(2) add.IH have "\<exists>M3'. A = B + M3' \<and> C - {#f x#} = image_mset f M3'"
       by auto
     then show ?thesis
       by (metis add.prems(1) add_diff_cancel_left' image_mset_Diff mset_subset_eq_add_left
           union_mset_add_mset_right)
   qed
 qed
 
 lemma image_mset_eq_plus_image_msetD:
   "image_mset f A = B + image_mset f C \<Longrightarrow> inj_on f (set_mset A \<union> set_mset C) \<Longrightarrow>
   \<exists>B'. A = B' + C \<and> B = image_mset f B'"
   unfolding add.commute[of B] add.commute[of _ C]
   by (rule image_mset_eq_image_mset_plusD; assumption)
 
 
 subsection \<open>Further conversions\<close>
 
 primrec mset :: "'a list \<Rightarrow> 'a multiset" where
   "mset [] = {#}" |
   "mset (a # x) = add_mset a (mset x)"
 
 lemma in_multiset_in_set:
   "x \<in># mset xs \<longleftrightarrow> x \<in> set xs"
   by (induct xs) simp_all
 
 lemma count_mset:
   "count (mset xs) x = length (filter (\<lambda>y. x = y) xs)"
   by (induct xs) simp_all
 
 lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])"
   by (induct x) auto
 
 lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])"
 by (induct x) auto
 
 lemma count_mset_gt_0: "x \<in> set xs \<Longrightarrow> count (mset xs) x > 0"
   by (induction xs) auto
 
 lemma count_mset_0_iff [simp]: "count (mset xs) x = 0 \<longleftrightarrow> x \<notin> set xs"
   by (induction xs) auto
 
 lemma mset_single_iff[iff]: "mset xs = {#x#} \<longleftrightarrow> xs = [x]"
   by (cases xs) auto
 
 lemma mset_single_iff_right[iff]: "{#x#} = mset xs \<longleftrightarrow> xs = [x]"
   by (cases xs) auto
 
 lemma set_mset_mset[simp]: "set_mset (mset xs) = set xs"
   by (induct xs) auto
 
 lemma set_mset_comp_mset [simp]: "set_mset \<circ> mset = set"
   by (simp add: fun_eq_iff)
 
 lemma size_mset [simp]: "size (mset xs) = length xs"
   by (induct xs) simp_all
 
 lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
   by (induct xs arbitrary: ys) auto
 
 lemma mset_filter[simp]: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
   by (induct xs) simp_all
 
 lemma mset_rev [simp]:
   "mset (rev xs) = mset xs"
   by (induct xs) simp_all
 
 lemma surj_mset: "surj mset"
   unfolding surj_def
 proof (rule allI)
   fix M
   show "\<exists>xs. M = mset xs"
     by (induction M) (auto intro: exI[of _ "_ # _"])
 qed
 
 lemma distinct_count_atmost_1:
   "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
 proof (induct x)
   case Nil then show ?case by simp
 next
   case (Cons x xs) show ?case (is "?lhs \<longleftrightarrow> ?rhs")
   proof
     assume ?lhs then show ?rhs using Cons by simp
   next
     assume ?rhs then have "x \<notin> set xs"
       by (simp split: if_splits)
     moreover from \<open>?rhs\<close> have "(\<forall>a. count (mset xs) a =
        (if a \<in> set xs then 1 else 0))"
       by (auto split: if_splits simp add: count_eq_zero_iff)
     ultimately show ?lhs using Cons by simp
   qed
 qed
 
 lemma mset_eq_setD:
   assumes "mset xs = mset ys"
   shows "set xs = set ys"
 proof -
   from assms have "set_mset (mset xs) = set_mset (mset ys)"
     by simp
   then show ?thesis by simp
 qed
 
 lemma set_eq_iff_mset_eq_distinct:
   \<open>distinct x \<Longrightarrow> distinct y \<Longrightarrow> set x = set y \<longleftrightarrow> mset x = mset y\<close>
   by (auto simp: multiset_eq_iff distinct_count_atmost_1)
 
 lemma set_eq_iff_mset_remdups_eq:
   \<open>set x = set y \<longleftrightarrow> mset (remdups x) = mset (remdups y)\<close>
-apply (rule iffI)
-apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
-apply (drule distinct_remdups [THEN distinct_remdups
-      [THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]])
-apply simp
-done
+  using set_eq_iff_mset_eq_distinct by fastforce
 
 lemma mset_eq_imp_distinct_iff:
   \<open>distinct xs \<longleftrightarrow> distinct ys\<close> if \<open>mset xs = mset ys\<close>
   using that by (auto simp add: distinct_count_atmost_1 dest: mset_eq_setD)
 
 lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
 proof (induct ls arbitrary: i)
   case Nil
   then show ?case by simp
 next
   case Cons
   then show ?case by (cases i) auto
 qed
 
 lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}"
   by (induct xs) (auto simp add: multiset_eq_iff)
 
 lemma mset_eq_length:
   assumes "mset xs = mset ys"
   shows "length xs = length ys"
   using assms by (metis size_mset)
 
 lemma mset_eq_length_filter:
   assumes "mset xs = mset ys"
   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
   using assms by (metis count_mset)
 
 lemma fold_multiset_equiv:
   \<open>List.fold f xs = List.fold f ys\<close>
     if f: \<open>\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x\<close>
     and \<open>mset xs = mset ys\<close>
 using f \<open>mset xs = mset ys\<close> [symmetric] proof (induction xs arbitrary: ys)
   case Nil
   then show ?case by simp
 next
   case (Cons x xs)
   then have *: \<open>set ys = set (x # xs)\<close>
     by (blast dest: mset_eq_setD)
   have \<open>\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x\<close>
     by (rule Cons.prems(1)) (simp_all add: *)
   moreover from * have \<open>x \<in> set ys\<close>
     by simp
   ultimately have \<open>List.fold f ys = List.fold f (remove1 x ys) \<circ> f x\<close>
     by (fact fold_remove1_split)
   moreover from Cons.prems have \<open>List.fold f xs = List.fold f (remove1 x ys)\<close>
     by (auto intro: Cons.IH)
   ultimately show ?case
     by simp
 qed
 
 lemma fold_permuted_eq:
   \<open>List.fold (\<odot>) xs z = List.fold (\<odot>) ys z\<close>
     if \<open>mset xs = mset ys\<close>
     and \<open>P z\<close> and P: \<open>\<And>x z. x \<in> set xs \<Longrightarrow> P z \<Longrightarrow> P (x \<odot> z)\<close>
     and f: \<open>\<And>x y z. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P z \<Longrightarrow> x \<odot> (y \<odot> z) = y \<odot> (x \<odot> z)\<close>
   for f (infixl \<open>\<odot>\<close> 70)
 using \<open>P z\<close> P f \<open>mset xs = mset ys\<close> [symmetric] proof (induction xs arbitrary: ys z)
   case Nil
   then show ?case by simp
 next
   case (Cons x xs)
   then have *: \<open>set ys = set (x # xs)\<close>
     by (blast dest: mset_eq_setD)
   have \<open>P z\<close>
     by (fact Cons.prems(1))
   moreover have \<open>\<And>x z. x \<in> set ys \<Longrightarrow> P z \<Longrightarrow> P (x \<odot> z)\<close>
     by (rule Cons.prems(2)) (simp_all add: *)
   moreover have \<open>\<And>x y z. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> P z \<Longrightarrow> x \<odot> (y \<odot> z) = y \<odot> (x \<odot> z)\<close>
     by (rule Cons.prems(3)) (simp_all add: *)
   moreover from * have \<open>x \<in> set ys\<close>
     by simp
   ultimately have \<open>fold (\<odot>) ys z = fold (\<odot>) (remove1 x ys) (x \<odot> z)\<close>
     by (induction ys arbitrary: z) auto
   moreover from Cons.prems have \<open>fold (\<odot>) xs (x \<odot> z) = fold (\<odot>) (remove1 x ys) (x \<odot> z)\<close>
     by (auto intro: Cons.IH)
   ultimately show ?case
     by simp
 qed
 
 lemma mset_shuffles: "zs \<in> shuffles xs ys \<Longrightarrow> mset zs = mset xs + mset ys"
   by (induction xs ys arbitrary: zs rule: shuffles.induct) auto
 
 lemma mset_insort [simp]: "mset (insort x xs) = add_mset x (mset xs)"
   by (induct xs) simp_all
 
 lemma mset_map[simp]: "mset (map f xs) = image_mset f (mset xs)"
   by (induct xs) simp_all
 
 global_interpretation mset_set: folding add_mset "{#}"
   defines mset_set = "folding_on.F add_mset {#}"
   by standard (simp add: fun_eq_iff)
 
 lemma sum_multiset_singleton [simp]: "sum (\<lambda>n. {#n#}) A = mset_set A"
   by (induction A rule: infinite_finite_induct) auto
 
 lemma count_mset_set [simp]:
   "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P")
   "\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q")
   "x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R")
 proof -
   have *: "count (mset_set A) x = 0" if "x \<notin> A" for A
   proof (cases "finite A")
     case False then show ?thesis by simp
   next
     case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto
   qed
   then show "PROP ?P" "PROP ?Q" "PROP ?R"
   by (auto elim!: Set.set_insert)
 qed \<comment> \<open>TODO: maybe define \<^const>\<open>mset_set\<close> also in terms of \<^const>\<open>Abs_multiset\<close>\<close>
 
 lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A"
   by (induct A rule: finite_induct) simp_all
 
 lemma mset_set_Union:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> mset_set (A \<union> B) = mset_set A + mset_set B"
   by (induction A rule: finite_induct) auto
 
 lemma filter_mset_mset_set [simp]:
   "finite A \<Longrightarrow> filter_mset P (mset_set A) = mset_set {x\<in>A. P x}"
 proof (induction A rule: finite_induct)
   case (insert x A)
   from insert.hyps have "filter_mset P (mset_set (insert x A)) =
       filter_mset P (mset_set A) + mset_set (if P x then {x} else {})"
     by simp
   also have "filter_mset P (mset_set A) = mset_set {x\<in>A. P x}"
     by (rule insert.IH)
   also from insert.hyps
     have "\<dots> + mset_set (if P x then {x} else {}) =
             mset_set ({x \<in> A. P x} \<union> (if P x then {x} else {}))" (is "_ = mset_set ?A")
      by (intro mset_set_Union [symmetric]) simp_all
   also from insert.hyps have "?A = {y\<in>insert x A. P y}" by auto
   finally show ?case .
 qed simp_all
 
 lemma mset_set_Diff:
   assumes "finite A" "B \<subseteq> A"
   shows  "mset_set (A - B) = mset_set A - mset_set B"
 proof -
   from assms have "mset_set ((A - B) \<union> B) = mset_set (A - B) + mset_set B"
     by (intro mset_set_Union) (auto dest: finite_subset)
   also from assms have "A - B \<union> B = A" by blast
   finally show ?thesis by simp
 qed
 
 lemma mset_set_set: "distinct xs \<Longrightarrow> mset_set (set xs) = mset xs"
   by (induction xs) simp_all
 
 lemma count_mset_set': "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
   by auto
 
 lemma subset_imp_msubset_mset_set: 
   assumes "A \<subseteq> B" "finite B"
   shows   "mset_set A \<subseteq># mset_set B"
 proof (rule mset_subset_eqI)
   fix x :: 'a
   from assms have "finite A" by (rule finite_subset)
   with assms show "count (mset_set A) x \<le> count (mset_set B) x"
     by (cases "x \<in> A"; cases "x \<in> B") auto
 qed
 
 lemma mset_set_set_mset_msubset: "mset_set (set_mset A) \<subseteq># A"
 proof (rule mset_subset_eqI)
   fix x show "count (mset_set (set_mset A)) x \<le> count A x"
     by (cases "x \<in># A") simp_all
 qed
 
 lemma mset_set_upto_eq_mset_upto:
   \<open>mset_set {..<n} = mset [0..<n]\<close>
   by (induction n) (auto simp: ac_simps lessThan_Suc)
 
 context linorder
 begin
 
 definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
 where
   "sorted_list_of_multiset M = fold_mset insort [] M"
 
 lemma sorted_list_of_multiset_empty [simp]:
   "sorted_list_of_multiset {#} = []"
   by (simp add: sorted_list_of_multiset_def)
 
 lemma sorted_list_of_multiset_singleton [simp]:
   "sorted_list_of_multiset {#x#} = [x]"
 proof -
   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
   show ?thesis by (simp add: sorted_list_of_multiset_def)
 qed
 
 lemma sorted_list_of_multiset_insert [simp]:
   "sorted_list_of_multiset (add_mset x M) = List.insort x (sorted_list_of_multiset M)"
 proof -
   interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
   show ?thesis by (simp add: sorted_list_of_multiset_def)
 qed
 
 end
 
 lemma mset_sorted_list_of_multiset[simp]: "mset (sorted_list_of_multiset M) = M"
   by (induct M) simp_all
 
 lemma sorted_list_of_multiset_mset[simp]: "sorted_list_of_multiset (mset xs) = sort xs"
   by (induct xs) simp_all
 
 lemma finite_set_mset_mset_set[simp]: "finite A \<Longrightarrow> set_mset (mset_set A) = A"
   by auto
 
 lemma mset_set_empty_iff: "mset_set A = {#} \<longleftrightarrow> A = {} \<or> infinite A"
   using finite_set_mset_mset_set by fastforce
 
 lemma infinite_set_mset_mset_set: "\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}"
   by simp
 
 lemma set_sorted_list_of_multiset [simp]:
   "set (sorted_list_of_multiset M) = set_mset M"
-by (induct M) (simp_all add: set_insort_key)
+  by (induct M) (simp_all add: set_insort_key)
 
 lemma sorted_list_of_mset_set [simp]:
   "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
-by (cases "finite A") (induct A rule: finite_induct, simp_all)
+  by (cases "finite A") (induct A rule: finite_induct, simp_all)
 
 lemma mset_upt [simp]: "mset [m..<n] = mset_set {m..<n}"
-  by (induction n) (simp_all add: atLeastLessThanSuc)
+  by (metis distinct_upt mset_set_set set_upt)
 
 lemma image_mset_map_of:
   "distinct (map fst xs) \<Longrightarrow> {#the (map_of xs i). i \<in># mset (map fst xs)#} = mset (map snd xs)"
 proof (induction xs)
   case (Cons x xs)
   have "{#the (map_of (x # xs) i). i \<in># mset (map fst (x # xs))#} =
           add_mset (snd x) {#the (if i = fst x then Some (snd x) else map_of xs i).
              i \<in># mset (map fst xs)#}" (is "_ = add_mset _ ?A") by simp
   also from Cons.prems have "?A = {#the (map_of xs i). i :# mset (map fst xs)#}"
     by (cases x, intro image_mset_cong) (auto simp: in_multiset_in_set)
   also from Cons.prems have "\<dots> = mset (map snd xs)" by (intro Cons.IH) simp_all
   finally show ?case by simp
 qed simp_all
 
 lemma msubset_mset_set_iff[simp]:
   assumes "finite A" "finite B"
   shows "mset_set A \<subseteq># mset_set B \<longleftrightarrow> A \<subseteq> B"
   using assms set_mset_mono subset_imp_msubset_mset_set by fastforce
 
 lemma mset_set_eq_iff[simp]:
   assumes "finite A" "finite B"
   shows "mset_set A = mset_set B \<longleftrightarrow> A = B"
   using assms by (fastforce dest: finite_set_mset_mset_set)
 
 lemma image_mset_mset_set: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
   assumes "inj_on f A"
   shows "image_mset f (mset_set A) = mset_set (f ` A)"
 proof cases
   assume "finite A"
   from this \<open>inj_on f A\<close> show ?thesis
     by (induct A) auto
 next
   assume "infinite A"
   from this \<open>inj_on f A\<close> have "infinite (f ` A)"
     using finite_imageD by blast
   from \<open>infinite A\<close> \<open>infinite (f ` A)\<close> show ?thesis by simp
 qed
 
 
 subsection \<open>More properties of the replicate, repeat, and image operations\<close>
 
 lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
   unfolding replicate_mset_def by (induct n) auto
 
 lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
-  by (auto split: if_splits)
+  by auto
 
 lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
   by (induct n, simp_all)
 
 lemma size_repeat_mset [simp]: "size (repeat_mset n A) = n * size A"
   by (induction n) auto
 
 lemma size_multiset_sum [simp]: "size (\<Sum>x\<in>A. f x :: 'a multiset) = (\<Sum>x\<in>A. size (f x))"
   by (induction A rule: infinite_finite_induct) auto
 
 lemma size_multiset_sum_list [simp]: "size (\<Sum>X\<leftarrow>Xs. X :: 'a multiset) = (\<Sum>X\<leftarrow>Xs. size X)"
   by (induction Xs) auto
 
 lemma count_le_replicate_mset_subset_eq: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<subseteq># M"
   by (auto simp add: mset_subset_eqI) (metis count_replicate_mset subseteq_mset_def)
 
 lemma replicate_count_mset_eq_filter_eq: "replicate (count (mset xs) k) k = filter (HOL.eq k) xs"
   by (induct xs) auto
 
 lemma replicate_mset_eq_empty_iff [simp]: "replicate_mset n a = {#} \<longleftrightarrow> n = 0"
   by (induct n) simp_all
 
 lemma replicate_mset_eq_iff:
   "replicate_mset m a = replicate_mset n b \<longleftrightarrow> m = 0 \<and> n = 0 \<or> m = n \<and> a = b"
   by (auto simp add: multiset_eq_iff)
 
 lemma repeat_mset_cancel1: "repeat_mset a A = repeat_mset a B \<longleftrightarrow> A = B \<or> a = 0"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_cancel2: "repeat_mset a A = repeat_mset b A \<longleftrightarrow> a = b \<or> A = {#}"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_eq_empty_iff: "repeat_mset n A = {#} \<longleftrightarrow> n = 0 \<or> A = {#}"
   by (cases n) auto
 
 lemma image_replicate_mset [simp]:
   "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
   by (induct n) simp_all
 
 lemma replicate_mset_msubseteq_iff:
   "replicate_mset m a \<subseteq># replicate_mset n b \<longleftrightarrow> m = 0 \<or> a = b \<and> m \<le> n"
   by (cases m)
     (auto simp: insert_subset_eq_iff simp flip: count_le_replicate_mset_subset_eq)
 
 lemma msubseteq_replicate_msetE:
   assumes "A \<subseteq># replicate_mset n a"
   obtains m where "m \<le> n" and "A = replicate_mset m a"
 proof (cases "n = 0")
   case True
   with assms that show thesis
     by simp
 next
   case False
   from assms have "set_mset A \<subseteq> set_mset (replicate_mset n a)"
     by (rule set_mset_mono)
   with False have "set_mset A \<subseteq> {a}"
     by simp
   then have "\<exists>m. A = replicate_mset m a"
   proof (induction A)
     case empty
     then show ?case
       by simp
   next
     case (add b A)
     then obtain m where "A = replicate_mset m a"
       by auto
     with add.prems show ?case
       by (auto intro: exI [of _ "Suc m"])
   qed
   then obtain m where A: "A = replicate_mset m a" ..
   with assms have "m \<le> n"
     by (auto simp add: replicate_mset_msubseteq_iff)
   then show thesis using A ..
 qed
 
 lemma count_image_mset_lt_imp_lt_raw:
   assumes
     "finite A" and
     "A = set_mset M \<union> set_mset N" and
     "count (image_mset f M) b < count (image_mset f N) b"
   shows "\<exists>x. f x = b \<and> count M x < count N x"
   using assms
 proof (induct A arbitrary: M N b rule: finite_induct)
   case (insert x F)
   note fin = this(1) and x_ni_f = this(2) and ih = this(3) and x_f_eq_m_n = this(4) and
     cnt_b = this(5)
 
   let ?Ma = "{#y \<in># M. y \<noteq> x#}"
   let ?Mb = "{#y \<in># M. y = x#}"
   let ?Na = "{#y \<in># N. y \<noteq> x#}"
   let ?Nb = "{#y \<in># N. y = x#}"
 
   have m_part: "M = ?Mb + ?Ma" and n_part: "N = ?Nb + ?Na"
     using multiset_partition by blast+
 
   have f_eq_ma_na: "F = set_mset ?Ma \<union> set_mset ?Na"
     using x_f_eq_m_n x_ni_f by auto
 
   show ?case
   proof (cases "count (image_mset f ?Ma) b < count (image_mset f ?Na) b")
     case cnt_ba: True
     obtain xa where "f xa = b" and "count ?Ma xa < count ?Na xa"
       using ih[OF f_eq_ma_na cnt_ba] by blast
     thus ?thesis
       by (metis count_filter_mset not_less0)
   next
     case cnt_ba: False
     have fx_eq_b: "f x = b"
       using cnt_b cnt_ba
       by (subst (asm) m_part, subst (asm) n_part,
           auto simp: filter_eq_replicate_mset split: if_splits)
     moreover have "count M x < count N x"
       using cnt_b cnt_ba
       by (subst (asm) m_part, subst (asm) n_part,
           auto simp: filter_eq_replicate_mset split: if_splits)
     ultimately show ?thesis
       by blast
   qed
 qed auto
 
 lemma count_image_mset_lt_imp_lt:
   assumes cnt_b: "count (image_mset f M) b < count (image_mset f N) b"
   shows "\<exists>x. f x = b \<and> count M x < count N x"
   by (rule count_image_mset_lt_imp_lt_raw[of "set_mset M \<union> set_mset N", OF _ refl cnt_b]) auto
 
 lemma count_image_mset_le_imp_lt_raw:
   assumes
     "finite A" and
     "A = set_mset M \<union> set_mset N" and
     "count (image_mset f M) (f a) + count N a < count (image_mset f N) (f a) + count M a"
   shows "\<exists>b. f b = f a \<and> count M b < count N b"
   using assms
 proof (induct A arbitrary: M N rule: finite_induct)
   case (insert x F)
   note fin = this(1) and x_ni_f = this(2) and ih = this(3) and x_f_eq_m_n = this(4) and
     cnt_lt = this(5)
 
   let ?Ma = "{#y \<in># M. y \<noteq> x#}"
   let ?Mb = "{#y \<in># M. y = x#}"
   let ?Na = "{#y \<in># N. y \<noteq> x#}"
   let ?Nb = "{#y \<in># N. y = x#}"
 
   have m_part: "M = ?Mb + ?Ma" and n_part: "N = ?Nb + ?Na"
     using multiset_partition by blast+
 
   have f_eq_ma_na: "F = set_mset ?Ma \<union> set_mset ?Na"
     using x_f_eq_m_n x_ni_f by auto
 
   show ?case
   proof (cases "f x = f a")
     case fx_ne_fa: False
 
     have cnt_fma_fa: "count (image_mset f ?Ma) (f a) = count (image_mset f M) (f a)"
       using fx_ne_fa by (subst (2) m_part) (auto simp: filter_eq_replicate_mset)
     have cnt_fna_fa: "count (image_mset f ?Na) (f a) = count (image_mset f N) (f a)"
       using fx_ne_fa by (subst (2) n_part) (auto simp: filter_eq_replicate_mset)
     have cnt_ma_a: "count ?Ma a = count M a"
       using fx_ne_fa by (subst (2) m_part) (auto simp: filter_eq_replicate_mset)
     have cnt_na_a: "count ?Na a = count N a"
       using fx_ne_fa by (subst (2) n_part) (auto simp: filter_eq_replicate_mset)
 
     obtain b where fb_eq_fa: "f b = f a" and cnt_b: "count ?Ma b < count ?Na b"
       using ih[OF f_eq_ma_na] cnt_lt unfolding cnt_fma_fa cnt_fna_fa cnt_ma_a cnt_na_a by blast
     have fx_ne_fb: "f x \<noteq> f b"
       using fb_eq_fa fx_ne_fa by simp
 
     have cnt_ma_b: "count ?Ma b = count M b"
       using fx_ne_fb by (subst (2) m_part) auto
     have cnt_na_b: "count ?Na b = count N b"
       using fx_ne_fb by (subst (2) n_part) auto
 
     show ?thesis
       using fb_eq_fa cnt_b unfolding cnt_ma_b cnt_na_b by blast
   next
     case fx_eq_fa: True
     show ?thesis
     proof (cases "x = a")
       case x_eq_a: True
       have "count (image_mset f ?Ma) (f a) + count ?Na a
         < count (image_mset f ?Na) (f a) + count ?Ma a"
         using cnt_lt x_eq_a by (subst (asm) (1 2) m_part, subst (asm) (1 2) n_part,
             auto simp: filter_eq_replicate_mset)
       thus ?thesis
         using ih[OF f_eq_ma_na] by (metis count_filter_mset nat_neq_iff)
     next
       case x_ne_a: False
       show ?thesis
       proof (cases "count M x < count N x")
         case True
         thus ?thesis
           using fx_eq_fa by blast
      next
         case False
         hence cnt_x: "count M x \<ge> count N x"
           by fastforce
         have "count M x + count (image_mset f ?Ma) (f a) + count ?Na a
           < count N x + count (image_mset f ?Na) (f a) + count ?Ma a"
           using cnt_lt x_ne_a fx_eq_fa by (subst (asm) (1 2) m_part, subst (asm) (1 2) n_part,
             auto simp: filter_eq_replicate_mset)
         hence "count (image_mset f ?Ma) (f a) + count ?Na a
           < count (image_mset f ?Na) (f a) + count ?Ma a"
           using cnt_x by linarith
         thus ?thesis
           using ih[OF f_eq_ma_na] by (metis count_filter_mset nat_neq_iff)
       qed
     qed
   qed
 qed auto
 
 lemma count_image_mset_le_imp_lt:
   assumes
     "count (image_mset f M) (f a) \<le> count (image_mset f N) (f a)" and
     "count M a > count N a"
   shows "\<exists>b. f b = f a \<and> count M b < count N b"
   using assms by (auto intro: count_image_mset_le_imp_lt_raw[of "set_mset M \<union> set_mset N"])
 
 lemma size_filter_unsat_elem:
   assumes "x \<in># M" and "\<not> P x"
   shows "size {#x \<in># M. P x#} < size M"
 proof -
   have "size (filter_mset P M) \<noteq> size M"
-    using assms by (metis add.right_neutral add_diff_cancel_left' count_filter_mset mem_Collect_eq
-      multiset_partition nonempty_has_size set_mset_def size_union)
+    using assms
+    by (metis dual_order.strict_iff_order filter_mset_eq_conv mset_subset_size subset_mset.nless_le)
   then show ?thesis
     by (meson leD nat_neq_iff size_filter_mset_lesseq)
 qed
 
 lemma size_filter_ne_elem: "x \<in># M \<Longrightarrow> size {#y \<in># M. y \<noteq> x#} < size M"
   by (simp add: size_filter_unsat_elem[of x M "\<lambda>y. y \<noteq> x"])
 
 lemma size_eq_ex_count_lt:
   assumes
     sz_m_eq_n: "size M = size N" and
     m_eq_n: "M \<noteq> N"
   shows "\<exists>x. count M x < count N x"
 proof -
   obtain x where "count M x \<noteq> count N x"
     using m_eq_n by (meson multiset_eqI)
   moreover
   {
     assume "count M x < count N x"
     hence ?thesis
       by blast
   }
   moreover
   {
     assume cnt_x: "count M x > count N x"
 
     have "size {#y \<in># M. y = x#} + size {#y \<in># M. y \<noteq> x#} =
       size {#y \<in># N. y = x#} + size {#y \<in># N. y \<noteq> x#}"
       using sz_m_eq_n multiset_partition by (metis size_union)
     hence sz_m_minus_x: "size {#y \<in># M. y \<noteq> x#} < size {#y \<in># N. y \<noteq> x#}"
       using cnt_x by (simp add: filter_eq_replicate_mset)
     then obtain y where "count {#y \<in># M. y \<noteq> x#} y < count {#y \<in># N. y \<noteq> x#} y"
       using size_lt_imp_ex_count_lt[OF sz_m_minus_x] by blast
     hence "count M y < count N y"
       by (metis count_filter_mset less_asym)
     hence ?thesis
       by blast
   }
   ultimately show ?thesis
     by fastforce
 qed
 
 
 subsection \<open>Big operators\<close>
 
 locale comm_monoid_mset = comm_monoid
 begin
 
 interpretation comp_fun_commute f
   by standard (simp add: fun_eq_iff left_commute)
 
 interpretation comp?: comp_fun_commute "f \<circ> g"
   by (fact comp_comp_fun_commute)
 
 context
 begin
 
 definition F :: "'a multiset \<Rightarrow> 'a"
   where eq_fold: "F M = fold_mset f \<^bold>1 M"
 
 lemma empty [simp]: "F {#} = \<^bold>1"
   by (simp add: eq_fold)
 
 lemma singleton [simp]: "F {#x#} = x"
 proof -
   interpret comp_fun_commute
     by standard (simp add: fun_eq_iff left_commute)
   show ?thesis by (simp add: eq_fold)
 qed
 
 lemma union [simp]: "F (M + N) = F M \<^bold>* F N"
 proof -
   interpret comp_fun_commute f
     by standard (simp add: fun_eq_iff left_commute)
   show ?thesis
     by (induct N) (simp_all add: left_commute eq_fold)
 qed
 
 lemma add_mset [simp]: "F (add_mset x N) = x \<^bold>* F N"
   unfolding add_mset_add_single[of x N] union by (simp add: ac_simps)
 
 lemma insert [simp]:
   shows "F (image_mset g (add_mset x A)) = g x \<^bold>* F (image_mset g A)"
   by (simp add: eq_fold)
 
 lemma remove:
   assumes "x \<in># A"
   shows "F A = x \<^bold>* F (A - {#x#})"
   using multi_member_split[OF assms] by auto
 
 lemma neutral:
   "\<forall>x\<in>#A. x = \<^bold>1 \<Longrightarrow> F A = \<^bold>1"
   by (induct A) simp_all
 
 lemma neutral_const [simp]:
   "F (image_mset (\<lambda>_. \<^bold>1) A) = \<^bold>1"
   by (simp add: neutral)
 
 private lemma F_image_mset_product:
   "F {#g x j \<^bold>* F {#g i j. i \<in># A#}. j \<in># B#} =
     F (image_mset (g x) B) \<^bold>* F {#F {#g i j. i \<in># A#}. j \<in># B#}"
   by (induction B) (simp_all add: left_commute semigroup.assoc semigroup_axioms)
 
 lemma swap:
   "F (image_mset (\<lambda>i. F (image_mset (g i) B)) A) =
     F (image_mset (\<lambda>j. F (image_mset (\<lambda>i. g i j) A)) B)"
   apply (induction A, simp)
   apply (induction B, auto simp add: F_image_mset_product ac_simps)
   done
 
 lemma distrib: "F (image_mset (\<lambda>x. g x \<^bold>* h x) A) = F (image_mset g A) \<^bold>* F (image_mset h A)"
   by (induction A) (auto simp: ac_simps)
 
 lemma union_disjoint:
   "A \<inter># B = {#} \<Longrightarrow> F (image_mset g (A \<union># B)) = F (image_mset g A) \<^bold>* F (image_mset g B)"
   by (induction A) (auto simp: ac_simps)
 
 end
 end
 
 lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute ((+) :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
   by standard (simp add: add_ac comp_def)
 
 declare comp_fun_commute.fold_mset_add_mset[OF comp_fun_commute_plus_mset, simp]
 
 lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (+) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
   by (induct NN) auto
 
 context comm_monoid_add
 begin
 
 sublocale sum_mset: comm_monoid_mset plus 0
   defines sum_mset = sum_mset.F ..
 
 lemma sum_unfold_sum_mset:
   "sum f A = sum_mset (image_mset f (mset_set A))"
   by (cases "finite A") (induct A rule: finite_induct, simp_all)
 
 end
 
 notation sum_mset ("\<Sum>\<^sub>#")
 
 syntax (ASCII)
   "_sum_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
 syntax
   "_sum_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
 translations
   "\<Sum>i \<in># A. b" \<rightleftharpoons> "CONST sum_mset (CONST image_mset (\<lambda>i. b) A)"
 
 context comm_monoid_add
 begin
 
 lemma sum_mset_sum_list:
   "sum_mset (mset xs) = sum_list xs"
   by (induction xs) auto
 
 end
 
 context canonically_ordered_monoid_add
 begin
 
 lemma sum_mset_0_iff [simp]:
   "sum_mset M = 0  \<longleftrightarrow> (\<forall>x \<in> set_mset M. x = 0)"
   by (induction M) auto
 
 end
 
 context ordered_comm_monoid_add
 begin
 
 lemma sum_mset_mono:
   "sum_mset (image_mset f K) \<le> sum_mset (image_mset g K)"
   if "\<And>i. i \<in># K \<Longrightarrow> f i \<le> g i"
   using that by (induction K) (simp_all add: add_mono)
 
 end
 
 context cancel_comm_monoid_add
 begin
 
 lemma sum_mset_diff:
   "sum_mset (M - N) = sum_mset M - sum_mset N" if "N \<subseteq># M" for M N :: "'a multiset"
   using that by (auto simp add: subset_mset.le_iff_add)
 
 end
 
 context semiring_0
 begin
 
 lemma sum_mset_distrib_left:
   "c * (\<Sum>x \<in># M. f x) = (\<Sum>x \<in># M. c * f(x))"
   by (induction M) (simp_all add: algebra_simps)
 
 lemma sum_mset_distrib_right:
   "(\<Sum>x \<in># M. f x) * c = (\<Sum>x \<in># M. f x * c)"
   by (induction M) (simp_all add: algebra_simps)
 
 end
 
 lemma sum_mset_product:
   fixes f :: "'a::{comm_monoid_add,times} \<Rightarrow> 'b::semiring_0"
   shows "(\<Sum>i \<in># A. f i) * (\<Sum>i \<in># B. g i) = (\<Sum>i\<in>#A. \<Sum>j\<in>#B. f i * g j)"
   by (subst sum_mset.swap) (simp add: sum_mset_distrib_left sum_mset_distrib_right)
 
 context semiring_1
 begin
 
 lemma sum_mset_replicate_mset [simp]:
   "sum_mset (replicate_mset n a) = of_nat n * a"
   by (induction n) (simp_all add: algebra_simps)
 
 lemma sum_mset_delta:
   "sum_mset (image_mset (\<lambda>x. if x = y then c else 0) A) = c * of_nat (count A y)"
   by (induction A) (simp_all add: algebra_simps)
 
 lemma sum_mset_delta':
   "sum_mset (image_mset (\<lambda>x. if y = x then c else 0) A) = c * of_nat (count A y)"
   by (induction A) (simp_all add: algebra_simps)
 
 end
 
 lemma of_nat_sum_mset [simp]:
   "of_nat (sum_mset A) = sum_mset (image_mset of_nat A)"
   by (induction A) auto
 
 lemma size_eq_sum_mset:
   "size M = (\<Sum>a\<in>#M. 1)"
   using image_mset_const_eq [of "1::nat" M] by simp
 
 lemma size_mset_set [simp]:
   "size (mset_set A) = card A"
   by (simp only: size_eq_sum_mset card_eq_sum sum_unfold_sum_mset)
 
 lemma sum_mset_constant [simp]:
   fixes y :: "'b::semiring_1"
   shows \<open>(\<Sum>x\<in>#A. y) = of_nat (size A) * y\<close>
   by (induction A) (auto simp: algebra_simps)
 
 lemma set_mset_Union_mset[simp]: "set_mset (\<Sum>\<^sub># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
   by (induct MM) auto
 
 lemma in_Union_mset_iff[iff]: "x \<in># \<Sum>\<^sub># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
   by (induct MM) auto
 
 lemma count_sum:
   "count (sum f A) x = sum (\<lambda>a. count (f a) x) A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma sum_eq_empty_iff:
   assumes "finite A"
   shows "sum f A = {#} \<longleftrightarrow> (\<forall>a\<in>A. f a = {#})"
   using assms by induct simp_all
 
 lemma Union_mset_empty_conv[simp]: "\<Sum>\<^sub># M = {#} \<longleftrightarrow> (\<forall>i\<in>#M. i = {#})"
   by (induction M) auto
 
 lemma Union_image_single_mset[simp]: "\<Sum>\<^sub># (image_mset (\<lambda>x. {#x#}) m) = m"
   by(induction m) auto
 
 lemma size_multiset_sum_mset [simp]: "size (\<Sum>X\<in>#A. X :: 'a multiset) = (\<Sum>X\<in>#A. size X)"
   by (induction A) auto
 
 context comm_monoid_mult
 begin
 
 sublocale prod_mset: comm_monoid_mset times 1
   defines prod_mset = prod_mset.F ..
 
 lemma prod_mset_empty:
   "prod_mset {#} = 1"
   by (fact prod_mset.empty)
 
 lemma prod_mset_singleton:
   "prod_mset {#x#} = x"
   by (fact prod_mset.singleton)
 
 lemma prod_mset_Un:
   "prod_mset (A + B) = prod_mset A * prod_mset B"
   by (fact prod_mset.union)
 
 lemma prod_mset_prod_list:
   "prod_mset (mset xs) = prod_list xs"
   by (induct xs) auto
 
 lemma prod_mset_replicate_mset [simp]:
   "prod_mset (replicate_mset n a) = a ^ n"
   by (induct n) simp_all
 
 lemma prod_unfold_prod_mset:
   "prod f A = prod_mset (image_mset f (mset_set A))"
   by (cases "finite A") (induct A rule: finite_induct, simp_all)
 
 lemma prod_mset_multiplicity:
   "prod_mset M = prod (\<lambda>x. x ^ count M x) (set_mset M)"
   by (simp add: fold_mset_def prod.eq_fold prod_mset.eq_fold funpow_times_power comp_def)
 
 lemma prod_mset_delta: "prod_mset (image_mset (\<lambda>x. if x = y then c else 1) A) = c ^ count A y"
   by (induction A) simp_all
 
 lemma prod_mset_delta': "prod_mset (image_mset (\<lambda>x. if y = x then c else 1) A) = c ^ count A y"
   by (induction A) simp_all
 
 lemma prod_mset_subset_imp_dvd:
   assumes "A \<subseteq># B"
   shows   "prod_mset A dvd prod_mset B"
 proof -
   from assms have "B = (B - A) + A" by (simp add: subset_mset.diff_add)
   also have "prod_mset \<dots> = prod_mset (B - A) * prod_mset A" by simp
   also have "prod_mset A dvd \<dots>" by simp
   finally show ?thesis .
 qed
 
 lemma dvd_prod_mset:
   assumes "x \<in># A"
   shows "x dvd prod_mset A"
   using assms prod_mset_subset_imp_dvd [of "{#x#}" A] by simp
 
 end
 
 notation prod_mset ("\<Prod>\<^sub>#")
 
 syntax (ASCII)
   "_prod_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
 syntax
   "_prod_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
 translations
   "\<Prod>i \<in># A. b" \<rightleftharpoons> "CONST prod_mset (CONST image_mset (\<lambda>i. b) A)"
 
 lemma prod_mset_constant [simp]: "(\<Prod>_\<in>#A. c) = c ^ size A"
   by (simp add: image_mset_const_eq)
 
 lemma (in semidom) prod_mset_zero_iff [iff]:
   "prod_mset A = 0 \<longleftrightarrow> 0 \<in># A"
   by (induct A) auto
 
 lemma (in semidom_divide) prod_mset_diff:
   assumes "B \<subseteq># A" and "0 \<notin># B"
   shows "prod_mset (A - B) = prod_mset A div prod_mset B"
 proof -
   from assms obtain C where "A = B + C"
     by (metis subset_mset.add_diff_inverse)
   with assms show ?thesis by simp
 qed
 
 lemma (in semidom_divide) prod_mset_minus:
   assumes "a \<in># A" and "a \<noteq> 0"
   shows "prod_mset (A - {#a#}) = prod_mset A div a"
   using assms prod_mset_diff [of "{#a#}" A] by auto
 
 lemma (in normalization_semidom) normalize_prod_mset_normalize:
   "normalize (prod_mset (image_mset normalize A)) = normalize (prod_mset A)"
 proof (induction A)
   case (add x A)
   have "normalize (prod_mset (image_mset normalize (add_mset x A))) =
           normalize (x * normalize (prod_mset (image_mset normalize A)))"
     by simp
   also note add.IH
   finally show ?case by simp
 qed auto
 
 lemma (in algebraic_semidom) is_unit_prod_mset_iff:
   "is_unit (prod_mset A) \<longleftrightarrow> (\<forall>x \<in># A. is_unit x)"
   by (induct A) (auto simp: is_unit_mult_iff)
 
 lemma (in normalization_semidom_multiplicative) normalize_prod_mset:
   "normalize (prod_mset A) = prod_mset (image_mset normalize A)"
   by (induct A) (simp_all add: normalize_mult)
 
 lemma (in normalization_semidom_multiplicative) normalized_prod_msetI:
   assumes "\<And>a. a \<in># A \<Longrightarrow> normalize a = a"
   shows "normalize (prod_mset A) = prod_mset A"
 proof -
   from assms have "image_mset normalize A = A"
     by (induct A) simp_all
   then show ?thesis by (simp add: normalize_prod_mset)
 qed
 
 lemma image_prod_mset_multiplicity:
   "prod_mset (image_mset f M) = prod (\<lambda>x. f x ^ count M x) (set_mset M)"
 proof (induction M)
   case (add x M)
   show ?case
   proof (cases "x \<in> set_mset M")
     case True
     have "(\<Prod>y\<in>set_mset (add_mset x M). f y ^ count (add_mset x M) y) = 
             (\<Prod>y\<in>set_mset M. (if y = x then f x else 1) * f y ^ count M y)"
       using True add by (intro prod.cong) auto
     also have "\<dots> = f x * (\<Prod>y\<in>set_mset M. f y ^ count M y)"
       using True by (subst prod.distrib) auto
     also note add.IH [symmetric]
     finally show ?thesis using True by simp
   next
     case False
     hence "(\<Prod>y\<in>set_mset (add_mset x M). f y ^ count (add_mset x M) y) =
               f x * (\<Prod>y\<in>set_mset M. f y ^ count (add_mset x M) y)"
       by (auto simp: not_in_iff)
     also have "(\<Prod>y\<in>set_mset M. f y ^ count (add_mset x M) y) = 
                  (\<Prod>y\<in>set_mset M. f y ^ count M y)"
       using False by (intro prod.cong) auto
     also note add.IH [symmetric]
     finally show ?thesis by simp
   qed
 qed auto
 
 subsection \<open>Multiset as order-ignorant lists\<close>
 
 context linorder
 begin
 
 lemma mset_insort [simp]:
   "mset (insort_key k x xs) = add_mset x (mset xs)"
   by (induct xs) simp_all
 
 lemma mset_sort [simp]:
   "mset (sort_key k xs) = mset xs"
   by (induct xs) simp_all
 
 text \<open>
   This lemma shows which properties suffice to show that a function
   \<open>f\<close> with \<open>f xs = ys\<close> behaves like sort.
 \<close>
 
 lemma properties_for_sort_key:
   assumes "mset ys = mset xs"
     and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
     and "sorted (map f ys)"
   shows "sort_key f xs = ys"
   using assms
 proof (induct xs arbitrary: ys)
   case Nil then show ?case by simp
 next
   case (Cons x xs)
   from Cons.prems(2) have
     "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
     by (simp add: filter_remove1)
   with Cons.prems have "sort_key f xs = remove1 x ys"
     by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
   moreover from Cons.prems have "x \<in># mset ys"
     by auto
   then have "x \<in> set ys"
     by simp
   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
 qed
 
 lemma properties_for_sort:
   assumes multiset: "mset ys = mset xs"
     and "sorted ys"
   shows "sort xs = ys"
 proof (rule properties_for_sort_key)
   from multiset show "mset ys = mset xs" .
   from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp
   from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k
     by (rule mset_eq_length_filter)
   then have "replicate (length (filter (\<lambda>y. k = y) ys)) k =
     replicate (length (filter (\<lambda>x. k = x) xs)) k" for k
     by simp
   then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k
     by (simp add: replicate_length_filter)
 qed
 
 lemma sort_key_inj_key_eq:
   assumes mset_equal: "mset xs = mset ys"
     and "inj_on f (set xs)"
     and "sorted (map f ys)"
   shows "sort_key f xs = ys"
 proof (rule properties_for_sort_key)
   from mset_equal
   show "mset ys = mset xs" by simp
   from \<open>sorted (map f ys)\<close>
   show "sorted (map f ys)" .
   show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k
   proof -
     from mset_equal
     have set_equal: "set xs = set ys" by (rule mset_eq_setD)
     with that have "insert k (set ys) = set ys" by auto
     with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))"
       by (simp add: set_equal)
     from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys"
       by (auto intro!: inj_on_filter_key_eq)
     also have "\<dots> = replicate (count (mset ys) k) k"
       by (simp add: replicate_count_mset_eq_filter_eq)
     also have "\<dots> = replicate (count (mset xs) k) k"
       using mset_equal by simp
     also have "\<dots> = filter (HOL.eq k) xs"
       by (simp add: replicate_count_mset_eq_filter_eq)
     also have "\<dots> = [x\<leftarrow>xs . f k = f x]"
       using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal)
     finally show ?thesis .
   qed
 qed
 
 lemma sort_key_eq_sort_key:
   assumes "mset xs = mset ys"
     and "inj_on f (set xs)"
   shows "sort_key f xs = sort_key f ys"
   by (rule sort_key_inj_key_eq) (simp_all add: assms)
 
 lemma sort_key_by_quicksort:
   "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
     @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
     @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
 proof (rule properties_for_sort_key)
   show "mset ?rhs = mset ?lhs"
     by (rule multiset_eqI) auto
   show "sorted (map f ?rhs)"
     by (auto simp add: sorted_append intro: sorted_map_same)
 next
   fix l
   assume "l \<in> set ?rhs"
   let ?pivot = "f (xs ! (length xs div 2))"
   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
   have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
   note *** = this [of "(<)"] this [of "(>)"] this [of "(=)"]
   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
   proof (cases "f l" ?pivot rule: linorder_cases)
     case less
     then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
     with less show ?thesis
       by (simp add: filter_sort [symmetric] ** ***)
   next
     case equal then show ?thesis
       by (simp add: * less_le)
   next
     case greater
     then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
     with greater show ?thesis
       by (simp add: filter_sort [symmetric] ** ***)
   qed
 qed
 
 lemma sort_by_quicksort:
   "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
     @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
     @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
   using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
 
 text \<open>A stable parameterized quicksort\<close>
 
 definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
   "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
 
 lemma part_code [code]:
   "part f pivot [] = ([], [], [])"
   "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
      if x' < pivot then (x # lts, eqs, gts)
      else if x' > pivot then (lts, eqs, x # gts)
      else (lts, x # eqs, gts))"
   by (auto simp add: part_def Let_def split_def)
 
 lemma sort_key_by_quicksort_code [code]:
   "sort_key f xs =
     (case xs of
       [] \<Rightarrow> []
     | [x] \<Rightarrow> xs
     | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
     | _ \<Rightarrow>
         let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
         in sort_key f lts @ eqs @ sort_key f gts)"
 proof (cases xs)
   case Nil then show ?thesis by simp
 next
   case (Cons _ ys) note hyps = Cons show ?thesis
   proof (cases ys)
     case Nil with hyps show ?thesis by simp
   next
     case (Cons _ zs) note hyps = hyps Cons show ?thesis
     proof (cases zs)
       case Nil with hyps show ?thesis by auto
     next
       case Cons
       from sort_key_by_quicksort [of f xs]
       have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
         in sort_key f lts @ eqs @ sort_key f gts)"
       by (simp only: split_def Let_def part_def fst_conv snd_conv)
       with hyps Cons show ?thesis by (simp only: list.cases)
     qed
   qed
 qed
 
 end
 
 hide_const (open) part
 
 lemma mset_remdups_subset_eq: "mset (remdups xs) \<subseteq># mset xs"
   by (induct xs) (auto intro: subset_mset.order_trans)
 
 lemma mset_update:
   "i < length ls \<Longrightarrow> mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
 proof (induct ls arbitrary: i)
   case Nil then show ?case by simp
 next
   case (Cons x xs)
   show ?case
   proof (cases i)
     case 0 then show ?thesis by simp
   next
     case (Suc i')
     with Cons show ?thesis
       by (cases \<open>x = xs ! i'\<close>) auto
   qed
 qed
 
 lemma mset_swap:
   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
     mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
   by (cases "i = j") (simp_all add: mset_update nth_mem_mset)
 
 lemma mset_eq_finite:
   \<open>finite {ys. mset ys = mset xs}\<close>
 proof -
   have \<open>{ys. mset ys = mset xs} \<subseteq> {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
     by (auto simp add: dest: mset_eq_setD mset_eq_length)
   moreover have \<open>finite {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
     using finite_lists_length_le by blast
   ultimately show ?thesis
     by (rule finite_subset)
 qed
 
 
 subsection \<open>The multiset order\<close>
 
 definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
   "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and>
       (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
 
 definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
   "mult r = (mult1 r)\<^sup>+"
 
 definition multp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multp r M N \<longleftrightarrow> (M, N) \<in> mult {(x, y). r x y}"
 
 declare multp_def[pred_set_conv]
 
 lemma mult1I:
   assumes "M = add_mset a M0" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
   shows "(N, M) \<in> mult1 r"
   using assms unfolding mult1_def by blast
 
 lemma mult1E:
   assumes "(N, M) \<in> mult1 r"
   obtains a M0 K where "M = add_mset a M0" "N = M0 + K" "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
   using assms unfolding mult1_def by blast
 
 lemma mono_mult1:
   assumes "r \<subseteq> r'" shows "mult1 r \<subseteq> mult1 r'"
   unfolding mult1_def using assms by blast
 
 lemma mono_mult:
   assumes "r \<subseteq> r'" shows "mult r \<subseteq> mult r'"
   unfolding mult_def using mono_mult1[OF assms] trancl_mono by blast
 
 lemma mono_multp[mono]: "r \<le> r' \<Longrightarrow> multp r \<le> multp r'"
   unfolding le_fun_def le_bool_def
 proof (intro allI impI)
   fix M N :: "'a multiset"
   assume "\<forall>x xa. r x xa \<longrightarrow> r' x xa"
   hence "{(x, y). r x y} \<subseteq> {(x, y). r' x y}"
     by blast
   thus "multp r M N \<Longrightarrow> multp r' M N"
     unfolding multp_def
     by (fact mono_mult[THEN subsetD, rotated])
 qed
 
 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
   by (simp add: mult1_def)
 
 
 subsubsection \<open>Well-foundedness\<close>
 
 lemma less_add:
   assumes mult1: "(N, add_mset a M0) \<in> mult1 r"
   shows
     "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = add_mset a M) \<or>
      (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
 proof -
   let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
   let ?R = "\<lambda>N M. \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> ?r K a"
   obtain a' M0' K where M0: "add_mset a M0 = add_mset a' M0'"
     and N: "N = M0' + K"
     and r: "?r K a'"
     using mult1 unfolding mult1_def by auto
   show ?thesis (is "?case1 \<or> ?case2")
   proof -
     from M0 consider "M0 = M0'" "a = a'"
       | K' where "M0 = add_mset a' K'" "M0' = add_mset a K'"
       by atomize_elim (simp only: add_eq_conv_ex)
     then show ?thesis
     proof cases
       case 1
       with N r have "?r K a \<and> N = M0 + K" by simp
       then have ?case2 ..
       then show ?thesis ..
     next
       case 2
       from N 2(2) have n: "N = add_mset a (K' + K)" by simp
       with r 2(1) have "?R (K' + K) M0" by blast
       with n have ?case1 by (simp add: mult1_def)
       then show ?thesis ..
     qed
   qed
 qed
 
 lemma all_accessible:
   assumes "wf r"
   shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
 proof
   let ?R = "mult1 r"
   let ?W = "Wellfounded.acc ?R"
   {
     fix M M0 a
     assume M0: "M0 \<in> ?W"
       and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. add_mset b M \<in> ?W)"
       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> add_mset a M \<in> ?W"
     have "add_mset a M0 \<in> ?W"
     proof (rule accI [of "add_mset a M0"])
       fix N
       assume "(N, add_mset a M0) \<in> ?R"
       then consider M where "(M, M0) \<in> ?R" "N = add_mset a M"
         | K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K"
         by atomize_elim (rule less_add)
       then show "N \<in> ?W"
       proof cases
         case 1
         from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> add_mset a M \<in> ?W" ..
         from this and \<open>(M, M0) \<in> ?R\<close> have "add_mset a M \<in> ?W" ..
         then show "N \<in> ?W" by (simp only: \<open>N = add_mset a M\<close>)
       next
         case 2
         from this(1) have "M0 + K \<in> ?W"
         proof (induct K)
           case empty
           from M0 show "M0 + {#} \<in> ?W" by simp
         next
           case (add x K)
           from add.prems have "(x, a) \<in> r" by simp
           with wf_hyp have "\<forall>M \<in> ?W. add_mset x M \<in> ?W" by blast
           moreover from add have "M0 + K \<in> ?W" by simp
           ultimately have "add_mset x (M0 + K) \<in> ?W" ..
           then show "M0 + (add_mset x K) \<in> ?W" by simp
         qed
         then show "N \<in> ?W" by (simp only: 2(2))
       qed
     qed
   } note tedious_reasoning = this
 
   show "M \<in> ?W" for M
   proof (induct M)
     show "{#} \<in> ?W"
     proof (rule accI)
       fix b assume "(b, {#}) \<in> ?R"
       with not_less_empty show "b \<in> ?W" by contradiction
     qed
 
     fix M a assume "M \<in> ?W"
     from \<open>wf r\<close> have "\<forall>M \<in> ?W. add_mset a M \<in> ?W"
     proof induct
       fix a
       assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. add_mset b M \<in> ?W)"
       show "\<forall>M \<in> ?W. add_mset a M \<in> ?W"
       proof
         fix M assume "M \<in> ?W"
         then show "add_mset a M \<in> ?W"
           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
       qed
     qed
     from this and \<open>M \<in> ?W\<close> show "add_mset a M \<in> ?W" ..
   qed
 qed
 
 lemma wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
   by (rule acc_wfI) (rule all_accessible)
 
 lemma wf_mult: "wf r \<Longrightarrow> wf (mult r)"
   unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
 
 lemma wfP_multp: "wfP r \<Longrightarrow> wfP (multp r)"
   unfolding multp_def wfP_def
   by (simp add: wf_mult)
 
 
 subsubsection \<open>Closure-free presentation\<close>
 
 text \<open>One direction.\<close>
 lemma mult_implies_one_step:
   assumes
     trans: "trans r" and
     MN: "(M, N) \<in> mult r"
   shows "\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)"
   using MN unfolding mult_def mult1_def
 proof (induction rule: converse_trancl_induct)
   case (base y)
   then show ?case by force
 next
   case (step y z) note yz = this(1) and zN = this(2) and N_decomp = this(3)
   obtain I J K where
     N: "N = I + J" "z = I + K" "J \<noteq> {#}" "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
     using N_decomp by blast
   obtain a M0 K' where
     z: "z = add_mset a M0" and y: "y = M0 + K'" and K: "\<forall>b. b \<in># K' \<longrightarrow> (b, a) \<in> r"
     using yz by blast
   show ?case
   proof (cases "a \<in># K")
     case True
     moreover have "\<exists>j\<in>#J. (k, j) \<in> r" if "k \<in># K'" for k
       using K N trans True by (meson that transE)
     ultimately show ?thesis
       by (rule_tac x = I in exI, rule_tac x = J in exI, rule_tac x = "(K - {#a#}) + K'" in exI)
         (use z y N in \<open>auto simp del: subset_mset.add_diff_assoc2 dest: in_diffD\<close>)
   next
     case False
     then have "a \<in># I" by (metis N(2) union_iff union_single_eq_member z)
     moreover have "M0 = I + K - {#a#}"
       using N(2) z by force
     ultimately show ?thesis
       by (rule_tac x = "I - {#a#}" in exI, rule_tac x = "add_mset a J" in exI,
           rule_tac x = "K + K'" in exI)
         (use z y N False K in \<open>auto simp: add.assoc\<close>)
   qed
 qed
 
 lemma multp_implies_one_step:
   "transp R \<Longrightarrow> multp R M N \<Longrightarrow> \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>x\<in>#J. R k x)"
   by (rule mult_implies_one_step[to_pred])
 
 lemma one_step_implies_mult:
   assumes
     "J \<noteq> {#}" and
     "\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r"
   shows "(I + K, I + J) \<in> mult r"
   using assms
 proof (induction "size J" arbitrary: I J K)
   case 0
   then show ?case by auto
 next
   case (Suc n) note IH = this(1) and size_J = this(2)[THEN sym]
   obtain J' a where J: "J = add_mset a J'"
     using size_J by (blast dest: size_eq_Suc_imp_eq_union)
   show ?case
   proof (cases "J' = {#}")
     case True
     then show ?thesis
       using J Suc by (fastforce simp add: mult_def mult1_def)
   next
     case [simp]: False
     have K: "K = {#x \<in># K. (x, a) \<in> r#} + {#x \<in># K. (x, a) \<notin> r#}"
       by simp
     have "(I + K, (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r"
       using IH[of J' "{# x \<in># K. (x, a) \<notin> r#}" "I + {# x \<in># K. (x, a) \<in> r#}"]
         J Suc.prems K size_J by (auto simp: ac_simps)
     moreover have "(I + {#x \<in># K. (x, a) \<in> r#} + J', I + J) \<in> mult r"
       by (fastforce simp: J mult1_def mult_def)
     ultimately show ?thesis
       unfolding mult_def by simp
   qed
 qed
 
 lemma one_step_implies_multp:
   "J \<noteq> {#} \<Longrightarrow> \<forall>k\<in>#K. \<exists>j\<in>#J. R k j \<Longrightarrow> multp R (I + K) (I + J)"
   by (rule one_step_implies_mult[of _ _ "{(x, y). r x y}" for r, folded multp_def, simplified])
 
 lemma subset_implies_mult:
   assumes sub: "A \<subset># B"
   shows "(A, B) \<in> mult r"
 proof -
   have ApBmA: "A + (B - A) = B"
     using sub by simp
   have BmA: "B - A \<noteq> {#}"
     using sub by (simp add: Diff_eq_empty_iff_mset subset_mset.less_le_not_le)
   thus ?thesis
     by (rule one_step_implies_mult[of "B - A" "{#}" _ A, unfolded ApBmA, simplified])
 qed
 
 lemma subset_implies_multp: "A \<subset># B \<Longrightarrow> multp r A B"
   by (rule subset_implies_mult[of _ _ "{(x, y). r x y}" for r, folded multp_def])
 
 lemma multp_repeat_mset_repeat_msetI:
   assumes "transp R" and "multp R A B" and "n \<noteq> 0"
   shows "multp R (repeat_mset n A) (repeat_mset n  B)"
 proof -
   from \<open>transp R\<close> \<open>multp R A B\<close> obtain I J K where
     "B = I + J" and "A = I + K" and "J \<noteq> {#}" and "\<forall>k \<in># K. \<exists>x \<in># J. R k x"
     by (auto dest: multp_implies_one_step)
 
   have repeat_n_A_eq: "repeat_mset n A = repeat_mset n I + repeat_mset n K"
     using \<open>A = I + K\<close> by simp
 
   have repeat_n_B_eq: "repeat_mset n B = repeat_mset n I + repeat_mset n J"
     using \<open>B = I + J\<close> by simp
 
   show ?thesis
     unfolding repeat_n_A_eq repeat_n_B_eq
   proof (rule one_step_implies_multp)
     from \<open>n \<noteq> 0\<close> show "repeat_mset n J \<noteq> {#}"
       using \<open>J \<noteq> {#}\<close>
       by (simp add: repeat_mset_eq_empty_iff)
   next
     show "\<forall>k \<in># repeat_mset n K. \<exists>j \<in># repeat_mset n J. R k j"
       using \<open>\<forall>k \<in># K. \<exists>x \<in># J. R k x\<close>
       by (metis count_greater_zero_iff nat_0_less_mult_iff repeat_mset.rep_eq)
   qed
 qed
 
 
 subsubsection \<open>Monotonicity\<close>
 
 lemma multp_mono_strong:
   assumes "multp R M1 M2" and "transp R" and
     S_if_R: "\<And>x y. x \<in> set_mset M1 \<Longrightarrow> y \<in> set_mset M2 \<Longrightarrow> R x y \<Longrightarrow> S x y"
   shows "multp S M1 M2"
 proof -
   obtain I J K where "M2 = I + J" and "M1 = I + K" and "J \<noteq> {#}" and "\<forall>k\<in>#K. \<exists>x\<in>#J. R k x"
     using multp_implies_one_step[OF \<open>transp R\<close> \<open>multp R M1 M2\<close>] by auto
   show ?thesis
     unfolding \<open>M2 = I + J\<close> \<open>M1 = I + K\<close>
   proof (rule one_step_implies_multp[OF \<open>J \<noteq> {#}\<close>])
     show "\<forall>k\<in>#K. \<exists>j\<in>#J. S k j"
       using S_if_R
       by (metis \<open>M1 = I + K\<close> \<open>M2 = I + J\<close> \<open>\<forall>k\<in>#K. \<exists>x\<in>#J. R k x\<close> union_iff)
   qed
 qed
 
 lemma mult_mono_strong:
   assumes "(M1, M2) \<in> mult r" and "trans r" and
     S_if_R: "\<And>x y. x \<in> set_mset M1 \<Longrightarrow> y \<in> set_mset M2 \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s"
   shows "(M1, M2) \<in> mult s"
   using assms multp_mono_strong[of "\<lambda>x y. (x, y) \<in> r" M1 M2 "\<lambda>x y. (x, y) \<in> s",
       unfolded multp_def transp_trans_eq, simplified]
   by blast
 
 lemma monotone_on_multp_multp_image_mset:
   assumes "monotone_on A orda ordb f" and "transp orda"
   shows "monotone_on {M. set_mset M \<subseteq> A} (multp orda) (multp ordb) (image_mset f)"
 proof (rule monotone_onI)
   fix M1 M2
   assume
     M1_in: "M1 \<in> {M. set_mset M \<subseteq> A}" and
     M2_in: "M2 \<in> {M. set_mset M \<subseteq> A}" and
     M1_lt_M2: "multp orda M1 M2"
 
   from multp_implies_one_step[OF \<open>transp orda\<close> M1_lt_M2] obtain I J K where
     M2_eq: "M2 = I + J" and
     M1_eq: "M1 = I + K" and
     J_neq_mempty: "J \<noteq> {#}" and
     ball_K_less: "\<forall>k\<in>#K. \<exists>x\<in>#J. orda k x"
     by metis
 
   have "multp ordb (image_mset f I + image_mset f K) (image_mset f I + image_mset f J)"
   proof (intro one_step_implies_multp ballI)
     show "image_mset f J \<noteq> {#}"
       using J_neq_mempty by simp
   next
     fix k' assume "k'\<in>#image_mset f K"
     then obtain k where "k' = f k" and k_in: "k \<in># K"
       by auto
     then obtain j where j_in: "j\<in>#J" and "orda k j"
       using ball_K_less by auto
 
     have "ordb (f k) (f j)"
     proof (rule \<open>monotone_on A orda ordb f\<close>[THEN monotone_onD, OF _ _ \<open>orda k j\<close>])
       show "k \<in> A"
         using M1_eq M1_in k_in by auto
     next
       show "j \<in> A"
         using M2_eq M2_in j_in by auto
     qed
     thus "\<exists>j\<in>#image_mset f J. ordb k' j"
       using \<open>j \<in># J\<close> \<open>k' = f k\<close> by auto
   qed
   thus "multp ordb (image_mset f M1) (image_mset f M2)"
     by (simp add: M1_eq M2_eq)
 qed
 
 lemma monotone_multp_multp_image_mset:
   assumes "monotone orda ordb f" and "transp orda"
   shows "monotone (multp orda) (multp ordb) (image_mset f)"
   by (rule monotone_on_multp_multp_image_mset[OF assms, simplified])
 
 lemma multp_image_mset_image_msetI:
   assumes "multp (\<lambda>x y. R (f x) (f y)) M1 M2" and "transp R"
   shows "multp R (image_mset f M1) (image_mset f M2)"
 proof -
   from \<open>transp R\<close> have "transp (\<lambda>x y. R (f x) (f y))"
     by (auto intro: transpI dest: transpD)
   with \<open>multp (\<lambda>x y. R (f x) (f y)) M1 M2\<close> obtain I J K where
     "M2 = I + J" and "M1 = I + K" and "J \<noteq> {#}" and "\<forall>k\<in>#K. \<exists>x\<in>#J. R (f k) (f x)"
     using multp_implies_one_step by blast
 
   have "multp R (image_mset f I + image_mset f K) (image_mset f I + image_mset f J)"
   proof (rule one_step_implies_multp)
     show "image_mset f J \<noteq> {#}"
       by (simp add: \<open>J \<noteq> {#}\<close>)
   next
     show "\<forall>k\<in>#image_mset f K. \<exists>j\<in>#image_mset f J. R k j"
       by (simp add: \<open>\<forall>k\<in>#K. \<exists>x\<in>#J. R (f k) (f x)\<close>)
   qed
   thus ?thesis
     by (simp add: \<open>M1 = I + K\<close> \<open>M2 = I + J\<close>)
 qed
 
 lemma multp_image_mset_image_msetD:
   assumes
     "multp R (image_mset f A) (image_mset f B)" and
     "transp R" and
     inj_on_f: "inj_on f (set_mset A \<union> set_mset B)"
   shows "multp (\<lambda>x y. R (f x) (f y)) A B"
 proof -
   from assms(1,2) obtain I J K where
     f_B_eq: "image_mset f B = I + J" and
     f_A_eq: "image_mset f A = I + K" and
     J_neq_mempty: "J \<noteq> {#}" and
     ball_K_less: "\<forall>k\<in>#K. \<exists>x\<in>#J. R k x"
     by (auto dest: multp_implies_one_step)
 
   from f_B_eq obtain I' J' where
     B_def: "B = I' + J'" and I_def: "I = image_mset f I'" and J_def: "J = image_mset f J'"
     using image_mset_eq_plusD by blast
 
   from inj_on_f have inj_on_f': "inj_on f (set_mset A \<union> set_mset I')"
     by (rule inj_on_subset) (auto simp add: B_def)
 
   from f_A_eq obtain K' where
     A_def: "A = I' + K'" and K_def: "K = image_mset f K'"
     by (auto simp: I_def dest: image_mset_eq_image_mset_plusD[OF _ inj_on_f'])
 
   show ?thesis
     unfolding A_def B_def
   proof (intro one_step_implies_multp ballI)
     from J_neq_mempty show "J' \<noteq> {#}"
       by (simp add: J_def)
   next
     fix k assume "k \<in># K'"
     with ball_K_less obtain j' where "j' \<in># J" and "R (f k) j'"
       using K_def by auto
     moreover then obtain j where "j \<in># J'" and "f j = j'"
       using J_def by auto
     ultimately show "\<exists>j\<in>#J'. R (f k) (f j)"
       by blast
   qed
 qed
 
 
 subsubsection \<open>The multiset extension is cancellative for multiset union\<close>
 
 lemma mult_cancel:
   assumes "trans s" and "irrefl_on (set_mset Z) s"
   shows "(X + Z, Y + Z) \<in> mult s \<longleftrightarrow> (X, Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
 proof
   assume ?L thus ?R
     using \<open>irrefl_on (set_mset Z) s\<close>
   proof (induct Z)
     case (add z Z)
     obtain X' Y' Z' where *: "add_mset z X + Z = Z' + X'" "add_mset z Y + Z = Z' + Y'" "Y' \<noteq> {#}"
       "\<forall>x \<in> set_mset X'. \<exists>y \<in> set_mset Y'. (x, y) \<in> s"
       using mult_implies_one_step[OF \<open>trans s\<close> add(2)] by auto
     consider Z2 where "Z' = add_mset z Z2" | X2 Y2 where "X' = add_mset z X2" "Y' = add_mset z Y2"
       using *(1,2) by (metis add_mset_remove_trivial_If insert_iff set_mset_add_mset_insert union_iff)
     thus ?case
     proof (cases)
       case 1 thus ?thesis
         using * one_step_implies_mult[of Y' X' s Z2] add(3)
         by (auto simp: add.commute[of _ "{#_#}"] add.assoc intro: add(1) elim: irrefl_on_subset)
     next
       case 2 then obtain y where "y \<in> set_mset Y2" "(z, y) \<in> s"
         using *(4) \<open>irrefl_on (set_mset (add_mset z Z)) s\<close>
         by (auto simp: irrefl_on_def)
       moreover from this transD[OF \<open>trans s\<close> _ this(2)]
       have "x' \<in> set_mset X2 \<Longrightarrow> \<exists>y \<in> set_mset Y2. (x', y) \<in> s" for x'
         using 2 *(4)[rule_format, of x'] by auto
       ultimately show ?thesis
         using * one_step_implies_mult[of Y2 X2 s Z'] 2 add(3)
         by (force simp: add.commute[of "{#_#}"] add.assoc[symmetric] intro: add(1)
             elim: irrefl_on_subset)
     qed
   qed auto
 next
   assume ?R then obtain I J K
     where "Y = I + J" "X = I + K" "J \<noteq> {#}" "\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> s"
     using mult_implies_one_step[OF \<open>trans s\<close>] by blast
   thus ?L using one_step_implies_mult[of J K s "I + Z"] by (auto simp: ac_simps)
 qed
 
 lemma multp_cancel:
   "transp R \<Longrightarrow> irreflp_on (set_mset Z) R \<Longrightarrow> multp R (X + Z) (Y + Z) \<longleftrightarrow> multp R X Y"
   by (rule mult_cancel[to_pred])
 
 lemma mult_cancel_add_mset:
   "trans r \<Longrightarrow> irrefl_on {z} r \<Longrightarrow>
     ((add_mset z X, add_mset z Y) \<in> mult r) = ((X, Y) \<in> mult r)"
   by (rule mult_cancel[of _ "{#_#}", simplified])
 
 lemma multp_cancel_add_mset:
   "transp R \<Longrightarrow> irreflp_on {z} R \<Longrightarrow> multp R (add_mset z X) (add_mset z Y) = multp R X Y"
   by (rule mult_cancel_add_mset[to_pred, folded bot_set_def])
 
 lemma mult_cancel_max0:
   assumes "trans s" and "irrefl_on (set_mset X \<inter> set_mset Y) s"
   shows "(X, Y) \<in> mult s \<longleftrightarrow> (X - X \<inter># Y, Y - X \<inter># Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
 proof -
   have "(X - X \<inter># Y + X \<inter># Y, Y - X \<inter># Y + X \<inter># Y) \<in> mult s \<longleftrightarrow> (X - X \<inter># Y, Y - X \<inter># Y) \<in> mult s"
   proof (rule mult_cancel)
     from assms show "trans s"
       by simp
   next
     from assms show "irrefl_on (set_mset (X \<inter># Y)) s"
       by simp
   qed
   moreover have "X - X \<inter># Y + X \<inter># Y = X" "Y - X \<inter># Y + X \<inter># Y = Y"
     by (auto simp flip: count_inject)
   ultimately show ?thesis
     by simp
 qed
 
 lemma mult_cancel_max:
   "trans r \<Longrightarrow> irrefl_on (set_mset X \<inter> set_mset Y) r \<Longrightarrow>
     (X, Y) \<in> mult r \<longleftrightarrow> (X - Y, Y - X) \<in> mult r"
   by (rule mult_cancel_max0[simplified])
 
 lemma multp_cancel_max:
   "transp R \<Longrightarrow> irreflp_on (set_mset X \<inter> set_mset Y) R \<Longrightarrow> multp R X Y \<longleftrightarrow> multp R (X - Y) (Y - X)"
   by (rule mult_cancel_max[to_pred])
 
 
 subsubsection \<open>Strict partial-order properties\<close>
 
 lemma mult1_lessE:
   assumes "(N, M) \<in> mult1 {(a, b). r a b}" and "asymp r"
   obtains a M0 K where "M = add_mset a M0" "N = M0 + K"
     "a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> r b a"
 proof -
   from assms obtain a M0 K where "M = add_mset a M0" "N = M0 + K" and
     *: "b \<in># K \<Longrightarrow> r b a" for b by (blast elim: mult1E)
   moreover from * [of a] have "a \<notin># K"
     using \<open>asymp r\<close> by (meson asympD)
   ultimately show thesis by (auto intro: that)
 qed
 
 lemma trans_on_mult:
   assumes "trans_on A r" and "\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A"
   shows "trans_on B (mult r)"
   using assms by (metis mult_def subset_UNIV trans_on_subset trans_trancl)
 
 lemma trans_mult: "trans r \<Longrightarrow> trans (mult r)"
   using trans_on_mult[of UNIV r UNIV, simplified] .
 
 lemma transp_on_multp:
   assumes "transp_on A r" and "\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A"
   shows "transp_on B (multp r)"
   by (metis mult_def multp_def transD trans_trancl transp_onI)
 
 lemma transp_multp: "transp r \<Longrightarrow> transp (multp r)"
   using transp_on_multp[of UNIV r UNIV, simplified] .
 
 lemma irrefl_mult:
   assumes "trans r" "irrefl r"
   shows "irrefl (mult r)"
 proof (intro irreflI notI)
   fix M
   assume "(M, M) \<in> mult r"
   then obtain I J K where "M = I + J" and "M = I + K"
     and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> r)"
     using mult_implies_one_step[OF \<open>trans r\<close>] by blast
   then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. (k, j) \<in> r" by auto
   have "finite (set_mset K)" by simp
   hence "set_mset K = {}"
     using **
   proof (induction rule: finite_induct)
     case empty
     thus ?case by simp
   next
     case (insert x F)
     have False
       using \<open>irrefl r\<close>[unfolded irrefl_def, rule_format]
       using \<open>trans r\<close>[THEN transD]
       by (metis equals0D insert.IH insert.prems insertE insertI1 insertI2)
     thus ?case ..
   qed
   with * show False by simp
 qed
 
 lemma irreflp_multp: "transp R \<Longrightarrow> irreflp R \<Longrightarrow> irreflp (multp R)"
   by (rule irrefl_mult[of "{(x, y). r x y}" for r,
     folded transp_trans_eq irreflp_irrefl_eq, simplified, folded multp_def])
 
 instantiation multiset :: (preorder) order begin
 
 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"
   where "M < N \<longleftrightarrow> multp (<) M N"
 
 definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"
   where "less_eq_multiset M N \<longleftrightarrow> M < N \<or> M = N"
 
 instance
 proof intro_classes
   fix M N :: "'a multiset"
   show "(M < N) = (M \<le> N \<and> \<not> N \<le> M)"
     unfolding less_eq_multiset_def less_multiset_def
     by (metis irreflp_def irreflp_on_less irreflp_multp transpE transp_on_less transp_multp)
 next
   fix M :: "'a multiset"
   show "M \<le> M"
     unfolding less_eq_multiset_def
     by simp
 next
   fix M1 M2 M3 :: "'a multiset"
   show "M1 \<le> M2 \<Longrightarrow> M2 \<le> M3 \<Longrightarrow> M1 \<le> M3"
     unfolding less_eq_multiset_def less_multiset_def
     using transp_multp[OF transp_on_less, THEN transpD]
     by blast
 next
   fix M N :: "'a multiset"
   show "M \<le> N \<Longrightarrow> N \<le> M \<Longrightarrow> M = N"
     unfolding less_eq_multiset_def less_multiset_def
     using transp_multp[OF transp_on_less, THEN transpD]
     using irreflp_multp[OF transp_on_less irreflp_on_less, unfolded irreflp_def, rule_format]
     by blast
 qed
 
 end
 
 lemma mset_le_irrefl [elim!]:
   fixes M :: "'a::preorder multiset"
   shows "M < M \<Longrightarrow> R"
   by simp
 
 lemma wfP_less_multiset[simp]:
   assumes wfP_less: "wfP ((<) :: ('a :: preorder) \<Rightarrow> 'a \<Rightarrow> bool)"
   shows "wfP ((<) :: 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool)"
   unfolding less_multiset_def
   using wfP_multp[OF wfP_less] .
 
 
 subsubsection \<open>Strict total-order properties\<close>
 
 lemma total_on_mult:
   assumes "total_on A r" and "trans r" and "\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A"
   shows "total_on B (mult r)"
 proof (rule total_onI)
   fix M1 M2 assume "M1 \<in> B" and "M2 \<in> B" and "M1 \<noteq> M2"
   let ?I = "M1 \<inter># M2"
   show "(M1, M2) \<in> mult r \<or> (M2, M1) \<in> mult r"
   proof (cases "M1 - ?I = {#} \<or> M2 - ?I = {#}")
     case True
     with \<open>M1 \<noteq> M2\<close> show ?thesis
       by (metis Diff_eq_empty_iff_mset diff_intersect_left_idem diff_intersect_right_idem
           subset_implies_mult subset_mset.less_le)
   next
     case False
     from assms(1) have "total_on (set_mset (M1 - ?I)) r"
       by (meson \<open>M1 \<in> B\<close> assms(3) diff_subset_eq_self set_mset_mono total_on_subset)
     with False obtain greatest1 where
       greatest1_in: "greatest1 \<in># M1 - ?I" and
       greatest1_greatest: "\<forall>x \<in># M1 - ?I. greatest1 \<noteq> x \<longrightarrow> (x, greatest1) \<in> r"
       using Multiset.bex_greatest_element[to_set, of "M1 - ?I" r]
       by (metis assms(2) subset_UNIV trans_on_subset)
 
     from assms(1) have "total_on (set_mset (M2 - ?I)) r"
       by (meson \<open>M2 \<in> B\<close> assms(3) diff_subset_eq_self set_mset_mono total_on_subset)
     with False obtain greatest2 where
       greatest2_in: "greatest2 \<in># M2 - ?I" and
       greatest2_greatest: "\<forall>x \<in># M2 - ?I. greatest2 \<noteq> x \<longrightarrow> (x, greatest2) \<in> r"
       using Multiset.bex_greatest_element[to_set, of "M2 - ?I" r]
       by (metis assms(2) subset_UNIV trans_on_subset)
 
     have "greatest1 \<noteq> greatest2"
       using greatest1_in \<open>greatest2 \<in># M2 - ?I\<close>
       by (metis diff_intersect_left_idem diff_intersect_right_idem dual_order.eq_iff in_diff_count
           in_diff_countE le_add_same_cancel2 less_irrefl zero_le)
     hence "(greatest1, greatest2) \<in> r \<or> (greatest2, greatest1) \<in> r"
       using \<open>total_on A r\<close>[unfolded total_on_def, rule_format, of greatest1 greatest2]
         \<open>M1 \<in> B\<close> \<open>M2 \<in> B\<close> greatest1_in greatest2_in assms(3)
       by (meson in_diffD in_mono)
     thus ?thesis
     proof (elim disjE)
       assume "(greatest1, greatest2) \<in> r"
       have "(?I + (M1 - ?I), ?I + (M2 - ?I)) \<in> mult r"
       proof (rule one_step_implies_mult[of "M2 - ?I" "M1 - ?I" r ?I])
         show "M2 - ?I \<noteq> {#}"
           using False by force
       next
         show "\<forall>k\<in>#M1 - ?I. \<exists>j\<in>#M2 - ?I. (k, j) \<in> r"
           using \<open>(greatest1, greatest2) \<in> r\<close> greatest2_in greatest1_greatest
           by (metis assms(2) transD)
       qed
       hence "(M1, M2) \<in> mult r"
         by (metis subset_mset.add_diff_inverse subset_mset.inf.cobounded1
             subset_mset.inf.cobounded2)
       thus "(M1, M2) \<in> mult r \<or> (M2, M1) \<in> mult r" ..
     next
       assume "(greatest2, greatest1) \<in> r"
       have "(?I + (M2 - ?I), ?I + (M1 - ?I)) \<in> mult r"
       proof (rule one_step_implies_mult[of "M1 - ?I" "M2 - ?I" r ?I])
         show "M1 - M1 \<inter># M2 \<noteq> {#}"
           using False by force
       next
         show "\<forall>k\<in>#M2 - ?I. \<exists>j\<in>#M1 - ?I. (k, j) \<in> r"
           using \<open>(greatest2, greatest1) \<in> r\<close> greatest1_in greatest2_greatest
           by (metis assms(2) transD)
       qed
       hence "(M2, M1) \<in> mult r"
         by (metis subset_mset.add_diff_inverse subset_mset.inf.cobounded1
             subset_mset.inf.cobounded2)
       thus "(M1, M2) \<in> mult r \<or> (M2, M1) \<in> mult r" ..
     qed
   qed
 qed
 
 lemma total_mult: "total r \<Longrightarrow> trans r \<Longrightarrow> total (mult r)"
   by (rule total_on_mult[of UNIV r UNIV, simplified])
 
 lemma totalp_on_multp:
   "totalp_on A R \<Longrightarrow> transp R \<Longrightarrow> (\<And>M. M \<in> B \<Longrightarrow> set_mset M \<subseteq> A) \<Longrightarrow> totalp_on B (multp R)"
   using total_on_mult[of A "{(x,y). R x y}" B, to_pred]
   by (simp add: multp_def total_on_def totalp_on_def)
 
 lemma totalp_multp: "totalp R \<Longrightarrow> transp R \<Longrightarrow> totalp (multp R)"
   by (rule totalp_on_multp[of UNIV R UNIV, simplified])
 
 
 subsection \<open>Quasi-executable version of the multiset extension\<close>
 
 text \<open>
   Predicate variants of \<open>mult\<close> and the reflexive closure of \<open>mult\<close>, which are
   executable whenever the given predicate \<open>P\<close> is. Together with the standard
   code equations for \<open>(\<inter>#\<close>) and \<open>(-\<close>) this should yield quadratic
   (with respect to calls to \<open>P\<close>) implementations of \<open>multp_code\<close> and \<open>multeqp_code\<close>.
 \<close>
 
 definition multp_code :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multp_code P N M =
     (let Z = M \<inter># N; X = M - Z in
     X \<noteq> {#} \<and> (let Y = N - Z in (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x)))"
 
 definition multeqp_code :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multeqp_code P N M =
     (let Z = M \<inter># N; X = M - Z; Y = N - Z in
     (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x))"
 
 lemma multp_code_iff_mult:
   assumes "irrefl_on (set_mset N \<inter> set_mset M) R" and "trans R" and
     [simp]: "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
   shows "multp_code P N M \<longleftrightarrow> (N, M) \<in> mult R" (is "?L \<longleftrightarrow> ?R")
 proof -
   have *: "M \<inter># N + (N - M \<inter># N) = N" "M \<inter># N + (M - M \<inter># N) = M"
     "(M - M \<inter># N) \<inter># (N - M \<inter># N) = {#}" by (auto simp flip: count_inject)
   show ?thesis
   proof
     assume ?L thus ?R
       using one_step_implies_mult[of "M - M \<inter># N" "N - M \<inter># N" R "M \<inter># N"] *
       by (auto simp: multp_code_def Let_def)
   next
     { fix I J K :: "'a multiset" assume "(I + J) \<inter># (I + K) = {#}"
       then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty)
     } note [dest!] = this
     assume ?R thus ?L
       using mult_cancel_max
       using mult_implies_one_step[OF assms(2), of "N - M \<inter># N" "M - M \<inter># N"]
         mult_cancel_max[OF assms(2,1)] * by (auto simp: multp_code_def)
   qed
 qed
 
 lemma multp_code_iff_multp:
   "irreflp_on (set_mset M \<inter> set_mset N) R \<Longrightarrow> transp R \<Longrightarrow> multp_code R M N \<longleftrightarrow> multp R M N"
   using multp_code_iff_mult[simplified, to_pred, of M N R R] by simp
 
 lemma multp_code_eq_multp:
   assumes "irreflp R" and "transp R"
   shows "multp_code R = multp R"
 proof (intro ext)
   fix M N
   show "multp_code R M N = multp R M N"
   proof (rule multp_code_iff_multp)
     from assms show "irreflp_on (set_mset M \<inter> set_mset N) R"
       by (auto intro: irreflp_on_subset)
   next
     from assms show "transp R"
       by simp
   qed
 qed
 
 lemma multeqp_code_iff_reflcl_mult:
   assumes "irrefl_on (set_mset N \<inter> set_mset M) R" and "trans R" and "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
   shows "multeqp_code P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>="
 proof -
   { assume "N \<noteq> M" "M - M \<inter># N = {#}"
     then obtain y where "count N y \<noteq> count M y" by (auto simp flip: count_inject)
     then have "\<exists>y. count M y < count N y" using \<open>M - M \<inter># N = {#}\<close>
       by (auto simp flip: count_inject dest!: le_neq_implies_less fun_cong[of _ _ y])
   }
   then have "multeqp_code P N M \<longleftrightarrow> multp_code P N M \<or> N = M"
     by (auto simp: multeqp_code_def multp_code_def Let_def in_diff_count)
   thus ?thesis
     using multp_code_iff_mult[OF assms] by simp
 qed
 
 lemma multeqp_code_iff_reflclp_multp:
   "irreflp_on (set_mset M \<inter> set_mset N) R \<Longrightarrow> transp R \<Longrightarrow> multeqp_code R M N \<longleftrightarrow> (multp R)\<^sup>=\<^sup>= M N"
   using multeqp_code_iff_reflcl_mult[simplified, to_pred, of M N R R] by simp
 
 lemma multeqp_code_eq_reflclp_multp:
   assumes "irreflp R" and "transp R"
   shows "multeqp_code R = (multp R)\<^sup>=\<^sup>="
 proof (intro ext)
   fix M N
   show "multeqp_code R M N \<longleftrightarrow> (multp R)\<^sup>=\<^sup>= M N"
   proof (rule multeqp_code_iff_reflclp_multp)
     from assms show "irreflp_on (set_mset M \<inter> set_mset N) R"
       by (auto intro: irreflp_on_subset)
   next
     from assms show "transp R"
       by simp
   qed
 qed
 
 
 subsubsection \<open>Monotonicity of multiset union\<close>
 
 lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
   by (force simp: mult1_def)
 
 lemma union_le_mono2: "B < D \<Longrightarrow> C + B < C + (D::'a::preorder multiset)"
-apply (unfold less_multiset_def multp_def mult_def)
-apply (erule trancl_induct)
- apply (blast intro: mult1_union)
-apply (blast intro: mult1_union trancl_trans)
-done
+  unfolding less_multiset_def multp_def mult_def
+  by (induction rule: trancl_induct; blast intro: mult1_union trancl_trans)
 
 lemma union_le_mono1: "B < D \<Longrightarrow> B + C < D + (C::'a::preorder multiset)"
-apply (subst add.commute [of B C])
-apply (subst add.commute [of D C])
-apply (erule union_le_mono2)
-done
+  by (metis add.commute union_le_mono2)
 
 lemma union_less_mono:
   fixes A B C D :: "'a::preorder multiset"
   shows "A < C \<Longrightarrow> B < D \<Longrightarrow> A + B < C + D"
   by (blast intro!: union_le_mono1 union_le_mono2 less_trans)
 
 instantiation multiset :: (preorder) ordered_ab_semigroup_add
 begin
 instance
   by standard (auto simp add: less_eq_multiset_def intro: union_le_mono2)
 end
 
 
 subsubsection \<open>Termination proofs with multiset orders\<close>
 
 lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
   and multi_member_this: "x \<in># {# x #} + XS"
   and multi_member_last: "x \<in># {# x #}"
   by auto
 
 definition "ms_strict = mult pair_less"
 definition "ms_weak = ms_strict \<union> Id"
 
 lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
-unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
-by (auto intro: wf_mult1 wf_trancl simp: mult_def)
+  unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
+  by (auto intro: wf_mult1 wf_trancl simp: mult_def)
 
 lemma smsI:
   "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
   unfolding ms_strict_def
 by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
 
 lemma wmsI:
   "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}
   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
 unfolding ms_weak_def ms_strict_def
 by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
 
 inductive pw_leq
 where
   pw_leq_empty: "pw_leq {#} {#}"
 | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
 
 lemma pw_leq_lstep:
   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
 by (drule pw_leq_step) (rule pw_leq_empty, simp)
 
 lemma pw_leq_split:
   assumes "pw_leq X Y"
   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
   using assms
 proof induct
   case pw_leq_empty thus ?case by auto
 next
   case (pw_leq_step x y X Y)
   then obtain A B Z where
     [simp]: "X = A + Z" "Y = B + Z"
       and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
     by auto
   from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less"
     unfolding pair_leq_def by auto
   thus ?case
   proof cases
     case [simp]: 1
     have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and>
       ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
       by auto
     thus ?thesis by blast
   next
     case 2
     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
     have "{#x#} + X = ?A' + Z"
       "{#y#} + Y = ?B' + Z"
       by auto
     moreover have
       "(set_mset ?A', set_mset ?B') \<in> max_strict"
       using 1 2 unfolding max_strict_def
       by (auto elim!: max_ext.cases)
     ultimately show ?thesis by blast
   qed
 qed
 
 lemma
   assumes pwleq: "pw_leq Z Z'"
   shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
     and ms_weakI1:  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
     and ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
 proof -
   from pw_leq_split[OF pwleq]
   obtain A' B' Z''
     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
     and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
     by blast
   {
     assume max: "(set_mset A, set_mset B) \<in> max_strict"
     from mx_or_empty
     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
     proof
       assume max': "(set_mset A', set_mset B') \<in> max_strict"
       with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict"
         by (auto simp: max_strict_def intro: max_ext_additive)
       thus ?thesis by (rule smsI)
     next
       assume [simp]: "A' = {#} \<and> B' = {#}"
       show ?thesis by (rule smsI) (auto intro: max)
     qed
     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps)
     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
   }
   from mx_or_empty
   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add: ac_simps)
 qed
 
 lemma empty_neutral: "{#} + x = x" "x + {#} = x"
 and nonempty_plus: "{# x #} + rs \<noteq> {#}"
 and nonempty_single: "{# x #} \<noteq> {#}"
 by auto
 
 setup \<open>
   let
     fun msetT T = \<^Type>\<open>multiset T\<close>;
 
     fun mk_mset T [] = \<^instantiate>\<open>'a = T in term \<open>{#}\<close>\<close>
       | mk_mset T [x] = \<^instantiate>\<open>'a = T and x in term \<open>{#x#}\<close>\<close>
       | mk_mset T (x :: xs) = \<^Const>\<open>plus \<open>msetT T\<close> for \<open>mk_mset T [x]\<close> \<open>mk_mset T xs\<close>\<close>
 
     fun mset_member_tac ctxt m i =
       if m <= 0 then
         resolve_tac ctxt @{thms multi_member_this} i ORELSE
         resolve_tac ctxt @{thms multi_member_last} i
       else
         resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i
 
     fun mset_nonempty_tac ctxt =
       resolve_tac ctxt @{thms nonempty_plus} ORELSE'
       resolve_tac ctxt @{thms nonempty_single}
 
     fun regroup_munion_conv ctxt =
       Function_Lib.regroup_conv ctxt \<^const_abbrev>\<open>empty_mset\<close> \<^const_name>\<open>plus\<close>
         (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))
 
     fun unfold_pwleq_tac ctxt i =
       (resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st))
         ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i)
         ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i)
 
     val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union},
                         @{thm Un_insert_left}, @{thm Un_empty_left}]
   in
     ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
     {
       msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
       mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
       mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps,
       smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
       reduction_pair = @{thm ms_reduction_pair}
     })
   end
 \<close>
 
 
 subsection \<open>Legacy theorem bindings\<close>
 
 lemmas multi_count_eq = multiset_eq_iff [symmetric]
 
 lemma union_commute: "M + N = N + (M::'a multiset)"
   by (fact add.commute)
 
 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
   by (fact add.assoc)
 
 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
   by (fact add.left_commute)
 
 lemmas union_ac = union_assoc union_commute union_lcomm add_mset_commute
 
 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
   by (fact add_right_cancel)
 
 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
   by (fact add_left_cancel)
 
 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
   by (fact add_left_imp_eq)
 
 lemma mset_subset_trans: "(M::'a multiset) \<subset># K \<Longrightarrow> K \<subset># N \<Longrightarrow> M \<subset># N"
   by (fact subset_mset.less_trans)
 
 lemma multiset_inter_commute: "A \<inter># B = B \<inter># A"
   by (fact subset_mset.inf.commute)
 
 lemma multiset_inter_assoc: "A \<inter># (B \<inter># C) = A \<inter># B \<inter># C"
   by (fact subset_mset.inf.assoc [symmetric])
 
 lemma multiset_inter_left_commute: "A \<inter># (B \<inter># C) = B \<inter># (A \<inter># C)"
   by (fact subset_mset.inf.left_commute)
 
 lemmas multiset_inter_ac =
   multiset_inter_commute
   multiset_inter_assoc
   multiset_inter_left_commute
 
 lemma mset_le_not_refl: "\<not> M < (M::'a::preorder multiset)"
   by (fact less_irrefl)
 
 lemma mset_le_trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < (N::'a::preorder multiset)"
   by (fact less_trans)
 
 lemma mset_le_not_sym: "M < N \<Longrightarrow> \<not> N < (M::'a::preorder multiset)"
   by (fact less_not_sym)
 
 lemma mset_le_asym: "M < N \<Longrightarrow> (\<not> P \<Longrightarrow> N < (M::'a::preorder multiset)) \<Longrightarrow> P"
   by (fact less_asym)
 
 declaration \<open>
   let
     fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ $ t') =
           let
             val (maybe_opt, ps) =
               Nitpick_Model.dest_plain_fun t'
               ||> (~~)
               ||> map (apsnd (snd o HOLogic.dest_number))
             fun elems_for t =
               (case AList.lookup (=) ps t of
                 SOME n => replicate n t
               | NONE => [Const (maybe_name, elem_T --> elem_T) $ t])
           in
             (case maps elems_for (all_values elem_T) @
                  (if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of
               [] => \<^Const>\<open>Groups.zero T\<close>
             | ts => foldl1 (fn (s, t) => \<^Const>\<open>add_mset elem_T for s t\<close>) ts)
           end
       | multiset_postproc _ _ _ _ t = t
   in Nitpick_Model.register_term_postprocessor \<^typ>\<open>'a multiset\<close> multiset_postproc end
 \<close>
 
 
 subsection \<open>Naive implementation using lists\<close>
 
 code_datatype mset
 
 lemma [code]: "{#} = mset []"
   by simp
 
 lemma [code]: "add_mset x (mset xs) = mset (x # xs)"
   by simp
 
 lemma [code]: "Multiset.is_empty (mset xs) \<longleftrightarrow> List.null xs"
   by (simp add: Multiset.is_empty_def List.null_def)
 
 lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)"
   by simp
 
 lemma [code]: "image_mset f (mset xs) = mset (map f xs)"
   by simp
 
 lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
   by simp
 
 lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"
   by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute diff_diff_add)
 
 lemma [code]:
   "mset xs \<inter># mset ys =
     mset (snd (fold (\<lambda>x (ys, zs).
       if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
 proof -
   have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
     if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
       (mset xs \<inter># mset ys) + mset zs"
     by (induct xs arbitrary: ys)
       (auto simp add: inter_add_right1 inter_add_right2 ac_simps)
   then show ?thesis by simp
 qed
 
 lemma [code]:
   "mset xs \<union># mset ys =
     mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
 proof -
   have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
       (mset xs \<union># mset ys) + mset zs"
     by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
   then show ?thesis by simp
 qed
 
 declare in_multiset_in_set [code_unfold]
 
 lemma [code]: "count (mset xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
 proof -
   have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (mset xs) x + n"
     by (induct xs) simp_all
   then show ?thesis by simp
 qed
 
 declare set_mset_mset [code]
 
 declare sorted_list_of_multiset_mset [code]
 
 lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close>
   "mset_set A = mset (sorted_list_of_set A)"
-  apply (cases "finite A")
-  apply simp_all
-  apply (induct A rule: finite_induct)
-  apply simp_all
-  done
+  by (metis mset_sorted_list_of_multiset sorted_list_of_mset_set)
 
 declare size_mset [code]
 
 fun subset_eq_mset_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
   "subset_eq_mset_impl [] ys = Some (ys \<noteq> [])"
 | "subset_eq_mset_impl (Cons x xs) ys = (case List.extract ((=) x) ys of
      None \<Rightarrow> None
    | Some (ys1,_,ys2) \<Rightarrow> subset_eq_mset_impl xs (ys1 @ ys2))"
 
 lemma subset_eq_mset_impl: "(subset_eq_mset_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<subseteq># mset ys) \<and>
   (subset_eq_mset_impl xs ys = Some True \<longleftrightarrow> mset xs \<subset># mset ys) \<and>
   (subset_eq_mset_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
 proof (induct xs arbitrary: ys)
   case (Nil ys)
   show ?case by (auto simp: subset_mset.zero_less_iff_neq_zero)
 next
   case (Cons x xs ys)
   show ?case
   proof (cases "List.extract ((=) x) ys")
     case None
     hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
     {
       assume "mset (x # xs) \<subseteq># mset ys"
       from set_mset_mono[OF this] x have False by simp
     } note nle = this
     moreover
     {
       assume "mset (x # xs) \<subset># mset ys"
       hence "mset (x # xs) \<subseteq># mset ys" by auto
       from nle[OF this] have False .
     }
     ultimately show ?thesis using None by auto
   next
     case (Some res)
     obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
     note Some = Some[unfolded res]
     from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
     hence id: "mset ys = add_mset x (mset (ys1 @ ys2))"
       by auto
     show ?thesis unfolding subset_eq_mset_impl.simps
-      unfolding Some option.simps split
-      unfolding id
-      using Cons[of "ys1 @ ys2"]
-      unfolding subset_mset_def subseteq_mset_def by auto
+      by (simp add: Some id Cons)
   qed
 qed
 
 lemma [code]: "mset xs \<subseteq># mset ys \<longleftrightarrow> subset_eq_mset_impl xs ys \<noteq> None"
-  using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)
+  by (simp add: subset_eq_mset_impl)
 
 lemma [code]: "mset xs \<subset># mset ys \<longleftrightarrow> subset_eq_mset_impl xs ys = Some True"
-  using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)
+  using subset_eq_mset_impl by blast
 
 instantiation multiset :: (equal) equal
 begin
 
 definition
   [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
 lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> subset_eq_mset_impl xs ys = Some False"
   unfolding equal_multiset_def
   using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)
 
 instance
   by standard (simp add: equal_multiset_def)
 
 end
 
 declare sum_mset_sum_list [code]
 
 lemma [code]: "prod_mset (mset xs) = fold times xs 1"
 proof -
   have "\<And>x. fold times xs x = prod_mset (mset xs) * x"
     by (induct xs) (simp_all add: ac_simps)
   then show ?thesis by simp
 qed
 
 text \<open>
   Exercise for the casual reader: add implementations for \<^term>\<open>(\<le>)\<close>
   and \<^term>\<open>(<)\<close> (multiset order).
 \<close>
 
 text \<open>Quickcheck generators\<close>
 
 context
   includes term_syntax
 begin
 
 definition
   msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
   [code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs"
 
 end
 
 instantiation multiset :: (random) random
 begin
 
 context
   includes state_combinator_syntax
 begin
 
 definition
   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
 
 instance ..
 
 end
 
 end
 
 instantiation multiset :: (full_exhaustive) full_exhaustive
 begin
 
 definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
 where
   "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"
 
 instance ..
 
 end
 
 hide_const (open) msetify
 
 
 subsection \<open>BNF setup\<close>
 
 definition rel_mset where
   "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"
 
 lemma mset_zip_take_Cons_drop_twice:
   assumes "length xs = length ys" "j \<le> length xs"
   shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
     add_mset (x,y) (mset (zip xs ys))"
   using assms
 proof (induct xs ys arbitrary: x y j rule: list_induct2)
   case Nil
   thus ?case
     by simp
 next
   case (Cons x xs y ys)
   thus ?case
   proof (cases "j = 0")
     case True
     thus ?thesis
       by simp
   next
     case False
     then obtain k where k: "j = Suc k"
       by (cases j) simp
     hence "k \<le> length xs"
       using Cons.prems by auto
     hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
       add_mset (x,y) (mset (zip xs ys))"
       by (rule Cons.hyps(2))
     thus ?thesis
       unfolding k by auto
   qed
 qed
 
 lemma ex_mset_zip_left:
   assumes "length xs = length ys" "mset xs' = mset xs"
   shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
 using assms
 proof (induct xs ys arbitrary: xs' rule: list_induct2)
   case Nil
   thus ?case
     by auto
 next
   case (Cons x xs y ys xs')
   obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
     by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD)
 
   define xsa where "xsa = take j xs' @ drop (Suc j) xs'"
   have "mset xs' = {#x#} + mset xsa"
     unfolding xsa_def using j_len nth_j
     by (metis Cons_nth_drop_Suc union_mset_add_mset_right add_mset_remove_trivial add_diff_cancel_left'
         append_take_drop_id mset.simps(2) mset_append)
   hence ms_x: "mset xsa = mset xs"
     by (simp add: Cons.prems)
   then obtain ysa where
     len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
     using Cons.hyps(2) by blast
 
   define ys' where "ys' = take j ysa @ y # drop j ysa"
   have xs': "xs' = take j xsa @ x # drop j xsa"
     using ms_x j_len nth_j Cons.prems xsa_def
     by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
       length_drop size_mset)
   have j_len': "j \<le> length xsa"
     using j_len xs' xsa_def
     by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
   have "length ys' = length xs'"
     unfolding ys'_def using Cons.prems len_a ms_x
     by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length)
   moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))"
     unfolding xs' ys'_def
     by (rule trans[OF mset_zip_take_Cons_drop_twice])
       (auto simp: len_a ms_a j_len')
   ultimately show ?case
     by blast
 qed
 
 lemma list_all2_reorder_left_invariance:
   assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
   shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
 proof -
   have len: "length xs = length ys"
     using rel list_all2_conv_all_nth by auto
   obtain ys' where
     len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)"
     using len ms_x by (metis ex_mset_zip_left)
   have "list_all2 R xs' ys'"
     using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD)
   moreover have "mset ys' = mset ys"
     using len len' ms_xy map_snd_zip mset_map by metis
   ultimately show ?thesis
     by blast
 qed
 
 lemma ex_mset: "\<exists>xs. mset xs = X"
   by (induct X) (simp, metis mset.simps(2))
 
 inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool" 
 where
   "pred_mset P {#}"
 | "\<lbrakk>P a; pred_mset P M\<rbrakk> \<Longrightarrow> pred_mset P (add_mset a M)"
 
 lemma pred_mset_iff: \<comment> \<open>TODO: alias for \<^const>\<open>Multiset.Ball\<close>\<close>
   \<open>pred_mset P M \<longleftrightarrow> Multiset.Ball M P\<close>  (is \<open>?P \<longleftrightarrow> ?Q\<close>)
 proof
   assume ?P
   then show ?Q by induction simp_all
 next
   assume ?Q
   then show ?P
     by (induction M) (auto intro: pred_mset.intros)
 qed
 
 bnf "'a multiset"
   map: image_mset
   sets: set_mset
   bd: natLeq
   wits: "{#}"
   rel: rel_mset
   pred: pred_mset
 proof -
-  show "image_mset id = id"
-    by (rule image_mset.id)
   show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g
     unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
   show "(\<And>z. z \<in> set_mset X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X" for f g X
     by (induct X) simp_all
-  show "set_mset \<circ> image_mset f = (`) f \<circ> set_mset" for f
-    by auto
   show "card_order natLeq"
     by (rule natLeq_card_order)
   show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
     by (rule natLeq_cinfinite)
   show "regularCard natLeq"
     by (rule regularCard_natLeq)
   show "ordLess2 (card_of (set_mset X)) natLeq" for X
     by transfer
       (auto simp: finite_iff_ordLess_natLeq[symmetric])
   show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S
     unfolding rel_mset_def[abs_def] OO_def
-    apply clarify
-    subgoal for X Z Y xs ys' ys zs
-      apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys])
-      apply (auto intro: list_all2_trans)
-      done
-    done
+    by (smt (verit, ccfv_SIG) list_all2_reorder_left_invariance list_all2_trans predicate2I)
   show "rel_mset R =
     (\<lambda>x y. \<exists>z. set_mset z \<subseteq> {(x, y). R x y} \<and>
     image_mset fst z = x \<and> image_mset snd z = y)" for R
     unfolding rel_mset_def[abs_def]
-    apply (rule ext)+
-    apply safe
-     apply (rule_tac x = "mset (zip xs ys)" in exI;
-       auto simp: in_set_zip list_all2_iff simp flip: mset_map)
-    apply (rename_tac XY)
-    apply (cut_tac X = XY in ex_mset)
-    apply (erule exE)
-    apply (rename_tac xys)
-    apply (rule_tac x = "map fst xys" in exI)
-    apply (auto simp: mset_map)
-    apply (rule_tac x = "map snd xys" in exI)
-    apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd)
-    done
-  show "z \<in> set_mset {#} \<Longrightarrow> False" for z
-    by auto
+    by (metis (no_types, lifting) ex_mset list.in_rel mem_Collect_eq mset_map set_mset_mset)
   show "pred_mset P = (\<lambda>x. Ball (set_mset x) P)" for P
     by (simp add: fun_eq_iff pred_mset_iff)
-qed
+qed auto
 
 inductive rel_mset' :: \<open>('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset \<Rightarrow> bool\<close>
 where
   Zero[intro]: "rel_mset' R {#} {#}"
 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (add_mset a M) (add_mset b N)"
 
 lemma rel_mset_Zero: "rel_mset R {#} {#}"
 unfolding rel_mset_def Grp_def by auto
 
 declare multiset.count[simp]
 declare count_Abs_multiset[simp]
 declare multiset.count_inverse[simp]
 
 lemma rel_mset_Plus:
   assumes ab: "R a b"
     and MN: "rel_mset R M N"
   shows "rel_mset R (add_mset a M) (add_mset b N)"
 proof -
   have "\<exists>ya. add_mset a (image_mset fst y) = image_mset fst ya \<and>
     add_mset b (image_mset snd y) = image_mset snd ya \<and>
     set_mset ya \<subseteq> {(x, y). R x y}"
     if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y
     using that by (intro exI[of _ "add_mset (a,b) y"]) auto
   thus ?thesis
   using assms
   unfolding multiset.rel_compp_Grp Grp_def by blast
 qed
 
 lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
   by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)
 
 lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N"
   unfolding multiset.rel_compp_Grp Grp_def by auto
 
 lemma rel_mset_Zero_iff [simp]:
   shows "rel_mset rel {#} Y \<longleftrightarrow> Y = {#}" and "rel_mset rel X {#} \<longleftrightarrow> X = {#}"
   by (auto simp add: rel_mset_Zero dest: rel_mset_size)
 
 lemma multiset_induct2[case_names empty addL addR]:
   assumes empty: "P {#} {#}"
     and addL: "\<And>a M N. P M N \<Longrightarrow> P (add_mset a M) N"
     and addR: "\<And>a M N. P M N \<Longrightarrow> P M (add_mset a N)"
   shows "P M N"
-apply(induct N rule: multiset_induct)
-  apply(induct M rule: multiset_induct, rule empty, erule addL)
-  apply(induct M rule: multiset_induct, erule addR, erule addR)
-done
+  by (induct N rule: multiset_induct; induct M rule: multiset_induct) (auto simp: assms)
 
 lemma multiset_induct2_size[consumes 1, case_names empty add]:
   assumes c: "size M = size N"
     and empty: "P {#} {#}"
     and add: "\<And>a b M N a b. P M N \<Longrightarrow> P (add_mset a M) (add_mset b N)"
   shows "P M N"
   using c
 proof (induct M arbitrary: N rule: measure_induct_rule[of size])
   case (less M)
   show ?case
   proof(cases "M = {#}")
     case True hence "N = {#}" using less.prems by auto
     thus ?thesis using True empty by auto
   next
     case False then obtain M1 a where M: "M = add_mset a M1" by (metis multi_nonempty_split)
     have "N \<noteq> {#}" using False less.prems by auto
     then obtain N1 b where N: "N = add_mset b N1" by (metis multi_nonempty_split)
     have "size M1 = size N1" using less.prems unfolding M N by auto
     thus ?thesis using M N less.hyps add by auto
   qed
 qed
 
 lemma msed_map_invL:
   assumes "image_mset f (add_mset a M) = N"
   shows "\<exists>N1. N = add_mset (f a) N1 \<and> image_mset f M = N1"
 proof -
   have "f a \<in># N"
     using assms multiset.set_map[of f "add_mset a M"] by auto
   then obtain N1 where N: "N = add_mset (f a) N1" using multi_member_split by metis
   have "image_mset f M = N1" using assms unfolding N by simp
   thus ?thesis using N by blast
 qed
 
 lemma msed_map_invR:
   assumes "image_mset f M = add_mset b N"
   shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> image_mset f M1 = N"
 proof -
   obtain a where a: "a \<in># M" and fa: "f a = b"
     using multiset.set_map[of f M] unfolding assms
     by (metis image_iff union_single_eq_member)
   then obtain M1 where M: "M = add_mset a M1" using multi_member_split by metis
   have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
   thus ?thesis using M fa by blast
 qed
 
 lemma msed_rel_invL:
   assumes "rel_mset R (add_mset a M) N"
   shows "\<exists>N1 b. N = add_mset b N1 \<and> R a b \<and> rel_mset R M N1"
 proof -
   obtain K where KM: "image_mset fst K = add_mset a M"
     and KN: "image_mset snd K = N" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
     using assms
     unfolding multiset.rel_compp_Grp Grp_def by auto
   obtain K1 ab where K: "K = add_mset ab K1" and a: "fst ab = a"
     and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
   obtain N1 where N: "N = add_mset (snd ab) N1" and K1N1: "image_mset snd K1 = N1"
     using msed_map_invL[OF KN[unfolded K]] by auto
   have Rab: "R a (snd ab)" using sK a unfolding K by auto
   have "rel_mset R M N1" using sK K1M K1N1
     unfolding K multiset.rel_compp_Grp Grp_def by auto
   thus ?thesis using N Rab by auto
 qed
 
 lemma msed_rel_invR:
   assumes "rel_mset R M (add_mset b N)"
   shows "\<exists>M1 a. M = add_mset a M1 \<and> R a b \<and> rel_mset R M1 N"
 proof -
   obtain K where KN: "image_mset snd K = add_mset b N"
     and KM: "image_mset fst K = M" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
     using assms
     unfolding multiset.rel_compp_Grp Grp_def by auto
   obtain K1 ab where K: "K = add_mset ab K1" and b: "snd ab = b"
     and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
   obtain M1 where M: "M = add_mset (fst ab) M1" and K1M1: "image_mset fst K1 = M1"
     using msed_map_invL[OF KM[unfolded K]] by auto
   have Rab: "R (fst ab) b" using sK b unfolding K by auto
   have "rel_mset R M1 N" using sK K1N K1M1
     unfolding K multiset.rel_compp_Grp Grp_def by auto
   thus ?thesis using M Rab by auto
 qed
 
 lemma rel_mset_imp_rel_mset':
   assumes "rel_mset R M N"
   shows "rel_mset' R M N"
 using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
   case (less M)
   have c: "size M = size N" using rel_mset_size[OF less.prems] .
   show ?case
   proof(cases "M = {#}")
     case True hence "N = {#}" using c by simp
     thus ?thesis using True rel_mset'.Zero by auto
   next
     case False then obtain M1 a where M: "M = add_mset a M1" by (metis multi_nonempty_split)
     obtain N1 b where N: "N = add_mset b N1" and R: "R a b" and ms: "rel_mset R M1 N1"
       using msed_rel_invL[OF less.prems[unfolded M]] by auto
     have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
     thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
   qed
 qed
 
 lemma rel_mset_rel_mset': "rel_mset R M N = rel_mset' R M N"
   using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto
 
 text \<open>The main end product for \<^const>\<open>rel_mset\<close>: inductive characterization:\<close>
 lemmas rel_mset_induct[case_names empty add, induct pred: rel_mset] =
   rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]
 
 
 subsection \<open>Size setup\<close>
 
 lemma size_multiset_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
   apply (rule ext)
   subgoal for x by (induct x) auto
   done
 
 setup \<open>
   BNF_LFP_Size.register_size_global \<^type_name>\<open>multiset\<close> \<^const_name>\<open>size_multiset\<close>
     @{thm size_multiset_overloaded_def}
     @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
       size_union}
     @{thms size_multiset_o_map}
 \<close>
 
 subsection \<open>Lemmas about Size\<close>
 
 lemma size_mset_SucE: "size A = Suc n \<Longrightarrow> (\<And>a B. A = {#a#} + B \<Longrightarrow> size B = n \<Longrightarrow> P) \<Longrightarrow> P"
   by (cases A) (auto simp add: ac_simps)
 
 lemma size_Suc_Diff1: "x \<in># M \<Longrightarrow> Suc (size (M - {#x#})) = size M"
   using arg_cong[OF insert_DiffM, of _ _ size] by simp
 
 lemma size_Diff_singleton: "x \<in># M \<Longrightarrow> size (M - {#x#}) = size M - 1"
   by (simp flip: size_Suc_Diff1)
 
 lemma size_Diff_singleton_if: "size (A - {#x#}) = (if x \<in># A then size A - 1 else size A)"
   by (simp add: diff_single_trivial size_Diff_singleton)
 
 lemma size_Un_Int: "size A + size B = size (A \<union># B) + size (A \<inter># B)"
   by (metis inter_subset_eq_union size_union subset_mset.diff_add union_diff_inter_eq_sup)
 
 lemma size_Un_disjoint: "A \<inter># B = {#} \<Longrightarrow> size (A \<union># B) = size A + size B"
   using size_Un_Int[of A B] by simp
 
 lemma size_Diff_subset_Int: "size (M - M') = size M - size (M \<inter># M')"
   by (metis diff_intersect_left_idem size_Diff_submset subset_mset.inf_le1)
 
 lemma diff_size_le_size_Diff: "size (M :: _ multiset) - size M' \<le> size (M - M')"
   by (simp add: diff_le_mono2 size_Diff_subset_Int size_mset_mono)
 
 lemma size_Diff1_less: "x\<in># M \<Longrightarrow> size (M - {#x#}) < size M"
   by (rule Suc_less_SucD) (simp add: size_Suc_Diff1)
 
 lemma size_Diff2_less: "x\<in># M \<Longrightarrow> y\<in># M \<Longrightarrow> size (M - {#x#} - {#y#}) < size M"
   by (metis less_imp_diff_less size_Diff1_less size_Diff_subset_Int)
 
 lemma size_Diff1_le: "size (M - {#x#}) \<le> size M"
   by (cases "x \<in># M") (simp_all add: size_Diff1_less less_imp_le diff_single_trivial)
 
 lemma size_psubset: "M \<subseteq># M' \<Longrightarrow> size M < size M' \<Longrightarrow> M \<subset># M'"
   using less_irrefl subset_mset_def by blast
 
 lifting_update multiset.lifting
 lifting_forget multiset.lifting
 
 hide_const (open) wcount
 
 end
diff --git a/src/HOL/Library/Poly_Mapping.thy b/src/HOL/Library/Poly_Mapping.thy
--- a/src/HOL/Library/Poly_Mapping.thy
+++ b/src/HOL/Library/Poly_Mapping.thy
@@ -1,1878 +1,1849 @@
 (* Author: Andreas Lochbihler, ETH Zurich
    Author: Florian Haftmann, TU Muenchen
    with some material ported from HOL Light by LCP
 *)
 
 section \<open>Polynomial mapping: combination of almost everywhere zero functions with an algebraic view\<close>
 
 theory Poly_Mapping
 imports Groups_Big_Fun Fun_Lexorder More_List
 begin
 
 subsection \<open>Preliminary: auxiliary operations for \emph{almost everywhere zero}\<close>
 
 text \<open>
   A central notion for polynomials are functions being \emph{almost everywhere zero}.
   For these we provide some auxiliary definitions and lemmas.
 \<close>
 
 lemma finite_mult_not_eq_zero_leftI:
   fixes f :: "'b \<Rightarrow> 'a :: mult_zero"
   assumes "finite {a. f a \<noteq> 0}"
   shows "finite {a. g a * f a \<noteq> 0}"
 proof -
   have "{a. g a * f a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto
   then show ?thesis using assms by (rule finite_subset)
 qed
 
 lemma finite_mult_not_eq_zero_rightI:
   fixes f :: "'b \<Rightarrow> 'a :: mult_zero"
   assumes "finite {a. f a \<noteq> 0}"
   shows "finite {a. f a * g a \<noteq> 0}"
 proof -
   have "{a. f a * g a \<noteq> 0} \<subseteq> {a. f a \<noteq> 0}" by auto
   then show ?thesis using assms by (rule finite_subset)
 qed
 
 lemma finite_mult_not_eq_zero_prodI:
   fixes f g :: "'a \<Rightarrow> 'b::semiring_0"
   assumes "finite {a. f a \<noteq> 0}" (is "finite ?F")
   assumes "finite {b. g b \<noteq> 0}" (is "finite ?G")
   shows "finite {(a, b). f a * g b \<noteq> 0}"
 proof -
   from assms have "finite (?F \<times> ?G)"
     by blast
   then have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}"
     by simp
   then show ?thesis
     by (rule rev_finite_subset) auto
 qed
 
 lemma finite_not_eq_zero_sumI:
   fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0"
   assumes "finite {a. f a \<noteq> 0}" (is "finite ?F")
   assumes "finite {b. g b \<noteq> 0}" (is "finite ?G")
   shows "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?FG")
 proof -
   from assms have "finite (?F \<times> ?G)"
     by (simp add: finite_cartesian_product_iff)
   then have "finite (case_prod plus ` (?F \<times> ?G))"
     by (rule finite_imageI)
   also have "case_prod plus ` (?F \<times> ?G) = ?FG"
     by auto
   finally show ?thesis
     by simp
 qed
 
 lemma finite_mult_not_eq_zero_sumI:
   fixes f g :: "'a::monoid_add \<Rightarrow> 'b::semiring_0"
   assumes "finite {a. f a \<noteq> 0}"
   assumes "finite {b. g b \<noteq> 0}"
   shows "finite {a + b | a b. f a * g b \<noteq> 0}"
 proof -
   from assms
   have "finite {a + b | a b. f a \<noteq> 0 \<and> g b \<noteq> 0}"
     by (rule finite_not_eq_zero_sumI)
   then show ?thesis
     by (rule rev_finite_subset) (auto dest: mult_not_zero)
 qed
 
 lemma finite_Sum_any_not_eq_zero_weakenI:
   assumes "finite {a. \<exists>b. f a b \<noteq> 0}"
   shows "finite {a. Sum_any (f a) \<noteq> 0}"
 proof -
   have "{a. Sum_any (f a) \<noteq> 0} \<subseteq> {a. \<exists>b. f a b \<noteq> 0}"
     by (auto elim: Sum_any.not_neutral_obtains_not_neutral)
   then show ?thesis using assms by (rule finite_subset)
 qed
 
 context zero
 begin
 
 definition "when" :: "'a \<Rightarrow> bool \<Rightarrow> 'a" (infixl "when" 20)
 where
   "(a when P) = (if P then a else 0)"
 
 text \<open>
   Case distinctions always complicate matters, particularly when
   nested.  The @{const "when"} operation allows to minimise these
   if @{term 0} is the false-case value and makes proof obligations
   much more readable.
 \<close>
 
 lemma "when" [simp]:
   "P \<Longrightarrow> (a when P) = a"
   "\<not> P \<Longrightarrow> (a when P) = 0"
   by (simp_all add: when_def)
 
 lemma when_simps [simp]:
   "(a when True) = a"
   "(a when False) = 0"
   by simp_all
 
 lemma when_cong:
   assumes "P \<longleftrightarrow> Q"
     and "Q \<Longrightarrow> a = b"
   shows "(a when P) = (b when Q)"
   using assms by (simp add: when_def)
 
 lemma zero_when [simp]:
   "(0 when P) = 0"
   by (simp add: when_def)
 
 lemma when_when:
   "(a when P when Q) = (a when P \<and> Q)"
   by (cases Q) simp_all
 
 lemma when_commute:
   "(a when Q when P) = (a when P when Q)"
   by (simp add: when_when conj_commute)
 
 lemma when_neq_zero [simp]:
   "(a when P) \<noteq> 0 \<longleftrightarrow> P \<and> a \<noteq> 0"
   by (cases P) simp_all
 
 end
 
 context monoid_add
 begin
 
 lemma when_add_distrib:
   "(a + b when P) = (a when P) + (b when P)"
   by (simp add: when_def)
 
 end
 
 context semiring_1
 begin
 
 lemma zero_power_eq:
   "0 ^ n = (1 when n = 0)"
   by (simp add: power_0_left)
 
 end
 
 context comm_monoid_add
 begin
 
 lemma Sum_any_when_equal [simp]:
   "(\<Sum>a. (f a when a = b)) = f b"
   by (simp add: when_def)
 
 lemma Sum_any_when_equal' [simp]:
   "(\<Sum>a. (f a when b = a)) = f b"
   by (simp add: when_def)
 
 lemma Sum_any_when_independent:
   "(\<Sum>a. g a when P) = ((\<Sum>a. g a) when P)"
   by (cases P) simp_all
 
 lemma Sum_any_when_dependent_prod_right:
   "(\<Sum>(a, b). g a when b = h a) = (\<Sum>a. g a)"
 proof -
   have "inj_on (\<lambda>a. (a, h a)) {a. g a \<noteq> 0}"
     by (rule inj_onI) auto
   then show ?thesis unfolding Sum_any.expand_set
     by (rule sum.reindex_cong) auto
 qed
 
 lemma Sum_any_when_dependent_prod_left:
   "(\<Sum>(a, b). g b when a = h b) = (\<Sum>b. g b)"
 proof -
   have "(\<Sum>(a, b). g b when a = h b) = (\<Sum>(b, a). g b when a = h b)"
     by (rule Sum_any.reindex_cong [of prod.swap]) (simp_all add: fun_eq_iff)
   then show ?thesis by (simp add: Sum_any_when_dependent_prod_right)
 qed
 
 end
 
 context cancel_comm_monoid_add
 begin
 
 lemma when_diff_distrib:
   "(a - b when P) = (a when P) - (b when P)"
   by (simp add: when_def)
 
 end
 
 context group_add
 begin
 
 lemma when_uminus_distrib:
   "(- a when P) = - (a when P)"
   by (simp add: when_def)
 
 end
 
 context mult_zero
 begin
 
 lemma mult_when:
   "a * (b when P) = (a * b when P)"
   by (cases P) simp_all
 
 lemma when_mult:
   "(a when P) * b = (a * b when P)"
   by (cases P) simp_all
 
 end
 
 
 subsection \<open>Type definition\<close>
 
 text \<open>
   The following type is of central importance:
 \<close>
 
 typedef (overloaded) ('a, 'b) poly_mapping ("(_ \<Rightarrow>\<^sub>0 /_)" [1, 0] 0) =
   "{f :: 'a \<Rightarrow> 'b::zero. finite {x. f x \<noteq> 0}}"
   morphisms lookup Abs_poly_mapping
 proof -
   have "(\<lambda>_::'a. (0 :: 'b)) \<in> ?poly_mapping" by simp
   then show ?thesis by (blast intro!: exI)
 qed
 
 declare lookup_inverse [simp]
 declare lookup_inject [simp]
 
 lemma lookup_Abs_poly_mapping [simp]:
   "finite {x. f x \<noteq> 0} \<Longrightarrow> lookup (Abs_poly_mapping f) = f"
   using Abs_poly_mapping_inverse [of f] by simp
 
 lemma finite_lookup [simp]:
   "finite {k. lookup f k \<noteq> 0}"
   using poly_mapping.lookup [of f] by simp
 
 lemma finite_lookup_nat [simp]:
   fixes f :: "'a \<Rightarrow>\<^sub>0 nat"
   shows "finite {k. 0 < lookup f k}"
   using poly_mapping.lookup [of f] by simp
 
 lemma poly_mapping_eqI:
   assumes "\<And>k. lookup f k = lookup g k"
   shows "f = g"
   using assms unfolding poly_mapping.lookup_inject [symmetric]
   by blast
 
 lemma poly_mapping_eq_iff: "a = b \<longleftrightarrow> lookup a = lookup b"
   by auto
 
 text \<open>
   We model the universe of functions being \emph{almost everywhere zero}
   by means of a separate type @{typ "('a, 'b) poly_mapping"}.
   For convenience we provide a suggestive infix syntax which
   is a variant of the usual function space syntax.  Conversion between both types
   happens through the morphisms
   \begin{quote}
     @{term_type lookup}
   \end{quote}
   \begin{quote}
     @{term_type Abs_poly_mapping}
   \end{quote}
   satisfying
   \begin{quote}
     @{thm lookup_inverse}
   \end{quote}
   \begin{quote}
     @{thm lookup_Abs_poly_mapping}
   \end{quote}
   Luckily, we have rarely to deal with those low-level morphisms explicitly
   but rely on Isabelle's \emph{lifting} package with its method \<open>transfer\<close>
   and its specification tool \<open>lift_definition\<close>.
 \<close>
 
 setup_lifting type_definition_poly_mapping
 code_datatype Abs_poly_mapping\<comment>\<open>FIXME? workaround for preventing \<open>code_abstype\<close> setup\<close>
 
 text \<open>
   @{typ "'a \<Rightarrow>\<^sub>0 'b"} serves distinctive purposes:
   \begin{enumerate}
     \item A clever nesting as @{typ "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"}
       later in theory \<open>MPoly\<close> gives a suitable
       representation type for polynomials \emph{almost for free}:
       Interpreting @{typ "nat \<Rightarrow>\<^sub>0 nat"} as a mapping from variable identifiers
       to exponents yields monomials, and the whole type maps monomials
       to coefficients.  Lets call this the \emph{ultimate interpretation}.
     \item A further more specialised type isomorphic to @{typ "nat \<Rightarrow>\<^sub>0 'a"}
       is apt to direct implementation using code generation
       \<^cite>\<open>"Haftmann-Nipkow:2010:code"\<close>.
   \end{enumerate}
   Note that despite the names \emph{mapping} and \emph{lookup} suggest something
   implementation-near, it is best to keep @{typ "'a \<Rightarrow>\<^sub>0 'b"} as an abstract
   \emph{algebraic} type
   providing operations like \emph{addition}, \emph{multiplication} without any notion
   of key-order, data structures etc.  This implementations-specific notions are
   easily introduced later for particular implementations but do not provide any
   gain for specifying logical properties of polynomials.
 \<close>
 
 subsection \<open>Additive structure\<close>
 
 text \<open>
   The additive structure covers the usual operations \<open>0\<close>, \<open>+\<close> and
   (unary and binary) \<open>-\<close>.  Recalling the ultimate interpretation, it
   is obvious that these have just lift the corresponding operations on values
   to mappings.
 
   Isabelle has a rich hierarchy of algebraic type classes, and in such situations
   of pointwise lifting a typical pattern is to have instantiations for a considerable
   number of type classes.
 
   The operations themselves are specified using \<open>lift_definition\<close>, where
   the proofs of the \emph{almost everywhere zero} property can be significantly involved.
 
   The @{const lookup} operation is supposed to be usable explicitly (unless in
   other situations where the morphisms between types are somehow internal
   to the \emph{lifting} package).  Hence it is good style to provide explicit rewrite
   rules how @{const lookup} acts on operations immediately.
 \<close>
 
 instantiation poly_mapping :: (type, zero) zero
 begin
 
 lift_definition zero_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b"
   is "\<lambda>k. 0"
   by simp
 
 instance ..
 
 end
 
 lemma Abs_poly_mapping [simp]: "Abs_poly_mapping (\<lambda>k. 0) = 0"
   by (simp add: zero_poly_mapping.abs_eq)
 
 lemma lookup_zero [simp]: "lookup 0 k = 0"
   by transfer rule
 
 instantiation poly_mapping :: (type, monoid_add) monoid_add
 begin
 
 lift_definition plus_poly_mapping ::
     "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
   is "\<lambda>f1 f2 k. f1 k + f2 k"
 proof -
   fix f1 f2 :: "'a \<Rightarrow> 'b"
   assume "finite {k. f1 k \<noteq> 0}"
     and "finite {k. f2 k \<noteq> 0}"
   then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto
   moreover have "{x. f1 x + f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}"
     by auto
   ultimately show "finite {x. f1 x + f2 x \<noteq> 0}"
     by (blast intro: finite_subset)
 qed
 
 instance
   by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
 
 end
 
 lemma lookup_add:
   "lookup (f + g) k = lookup f k + lookup g k"
   by transfer rule
 
 instance poly_mapping :: (type, comm_monoid_add) comm_monoid_add
   by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
 
 lemma lookup_sum: "lookup (sum pp X) i = sum (\<lambda>x. lookup (pp x) i) X"
   by (induction rule: infinite_finite_induct) (auto simp: lookup_add)
 
 (*instance poly_mapping :: (type, "{monoid_add, cancel_semigroup_add}") cancel_semigroup_add
   by intro_classes (transfer, simp add: fun_eq_iff)+*)
 
 instantiation poly_mapping :: (type, cancel_comm_monoid_add) cancel_comm_monoid_add
 begin
 
 lift_definition minus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
   is "\<lambda>f1 f2 k. f1 k - f2 k"
 proof -
   fix f1 f2 :: "'a \<Rightarrow> 'b"
   assume "finite {k. f1 k \<noteq> 0}"
     and "finite {k. f2 k \<noteq> 0}"
   then have "finite ({k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0})" by auto
   moreover have "{x. f1 x - f2 x \<noteq> 0} \<subseteq> {k. f1 k \<noteq> 0} \<union> {k. f2 k \<noteq> 0}"
     by auto
   ultimately show "finite {x. f1 x - f2 x \<noteq> 0}" by (blast intro: finite_subset)
 qed
 
 instance
   by intro_classes (transfer, simp add: fun_eq_iff diff_diff_add)+
 
 end
 
 instantiation poly_mapping :: (type, ab_group_add) ab_group_add
 begin
 
 lift_definition uminus_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
   is uminus
   by simp
 
 
 instance
   by intro_classes (transfer, simp add: fun_eq_iff ac_simps)+
 
 end
 
 lemma lookup_uminus [simp]:
   "lookup (- f) k = - lookup f k"
   by transfer simp
 
 lemma lookup_minus:
   "lookup (f - g) k = lookup f k - lookup g k"
   by transfer rule
 
 
 subsection \<open>Multiplicative structure\<close>
 
 instantiation poly_mapping :: (zero, zero_neq_one) zero_neq_one
 begin
 
 lift_definition one_poly_mapping :: "'a \<Rightarrow>\<^sub>0 'b"
   is "\<lambda>k. 1 when k = 0"
   by simp
 
 instance
   by intro_classes (transfer, simp add: fun_eq_iff)
 
 end
 
 lemma lookup_one:
   "lookup 1 k = (1 when k = 0)"
   by transfer rule
 
 lemma lookup_one_zero [simp]:
   "lookup 1 0 = 1"
   by transfer simp
 
 definition prod_fun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a::monoid_add \<Rightarrow> 'b::semiring_0"
 where
   "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q)))"
 
 lemma prod_fun_unfold_prod:
   fixes f g :: "'a :: monoid_add \<Rightarrow> 'b::semiring_0"
   assumes fin_f: "finite {a. f a \<noteq> 0}"
   assumes fin_g: "finite {b. g b \<noteq> 0}"
   shows "prod_fun f g k = (\<Sum>(a, b). f a * g b when k = a + b)"
 proof -
   let ?C = "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"
   from fin_f fin_g have "finite ?C" by blast
   moreover have "{a. \<exists>b. (f a * g b when k = a + b) \<noteq> 0} \<times>
     {b. \<exists>a. (f a * g b when k = a + b) \<noteq> 0} \<subseteq> {a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"
     by auto
   ultimately show ?thesis using fin_g
-    by (auto simp add: prod_fun_def
+    by (auto simp: prod_fun_def
       Sum_any.cartesian_product [of "{a. f a \<noteq> 0} \<times> {b. g b \<noteq> 0}"] Sum_any_right_distrib mult_when)
 qed
 
 lemma finite_prod_fun:
   fixes f1 f2 :: "'a :: monoid_add \<Rightarrow> 'b :: semiring_0"
   assumes fin1: "finite {l. f1 l \<noteq> 0}"
   and fin2: "finite {q. f2 q \<noteq> 0}"
   shows "finite {k. prod_fun f1 f2 k \<noteq> 0}"
 proof -
   have *: "finite {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}"
     using assms by simp
   { fix k l
     have "{q. (f2 q when k = l + q) \<noteq> 0} \<subseteq> {q. f2 q \<noteq> 0 \<and> k = l + q}" by auto
     with fin2 have "sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} = (\<Sum>q. (f2 q when k = l + q))"
       by (simp add: Sum_any.expand_superset [of "{q. f2 q \<noteq> 0 \<and> k = l + q}"]) }
   note aux = this
   have "{k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}
     \<subseteq> {k. (\<exists>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}"
     by (auto elim!: Sum_any.not_neutral_obtains_not_neutral)
   also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q} \<noteq> 0)}"
     by (auto dest: mult_not_zero)
   also have "\<dots> \<subseteq> {k. (\<exists>l. f1 l \<noteq> 0 \<and> (\<exists>q. f2 q \<noteq> 0 \<and> k = l + q))}"
     by (auto elim!: sum.not_neutral_contains_not_neutral)
   finally have "finite {k. (\<Sum>l. f1 l * sum f2 {q. f2 q \<noteq> 0 \<and> k = l + q}) \<noteq> 0}"
     using * by (rule finite_subset)
   with aux have "finite {k. (\<Sum>l. f1 l * (\<Sum>q. (f2 q when k = l + q))) \<noteq> 0}"
     by simp
   with fin2 show ?thesis
    by (simp add: prod_fun_def)
 qed
 
 instantiation poly_mapping :: (monoid_add, semiring_0) semiring_0
 begin
 
 lift_definition times_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
   is prod_fun
 by(rule finite_prod_fun)
 
 instance
 proof
   fix a b c :: "'a \<Rightarrow>\<^sub>0 'b"
   show "a * b * c = a * (b * c)"
   proof transfer
     fix f g h :: "'a \<Rightarrow> 'b"
     assume fin_f: "finite {a. f a \<noteq> 0}" (is "finite ?F")
     assume fin_g: "finite {b. g b \<noteq> 0}" (is "finite ?G")
     assume fin_h: "finite {c. h c \<noteq> 0}" (is "finite ?H")
     from fin_f fin_g have fin_fg: "finite {(a, b). f a * g b \<noteq> 0}" (is "finite ?FG")
       by (rule finite_mult_not_eq_zero_prodI)
     from fin_g fin_h have fin_gh: "finite {(b, c). g b * h c \<noteq> 0}" (is "finite ?GH")
       by (rule finite_mult_not_eq_zero_prodI)
     from fin_f fin_g have fin_fg': "finite {a + b | a b. f a * g b \<noteq> 0}" (is "finite ?FG'")
       by (rule finite_mult_not_eq_zero_sumI)
     then have fin_fg'': "finite {d. (\<Sum>(a, b). f a * g b when d = a + b) \<noteq> 0}"
       by (auto intro: finite_Sum_any_not_eq_zero_weakenI)
     from fin_g fin_h have fin_gh': "finite {b + c | b c. g b * h c \<noteq> 0}" (is "finite ?GH'")
       by (rule finite_mult_not_eq_zero_sumI)
     then have fin_gh'': "finite {d. (\<Sum>(b, c). g b * h c when d = b + c) \<noteq> 0}"
       by (auto intro: finite_Sum_any_not_eq_zero_weakenI)
     show "prod_fun (prod_fun f g) h = prod_fun f (prod_fun g h)" (is "?lhs = ?rhs")
     proof
       fix k
       from fin_f fin_g fin_h fin_fg''
       have "?lhs k = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b when ab = a + b) * h c when k = ab + c)"
         by (simp add: prod_fun_unfold_prod)
       also have "\<dots> = (\<Sum>(ab, c). (\<Sum>(a, b). f a * g b * h c when k = ab + c when ab = a + b))"
-        apply (subst Sum_any_left_distrib)
-        using fin_fg apply (simp add: split_def)
-        apply (subst Sum_any_when_independent [symmetric])
-        apply (simp add: when_when when_mult mult_when split_def conj_commute)
+        using fin_fg 
+        apply (simp add: Sum_any_left_distrib split_def flip: Sum_any_when_independent)
+        apply (simp add: when_when when_mult mult_when conj_commute)
         done
       also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = ab + c when ab = a + b)"
         apply (subst Sum_any.cartesian_product2 [of "(?FG' \<times> ?H) \<times> ?FG"])
-        apply (auto simp add: finite_cartesian_product_iff fin_fg fin_fg' fin_h dest: mult_not_zero)
+        apply (auto simp: finite_cartesian_product_iff fin_fg fin_fg' fin_h dest: mult_not_zero)
         done
       also have "\<dots> = (\<Sum>(ab, c, a, b). f a * g b * h c when k = a + b + c when ab = a + b)"
         by (rule Sum_any.cong) (simp add: split_def when_def)
       also have "\<dots> = (\<Sum>(ab, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c)
         when ab = (case cab of (c, a, b) \<Rightarrow> a + b))"
         by (simp add: split_def)
       also have "\<dots> = (\<Sum>(c, a, b). f a * g b * h c when k = a + b + c)"
         by (simp add: Sum_any_when_dependent_prod_left)
       also have "\<dots> = (\<Sum>(bc, cab). (case cab of (c, a, b) \<Rightarrow> f a * g b * h c when k = a + b + c)
         when bc = (case cab of (c, a, b) \<Rightarrow> b + c))"
         by (simp add: Sum_any_when_dependent_prod_left)
       also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when k = a + b + c when bc = b + c)"
         by (simp add: split_def)
       also have "\<dots> = (\<Sum>(bc, c, a, b). f a * g b * h c when bc = b + c when k = a + bc)"
         by (rule Sum_any.cong) (simp add: split_def when_def ac_simps)
       also have "\<dots> = (\<Sum>(a, bc, b, c). f a * g b * h c when bc = b + c when k = a + bc)"
       proof -
         have "bij (\<lambda>(a, d, b, c). (d, c, a, b))"
           by (auto intro!: bijI injI surjI [of _ "\<lambda>(d, c, a, b). (a, d, b, c)"] simp add: split_def)
         then show ?thesis
           by (rule Sum_any.reindex_cong) auto
       qed
       also have "\<dots> = (\<Sum>(a, bc). (\<Sum>(b, c). f a * g b * h c when bc = b + c when k = a + bc))"
         apply (subst Sum_any.cartesian_product2 [of "(?F \<times> ?GH') \<times> ?GH"])
-        apply (auto simp add: finite_cartesian_product_iff fin_f fin_gh fin_gh' ac_simps dest: mult_not_zero)
+        apply (auto simp: finite_cartesian_product_iff fin_f fin_gh fin_gh' ac_simps dest: mult_not_zero)
         done
       also have "\<dots> = (\<Sum>(a, bc). f a * (\<Sum>(b, c). g b * h c when bc = b + c) when k = a + bc)"
         apply (subst Sum_any_right_distrib)
         using fin_gh apply (simp add: split_def)
         apply (subst Sum_any_when_independent [symmetric])
         apply (simp add: when_when when_mult mult_when split_def ac_simps)
         done
       also from fin_f fin_g fin_h fin_gh''
       have "\<dots> = ?rhs k"
         by (simp add: prod_fun_unfold_prod)
       finally show "?lhs k = ?rhs k" .
     qed
   qed
   show "(a + b) * c = a * c + b * c"
   proof transfer
     fix f g h :: "'a \<Rightarrow> 'b"
     assume fin_f: "finite {k. f k \<noteq> 0}"
     assume fin_g: "finite {k. g k \<noteq> 0}"
     assume fin_h: "finite {k. h k \<noteq> 0}"
     show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)"
       apply (rule ext)
-      apply (auto simp add: prod_fun_def algebra_simps)
-      apply (subst Sum_any.distrib)
-      using fin_f fin_g apply (auto intro: finite_mult_not_eq_zero_rightI)
-      done
+      apply (simp add: prod_fun_def algebra_simps)
+      by (simp add: Sum_any.distrib fin_f fin_g finite_mult_not_eq_zero_rightI)
   qed
   show "a * (b + c) = a * b + a * c"
   proof transfer
     fix f g h :: "'a \<Rightarrow> 'b"
     assume fin_f: "finite {k. f k \<noteq> 0}"
     assume fin_g: "finite {k. g k \<noteq> 0}"
     assume fin_h: "finite {k. h k \<noteq> 0}"
     show "prod_fun f (\<lambda>k. g k + h k) = (\<lambda>k. prod_fun f g k + prod_fun f h k)"
       apply (rule ext)
-      apply (auto simp add: prod_fun_def Sum_any.distrib algebra_simps when_add_distrib)
-      apply (subst Sum_any.distrib)
-      apply (simp_all add: algebra_simps)
-      apply (auto intro: fin_g fin_h)
-      apply (subst Sum_any.distrib)
-      apply (simp_all add: algebra_simps)
-      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
-      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
-      done
+      apply (auto simp: prod_fun_def Sum_any.distrib algebra_simps when_add_distrib fin_g fin_h)
+      by (simp add: Sum_any.distrib fin_f finite_mult_not_eq_zero_rightI)
   qed
   show "0 * a = 0"
     by transfer (simp add: prod_fun_def [abs_def])
   show "a * 0 = 0"
     by transfer (simp add: prod_fun_def [abs_def])
 qed
 
 end
 
 lemma lookup_mult:
   "lookup (f * g) k = (\<Sum>l. lookup f l * (\<Sum>q. lookup g q when k = l + q))"
   by transfer (simp add: prod_fun_def)
 
 instance poly_mapping :: (comm_monoid_add, comm_semiring_0) comm_semiring_0
 proof
   fix a b c :: "'a \<Rightarrow>\<^sub>0 'b"
   show "a * b = b * a"
   proof transfer
     fix f g :: "'a \<Rightarrow> 'b"
     assume fin_f: "finite {a. f a \<noteq> 0}"
     assume fin_g: "finite {b. g b \<noteq> 0}"
     show "prod_fun f g = prod_fun g f"
     proof
       fix k
       have fin1: "\<And>l. finite {a. (f a when k = l + a) \<noteq> 0}"
         using fin_f by auto
       have fin2: "\<And>l. finite {b. (g b when k = l + b) \<noteq> 0}"
         using fin_g by auto
       from fin_f fin_g have "finite {(a, b). f a \<noteq> 0 \<and> g b \<noteq> 0}" (is "finite ?AB")
         by simp
-      show "prod_fun f g k = prod_fun g f k"
-        apply (simp add: prod_fun_def)
-        apply (subst Sum_any_right_distrib)
-        apply (rule fin2)
-        apply (subst Sum_any_right_distrib)
-         apply (rule fin1)
-        apply (subst Sum_any.swap [of ?AB])
-        apply (fact \<open>finite ?AB\<close>)
-        apply (auto simp add: mult_when ac_simps)
-        done
+      have "(\<Sum>a. \<Sum>n. f a * (g n when k = a + n)) = (\<Sum>a. \<Sum>n. g a * (f n when k = a + n))"
+        by (subst Sum_any.swap [OF \<open>finite ?AB\<close>]) (auto simp: mult_when ac_simps)
+      then show "prod_fun f g k = prod_fun g f k"
+        by (simp add: prod_fun_def Sum_any_right_distrib [OF fin2] Sum_any_right_distrib [OF fin1])
     qed
   qed
   show "(a + b) * c = a * c + b * c"
   proof transfer
     fix f g h :: "'a \<Rightarrow> 'b"
     assume fin_f: "finite {k. f k \<noteq> 0}"
     assume fin_g: "finite {k. g k \<noteq> 0}"
     assume fin_h: "finite {k. h k \<noteq> 0}"
     show "prod_fun (\<lambda>k. f k + g k) h = (\<lambda>k. prod_fun f h k + prod_fun g h k)"
-      apply (auto simp add: prod_fun_def fun_eq_iff algebra_simps)
-      apply (subst Sum_any.distrib)
-      using fin_f apply (rule finite_mult_not_eq_zero_rightI)
-      using fin_g apply (rule finite_mult_not_eq_zero_rightI)
-      apply simp_all
-      done
+      by (auto simp: prod_fun_def fun_eq_iff algebra_simps 
+            Sum_any.distrib fin_f fin_g finite_mult_not_eq_zero_rightI)
   qed
 qed
 
 instance poly_mapping :: (monoid_add, semiring_0_cancel) semiring_0_cancel
   ..
 
 instance poly_mapping :: (comm_monoid_add, comm_semiring_0_cancel) comm_semiring_0_cancel
   ..
 
 instance poly_mapping :: (monoid_add, semiring_1) semiring_1
 proof
   fix a :: "'a \<Rightarrow>\<^sub>0 'b"
   show "1 * a = a"
     by transfer (simp add: prod_fun_def [abs_def] when_mult)
   show "a * 1 = a"
     apply transfer
     apply (simp add: prod_fun_def [abs_def] Sum_any_right_distrib Sum_any_left_distrib mult_when)
     apply (subst when_commute)
     apply simp
     done
 qed
 
 instance poly_mapping :: (comm_monoid_add, comm_semiring_1) comm_semiring_1
 proof
   fix a :: "'a \<Rightarrow>\<^sub>0 'b"
   show "1 * a = a"
     by transfer (simp add: prod_fun_def [abs_def])
 qed
 
 instance poly_mapping :: (monoid_add, semiring_1_cancel) semiring_1_cancel
   ..
 
 instance poly_mapping :: (monoid_add, ring) ring
   ..
 
 instance poly_mapping :: (comm_monoid_add, comm_ring) comm_ring
   ..
 
 instance poly_mapping :: (monoid_add, ring_1) ring_1
   ..
 
 instance poly_mapping :: (comm_monoid_add, comm_ring_1) comm_ring_1
   ..
 
 
 subsection \<open>Single-point mappings\<close>
 
 lift_definition single :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero"
   is "\<lambda>k v k'. (v when k = k')"
   by simp
 
 lemma inj_single [iff]:
   "inj (single k)"
 proof (rule injI, transfer)
   fix k :: 'b and a b :: "'a::zero"
   assume "(\<lambda>k'. a when k = k') = (\<lambda>k'. b when k = k')"
   then have "(\<lambda>k'. a when k = k') k = (\<lambda>k'. b when k = k') k"
     by (rule arg_cong)
   then show "a = b" by simp
 qed
 
 lemma lookup_single:
   "lookup (single k v) k' = (v when k = k')"
-  by transfer rule
+  by (simp add: single.rep_eq)
 
 lemma lookup_single_eq [simp]:
   "lookup (single k v) k = v"
   by transfer simp
 
 lemma lookup_single_not_eq:
   "k \<noteq> k' \<Longrightarrow> lookup (single k v) k' = 0"
   by transfer simp
 
 lemma single_zero [simp]:
   "single k 0 = 0"
   by transfer simp
 
 lemma single_one [simp]:
   "single 0 1 = 1"
   by transfer simp
 
 lemma single_add:
   "single k (a + b) = single k a + single k b"
   by transfer (simp add: fun_eq_iff when_add_distrib)
 
 lemma single_uminus:
   "single k (- a) = - single k a"
   by transfer (simp add: fun_eq_iff when_uminus_distrib)
 
 lemma single_diff:
   "single k (a - b) = single k a - single k b"
   by transfer (simp add: fun_eq_iff when_diff_distrib)
 
 lemma single_numeral [simp]:
   "single 0 (numeral n) = numeral n"
   by (induct n) (simp_all only: numeral.simps numeral_add single_zero single_one single_add)
 
 lemma lookup_numeral:
   "lookup (numeral n) k = (numeral n when k = 0)"
 proof -
   have "lookup (numeral n) k = lookup (single 0 (numeral n)) k"
     by simp
   then show ?thesis unfolding lookup_single by simp
 qed
 
 lemma single_of_nat [simp]:
   "single 0 (of_nat n) = of_nat n"
   by (induct n) (simp_all add: single_add)
 
 lemma lookup_of_nat:
   "lookup (of_nat n) k = (of_nat n when k = 0)"
 proof -
   have "lookup (of_nat n) k = lookup (single 0 (of_nat n)) k"
     by simp
   then show ?thesis unfolding lookup_single by simp
 qed
 
 lemma of_nat_single:
   "of_nat = single 0 \<circ> of_nat"
   by (simp add: fun_eq_iff)
 
 lemma mult_single:
   "single k a * single l b = single (k + l) (a * b)"
 proof transfer
   fix k l :: 'a and a b :: 'b
   show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') = (\<lambda>k'. a * b when k + l = k')"
   proof
     fix k'
     have "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' = (\<Sum>n. a * b when l = n when k' = k + n)"
       by (simp add: prod_fun_def Sum_any_right_distrib mult_when when_mult)
     also have "\<dots> = (\<Sum>n. a * b when k' = k + n when l = n)"
       by (simp add: when_when conj_commute)
     also have "\<dots> = (a * b when k' = k + l)"
       by simp
     also have "\<dots> = (a * b when k + l = k')"
       by (simp add: when_def)
     finally show "prod_fun (\<lambda>k'. a when k = k') (\<lambda>k'. b when l = k') k' =
       (\<lambda>k'. a * b when k + l = k') k'" .
   qed
 qed
 
 instance poly_mapping :: (monoid_add, semiring_char_0) semiring_char_0
   by intro_classes (auto intro: inj_compose inj_of_nat simp add: of_nat_single)
 
 instance poly_mapping :: (monoid_add, ring_char_0) ring_char_0
   ..
 
 lemma single_of_int [simp]:
   "single 0 (of_int k) = of_int k"
   by (cases k) (simp_all add: single_diff single_uminus)
 
 lemma lookup_of_int:
   "lookup (of_int l) k = (of_int l when k = 0)"
-proof -
-  have "lookup (of_int l) k = lookup (single 0 (of_int l)) k"
-    by simp
-  then show ?thesis unfolding lookup_single by simp
-qed
+  by (metis lookup_single_not_eq single.rep_eq single_of_int)
 
 
 subsection \<open>Integral domains\<close>
 
 instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", semiring_no_zero_divisors) semiring_no_zero_divisors
   text \<open>The @{class "linorder"} constraint is a pragmatic device for the proof --- maybe it can be dropped\<close>
 proof
   fix f g :: "'a \<Rightarrow>\<^sub>0 'b"
   assume "f \<noteq> 0" and "g \<noteq> 0"
   then show "f * g \<noteq> 0"
   proof transfer
     fix f g :: "'a \<Rightarrow> 'b"
     define F where "F = {a. f a \<noteq> 0}"
     moreover define G where "G = {a. g a \<noteq> 0}"
     ultimately have [simp]:
       "\<And>a. f a \<noteq> 0 \<longleftrightarrow> a \<in> F"
       "\<And>b. g b \<noteq> 0 \<longleftrightarrow> b \<in> G"
       by simp_all
     assume "finite {a. f a \<noteq> 0}"
     then have [simp]: "finite F"
       by simp
     assume "finite {a. g a \<noteq> 0}"
     then have [simp]: "finite G"
       by simp
     assume "f \<noteq> (\<lambda>a. 0)"
     then obtain a where "f a \<noteq> 0"
-      by (auto simp add: fun_eq_iff)
+      by (auto simp: fun_eq_iff)
     assume "g \<noteq> (\<lambda>b. 0)"
     then obtain b where "g b \<noteq> 0"
-      by (auto simp add: fun_eq_iff)
+      by (auto simp: fun_eq_iff)
     from \<open>f a \<noteq> 0\<close> and \<open>g b \<noteq> 0\<close> have "F \<noteq> {}" and "G \<noteq> {}"
       by auto
     note Max_F = \<open>finite F\<close> \<open>F \<noteq> {}\<close>
     note Max_G = \<open>finite G\<close> \<open>G \<noteq> {}\<close>
     from Max_F and Max_G have [simp]:
       "Max F \<in> F"
       "Max G \<in> G"
       by auto
     from Max_F Max_G have [dest!]:
       "\<And>a. a \<in> F \<Longrightarrow> a \<le> Max F"
       "\<And>b. b \<in> G \<Longrightarrow> b \<le> Max G"
       by auto
     define q where "q = Max F + Max G"
     have "(\<Sum>(a, b). f a * g b when q = a + b) =
       (\<Sum>(a, b). f a * g b when q = a + b when a \<in> F \<and> b \<in> G)"
-      by (rule Sum_any.cong) (auto simp add: split_def when_def q_def intro: ccontr)
+      by (rule Sum_any.cong) (auto simp: split_def when_def q_def intro: ccontr)
     also have "\<dots> =
       (\<Sum>(a, b). f a * g b when (Max F, Max G) = (a, b))"
     proof (rule Sum_any.cong)
       fix ab :: "'a \<times> 'a"
       obtain a b where [simp]: "ab = (a, b)"
         by (cases ab) simp_all
       have [dest!]:
         "a \<le> Max F \<Longrightarrow> Max F \<noteq> a \<Longrightarrow> a < Max F"
         "b \<le> Max G \<Longrightarrow> Max G \<noteq> b \<Longrightarrow> b < Max G"
         by auto
       show "(case ab of (a, b) \<Rightarrow> f a * g b when q = a + b when a \<in> F \<and> b \<in> G) =
          (case ab of (a, b) \<Rightarrow> f a * g b when (Max F, Max G) = (a, b))"
-        by (auto simp add: split_def when_def q_def dest: add_strict_mono [of a "Max F" b "Max G"])
+        by (auto simp: split_def when_def q_def dest: add_strict_mono [of a "Max F" b "Max G"])
     qed
     also have "\<dots> = (\<Sum>ab. (case ab of (a, b) \<Rightarrow> f a * g b) when
       (Max F, Max G) = ab)"
       unfolding split_def when_def by auto
     also have "\<dots> \<noteq> 0"
       by simp
     finally have "prod_fun f g q \<noteq> 0"
       by (simp add: prod_fun_unfold_prod)
     then show "prod_fun f g \<noteq> (\<lambda>k. 0)"
-      by (auto simp add: fun_eq_iff)
+      by (auto simp: fun_eq_iff)
   qed
 qed
 
 instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_no_zero_divisors) ring_no_zero_divisors
   ..
 
 instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", ring_1_no_zero_divisors) ring_1_no_zero_divisors
   ..
 
 instance poly_mapping :: ("{ordered_cancel_comm_monoid_add, linorder}", idom) idom
   ..
 
 
 subsection \<open>Mapping order\<close>
 
 instantiation poly_mapping :: (linorder, "{zero, linorder}") linorder
 begin
 
 lift_definition less_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"
   is less_fun
   .
 
 lift_definition less_eq_poly_mapping :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"
   is "\<lambda>f g. less_fun f g \<or> f = g"
   .
 
 instance proof (rule linorder.intro_of_class)
   show "class.linorder (less_eq :: (_ \<Rightarrow>\<^sub>0 _) \<Rightarrow> _) less"
   proof (rule linorder_strictI, rule order_strictI)
     fix f g h :: "'a \<Rightarrow>\<^sub>0 'b"
     show "f \<le> g \<longleftrightarrow> f < g \<or> f = g"
       by transfer (rule refl)
     show "\<not> f < f"
       by transfer (rule less_fun_irrefl)
     show "f < g \<or> f = g \<or> g < f"
     proof transfer
       fix f g :: "'a \<Rightarrow> 'b"
       assume "finite {k. f k \<noteq> 0}" and "finite {k. g k \<noteq> 0}"
       then have "finite ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})"
         by simp
       moreover have "{k. f k \<noteq> g k} \<subseteq> {k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0}"
         by auto
       ultimately have "finite {k. f k \<noteq> g k}"
         by (rule rev_finite_subset)
       then show "less_fun f g \<or> f = g \<or> less_fun g f"
         by (rule less_fun_trichotomy)
     qed
     assume "f < g" then show "\<not> g < f"
       by transfer (rule less_fun_asym)
     note \<open>f < g\<close> moreover assume "g < h"
       ultimately show "f < h"
       by transfer (rule less_fun_trans)
   qed
 qed
 
 end
 
 instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, linorder}") ordered_ab_semigroup_add
 proof (intro_classes, transfer)
   fix f g h :: "'a \<Rightarrow> 'b"
   assume *: "less_fun f g \<or> f = g"
   { assume "less_fun f g"
     then obtain k where "f k < g k" "(\<And>k'. k' < k \<Longrightarrow> f k' = g k')"
       by (blast elim!: less_funE)
     then have "h k + f k < h k + g k" "(\<And>k'. k' < k \<Longrightarrow> h k' + f k' = h k' + g k')"
       by simp_all
     then have "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k)"
       by (blast intro: less_funI)
   }
   with * show "less_fun (\<lambda>k. h k + f k) (\<lambda>k. h k + g k) \<or> (\<lambda>k. h k + f k) = (\<lambda>k. h k + g k)"
-    by (auto simp add: fun_eq_iff)
+    by (auto simp: fun_eq_iff)
 qed
 
 instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") linordered_cancel_ab_semigroup_add
   ..
 
 instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") ordered_comm_monoid_add
   ..
 
 instance poly_mapping :: (linorder, "{ordered_comm_monoid_add, ordered_ab_semigroup_add_imp_le, cancel_comm_monoid_add, linorder}") ordered_cancel_comm_monoid_add
   ..
 
 instance poly_mapping :: (linorder, linordered_ab_group_add) linordered_ab_group_add
   ..
 
 text \<open>
   For pragmatism we leave out the final elements in the hierarchy:
   @{class linordered_ring}, @{class linordered_ring_strict}, @{class linordered_idom};
   remember that the order instance is a mere technical device, not a deeper algebraic
   property.
 \<close>
 
 
 subsection \<open>Fundamental mapping notions\<close>
 
 lift_definition keys :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a set"
   is "\<lambda>f. {k. f k \<noteq> 0}" .
 
 lift_definition range :: "('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'b set"
   is "\<lambda>f :: 'a \<Rightarrow> 'b. Set.range f - {0}" .
 
 lemma finite_keys [simp]:
   "finite (keys f)"
   by transfer
 
 lemma not_in_keys_iff_lookup_eq_zero:
   "k \<notin> keys f \<longleftrightarrow> lookup f k = 0"
   by transfer simp
 
 lemma lookup_not_eq_zero_eq_in_keys:
   "lookup f k \<noteq> 0 \<longleftrightarrow> k \<in> keys f"
   by transfer simp
 
 lemma lookup_eq_zero_in_keys_contradict [dest]:
   "lookup f k = 0 \<Longrightarrow> \<not> k \<in> keys f"
   by (simp add: not_in_keys_iff_lookup_eq_zero)
 
 lemma finite_range [simp]: "finite (Poly_Mapping.range p)"
 proof transfer
   fix f :: "'b \<Rightarrow> 'a"
   assume *: "finite {x. f x \<noteq> 0}"
   have "Set.range f - {0} \<subseteq> f ` {x. f x \<noteq> 0}"
     by auto
   thus "finite (Set.range f - {0})"
     by(rule finite_subset)(rule finite_imageI[OF *])
 qed
 
 lemma in_keys_lookup_in_range [simp]:
   "k \<in> keys f \<Longrightarrow> lookup f k \<in> range f"
   by transfer simp
 
 lemma in_keys_iff: "x \<in> (keys s) = (lookup s x \<noteq> 0)"
   by (transfer, simp)
 
 lemma keys_zero [simp]:
   "keys 0 = {}"
   by transfer simp
 
 lemma range_zero [simp]:
   "range 0 = {}"
   by transfer auto
 
 lemma keys_add:
   "keys (f + g) \<subseteq> keys f \<union> keys g"
   by transfer auto
 
 lemma keys_one [simp]:
   "keys 1 = {0}"
   by transfer simp
 
 lemma range_one [simp]:
   "range 1 = {1}"
-  by transfer (auto simp add: when_def)
+  by transfer (auto simp: when_def)
 
 lemma keys_single [simp]:
   "keys (single k v) = (if v = 0 then {} else {k})"
   by transfer simp
 
 lemma range_single [simp]:
   "range (single k v) = (if v = 0 then {} else {v})"
-  by transfer (auto simp add: when_def)
+  by transfer (auto simp: when_def)
 
 lemma keys_mult:
   "keys (f * g) \<subseteq> {a + b | a b. a \<in> keys f \<and> b \<in> keys g}"
   apply transfer
-  apply (auto simp add: prod_fun_def dest!: mult_not_zero elim!: Sum_any.not_neutral_obtains_not_neutral)
-  apply blast
+  apply (force simp: prod_fun_def dest!: mult_not_zero elim!: Sum_any.not_neutral_obtains_not_neutral)
   done
 
 lemma setsum_keys_plus_distrib:
   assumes hom_0: "\<And>k. f k 0 = 0"
   and hom_plus: "\<And>k. k \<in> Poly_Mapping.keys p \<union> Poly_Mapping.keys q \<Longrightarrow> f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k) = f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k)"
   shows
   "(\<Sum>k\<in>Poly_Mapping.keys (p + q). f k (Poly_Mapping.lookup (p + q) k)) =
    (\<Sum>k\<in>Poly_Mapping.keys p. f k (Poly_Mapping.lookup p k)) +
    (\<Sum>k\<in>Poly_Mapping.keys q. f k (Poly_Mapping.lookup q k))"
   (is "?lhs = ?p + ?q")
 proof -
   let ?A = "Poly_Mapping.keys p \<union> Poly_Mapping.keys q"
   have "?lhs = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k + Poly_Mapping.lookup q k))"
-    apply(rule sum.mono_neutral_cong_left)
-       apply(simp_all add: Poly_Mapping.keys_add)
-     apply(transfer fixing: f)
-     apply(auto simp add: hom_0)[1]
-    apply(transfer fixing: f)
-    apply(auto simp add: hom_0)[1]
-    done
+    by(intro sum.mono_neutral_cong_left) (auto simp: sum.mono_neutral_cong_left hom_0 in_keys_iff lookup_add)
   also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k) + f k (Poly_Mapping.lookup q k))"
     by(rule sum.cong)(simp_all add: hom_plus)
   also have "\<dots> = (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup p k)) + (\<Sum>k\<in>?A. f k (Poly_Mapping.lookup q k))"
     (is "_ = ?p' + ?q'")
     by(simp add: sum.distrib)
   also have "?p' = ?p"
     by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right)
   also have "?q' = ?q"
     by (simp add: hom_0 in_keys_iff sum.mono_neutral_cong_right)
   finally show ?thesis .
 qed
 
 subsection \<open>Degree\<close>
 
 definition degree :: "(nat \<Rightarrow>\<^sub>0 'a::zero) \<Rightarrow> nat"
 where
   "degree f = Max (insert 0 (Suc ` keys f))"
 
 lemma degree_zero [simp]:
   "degree 0 = 0"
   unfolding degree_def by transfer simp
 
 lemma degree_one [simp]:
   "degree 1 = 1"
   unfolding degree_def by transfer simp
 
 lemma degree_single_zero [simp]:
   "degree (single k 0) = 0"
   unfolding degree_def by transfer simp
 
 lemma degree_single_not_zero [simp]:
   "v \<noteq> 0 \<Longrightarrow> degree (single k v) = Suc k"
   unfolding degree_def by transfer simp
 
 lemma degree_zero_iff [simp]:
   "degree f = 0 \<longleftrightarrow> f = 0"
 unfolding degree_def proof transfer
   fix f :: "nat \<Rightarrow> 'a"
   assume "finite {n. f n \<noteq> 0}"
   then have fin: "finite (insert 0 (Suc ` {n. f n \<noteq> 0}))" by auto
   show "Max (insert 0 (Suc ` {n. f n \<noteq> 0})) = 0 \<longleftrightarrow> f = (\<lambda>n. 0)" (is "?P \<longleftrightarrow> ?Q")
   proof
     assume ?P
     have "{n. f n \<noteq> 0} = {}"
     proof (rule ccontr)
       assume "{n. f n \<noteq> 0} \<noteq> {}"
       then obtain n where "n \<in> {n. f n \<noteq> 0}" by blast
       then have "{n. f n \<noteq> 0} = insert n {n. f n \<noteq> 0}" by auto
       then have "Suc ` {n. f n \<noteq> 0} = insert (Suc n) (Suc ` {n. f n \<noteq> 0})" by auto
       with \<open>?P\<close> have "Max (insert 0 (insert (Suc n) (Suc ` {n. f n \<noteq> 0}))) = 0" by simp
       then have "Max (insert (Suc n) (insert 0 (Suc ` {n. f n \<noteq> 0}))) = 0"
         by (simp add: insert_commute)
       with fin have "max (Suc n) (Max (insert 0 (Suc ` {n. f n \<noteq> 0}))) = 0"
         by simp
       then show False by simp
     qed
     then show ?Q by (simp add: fun_eq_iff)
   next
     assume ?Q then show ?P by simp
   qed
 qed
 
 lemma degree_greater_zero_in_keys:
   assumes "0 < degree f"
   shows "degree f - 1 \<in> keys f"
 proof -
   from assms have "keys f \<noteq> {}"
-    by (auto simp add: degree_def)
+    by (auto simp: degree_def)
   then show ?thesis unfolding degree_def
     by (simp add: mono_Max_commute [symmetric] mono_Suc)
 qed
 
 lemma in_keys_less_degree:
   "n \<in> keys f \<Longrightarrow> n < degree f"
-unfolding degree_def by transfer (auto simp add: Max_gr_iff)
+unfolding degree_def by transfer (auto simp: Max_gr_iff)
 
 lemma beyond_degree_lookup_zero:
   "degree f \<le> n \<Longrightarrow> lookup f n = 0"
   unfolding degree_def by transfer auto
 
 lemma degree_add:
   "degree (f + g) \<le> max (degree f) (Poly_Mapping.degree g)"
 unfolding degree_def proof transfer
   fix f g :: "nat \<Rightarrow> 'a"
   assume f: "finite {x. f x \<noteq> 0}"
   assume g: "finite {x. g x \<noteq> 0}"
   let ?f = "Max (insert 0 (Suc ` {k. f k \<noteq> 0}))"
   let ?g = "Max (insert 0 (Suc ` {k. g k \<noteq> 0}))"
   have "Max (insert 0 (Suc ` {k. f k + g k \<noteq> 0})) \<le> Max (insert 0 (Suc ` ({k. f k \<noteq> 0} \<union> {k. g k \<noteq> 0})))"
     by (rule Max.subset_imp) (insert f g, auto)
   also have "\<dots> = max ?f ?g"
     using f g by (simp_all add: image_Un Max_Un [symmetric])
   finally show "Max (insert 0 (Suc ` {k. f k + g k \<noteq> 0}))
     \<le> max (Max (insert 0 (Suc ` {k. f k \<noteq> 0}))) (Max (insert 0 (Suc ` {k. g k \<noteq> 0})))"
     .
 qed
 
 lemma sorted_list_of_set_keys:
   "sorted_list_of_set (keys f) = filter (\<lambda>k. k \<in> keys f) [0..<degree f]" (is "_ = ?r")
 proof -
   have "keys f = set ?r"
     by (auto dest: in_keys_less_degree)
   moreover have "sorted_list_of_set (set ?r) = ?r"
     unfolding sorted_list_of_set_sort_remdups
     by (simp add: remdups_filter filter_sort [symmetric])
   ultimately show ?thesis by simp
 qed
 
 
 subsection \<open>Inductive structure\<close>
 
 lift_definition update :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b"
   is "\<lambda>k v f. f(k := v)"
 proof -
   fix f :: "'a \<Rightarrow> 'b" and k' v
   assume "finite {k. f k \<noteq> 0}"
   then have "finite (insert k' {k. f k \<noteq> 0})"
     by simp
   then show "finite {k. (f(k' := v)) k \<noteq> 0}"
     by (rule rev_finite_subset) auto
 qed
 
 lemma update_induct [case_names const update]:
   assumes const': "P 0"
   assumes update': "\<And>f a b. a \<notin> keys f \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> P f \<Longrightarrow> P (update a b f)"
   shows "P f"
 proof -
   obtain g where "f = Abs_poly_mapping g" and "finite {a. g a \<noteq> 0}"
     by (cases f) simp_all
   define Q where "Q g = P (Abs_poly_mapping g)" for g
   from \<open>finite {a. g a \<noteq> 0}\<close> have "Q g"
   proof (induct g rule: finite_update_induct)
     case const with const' Q_def show ?case
       by simp
   next
     case (update a b g)
     from \<open>finite {a. g a \<noteq> 0}\<close> \<open>g a = 0\<close> have "a \<notin> keys (Abs_poly_mapping g)"
       by (simp add: Abs_poly_mapping_inverse keys.rep_eq)
     moreover note \<open>b \<noteq> 0\<close>
     moreover from \<open>Q g\<close> have "P (Abs_poly_mapping g)"
       by (simp add: Q_def)
     ultimately have "P (update a b (Abs_poly_mapping g))"
       by (rule update')
     also from \<open>finite {a. g a \<noteq> 0}\<close>
     have "update a b (Abs_poly_mapping g) = Abs_poly_mapping (g(a := b))"
       by (simp add: update.abs_eq eq_onp_same_args)
     finally show ?case
       by (simp add: Q_def fun_upd_def)
   qed
   then show ?thesis by (simp add: Q_def \<open>f = Abs_poly_mapping g\<close>)
 qed
 
 lemma lookup_update:
   "lookup (update k v f) k' = (if k = k' then v else lookup f k')"
   by transfer simp
 
 lemma keys_update:
   "keys (update k v f) = (if v = 0 then keys f - {k} else insert k (keys f))"
   by transfer auto
 
 
 subsection \<open>Quasi-functorial structure\<close>
 
 lift_definition map :: "('b::zero \<Rightarrow> 'c::zero)
   \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c::zero)"
   is "\<lambda>g f k. g (f k) when f k \<noteq> 0"
   by simp
 
 context
   fixes f :: "'b \<Rightarrow> 'a"
   assumes inj_f: "inj f"
 begin
 
 lift_definition map_key :: "('a \<Rightarrow>\<^sub>0 'c::zero) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'c"
   is "\<lambda>p. p \<circ> f"
 proof -
   fix g :: "'c \<Rightarrow> 'd" and p :: "'a \<Rightarrow> 'c"
   assume "finite {x. p x \<noteq> 0}"
   hence "finite (f ` {y. p (f y) \<noteq> 0})"
     by(rule finite_subset[rotated]) auto
   thus "finite {x. (p \<circ> f) x \<noteq> 0}" unfolding o_def
     by(rule finite_imageD)(rule subset_inj_on[OF inj_f], simp)
 qed
 
 end
 
 lemma map_key_compose:
   assumes [transfer_rule]: "inj f" "inj g"
   shows "map_key f (map_key g p) = map_key (g \<circ> f) p"
 proof -
   from assms have [transfer_rule]: "inj (g \<circ> f)"
     by(simp add: inj_compose)
   show ?thesis by transfer(simp add: o_assoc)
 qed
 
 lemma map_key_id:
   "map_key (\<lambda>x. x) p = p"
 proof -
   have [transfer_rule]: "inj (\<lambda>x. x)" by simp
   show ?thesis by transfer(simp add: o_def)
 qed
 
 context
   fixes f :: "'a \<Rightarrow> 'b"
   assumes inj_f [transfer_rule]: "inj f"
 begin
 
 lemma map_key_map:
   "map_key f (map g p) = map g (map_key f p)"
   by transfer (simp add: fun_eq_iff)
 
 lemma map_key_plus:
   "map_key f (p + q) = map_key f p + map_key f q"
   by transfer (simp add: fun_eq_iff)
 
 lemma keys_map_key:
   "keys (map_key f p) = f -` keys p"
   by transfer auto
 
 lemma map_key_zero [simp]:
   "map_key f 0 = 0"
   by transfer (simp add: fun_eq_iff)
 
 lemma map_key_single [simp]:
   "map_key f (single (f k) v) = single k v"
   by transfer (simp add: fun_eq_iff inj_onD [OF inj_f] when_def)
 
 end
 
 lemma mult_map_scale_conv_mult: "map ((*) s) p = single 0 s * p"
 proof(transfer fixing: s)
   fix p :: "'a \<Rightarrow> 'b"
   assume *: "finite {x. p x \<noteq> 0}"
   { fix x
     have "prod_fun (\<lambda>k'. s when 0 = k') p x =
           (\<Sum>l :: 'a. if l = 0 then s * (\<Sum>q. p q when x = q) else 0)"
-      by(auto simp add: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta)
+      by(auto simp: prod_fun_def when_def intro: Sum_any.cong simp del: Sum_any.delta)
     also have "\<dots> = (\<lambda>k. s * p k when p k \<noteq> 0) x" by(simp add: when_def)
     also note calculation }
   then show "(\<lambda>k. s * p k when p k \<noteq> 0) = prod_fun (\<lambda>k'. s when 0 = k') p"
     by(simp add: fun_eq_iff)
 qed
 
 lemma map_single [simp]:
   "(c = 0 \<Longrightarrow> f 0 = 0) \<Longrightarrow> map f (single x c) = single x (f c)"
-by transfer(auto simp add: fun_eq_iff when_def)
+  by transfer(auto simp: fun_eq_iff when_def)
 
 lemma map_eq_zero_iff: "map f p = 0 \<longleftrightarrow> (\<forall>k \<in> keys p. f (lookup p k) = 0)"
-by transfer(auto simp add: fun_eq_iff when_def)
+  by transfer(auto simp: fun_eq_iff when_def)
 
 subsection \<open>Canonical dense representation of @{typ "nat \<Rightarrow>\<^sub>0 'a"}\<close>
 
 abbreviation no_trailing_zeros :: "'a :: zero list \<Rightarrow> bool"
 where
   "no_trailing_zeros \<equiv> no_trailing ((=) 0)"
 
 lift_definition "nth" :: "'a list \<Rightarrow> (nat \<Rightarrow>\<^sub>0 'a::zero)"
   is "nth_default 0"
   by (fact finite_nth_default_neq_default)
 
 text \<open>
   The opposite direction is directly specified on (later)
   type \<open>nat_mapping\<close>.
 \<close>
 
 lemma nth_Nil [simp]:
   "nth [] = 0"
   by transfer (simp add: fun_eq_iff)
 
 lemma nth_singleton [simp]:
   "nth [v] = single 0 v"
 proof (transfer, rule ext)
   fix n :: nat and v :: 'a
   show "nth_default 0 [v] n = (v when 0 = n)"
-    by (auto simp add: nth_default_def nth_append)
+    by (auto simp: nth_default_def nth_append)
 qed
 
 lemma nth_replicate [simp]:
   "nth (replicate n 0 @ [v]) = single n v"
 proof (transfer, rule ext)
   fix m n :: nat and v :: 'a
   show "nth_default 0 (replicate n 0 @ [v]) m = (v when n = m)"
-    by (auto simp add: nth_default_def nth_append)
+    by (auto simp: nth_default_def nth_append)
 qed
 
 lemma nth_strip_while [simp]:
   "nth (strip_while ((=) 0) xs) = nth xs"
   by transfer (fact nth_default_strip_while_dflt)
 
 lemma nth_strip_while' [simp]:
   "nth (strip_while (\<lambda>k. k = 0) xs) = nth xs"
   by (subst eq_commute) (fact nth_strip_while)
 
 lemma nth_eq_iff:
   "nth xs = nth ys \<longleftrightarrow> strip_while (HOL.eq 0) xs = strip_while (HOL.eq 0) ys"
   by transfer (simp add: nth_default_eq_iff)
 
 lemma lookup_nth [simp]:
   "lookup (nth xs) = nth_default 0 xs"
   by (fact nth.rep_eq)
 
 lemma keys_nth [simp]:
   "keys (nth xs) =  fst ` {(n, v) \<in> set (enumerate 0 xs). v \<noteq> 0}"
 proof transfer
   fix xs :: "'a list"
   { fix n
     assume "nth_default 0 xs n \<noteq> 0"
     then have "n < length xs" and "xs ! n \<noteq> 0"
-      by (auto simp add: nth_default_def split: if_splits)
+      by (auto simp: nth_default_def split: if_splits)
     then have "(n, xs ! n) \<in> {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}" (is "?x \<in> ?A")
-      by (auto simp add: in_set_conv_nth enumerate_eq_zip)
+      by (auto simp: in_set_conv_nth enumerate_eq_zip)
     then have "fst ?x \<in> fst ` ?A"
       by blast
     then have "n \<in> fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
       by simp
   }
   then show "{k. nth_default 0 xs k \<noteq> 0} = fst ` {(n, v). (n, v) \<in> set (enumerate 0 xs) \<and> v \<noteq> 0}"
-    by (auto simp add: in_enumerate_iff_nth_default_eq)
+    by (auto simp: in_enumerate_iff_nth_default_eq)
 qed
 
 lemma range_nth [simp]:
   "range (nth xs) = set xs - {0}"
   by transfer simp
 
 lemma degree_nth:
   "no_trailing_zeros xs \<Longrightarrow> degree (nth xs) = length xs"
 unfolding degree_def proof transfer
   fix xs :: "'a list"
   assume *: "no_trailing_zeros xs"
   let ?A = "{n. nth_default 0 xs n \<noteq> 0}"
   let ?f = "nth_default 0 xs"
   let ?bound = "Max (insert 0 (Suc ` {n. ?f n \<noteq> 0}))"
   show "?bound = length xs"
   proof (cases "xs = []")
     case False
     with * obtain n where n: "n < length xs" "xs ! n \<noteq> 0"
       by (fastforce simp add: no_trailing_unfold last_conv_nth neq_Nil_conv)
     then have "?bound = Max (Suc ` {k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs})"
-      by (subst Max_insert) (auto simp add: nth_default_def)
+      by (subst Max_insert) (auto simp: nth_default_def)
     also let ?A = "{k. k < length xs \<and> xs ! k \<noteq> 0}"
     have "{k. (k < length xs \<longrightarrow> xs ! k \<noteq> (0::'a)) \<and> k < length xs} = ?A" by auto
     also have "Max (Suc ` ?A) = Suc (Max ?A)" using n
-      by (subst mono_Max_commute [where f = Suc, symmetric]) (auto simp add: mono_Suc)
+      by (subst mono_Max_commute [where f = Suc, symmetric]) (auto simp: mono_Suc)
     also {
       have "Max ?A \<in> ?A" using n Max_in [of ?A] by fastforce
       hence "Suc (Max ?A) \<le> length xs" by simp
       moreover from * False have "length xs - 1 \<in> ?A"
-        by(auto simp add: no_trailing_unfold last_conv_nth)
+        by(auto simp: no_trailing_unfold last_conv_nth)
       hence "length xs - 1 \<le> Max ?A" using Max_ge[of ?A "length xs - 1"] by auto
       hence "length xs \<le> Suc (Max ?A)" by simp
       ultimately have "Suc (Max ?A) = length xs" by simp }
     finally show ?thesis .
   qed simp
 qed
 
 lemma nth_trailing_zeros [simp]:
   "nth (xs @ replicate n 0) = nth xs"
   by transfer simp
 
 lemma nth_idem:
   "nth (List.map (lookup f) [0..<degree f]) = f"
   unfolding degree_def by transfer
-    (auto simp add: nth_default_def fun_eq_iff not_less)
+    (auto simp: nth_default_def fun_eq_iff not_less)
 
 lemma nth_idem_bound:
   assumes "degree f \<le> n"
   shows "nth (List.map (lookup f) [0..<n]) = f"
 proof -
   from assms obtain m where "n = degree f + m"
     by (blast dest: le_Suc_ex)
   then have "[0..<n] = [0..<degree f] @ [degree f..<degree f + m]"
     by (simp add: upt_add_eq_append [of 0])
   moreover have "List.map (lookup f) [degree f..<degree f + m] = replicate m 0"
-    by (rule replicate_eqI) (auto simp add: beyond_degree_lookup_zero)
+    by (rule replicate_eqI) (auto simp: beyond_degree_lookup_zero)
   ultimately show ?thesis by (simp add: nth_idem)
 qed
 
 
 subsection \<open>Canonical sparse representation of @{typ "'a \<Rightarrow>\<^sub>0 'b"}\<close>
 
 lift_definition the_value :: "('a \<times> 'b) list \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b::zero"
   is "\<lambda>xs k. case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v"
 proof -
   fix xs :: "('a \<times> 'b) list"
   have fin: "finite {k. \<exists>v. map_of xs k = Some v}"
     using finite_dom_map_of [of xs] unfolding dom_def by auto
   then show "finite {k. (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v) \<noteq> 0}"
     using fin by (simp split: option.split)
 qed
 
 definition items :: "('a::linorder \<Rightarrow>\<^sub>0 'b::zero) \<Rightarrow> ('a \<times> 'b) list"
 where
   "items f = List.map (\<lambda>k. (k, lookup f k)) (sorted_list_of_set (keys f))"
 
 text \<open>
   For the canonical sparse representation we provide both
   directions of morphisms since the specification of
   ordered association lists in theory \<open>OAList\<close>
   will support arbitrary linear orders @{class linorder}
   as keys, not just natural numbers @{typ nat}.
 \<close>
 
 lemma the_value_items [simp]:
   "the_value (items f) = f"
   unfolding items_def
   by transfer (simp add: fun_eq_iff map_of_map_restrict restrict_map_def)
 
 lemma lookup_the_value:
   "lookup (the_value xs) k = (case map_of xs k of None \<Rightarrow> 0 | Some v \<Rightarrow> v)"
   by transfer rule
 
 lemma items_the_value:
   assumes "sorted (List.map fst xs)" and "distinct (List.map fst xs)" and "0 \<notin> snd ` set xs"
   shows "items (the_value xs) = xs"
 proof -
   from assms have "sorted_list_of_set (set (List.map fst xs)) = List.map fst xs"
     unfolding sorted_list_of_set_sort_remdups by (simp add: distinct_remdups_id sort_key_id_if_sorted)
   moreover from assms have "keys (the_value xs) = fst ` set xs"
-    by transfer (auto simp add: image_def split: option.split dest: set_map_of_compr)
+    by transfer (auto simp: image_def split: option.split dest: set_map_of_compr)
   ultimately show ?thesis
     unfolding items_def using assms
-    by (auto simp add: lookup_the_value intro: map_idI)
+    by (auto simp: lookup_the_value intro: map_idI)
 qed
 
 lemma the_value_Nil [simp]:
   "the_value [] = 0"
   by transfer (simp add: fun_eq_iff)
 
 lemma the_value_Cons [simp]:
   "the_value (x # xs) = update (fst x) (snd x) (the_value xs)"
   by transfer (simp add: fun_eq_iff)
 
 lemma items_zero [simp]:
   "items 0 = []"
   unfolding items_def by simp
 
 lemma items_one [simp]:
   "items 1 = [(0, 1)]"
   unfolding items_def by transfer simp
 
 lemma items_single [simp]:
   "items (single k v) = (if v = 0 then [] else [(k, v)])"
   unfolding items_def by simp
 
 lemma in_set_items_iff [simp]:
   "(k, v) \<in> set (items f) \<longleftrightarrow> k \<in> keys f \<and> lookup f k = v"
   unfolding items_def by transfer auto
 
 
 subsection \<open>Size estimation\<close>
 
 context
   fixes f :: "'a \<Rightarrow> nat"
   and g :: "'b :: zero \<Rightarrow> nat"
 begin
 
 definition poly_mapping_size :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> nat"
 where
   "poly_mapping_size m = g 0 + (\<Sum>k \<in> keys m. Suc (f k + g (lookup m k)))"
 
 lemma poly_mapping_size_0 [simp]:
   "poly_mapping_size 0 = g 0"
   by (simp add: poly_mapping_size_def)
 
 lemma poly_mapping_size_single [simp]:
   "poly_mapping_size (single k v) = (if v = 0 then g 0 else g 0 + f k + g v + 1)"
   unfolding poly_mapping_size_def by transfer simp
 
 lemma keys_less_poly_mapping_size:
   "k \<in> keys m \<Longrightarrow> f k + g (lookup m k) < poly_mapping_size m"
 unfolding poly_mapping_size_def
 proof transfer
   fix k :: 'a and m :: "'a \<Rightarrow> 'b" and f :: "'a \<Rightarrow> nat" and g
   let ?keys = "{k. m k \<noteq> 0}"
   assume *: "finite ?keys" "k \<in> ?keys"
   then have "f k + g (m k) = (\<Sum>k' \<in> ?keys. f k' + g (m k') when k' = k)"
     by (simp add: sum.delta when_def)
   also have "\<dots> < (\<Sum>k' \<in> ?keys. Suc (f k' + g (m k')))" using *
-    by (intro sum_strict_mono) (auto simp add: when_def)
+    by (intro sum_strict_mono) (auto simp: when_def)
   also have "\<dots> \<le> g 0 + \<dots>" by simp
   finally have "f k + g (m k) < \<dots>" .
   then show "f k + g (m k) < g 0 + (\<Sum>k | m k \<noteq> 0. Suc (f k + g (m k)))"
     by simp
 qed
 
 lemma lookup_le_poly_mapping_size:
   "g (lookup m k) \<le> poly_mapping_size m"
 proof (cases "k \<in> keys m")
   case True
   with keys_less_poly_mapping_size [of k m]
   show ?thesis by simp
 next
   case False
   then show ?thesis
     by (simp add: Poly_Mapping.poly_mapping_size_def in_keys_iff)
 qed 
 
 lemma poly_mapping_size_estimation:
   "k \<in> keys m \<Longrightarrow> y \<le> f k + g (lookup m k) \<Longrightarrow> y < poly_mapping_size m"
   using keys_less_poly_mapping_size by (auto intro: le_less_trans)
 
 lemma poly_mapping_size_estimation2:
   assumes "v \<in> range m" and "y \<le> g v"
   shows "y < poly_mapping_size m"
 proof -
   from assms obtain k where *: "lookup m k = v" "v \<noteq> 0"
     by transfer blast
   from * have "k \<in> keys m"
     by (simp add: in_keys_iff)
   then show ?thesis
   proof (rule poly_mapping_size_estimation)
     from assms * have "y \<le> g (lookup m k)"
       by simp
     then show "y \<le> f k + g (lookup m k)"
       by simp
   qed
 qed
 
 end
 
 lemma poly_mapping_size_one [simp]:
   "poly_mapping_size f g 1 = g 0 + f 0 + g 1 + 1"
   unfolding poly_mapping_size_def by transfer simp
 
 lemma poly_mapping_size_cong [fundef_cong]:
   "m = m' \<Longrightarrow> g 0 = g' 0 \<Longrightarrow> (\<And>k. k \<in> keys m' \<Longrightarrow> f k = f' k)
     \<Longrightarrow> (\<And>v. v \<in> range m' \<Longrightarrow> g v = g' v)
     \<Longrightarrow> poly_mapping_size f g m = poly_mapping_size f' g' m'"
-  by (auto simp add: poly_mapping_size_def intro!: sum.cong)
+  by (auto simp: poly_mapping_size_def intro!: sum.cong)
 
 instantiation poly_mapping :: (type, zero) size
 begin
 
 definition "size = poly_mapping_size (\<lambda>_. 0) (\<lambda>_. 0)"
 
 instance ..
 
 end
 
 
 subsection \<open>Further mapping operations and properties\<close>
 
 text \<open>It is like in algebra: there are many definitions, some are also used\<close>
 
 lift_definition mapp ::
   "('a \<Rightarrow> 'b :: zero \<Rightarrow> 'c :: zero) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c)"
   is "\<lambda>f p k. (if k \<in> keys p then f k (lookup p k) else 0)"
   by simp
 
 lemma mapp_cong [fundef_cong]:
   "\<lbrakk> m = m'; \<And>k. k \<in> keys m' \<Longrightarrow> f k (lookup m' k) = f' k (lookup m' k) \<rbrakk>
   \<Longrightarrow> mapp f m = mapp f' m'"
-  by transfer (auto simp add: fun_eq_iff)
+  by transfer (auto simp: fun_eq_iff)
 
 lemma lookup_mapp:
   "lookup (mapp f p) k = (f k (lookup p k) when k \<in> keys p)"
   by (simp add: mapp.rep_eq)
 
 lemma keys_mapp_subset: "keys (mapp f p) \<subseteq> keys p"
   by (meson in_keys_iff mapp.rep_eq subsetI)
 
 subsection\<open>Free Abelian Groups Over a Type\<close>
 
 abbreviation frag_of :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int"
   where "frag_of c \<equiv> Poly_Mapping.single c (1::int)"
 
 lemma lookup_frag_of [simp]:
    "Poly_Mapping.lookup(frag_of c) = (\<lambda>x. if x = c then 1 else 0)"
   by (force simp add: lookup_single_not_eq)
 
 lemma frag_of_nonzero [simp]: "frag_of a \<noteq> 0"
 proof -
   let ?f = "\<lambda>x. if x = a then 1 else (0::int)"
   have "?f \<noteq> (\<lambda>x. 0::int)"
     by (auto simp: fun_eq_iff)
   then have "Poly_Mapping.lookup (Abs_poly_mapping ?f) \<noteq> Poly_Mapping.lookup (Abs_poly_mapping (\<lambda>x. 0))"
     by fastforce
   then show ?thesis
     by (metis lookup_single_eq lookup_zero)
 qed
 
 definition frag_cmul :: "int \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 int)"
   where "frag_cmul c a = Abs_poly_mapping (\<lambda>x. c * Poly_Mapping.lookup a x)"
 
 lemma frag_cmul_zero [simp]: "frag_cmul 0 x = 0"
   by (simp add: frag_cmul_def)
 
 lemma frag_cmul_zero2 [simp]: "frag_cmul c 0 = 0"
   by (simp add: frag_cmul_def)
 
 lemma frag_cmul_one [simp]: "frag_cmul 1 x = x"
   by (auto simp: frag_cmul_def Poly_Mapping.poly_mapping.lookup_inverse)
 
 lemma frag_cmul_minus_one [simp]: "frag_cmul (-1) x = -x"
   by (simp add: frag_cmul_def uminus_poly_mapping_def poly_mapping_eqI)
 
 lemma frag_cmul_cmul [simp]: "frag_cmul c (frag_cmul d x) = frag_cmul (c*d) x"
   by (simp add: frag_cmul_def mult_ac)
 
 lemma lookup_frag_cmul [simp]: "poly_mapping.lookup (frag_cmul c x) i = c * poly_mapping.lookup x i"
   by (simp add: frag_cmul_def)
 
 lemma minus_frag_cmul [simp]: "- frag_cmul k x = frag_cmul (-k) x"
   by (simp add: poly_mapping_eqI)
 
 lemma keys_frag_of: "Poly_Mapping.keys(frag_of a) = {a}"
   by simp
 
 lemma finite_cmul_nonzero: "finite {x. c * Poly_Mapping.lookup a x \<noteq> (0::int)}"
   by simp
 
 lemma keys_cmul: "Poly_Mapping.keys(frag_cmul c a) \<subseteq> Poly_Mapping.keys a"
   using finite_cmul_nonzero [of c a]
   by (metis lookup_frag_cmul mult_zero_right not_in_keys_iff_lookup_eq_zero subsetI)
   
 
 lemma keys_cmul_iff [iff]: "i \<in> Poly_Mapping.keys (frag_cmul c x) \<longleftrightarrow> i \<in> Poly_Mapping.keys x \<and> c \<noteq> 0"
-  apply auto
-  apply (meson subsetD keys_cmul)
   by (metis in_keys_iff lookup_frag_cmul mult_eq_0_iff)
 
 lemma keys_minus [simp]: "Poly_Mapping.keys(-a) = Poly_Mapping.keys a"
   by (metis (no_types, opaque_lifting) in_keys_iff lookup_uminus neg_equal_0_iff_equal subsetI subset_antisym)
 
 lemma keys_diff: 
   "Poly_Mapping.keys(a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b"
-  by (auto simp add: in_keys_iff lookup_minus)
+  by (auto simp: in_keys_iff lookup_minus)
 
 lemma keys_eq_empty [simp]: "Poly_Mapping.keys c = {} \<longleftrightarrow> c = 0"
   by (metis in_keys_iff keys_zero lookup_zero poly_mapping_eqI)
 
 lemma frag_cmul_eq_0_iff [simp]: "frag_cmul k c = 0 \<longleftrightarrow> k=0 \<or> c=0"
   by auto (metis subsetI subset_antisym keys_cmul_iff keys_eq_empty)
 
 lemma frag_of_eq: "frag_of x = frag_of y \<longleftrightarrow> x = y"
   by (metis lookup_single_eq lookup_single_not_eq zero_neq_one)
 
 lemma frag_cmul_distrib: "frag_cmul (c+d) a = frag_cmul c a + frag_cmul d a"
   by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)
 
 lemma frag_cmul_distrib2: "frag_cmul c (a+b) = frag_cmul c a + frag_cmul c b"
 proof -
   have "finite {x. poly_mapping.lookup a x + poly_mapping.lookup b x \<noteq> 0}"
     using keys_add [of a b]
     by (metis (no_types, lifting) finite_keys finite_subset keys.rep_eq lookup_add mem_Collect_eq subsetI)
   then show ?thesis
     by (simp add: frag_cmul_def plus_poly_mapping_def int_distrib)
 qed
 
 lemma frag_cmul_diff_distrib: "frag_cmul (a - b) c = frag_cmul a c - frag_cmul b c"
   by (auto simp: left_diff_distrib lookup_minus poly_mapping_eqI)
 
 lemma frag_cmul_sum:
      "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"
 proof (induction rule: infinite_finite_induct)
   case (insert i I)
   then show ?case
     by (auto simp: algebra_simps frag_cmul_distrib2)
 qed auto
 
 lemma keys_sum: "Poly_Mapping.keys(sum b I) \<subseteq> (\<Union>i \<in>I. Poly_Mapping.keys(b i))"
 proof (induction I rule: infinite_finite_induct)
   case (insert i I)
   then show ?case
     using keys_add [of "b i" "sum b I"] by auto
 qed auto
 
 
 definition frag_extend :: "('b \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 int"
   where "frag_extend b x \<equiv> (\<Sum>i \<in> Poly_Mapping.keys x. frag_cmul (Poly_Mapping.lookup x i) (b i))"
 
 lemma frag_extend_0 [simp]: "frag_extend b 0 = 0"
   by (simp add: frag_extend_def)
 
 lemma frag_extend_of [simp]: "frag_extend f (frag_of a) = f a"
   by (simp add: frag_extend_def)
 
 lemma frag_extend_cmul:
    "frag_extend f (frag_cmul c x) = frag_cmul c (frag_extend f x)"
   by (auto simp: frag_extend_def frag_cmul_sum intro: sum.mono_neutral_cong_left)
 
 lemma frag_extend_minus:
    "frag_extend f (- x) = - (frag_extend f x)"
   using frag_extend_cmul [of f "-1"] by simp
 
 lemma frag_extend_add:
   "frag_extend f (a+b) = (frag_extend f a) + (frag_extend f b)"
 proof -
   have *: "(\<Sum>i\<in>Poly_Mapping.keys a. frag_cmul (poly_mapping.lookup a i) (f i)) 
          = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i))"
           "(\<Sum>i\<in>Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i)) 
          = (\<Sum>i\<in>Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup b i) (f i))"
     by (auto simp: in_keys_iff intro: sum.mono_neutral_cong_left)
   have "frag_extend f (a+b) = (\<Sum>i\<in>Poly_Mapping.keys (a + b).
        frag_cmul (poly_mapping.lookup a i) (f i) + frag_cmul (poly_mapping.lookup b i) (f i)) "
     by (auto simp: frag_extend_def Poly_Mapping.lookup_add frag_cmul_distrib)
   also have "... = (\<Sum>i \<in> Poly_Mapping.keys a \<union> Poly_Mapping.keys b. frag_cmul (poly_mapping.lookup a i) (f i) 
                          + frag_cmul (poly_mapping.lookup b i) (f i))"
-    apply (rule sum.mono_neutral_cong_left)
-    using keys_add [of a b]
-     apply (auto simp add: in_keys_iff plus_poly_mapping.rep_eq frag_cmul_distrib [symmetric])
-    done
+  proof (rule sum.mono_neutral_cong_left)
+    show "\<forall>i\<in>keys a \<union> keys b - keys (a + b).
+       frag_cmul (lookup a i) (f i) + frag_cmul (lookup b i) (f i) = 0"
+      by (metis DiffD2 frag_cmul_distrib frag_cmul_zero in_keys_iff lookup_add)
+  qed (auto simp: keys_add)
   also have "... = (frag_extend f a) + (frag_extend f b)"
     by (auto simp: * sum.distrib frag_extend_def)
   finally show ?thesis .
 qed
 
 lemma frag_extend_diff:
    "frag_extend f (a-b) = (frag_extend f a) - (frag_extend f b)"
   by (metis (no_types, opaque_lifting) add_uminus_conv_diff frag_extend_add frag_extend_minus)
 
 lemma frag_extend_sum:
    "finite I \<Longrightarrow> frag_extend f (\<Sum>i\<in>I. g i) = sum (frag_extend f o g) I"
   by (induction I rule: finite_induct) (simp_all add: frag_extend_add)
 
 lemma frag_extend_eq:
    "(\<And>f. f \<in> Poly_Mapping.keys c \<Longrightarrow> g f = h f) \<Longrightarrow> frag_extend g c = frag_extend h c"
   by (simp add: frag_extend_def)
 
 lemma frag_extend_eq_0:
    "(\<And>x. x \<in> Poly_Mapping.keys c \<Longrightarrow> f x = 0) \<Longrightarrow> frag_extend f c = 0"
   by (simp add: frag_extend_def)
 
 lemma keys_frag_extend: "Poly_Mapping.keys(frag_extend f c) \<subseteq> (\<Union>x \<in> Poly_Mapping.keys c. Poly_Mapping.keys(f x))"
   unfolding frag_extend_def
   using keys_sum by fastforce
 
 lemma frag_expansion: "a = frag_extend frag_of a"
 proof -
   have *: "finite I
         \<Longrightarrow> Poly_Mapping.lookup (\<Sum>i\<in>I. frag_cmul (Poly_Mapping.lookup a i) (frag_of i)) j =
             (if j \<in> I then Poly_Mapping.lookup a j else 0)" for I j
     by (induction I rule: finite_induct) (auto simp: lookup_single lookup_add)
   show ?thesis
     unfolding frag_extend_def
     by (rule poly_mapping_eqI) (fastforce simp add: in_keys_iff *)
 qed
 
 lemma frag_closure_minus_cmul:
   assumes "P 0" and P: "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> P(x - y)" "P c"
   shows "P(frag_cmul k c)"
 proof -
   have "P (frag_cmul (int n) c)" for n
-    apply (induction n)
-     apply (simp_all add: assms frag_cmul_distrib)
-    by (metis add.left_neutral add_diff_cancel_right' add_uminus_conv_diff P)
+  proof (induction n)
+    case 0
+    then show ?case
+      by (simp add: assms)
+  next
+    case (Suc n)
+    then show ?case
+      by (metis assms diff_0 diff_minus_eq_add frag_cmul_distrib frag_cmul_one of_nat_Suc)
+  qed
   then show ?thesis
     by (metis (no_types, opaque_lifting) add_diff_eq assms(2) diff_add_cancel frag_cmul_distrib int_diff_cases)
 qed
 
 lemma frag_induction [consumes 1, case_names zero one diff]:
   assumes supp: "Poly_Mapping.keys c \<subseteq> S"
     and 0: "P 0" and sing: "\<And>x. x \<in> S \<Longrightarrow> P(frag_of x)"
     and diff: "\<And>a b. \<lbrakk>P a; P b\<rbrakk> \<Longrightarrow> P(a - b)"
   shows "P c"
 proof -
   have "P (\<Sum>i\<in>I. frag_cmul (poly_mapping.lookup c i) (frag_of i))"
     if "I \<subseteq> Poly_Mapping.keys c" for I
     using finite_subset [OF that finite_keys [of c]] that supp
   proof (induction I arbitrary: c rule: finite_induct)
     case empty
     then show ?case
       by (auto simp: 0)
   next
     case (insert i I c)
     have ab: "a+b = a - (0 - b)" for a b :: "'a \<Rightarrow>\<^sub>0 int"
       by simp
     have Pfrag: "P (frag_cmul (poly_mapping.lookup c i) (frag_of i))"
       by (metis "0" diff frag_closure_minus_cmul insert.prems insert_subset sing subset_iff)
-    show ?case
-      apply (simp add: insert.hyps)
-      apply (subst ab)
-      using insert apply (blast intro: assms Pfrag)
-      done
+    with insert show ?case
+      by (metis (mono_tags, lifting) "0" ab diff insert_subset sum.insert)
   qed
   then show ?thesis
-    by (subst frag_expansion) (auto simp add: frag_extend_def)
+    by (subst frag_expansion) (auto simp: frag_extend_def)
 qed
 
 lemma frag_extend_compose:
   "frag_extend f (frag_extend (frag_of o g) c) = frag_extend (f o g) c"
   using subset_UNIV
   by (induction c rule: frag_induction) (auto simp: frag_extend_diff)
 
 lemma frag_split:
   fixes c :: "'a \<Rightarrow>\<^sub>0 int"
   assumes "Poly_Mapping.keys c \<subseteq> S \<union> T"
   obtains d e where "Poly_Mapping.keys d \<subseteq> S" "Poly_Mapping.keys e \<subseteq> T" "d + e = c"
 proof 
   let ?d = "frag_extend (\<lambda>f. if f \<in> S then frag_of f else 0) c"
   let ?e = "frag_extend (\<lambda>f. if f \<in> S then 0 else frag_of f) c"
   show "Poly_Mapping.keys ?d \<subseteq> S" "Poly_Mapping.keys ?e \<subseteq> T"
     using assms by (auto intro!: order_trans [OF keys_frag_extend] split: if_split_asm)
   show "?d + ?e = c"
     using assms
   proof (induction c rule: frag_induction)
     case (diff a b)
     then show ?case
       by (metis (no_types, lifting) frag_extend_diff add_diff_eq diff_add_eq diff_add_eq_diff_diff_swap)
   qed auto
 qed
 
 hide_const (open) lookup single update keys range map map_key degree nth the_value items foldr mapp
 
 end
\ No newline at end of file