re-moved theory Fin_Fun to AFP
authorhaftmann
Mon Oct 26 11:19:24 2009 +0100 (2009-10-26)
changeset 33177edbd2c09176b
parent 33176 d6936fd7cda8
child 33181 50268fcec3ce
re-moved theory Fin_Fun to AFP
src/HOL/Library/Fin_Fun.thy
src/HOL/Library/Library.thy
     1.1 --- a/src/HOL/Library/Fin_Fun.thy	Mon Oct 26 09:03:57 2009 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,1599 +0,0 @@
     1.4 -
     1.5 -(* Author: Andreas Lochbihler, Uni Karlsruhe *)
     1.6 -
     1.7 -header {* Almost everywhere constant functions *}
     1.8 -
     1.9 -theory Fin_Fun
    1.10 -imports Main Infinite_Set Enum
    1.11 -begin
    1.12 -
    1.13 -text {*
    1.14 -  This theory defines functions which are constant except for finitely
    1.15 -  many points (FinFun) and introduces a type finfin along with a
    1.16 -  number of operators for them. The code generator is set up such that
    1.17 -  such functions can be represented as data in the generated code and
    1.18 -  all operators are executable.
    1.19 -
    1.20 -  For details, see Formalising FinFuns - Generating Code for Functions as Data by A. Lochbihler in TPHOLs 2009.
    1.21 -*}
    1.22 -
    1.23 -
    1.24 -subsection {* The @{text "map_default"} operation *}
    1.25 -
    1.26 -definition map_default :: "'b \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    1.27 -where "map_default b f a \<equiv> case f a of None \<Rightarrow> b | Some b' \<Rightarrow> b'"
    1.28 -
    1.29 -lemma map_default_delete [simp]:
    1.30 -  "map_default b (f(a := None)) = (map_default b f)(a := b)"
    1.31 -by(simp add: map_default_def expand_fun_eq)
    1.32 -
    1.33 -lemma map_default_insert:
    1.34 -  "map_default b (f(a \<mapsto> b')) = (map_default b f)(a := b')"
    1.35 -by(simp add: map_default_def expand_fun_eq)
    1.36 -
    1.37 -lemma map_default_empty [simp]: "map_default b empty = (\<lambda>a. b)"
    1.38 -by(simp add: expand_fun_eq map_default_def)
    1.39 -
    1.40 -lemma map_default_inject:
    1.41 -  fixes g g' :: "'a \<rightharpoonup> 'b"
    1.42 -  assumes infin_eq: "\<not> finite (UNIV :: 'a set) \<or> b = b'"
    1.43 -  and fin: "finite (dom g)" and b: "b \<notin> ran g"
    1.44 -  and fin': "finite (dom g')" and b': "b' \<notin> ran g'"
    1.45 -  and eq': "map_default b g = map_default b' g'"
    1.46 -  shows "b = b'" "g = g'"
    1.47 -proof -
    1.48 -  from infin_eq show bb': "b = b'"
    1.49 -  proof
    1.50 -    assume infin: "\<not> finite (UNIV :: 'a set)"
    1.51 -    from fin fin' have "finite (dom g \<union> dom g')" by auto
    1.52 -    with infin have "UNIV - (dom g \<union> dom g') \<noteq> {}" by(auto dest: finite_subset)
    1.53 -    then obtain a where a: "a \<notin> dom g \<union> dom g'" by auto
    1.54 -    hence "map_default b g a = b" "map_default b' g' a = b'" by(auto simp add: map_default_def)
    1.55 -    with eq' show "b = b'" by simp
    1.56 -  qed
    1.57 -
    1.58 -  show "g = g'"
    1.59 -  proof
    1.60 -    fix x
    1.61 -    show "g x = g' x"
    1.62 -    proof(cases "g x")
    1.63 -      case None
    1.64 -      hence "map_default b g x = b" by(simp add: map_default_def)
    1.65 -      with bb' eq' have "map_default b' g' x = b'" by simp
    1.66 -      with b' have "g' x = None" by(simp add: map_default_def ran_def split: option.split_asm)
    1.67 -      with None show ?thesis by simp
    1.68 -    next
    1.69 -      case (Some c)
    1.70 -      with b have cb: "c \<noteq> b" by(auto simp add: ran_def)
    1.71 -      moreover from Some have "map_default b g x = c" by(simp add: map_default_def)
    1.72 -      with eq' have "map_default b' g' x = c" by simp
    1.73 -      ultimately have "g' x = Some c" using b' bb' by(auto simp add: map_default_def split: option.splits)
    1.74 -      with Some show ?thesis by simp
    1.75 -    qed
    1.76 -  qed
    1.77 -qed
    1.78 -
    1.79 -subsection {* The finfun type *}
    1.80 -
    1.81 -typedef ('a,'b) finfun = "{f::'a\<Rightarrow>'b. \<exists>b. finite {a. f a \<noteq> b}}"
    1.82 -proof -
    1.83 -  have "\<exists>f. finite {x. f x \<noteq> undefined}"
    1.84 -  proof
    1.85 -    show "finite {x. (\<lambda>y. undefined) x \<noteq> undefined}" by auto
    1.86 -  qed
    1.87 -  then show ?thesis by auto
    1.88 -qed
    1.89 -
    1.90 -syntax
    1.91 -  "finfun"      :: "type \<Rightarrow> type \<Rightarrow> type"         ("(_ \<Rightarrow>\<^isub>f /_)" [22, 21] 21)
    1.92 -
    1.93 -lemma fun_upd_finfun: "y(a := b) \<in> finfun \<longleftrightarrow> y \<in> finfun"
    1.94 -proof -
    1.95 -  { fix b'
    1.96 -    have "finite {a'. (y(a := b)) a' \<noteq> b'} = finite {a'. y a' \<noteq> b'}"
    1.97 -    proof(cases "b = b'")
    1.98 -      case True
    1.99 -      hence "{a'. (y(a := b)) a' \<noteq> b'} = {a'. y a' \<noteq> b'} - {a}" by auto
   1.100 -      thus ?thesis by simp
   1.101 -    next
   1.102 -      case False
   1.103 -      hence "{a'. (y(a := b)) a' \<noteq> b'} = insert a {a'. y a' \<noteq> b'}" by auto
   1.104 -      thus ?thesis by simp
   1.105 -    qed }
   1.106 -  thus ?thesis unfolding finfun_def by blast
   1.107 -qed
   1.108 -
   1.109 -lemma const_finfun: "(\<lambda>x. a) \<in> finfun"
   1.110 -by(auto simp add: finfun_def)
   1.111 -
   1.112 -lemma finfun_left_compose:
   1.113 -  assumes "y \<in> finfun"
   1.114 -  shows "g \<circ> y \<in> finfun"
   1.115 -proof -
   1.116 -  from assms obtain b where "finite {a. y a \<noteq> b}"
   1.117 -    unfolding finfun_def by blast
   1.118 -  hence "finite {c. g (y c) \<noteq> g b}"
   1.119 -  proof(induct x\<equiv>"{a. y a \<noteq> b}" arbitrary: y)
   1.120 -    case empty
   1.121 -    hence "y = (\<lambda>a. b)" by(auto intro: ext)
   1.122 -    thus ?case by(simp)
   1.123 -  next
   1.124 -    case (insert x F)
   1.125 -    note IH = `\<And>y. F = {a. y a \<noteq> b} \<Longrightarrow> finite {c. g (y c) \<noteq> g b}`
   1.126 -    from `insert x F = {a. y a \<noteq> b}` `x \<notin> F`
   1.127 -    have F: "F = {a. (y(x := b)) a \<noteq> b}" by(auto)
   1.128 -    show ?case
   1.129 -    proof(cases "g (y x) = g b")
   1.130 -      case True
   1.131 -      hence "{c. g ((y(x := b)) c) \<noteq> g b} = {c. g (y c) \<noteq> g b}" by auto
   1.132 -      with IH[OF F] show ?thesis by simp
   1.133 -    next
   1.134 -      case False
   1.135 -      hence "{c. g (y c) \<noteq> g b} = insert x {c. g ((y(x := b)) c) \<noteq> g b}" by auto
   1.136 -      with IH[OF F] show ?thesis by(simp)
   1.137 -    qed
   1.138 -  qed
   1.139 -  thus ?thesis unfolding finfun_def by auto
   1.140 -qed
   1.141 -
   1.142 -lemma assumes "y \<in> finfun"
   1.143 -  shows fst_finfun: "fst \<circ> y \<in> finfun"
   1.144 -  and snd_finfun: "snd \<circ> y \<in> finfun"
   1.145 -proof -
   1.146 -  from assms obtain b c where bc: "finite {a. y a \<noteq> (b, c)}"
   1.147 -    unfolding finfun_def by auto
   1.148 -  have "{a. fst (y a) \<noteq> b} \<subseteq> {a. y a \<noteq> (b, c)}"
   1.149 -    and "{a. snd (y a) \<noteq> c} \<subseteq> {a. y a \<noteq> (b, c)}" by auto
   1.150 -  hence "finite {a. fst (y a) \<noteq> b}" 
   1.151 -    and "finite {a. snd (y a) \<noteq> c}" using bc by(auto intro: finite_subset)
   1.152 -  thus "fst \<circ> y \<in> finfun" "snd \<circ> y \<in> finfun"
   1.153 -    unfolding finfun_def by auto
   1.154 -qed
   1.155 -
   1.156 -lemma map_of_finfun: "map_of xs \<in> finfun"
   1.157 -unfolding finfun_def
   1.158 -by(induct xs)(auto simp add: Collect_neg_eq Collect_conj_eq Collect_imp_eq intro: finite_subset)
   1.159 -
   1.160 -lemma Diag_finfun: "(\<lambda>x. (f x, g x)) \<in> finfun \<longleftrightarrow> f \<in> finfun \<and> g \<in> finfun"
   1.161 -by(auto intro: finite_subset simp add: Collect_neg_eq Collect_imp_eq Collect_conj_eq finfun_def)
   1.162 -
   1.163 -lemma finfun_right_compose:
   1.164 -  assumes g: "g \<in> finfun" and inj: "inj f"
   1.165 -  shows "g o f \<in> finfun"
   1.166 -proof -
   1.167 -  from g obtain b where b: "finite {a. g a \<noteq> b}" unfolding finfun_def by blast
   1.168 -  moreover have "f ` {a. g (f a) \<noteq> b} \<subseteq> {a. g a \<noteq> b}" by auto
   1.169 -  moreover from inj have "inj_on f {a.  g (f a) \<noteq> b}" by(rule subset_inj_on) blast
   1.170 -  ultimately have "finite {a. g (f a) \<noteq> b}"
   1.171 -    by(blast intro: finite_imageD[where f=f] finite_subset)
   1.172 -  thus ?thesis unfolding finfun_def by auto
   1.173 -qed
   1.174 -
   1.175 -lemma finfun_curry:
   1.176 -  assumes fin: "f \<in> finfun"
   1.177 -  shows "curry f \<in> finfun" "curry f a \<in> finfun"
   1.178 -proof -
   1.179 -  from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast
   1.180 -  moreover have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force)
   1.181 -  hence "{a. curry f a \<noteq> (\<lambda>b. c)} = fst ` {ab. f ab \<noteq> c}"
   1.182 -    by(auto simp add: curry_def expand_fun_eq)
   1.183 -  ultimately have "finite {a. curry f a \<noteq> (\<lambda>b. c)}" by simp
   1.184 -  thus "curry f \<in> finfun" unfolding finfun_def by blast
   1.185 -  
   1.186 -  have "snd ` {ab. f ab \<noteq> c} = {b. \<exists>a. f (a, b) \<noteq> c}" by(force)
   1.187 -  hence "{b. f (a, b) \<noteq> c} \<subseteq> snd ` {ab. f ab \<noteq> c}" by auto
   1.188 -  hence "finite {b. f (a, b) \<noteq> c}" by(rule finite_subset)(rule finite_imageI[OF c])
   1.189 -  thus "curry f a \<in> finfun" unfolding finfun_def by auto
   1.190 -qed
   1.191 -
   1.192 -lemmas finfun_simp = 
   1.193 -  fst_finfun snd_finfun Abs_finfun_inverse Rep_finfun_inverse Abs_finfun_inject Rep_finfun_inject Diag_finfun finfun_curry
   1.194 -lemmas finfun_iff = const_finfun fun_upd_finfun Rep_finfun map_of_finfun
   1.195 -lemmas finfun_intro = finfun_left_compose fst_finfun snd_finfun
   1.196 -
   1.197 -lemma Abs_finfun_inject_finite:
   1.198 -  fixes x y :: "'a \<Rightarrow> 'b"
   1.199 -  assumes fin: "finite (UNIV :: 'a set)"
   1.200 -  shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y"
   1.201 -proof
   1.202 -  assume "Abs_finfun x = Abs_finfun y"
   1.203 -  moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def
   1.204 -    by(auto intro: finite_subset[OF _ fin])
   1.205 -  ultimately show "x = y" by(simp add: Abs_finfun_inject)
   1.206 -qed simp
   1.207 -
   1.208 -lemma Abs_finfun_inject_finite_class:
   1.209 -  fixes x y :: "('a :: finite) \<Rightarrow> 'b"
   1.210 -  shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y"
   1.211 -using finite_UNIV
   1.212 -by(simp add: Abs_finfun_inject_finite)
   1.213 -
   1.214 -lemma Abs_finfun_inj_finite:
   1.215 -  assumes fin: "finite (UNIV :: 'a set)"
   1.216 -  shows "inj (Abs_finfun :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b)"
   1.217 -proof(rule inj_onI)
   1.218 -  fix x y :: "'a \<Rightarrow> 'b"
   1.219 -  assume "Abs_finfun x = Abs_finfun y"
   1.220 -  moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def
   1.221 -    by(auto intro: finite_subset[OF _ fin])
   1.222 -  ultimately show "x = y" by(simp add: Abs_finfun_inject)
   1.223 -qed
   1.224 -
   1.225 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.226 -
   1.227 -lemma Abs_finfun_inverse_finite:
   1.228 -  fixes x :: "'a \<Rightarrow> 'b"
   1.229 -  assumes fin: "finite (UNIV :: 'a set)"
   1.230 -  shows "Rep_finfun (Abs_finfun x) = x"
   1.231 -proof -
   1.232 -  from fin have "x \<in> finfun"
   1.233 -    by(auto simp add: finfun_def intro: finite_subset)
   1.234 -  thus ?thesis by simp
   1.235 -qed
   1.236 -
   1.237 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.238 -
   1.239 -lemma Abs_finfun_inverse_finite_class:
   1.240 -  fixes x :: "('a :: finite) \<Rightarrow> 'b"
   1.241 -  shows "Rep_finfun (Abs_finfun x) = x"
   1.242 -using finite_UNIV by(simp add: Abs_finfun_inverse_finite)
   1.243 -
   1.244 -lemma finfun_eq_finite_UNIV: "finite (UNIV :: 'a set) \<Longrightarrow> (finfun :: ('a \<Rightarrow> 'b) set) = UNIV"
   1.245 -unfolding finfun_def by(auto intro: finite_subset)
   1.246 -
   1.247 -lemma finfun_finite_UNIV_class: "finfun = (UNIV :: ('a :: finite \<Rightarrow> 'b) set)"
   1.248 -by(simp add: finfun_eq_finite_UNIV)
   1.249 -
   1.250 -lemma map_default_in_finfun:
   1.251 -  assumes fin: "finite (dom f)"
   1.252 -  shows "map_default b f \<in> finfun"
   1.253 -unfolding finfun_def
   1.254 -proof(intro CollectI exI)
   1.255 -  from fin show "finite {a. map_default b f a \<noteq> b}"
   1.256 -    by(auto simp add: map_default_def dom_def Collect_conj_eq split: option.splits)
   1.257 -qed
   1.258 -
   1.259 -lemma finfun_cases_map_default:
   1.260 -  obtains b g where "f = Abs_finfun (map_default b g)" "finite (dom g)" "b \<notin> ran g"
   1.261 -proof -
   1.262 -  obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by(cases f)
   1.263 -  from y obtain b where b: "finite {a. y a \<noteq> b}" unfolding finfun_def by auto
   1.264 -  let ?g = "(\<lambda>a. if y a = b then None else Some (y a))"
   1.265 -  have "map_default b ?g = y" by(simp add: expand_fun_eq map_default_def)
   1.266 -  with f have "f = Abs_finfun (map_default b ?g)" by simp
   1.267 -  moreover from b have "finite (dom ?g)" by(auto simp add: dom_def)
   1.268 -  moreover have "b \<notin> ran ?g" by(auto simp add: ran_def)
   1.269 -  ultimately show ?thesis by(rule that)
   1.270 -qed
   1.271 -
   1.272 -
   1.273 -subsection {* Kernel functions for type @{typ "'a \<Rightarrow>\<^isub>f 'b"} *}
   1.274 -
   1.275 -definition finfun_const :: "'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("\<lambda>\<^isup>f/ _" [0] 1)
   1.276 -where [code del]: "(\<lambda>\<^isup>f b) = Abs_finfun (\<lambda>x. b)"
   1.277 -
   1.278 -definition finfun_update :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("_'(\<^sup>f/ _ := _')" [1000,0,0] 1000)
   1.279 -where [code del]: "f(\<^sup>fa := b) = Abs_finfun ((Rep_finfun f)(a := b))"
   1.280 -
   1.281 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.282 -
   1.283 -lemma finfun_update_twist: "a \<noteq> a' \<Longrightarrow> f(\<^sup>f a := b)(\<^sup>f a' := b') = f(\<^sup>f a' := b')(\<^sup>f a := b)"
   1.284 -by(simp add: finfun_update_def fun_upd_twist)
   1.285 -
   1.286 -lemma finfun_update_twice [simp]:
   1.287 -  "finfun_update (finfun_update f a b) a b' = finfun_update f a b'"
   1.288 -by(simp add: finfun_update_def)
   1.289 -
   1.290 -lemma finfun_update_const_same: "(\<lambda>\<^isup>f b)(\<^sup>f a := b) = (\<lambda>\<^isup>f b)"
   1.291 -by(simp add: finfun_update_def finfun_const_def expand_fun_eq)
   1.292 -
   1.293 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.294 -
   1.295 -subsection {* Code generator setup *}
   1.296 -
   1.297 -definition finfun_update_code :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b" ("_'(\<^sup>f\<^sup>c/ _ := _')" [1000,0,0] 1000)
   1.298 -where [simp, code del]: "finfun_update_code = finfun_update"
   1.299 -
   1.300 -code_datatype finfun_const finfun_update_code
   1.301 -
   1.302 -lemma finfun_update_const_code [code]:
   1.303 -  "(\<lambda>\<^isup>f b)(\<^sup>f a := b') = (if b = b' then (\<lambda>\<^isup>f b) else finfun_update_code (\<lambda>\<^isup>f b) a b')"
   1.304 -by(simp add: finfun_update_const_same)
   1.305 -
   1.306 -lemma finfun_update_update_code [code]:
   1.307 -  "(finfun_update_code f a b)(\<^sup>f a' := b') = (if a = a' then f(\<^sup>f a := b') else finfun_update_code (f(\<^sup>f a' := b')) a b)"
   1.308 -by(simp add: finfun_update_twist)
   1.309 -
   1.310 -
   1.311 -subsection {* Setup for quickcheck *}
   1.312 -
   1.313 -notation fcomp (infixl "o>" 60)
   1.314 -notation scomp (infixl "o\<rightarrow>" 60)
   1.315 -
   1.316 -definition (in term_syntax) valtermify_finfun_const ::
   1.317 -  "'b\<Colon>typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> ('a\<Colon>typerep \<Rightarrow>\<^isub>f 'b) \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
   1.318 -  "valtermify_finfun_const y = Code_Evaluation.valtermify finfun_const {\<cdot>} y"
   1.319 -
   1.320 -definition (in term_syntax) valtermify_finfun_update_code ::
   1.321 -  "'a\<Colon>typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> 'b\<Colon>typerep \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> ('a \<Rightarrow>\<^isub>f 'b) \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> ('a \<Rightarrow>\<^isub>f 'b) \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
   1.322 -  "valtermify_finfun_update_code x y f = Code_Evaluation.valtermify finfun_update_code {\<cdot>} f {\<cdot>} x {\<cdot>} y"
   1.323 -
   1.324 -instantiation finfun :: (random, random) random
   1.325 -begin
   1.326 -
   1.327 -primrec random_finfun_aux :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> Random.seed \<Rightarrow> ('a \<Rightarrow>\<^isub>f 'b \<times> (unit \<Rightarrow> Code_Evaluation.term)) \<times> Random.seed" where
   1.328 -    "random_finfun_aux 0 j = Quickcheck.collapse (Random.select_weight
   1.329 -       [(1, Quickcheck.random j o\<rightarrow> (\<lambda>y. Pair (valtermify_finfun_const y)))])"
   1.330 -  | "random_finfun_aux (Suc_code_numeral i) j = Quickcheck.collapse (Random.select_weight
   1.331 -       [(Suc_code_numeral i, Quickcheck.random j o\<rightarrow> (\<lambda>x. Quickcheck.random j o\<rightarrow> (\<lambda>y. random_finfun_aux i j o\<rightarrow> (\<lambda>f. Pair (valtermify_finfun_update_code x y f))))),
   1.332 -         (1, Quickcheck.random j o\<rightarrow> (\<lambda>y. Pair (valtermify_finfun_const y)))])"
   1.333 -
   1.334 -definition 
   1.335 -  "Quickcheck.random i = random_finfun_aux i i"
   1.336 -
   1.337 -instance ..
   1.338 -
   1.339 -end
   1.340 -
   1.341 -lemma random_finfun_aux_code [code]:
   1.342 -  "random_finfun_aux i j = Quickcheck.collapse (Random.select_weight
   1.343 -     [(i, Quickcheck.random j o\<rightarrow> (\<lambda>x. Quickcheck.random j o\<rightarrow> (\<lambda>y. random_finfun_aux (i - 1) j o\<rightarrow> (\<lambda>f. Pair (valtermify_finfun_update_code x y f))))),
   1.344 -       (1, Quickcheck.random j o\<rightarrow> (\<lambda>y. Pair (valtermify_finfun_const y)))])"
   1.345 -  apply (cases i rule: code_numeral.exhaust)
   1.346 -  apply (simp_all only: random_finfun_aux.simps code_numeral_zero_minus_one Suc_code_numeral_minus_one)
   1.347 -  apply (subst select_weight_cons_zero) apply (simp only:)
   1.348 -  done
   1.349 -
   1.350 -no_notation fcomp (infixl "o>" 60)
   1.351 -no_notation scomp (infixl "o\<rightarrow>" 60)
   1.352 -
   1.353 -
   1.354 -subsection {* @{text "finfun_update"} as instance of @{text "fun_left_comm"} *}
   1.355 -
   1.356 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.357 -
   1.358 -interpretation finfun_update: fun_left_comm "\<lambda>a f. f(\<^sup>f a :: 'a := b')"
   1.359 -proof
   1.360 -  fix a' a :: 'a
   1.361 -  fix b
   1.362 -  have "(Rep_finfun b)(a := b', a' := b') = (Rep_finfun b)(a' := b', a := b')"
   1.363 -    by(cases "a = a'")(auto simp add: fun_upd_twist)
   1.364 -  thus "b(\<^sup>f a := b')(\<^sup>f a' := b') = b(\<^sup>f a' := b')(\<^sup>f a := b')"
   1.365 -    by(auto simp add: finfun_update_def fun_upd_twist)
   1.366 -qed
   1.367 -
   1.368 -lemma fold_finfun_update_finite_univ:
   1.369 -  assumes fin: "finite (UNIV :: 'a set)"
   1.370 -  shows "fold (\<lambda>a f. f(\<^sup>f a := b')) (\<lambda>\<^isup>f b) (UNIV :: 'a set) = (\<lambda>\<^isup>f b')"
   1.371 -proof -
   1.372 -  { fix A :: "'a set"
   1.373 -    from fin have "finite A" by(auto intro: finite_subset)
   1.374 -    hence "fold (\<lambda>a f. f(\<^sup>f a := b')) (\<lambda>\<^isup>f b) A = Abs_finfun (\<lambda>a. if a \<in> A then b' else b)"
   1.375 -    proof(induct)
   1.376 -      case (insert x F)
   1.377 -      have "(\<lambda>a. if a = x then b' else (if a \<in> F then b' else b)) = (\<lambda>a. if a = x \<or> a \<in> F then b' else b)"
   1.378 -        by(auto intro: ext)
   1.379 -      with insert show ?case
   1.380 -        by(simp add: finfun_const_def fun_upd_def)(simp add: finfun_update_def Abs_finfun_inverse_finite[OF fin] fun_upd_def)
   1.381 -    qed(simp add: finfun_const_def) }
   1.382 -  thus ?thesis by(simp add: finfun_const_def)
   1.383 -qed
   1.384 -
   1.385 -
   1.386 -subsection {* Default value for FinFuns *}
   1.387 -
   1.388 -definition finfun_default_aux :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b"
   1.389 -where [code del]: "finfun_default_aux f = (if finite (UNIV :: 'a set) then undefined else THE b. finite {a. f a \<noteq> b})"
   1.390 -
   1.391 -lemma finfun_default_aux_infinite:
   1.392 -  fixes f :: "'a \<Rightarrow> 'b"
   1.393 -  assumes infin: "infinite (UNIV :: 'a set)"
   1.394 -  and fin: "finite {a. f a \<noteq> b}"
   1.395 -  shows "finfun_default_aux f = b"
   1.396 -proof -
   1.397 -  let ?B = "{a. f a \<noteq> b}"
   1.398 -  from fin have "(THE b. finite {a. f a \<noteq> b}) = b"
   1.399 -  proof(rule the_equality)
   1.400 -    fix b'
   1.401 -    assume "finite {a. f a \<noteq> b'}" (is "finite ?B'")
   1.402 -    with infin fin have "UNIV - (?B' \<union> ?B) \<noteq> {}" by(auto dest: finite_subset)
   1.403 -    then obtain a where a: "a \<notin> ?B' \<union> ?B" by auto
   1.404 -    thus "b' = b" by auto
   1.405 -  qed
   1.406 -  thus ?thesis using infin by(simp add: finfun_default_aux_def)
   1.407 -qed
   1.408 -
   1.409 -
   1.410 -lemma finite_finfun_default_aux:
   1.411 -  fixes f :: "'a \<Rightarrow> 'b"
   1.412 -  assumes fin: "f \<in> finfun"
   1.413 -  shows "finite {a. f a \<noteq> finfun_default_aux f}"
   1.414 -proof(cases "finite (UNIV :: 'a set)")
   1.415 -  case True thus ?thesis using fin
   1.416 -    by(auto simp add: finfun_def finfun_default_aux_def intro: finite_subset)
   1.417 -next
   1.418 -  case False
   1.419 -  from fin obtain b where b: "finite {a. f a \<noteq> b}" (is "finite ?B")
   1.420 -    unfolding finfun_def by blast
   1.421 -  with False show ?thesis by(simp add: finfun_default_aux_infinite)
   1.422 -qed
   1.423 -
   1.424 -lemma finfun_default_aux_update_const:
   1.425 -  fixes f :: "'a \<Rightarrow> 'b"
   1.426 -  assumes fin: "f \<in> finfun"
   1.427 -  shows "finfun_default_aux (f(a := b)) = finfun_default_aux f"
   1.428 -proof(cases "finite (UNIV :: 'a set)")
   1.429 -  case False
   1.430 -  from fin obtain b' where b': "finite {a. f a \<noteq> b'}" unfolding finfun_def by blast
   1.431 -  hence "finite {a'. (f(a := b)) a' \<noteq> b'}"
   1.432 -  proof(cases "b = b' \<and> f a \<noteq> b'") 
   1.433 -    case True
   1.434 -    hence "{a. f a \<noteq> b'} = insert a {a'. (f(a := b)) a' \<noteq> b'}" by auto
   1.435 -    thus ?thesis using b' by simp
   1.436 -  next
   1.437 -    case False
   1.438 -    moreover
   1.439 -    { assume "b \<noteq> b'"
   1.440 -      hence "{a'. (f(a := b)) a' \<noteq> b'} = insert a {a. f a \<noteq> b'}" by auto
   1.441 -      hence ?thesis using b' by simp }
   1.442 -    moreover
   1.443 -    { assume "b = b'" "f a = b'"
   1.444 -      hence "{a'. (f(a := b)) a' \<noteq> b'} = {a. f a \<noteq> b'}" by auto
   1.445 -      hence ?thesis using b' by simp }
   1.446 -    ultimately show ?thesis by blast
   1.447 -  qed
   1.448 -  with False b' show ?thesis by(auto simp del: fun_upd_apply simp add: finfun_default_aux_infinite)
   1.449 -next
   1.450 -  case True thus ?thesis by(simp add: finfun_default_aux_def)
   1.451 -qed
   1.452 -
   1.453 -definition finfun_default :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'b"
   1.454 -  where [code del]: "finfun_default f = finfun_default_aux (Rep_finfun f)"
   1.455 -
   1.456 -lemma finite_finfun_default: "finite {a. Rep_finfun f a \<noteq> finfun_default f}"
   1.457 -unfolding finfun_default_def by(simp add: finite_finfun_default_aux)
   1.458 -
   1.459 -lemma finfun_default_const: "finfun_default ((\<lambda>\<^isup>f b) :: 'a \<Rightarrow>\<^isub>f 'b) = (if finite (UNIV :: 'a set) then undefined else b)"
   1.460 -apply(auto simp add: finfun_default_def finfun_const_def finfun_default_aux_infinite)
   1.461 -apply(simp add: finfun_default_aux_def)
   1.462 -done
   1.463 -
   1.464 -lemma finfun_default_update_const:
   1.465 -  "finfun_default (f(\<^sup>f a := b)) = finfun_default f"
   1.466 -unfolding finfun_default_def finfun_update_def
   1.467 -by(simp add: finfun_default_aux_update_const)
   1.468 -
   1.469 -subsection {* Recursion combinator and well-formedness conditions *}
   1.470 -
   1.471 -definition finfun_rec :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow>\<^isub>f 'b) \<Rightarrow> 'c"
   1.472 -where [code del]:
   1.473 -  "finfun_rec cnst upd f \<equiv>
   1.474 -   let b = finfun_default f;
   1.475 -       g = THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g
   1.476 -   in fold (\<lambda>a. upd a (map_default b g a)) (cnst b) (dom g)"
   1.477 -
   1.478 -locale finfun_rec_wf_aux =
   1.479 -  fixes cnst :: "'b \<Rightarrow> 'c"
   1.480 -  and upd :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"
   1.481 -  assumes upd_const_same: "upd a b (cnst b) = cnst b"
   1.482 -  and upd_commute: "a \<noteq> a' \<Longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)"
   1.483 -  and upd_idemp: "b \<noteq> b' \<Longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
   1.484 -begin
   1.485 -
   1.486 -
   1.487 -lemma upd_left_comm: "fun_left_comm (\<lambda>a. upd a (f a))"
   1.488 -by(unfold_locales)(auto intro: upd_commute)
   1.489 -
   1.490 -lemma upd_upd_twice: "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
   1.491 -by(cases "b \<noteq> b'")(auto simp add: fun_upd_def upd_const_same upd_idemp)
   1.492 -
   1.493 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.494 -
   1.495 -lemma map_default_update_const:
   1.496 -  assumes fin: "finite (dom f)"
   1.497 -  and anf: "a \<notin> dom f"
   1.498 -  and fg: "f \<subseteq>\<^sub>m g"
   1.499 -  shows "upd a d  (fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)) =
   1.500 -         fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)"
   1.501 -proof -
   1.502 -  let ?upd = "\<lambda>a. upd a (map_default d g a)"
   1.503 -  let ?fr = "\<lambda>A. fold ?upd (cnst d) A"
   1.504 -  interpret gwf: fun_left_comm "?upd" by(rule upd_left_comm)
   1.505 -  
   1.506 -  from fin anf fg show ?thesis
   1.507 -  proof(induct A\<equiv>"dom f" arbitrary: f)
   1.508 -    case empty
   1.509 -    from `{} = dom f` have "f = empty" by(auto simp add: dom_def intro: ext)
   1.510 -    thus ?case by(simp add: finfun_const_def upd_const_same)
   1.511 -  next
   1.512 -    case (insert a' A)
   1.513 -    note IH = `\<And>f.  \<lbrakk> a \<notin> dom f; f \<subseteq>\<^sub>m g; A = dom f\<rbrakk> \<Longrightarrow> upd a d (?fr (dom f)) = ?fr (dom f)`
   1.514 -    note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A`
   1.515 -    note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g`
   1.516 -    
   1.517 -    from domf obtain b where b: "f a' = Some b" by auto
   1.518 -    let ?f' = "f(a' := None)"
   1.519 -    have "upd a d (?fr (insert a' A)) = upd a d (upd a' (map_default d g a') (?fr A))"
   1.520 -      by(subst gwf.fold_insert[OF fin a'nA]) rule
   1.521 -    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
   1.522 -    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
   1.523 -    also from anf domf have "a \<noteq> a'" by auto note upd_commute[OF this]
   1.524 -    also from domf a'nA anf fg have "a \<notin> dom ?f'" "?f' \<subseteq>\<^sub>m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def)
   1.525 -    note A also note IH[OF `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g` A]
   1.526 -    also have "upd a' (map_default d f a') (?fr (dom (f(a' := None)))) = ?fr (dom f)"
   1.527 -      unfolding domf[symmetric] gwf.fold_insert[OF fin a'nA] ga' unfolding A ..
   1.528 -    also have "insert a' (dom ?f') = dom f" using domf by auto
   1.529 -    finally show ?case .
   1.530 -  qed
   1.531 -qed
   1.532 -
   1.533 -lemma map_default_update_twice:
   1.534 -  assumes fin: "finite (dom f)"
   1.535 -  and anf: "a \<notin> dom f"
   1.536 -  and fg: "f \<subseteq>\<^sub>m g"
   1.537 -  shows "upd a d'' (upd a d' (fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))) =
   1.538 -         upd a d'' (fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))"
   1.539 -proof -
   1.540 -  let ?upd = "\<lambda>a. upd a (map_default d g a)"
   1.541 -  let ?fr = "\<lambda>A. fold ?upd (cnst d) A"
   1.542 -  interpret gwf: fun_left_comm "?upd" by(rule upd_left_comm)
   1.543 -  
   1.544 -  from fin anf fg show ?thesis
   1.545 -  proof(induct A\<equiv>"dom f" arbitrary: f)
   1.546 -    case empty
   1.547 -    from `{} = dom f` have "f = empty" by(auto simp add: dom_def intro: ext)
   1.548 -    thus ?case by(auto simp add: finfun_const_def finfun_update_def upd_upd_twice)
   1.549 -  next
   1.550 -    case (insert a' A)
   1.551 -    note IH = `\<And>f. \<lbrakk>a \<notin> dom f; f \<subseteq>\<^sub>m g; A = dom f\<rbrakk> \<Longrightarrow> upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (?fr (dom f))`
   1.552 -    note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A`
   1.553 -    note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g`
   1.554 -    
   1.555 -    from domf obtain b where b: "f a' = Some b" by auto
   1.556 -    let ?f' = "f(a' := None)"
   1.557 -    let ?b' = "case f a' of None \<Rightarrow> d | Some b \<Rightarrow> b"
   1.558 -    from domf have "upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (upd a d' (?fr (insert a' A)))" by simp
   1.559 -    also note gwf.fold_insert[OF fin a'nA]
   1.560 -    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
   1.561 -    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
   1.562 -    also from anf domf have ana': "a \<noteq> a'" by auto note upd_commute[OF this]
   1.563 -    also note upd_commute[OF ana']
   1.564 -    also from domf a'nA anf fg have "a \<notin> dom ?f'" "?f' \<subseteq>\<^sub>m g" and A: "A = dom ?f'" by(auto simp add: ran_def map_le_def)
   1.565 -    note A also note IH[OF `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g` A]
   1.566 -    also note upd_commute[OF ana'[symmetric]] also note ga'[symmetric] also note A[symmetric]
   1.567 -    also note gwf.fold_insert[symmetric, OF fin a'nA] also note domf
   1.568 -    finally show ?case .
   1.569 -  qed
   1.570 -qed
   1.571 -
   1.572 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.573 -
   1.574 -lemma map_default_eq_id [simp]: "map_default d ((\<lambda>a. Some (f a)) |` {a. f a \<noteq> d}) = f"
   1.575 -by(auto simp add: map_default_def restrict_map_def intro: ext)
   1.576 -
   1.577 -lemma finite_rec_cong1:
   1.578 -  assumes f: "fun_left_comm f" and g: "fun_left_comm g"
   1.579 -  and fin: "finite A"
   1.580 -  and eq: "\<And>a. a \<in> A \<Longrightarrow> f a = g a"
   1.581 -  shows "fold f z A = fold g z A"
   1.582 -proof -
   1.583 -  interpret f: fun_left_comm f by(rule f)
   1.584 -  interpret g: fun_left_comm g by(rule g)
   1.585 -  { fix B
   1.586 -    assume BsubA: "B \<subseteq> A"
   1.587 -    with fin have "finite B" by(blast intro: finite_subset)
   1.588 -    hence "B \<subseteq> A \<Longrightarrow> fold f z B = fold g z B"
   1.589 -    proof(induct)
   1.590 -      case empty thus ?case by simp
   1.591 -    next
   1.592 -      case (insert a B)
   1.593 -      note finB = `finite B` note anB = `a \<notin> B` note sub = `insert a B \<subseteq> A`
   1.594 -      note IH = `B \<subseteq> A \<Longrightarrow> fold f z B = fold g z B`
   1.595 -      from sub anB have BpsubA: "B \<subset> A" and BsubA: "B \<subseteq> A" and aA: "a \<in> A" by auto
   1.596 -      from IH[OF BsubA] eq[OF aA] finB anB
   1.597 -      show ?case by(auto)
   1.598 -    qed
   1.599 -    with BsubA have "fold f z B = fold g z B" by blast }
   1.600 -  thus ?thesis by blast
   1.601 -qed
   1.602 -
   1.603 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.604 -
   1.605 -lemma finfun_rec_upd [simp]:
   1.606 -  "finfun_rec cnst upd (f(\<^sup>f a' := b')) = upd a' b' (finfun_rec cnst upd f)"
   1.607 -proof -
   1.608 -  obtain b where b: "b = finfun_default f" by auto
   1.609 -  let ?the = "\<lambda>f g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g"
   1.610 -  obtain g where g: "g = The (?the f)" by blast
   1.611 -  obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by (cases f)
   1.612 -  from f y b have bfin: "finite {a. y a \<noteq> b}" by(simp add: finfun_default_def finite_finfun_default_aux)
   1.613 -
   1.614 -  let ?g = "(\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}"
   1.615 -  from bfin have fing: "finite (dom ?g)" by auto
   1.616 -  have bran: "b \<notin> ran ?g" by(auto simp add: ran_def restrict_map_def)
   1.617 -  have yg: "y = map_default b ?g" by simp
   1.618 -  have gg: "g = ?g" unfolding g
   1.619 -  proof(rule the_equality)
   1.620 -    from f y bfin show "?the f ?g"
   1.621 -      by(auto)(simp add: restrict_map_def ran_def split: split_if_asm)
   1.622 -  next
   1.623 -    fix g'
   1.624 -    assume "?the f g'"
   1.625 -    hence fin': "finite (dom g')" and ran': "b \<notin> ran g'"
   1.626 -      and eq: "Abs_finfun (map_default b ?g) = Abs_finfun (map_default b g')" using f yg by auto
   1.627 -    from fin' fing have "map_default b ?g \<in> finfun" "map_default b g' \<in> finfun" by(blast intro: map_default_in_finfun)+
   1.628 -    with eq have "map_default b ?g = map_default b g'" by simp
   1.629 -    with fing bran fin' ran' show "g' = ?g" by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
   1.630 -  qed
   1.631 -
   1.632 -  show ?thesis
   1.633 -  proof(cases "b' = b")
   1.634 -    case True
   1.635 -    note b'b = True
   1.636 -
   1.637 -    let ?g' = "(\<lambda>a. Some ((y(a' := b)) a)) |` {a. (y(a' := b)) a \<noteq> b}"
   1.638 -    from bfin b'b have fing': "finite (dom ?g')"
   1.639 -      by(auto simp add: Collect_conj_eq Collect_imp_eq intro: finite_subset)
   1.640 -    have brang': "b \<notin> ran ?g'" by(auto simp add: ran_def restrict_map_def)
   1.641 -
   1.642 -    let ?b' = "\<lambda>a. case ?g' a of None \<Rightarrow> b | Some b \<Rightarrow> b"
   1.643 -    let ?b = "map_default b ?g"
   1.644 -    from upd_left_comm upd_left_comm fing'
   1.645 -    have "fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g') = fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')"
   1.646 -      by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b b map_default_def)
   1.647 -    also interpret gwf: fun_left_comm "\<lambda>a. upd a (?b a)" by(rule upd_left_comm)
   1.648 -    have "fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g') = upd a' b' (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))"
   1.649 -    proof(cases "y a' = b")
   1.650 -      case True
   1.651 -      with b'b have g': "?g' = ?g" by(auto simp add: restrict_map_def intro: ext)
   1.652 -      from True have a'ndomg: "a' \<notin> dom ?g" by auto
   1.653 -      from f b'b b show ?thesis unfolding g'
   1.654 -        by(subst map_default_update_const[OF fing a'ndomg map_le_refl, symmetric]) simp
   1.655 -    next
   1.656 -      case False
   1.657 -      hence domg: "dom ?g = insert a' (dom ?g')" by auto
   1.658 -      from False b'b have a'ndomg': "a' \<notin> dom ?g'" by auto
   1.659 -      have "fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g')) = 
   1.660 -            upd a' (?b a') (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'))"
   1.661 -        using fing' a'ndomg' unfolding b'b by(rule gwf.fold_insert)
   1.662 -      hence "upd a' b (fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g'))) =
   1.663 -             upd a' b (upd a' (?b a') (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')))" by simp
   1.664 -      also from b'b have g'leg: "?g' \<subseteq>\<^sub>m ?g" by(auto simp add: restrict_map_def map_le_def)
   1.665 -      note map_default_update_twice[OF fing' a'ndomg' this, of b "?b a'" b]
   1.666 -      also note map_default_update_const[OF fing' a'ndomg' g'leg, of b]
   1.667 -      finally show ?thesis unfolding b'b domg[unfolded b'b] by(rule sym)
   1.668 -    qed
   1.669 -    also have "The (?the (f(\<^sup>f a' := b'))) = ?g'"
   1.670 -    proof(rule the_equality)
   1.671 -      from f y b b'b brang' fing' show "?the (f(\<^sup>f a' := b')) ?g'"
   1.672 -        by(auto simp del: fun_upd_apply simp add: finfun_update_def)
   1.673 -    next
   1.674 -      fix g'
   1.675 -      assume "?the (f(\<^sup>f a' := b')) g'"
   1.676 -      hence fin': "finite (dom g')" and ran': "b \<notin> ran g'"
   1.677 -        and eq: "f(\<^sup>f a' := b') = Abs_finfun (map_default b g')" 
   1.678 -        by(auto simp del: fun_upd_apply)
   1.679 -      from fin' fing' have "map_default b g' \<in> finfun" "map_default b ?g' \<in> finfun"
   1.680 -        by(blast intro: map_default_in_finfun)+
   1.681 -      with eq f b'b b have "map_default b ?g' = map_default b g'"
   1.682 -        by(simp del: fun_upd_apply add: finfun_update_def)
   1.683 -      with fing' brang' fin' ran' show "g' = ?g'"
   1.684 -        by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
   1.685 -    qed
   1.686 -    ultimately show ?thesis unfolding finfun_rec_def Let_def b gg[unfolded g b] using bfin b'b b
   1.687 -      by(simp only: finfun_default_update_const map_default_def)
   1.688 -  next
   1.689 -    case False
   1.690 -    note b'b = this
   1.691 -    let ?g' = "?g(a' \<mapsto> b')"
   1.692 -    let ?b' = "map_default b ?g'"
   1.693 -    let ?b = "map_default b ?g"
   1.694 -    from fing have fing': "finite (dom ?g')" by auto
   1.695 -    from bran b'b have bnrang': "b \<notin> ran ?g'" by(auto simp add: ran_def)
   1.696 -    have ffmg': "map_default b ?g' = y(a' := b')" by(auto intro: ext simp add: map_default_def restrict_map_def)
   1.697 -    with f y have f_Abs: "f(\<^sup>f a' := b') = Abs_finfun (map_default b ?g')" by(auto simp add: finfun_update_def)
   1.698 -    have g': "The (?the (f(\<^sup>f a' := b'))) = ?g'"
   1.699 -    proof
   1.700 -      from fing' bnrang' f_Abs show "?the (f(\<^sup>f a' := b')) ?g'" by(auto simp add: finfun_update_def restrict_map_def)
   1.701 -    next
   1.702 -      fix g' assume "?the (f(\<^sup>f a' := b')) g'"
   1.703 -      hence f': "f(\<^sup>f a' := b') = Abs_finfun (map_default b g')"
   1.704 -        and fin': "finite (dom g')" and brang': "b \<notin> ran g'" by auto
   1.705 -      from fing' fin' have "map_default b ?g' \<in> finfun" "map_default b g' \<in> finfun"
   1.706 -        by(auto intro: map_default_in_finfun)
   1.707 -      with f' f_Abs have "map_default b g' = map_default b ?g'" by simp
   1.708 -      with fin' brang' fing' bnrang' show "g' = ?g'"
   1.709 -        by(rule map_default_inject[OF disjI2[OF refl]])
   1.710 -    qed
   1.711 -    have dom: "dom (((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b})(a' \<mapsto> b')) = insert a' (dom ((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}))"
   1.712 -      by auto
   1.713 -    show ?thesis
   1.714 -    proof(cases "y a' = b")
   1.715 -      case True
   1.716 -      hence a'ndomg: "a' \<notin> dom ?g" by auto
   1.717 -      from f y b'b True have yff: "y = map_default b (?g' |` dom ?g)"
   1.718 -        by(auto simp add: restrict_map_def map_default_def intro!: ext)
   1.719 -      hence f': "f = Abs_finfun (map_default b (?g' |` dom ?g))" using f by simp
   1.720 -      interpret g'wf: fun_left_comm "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm)
   1.721 -      from upd_left_comm upd_left_comm fing
   1.722 -      have "fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g) = fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)"
   1.723 -        by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b True map_default_def)
   1.724 -      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric]
   1.725 -        unfolding g' g[symmetric] gg g'wf.fold_insert[OF fing a'ndomg, of "cnst b", folded dom]
   1.726 -        by -(rule arg_cong2[where f="upd a'"], simp_all add: map_default_def)
   1.727 -    next
   1.728 -      case False
   1.729 -      hence "insert a' (dom ?g) = dom ?g" by auto
   1.730 -      moreover {
   1.731 -        let ?g'' = "?g(a' := None)"
   1.732 -        let ?b'' = "map_default b ?g''"
   1.733 -        from False have domg: "dom ?g = insert a' (dom ?g'')" by auto
   1.734 -        from False have a'ndomg'': "a' \<notin> dom ?g''" by auto
   1.735 -        have fing'': "finite (dom ?g'')" by(rule finite_subset[OF _ fing]) auto
   1.736 -        have bnrang'': "b \<notin> ran ?g''" by(auto simp add: ran_def restrict_map_def)
   1.737 -        interpret gwf: fun_left_comm "\<lambda>a. upd a (?b a)" by(rule upd_left_comm)
   1.738 -        interpret g'wf: fun_left_comm "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm)
   1.739 -        have "upd a' b' (fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g''))) =
   1.740 -              upd a' b' (upd a' (?b a') (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')))"
   1.741 -          unfolding gwf.fold_insert[OF fing'' a'ndomg''] f ..
   1.742 -        also have g''leg: "?g |` dom ?g'' \<subseteq>\<^sub>m ?g" by(auto simp add: map_le_def)
   1.743 -        have "dom (?g |` dom ?g'') = dom ?g''" by auto
   1.744 -        note map_default_update_twice[where d=b and f = "?g |` dom ?g''" and a=a' and d'="?b a'" and d''=b' and g="?g",
   1.745 -                                     unfolded this, OF fing'' a'ndomg'' g''leg]
   1.746 -        also have b': "b' = ?b' a'" by(auto simp add: map_default_def)
   1.747 -        from upd_left_comm upd_left_comm fing''
   1.748 -        have "fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'') = fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g'')"
   1.749 -          by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b map_default_def)
   1.750 -        with b' have "upd a' b' (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')) =
   1.751 -                     upd a' (?b' a') (fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g''))" by simp
   1.752 -        also note g'wf.fold_insert[OF fing'' a'ndomg'', symmetric]
   1.753 -        finally have "upd a' b' (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g)) =
   1.754 -                   fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)"
   1.755 -          unfolding domg . }
   1.756 -      ultimately have "fold (\<lambda>a. upd a (?b' a)) (cnst b) (insert a' (dom ?g)) =
   1.757 -                    upd a' b' (fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))" by simp
   1.758 -      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] g[symmetric] g' dom[symmetric]
   1.759 -        using b'b gg by(simp add: map_default_insert)
   1.760 -    qed
   1.761 -  qed
   1.762 -qed
   1.763 -
   1.764 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.765 -
   1.766 -end
   1.767 -
   1.768 -locale finfun_rec_wf = finfun_rec_wf_aux + 
   1.769 -  assumes const_update_all:
   1.770 -  "finite (UNIV :: 'a set) \<Longrightarrow> fold (\<lambda>a. upd a b') (cnst b) (UNIV :: 'a set) = cnst b'"
   1.771 -begin
   1.772 -
   1.773 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.774 -
   1.775 -lemma finfun_rec_const [simp]:
   1.776 -  "finfun_rec cnst upd (\<lambda>\<^isup>f c) = cnst c"
   1.777 -proof(cases "finite (UNIV :: 'a set)")
   1.778 -  case False
   1.779 -  hence "finfun_default ((\<lambda>\<^isup>f c) :: 'a \<Rightarrow>\<^isub>f 'b) = c" by(simp add: finfun_default_const)
   1.780 -  moreover have "(THE g :: 'a \<rightharpoonup> 'b. (\<lambda>\<^isup>f c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g) = empty"
   1.781 -  proof
   1.782 -    show "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c empty) \<and> finite (dom empty) \<and> c \<notin> ran empty"
   1.783 -      by(auto simp add: finfun_const_def)
   1.784 -  next
   1.785 -    fix g :: "'a \<rightharpoonup> 'b"
   1.786 -    assume "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g"
   1.787 -    hence g: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default c g)" and fin: "finite (dom g)" and ran: "c \<notin> ran g" by blast+
   1.788 -    from g map_default_in_finfun[OF fin, of c] have "map_default c g = (\<lambda>a. c)"
   1.789 -      by(simp add: finfun_const_def)
   1.790 -    moreover have "map_default c empty = (\<lambda>a. c)" by simp
   1.791 -    ultimately show "g = empty" by-(rule map_default_inject[OF disjI2[OF refl] fin ran], auto)
   1.792 -  qed
   1.793 -  ultimately show ?thesis by(simp add: finfun_rec_def)
   1.794 -next
   1.795 -  case True
   1.796 -  hence default: "finfun_default ((\<lambda>\<^isup>f c) :: 'a \<Rightarrow>\<^isub>f 'b) = undefined" by(simp add: finfun_default_const)
   1.797 -  let ?the = "\<lambda>g :: 'a \<rightharpoonup> 'b. (\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g) \<and> finite (dom g) \<and> undefined \<notin> ran g"
   1.798 -  show ?thesis
   1.799 -  proof(cases "c = undefined")
   1.800 -    case True
   1.801 -    have the: "The ?the = empty"
   1.802 -    proof
   1.803 -      from True show "?the empty" by(auto simp add: finfun_const_def)
   1.804 -    next
   1.805 -      fix g'
   1.806 -      assume "?the g'"
   1.807 -      hence fg: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g')"
   1.808 -        and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all
   1.809 -      from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun)
   1.810 -      with fg have "map_default undefined g' = (\<lambda>a. c)"
   1.811 -        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1])
   1.812 -      with True show "g' = empty"
   1.813 -        by -(rule map_default_inject(2)[OF _ fin g], auto)
   1.814 -    qed
   1.815 -    show ?thesis unfolding finfun_rec_def using `finite UNIV` True
   1.816 -      unfolding Let_def the default by(simp)
   1.817 -  next
   1.818 -    case False
   1.819 -    have the: "The ?the = (\<lambda>a :: 'a. Some c)"
   1.820 -    proof
   1.821 -      from False True show "?the (\<lambda>a :: 'a. Some c)"
   1.822 -        by(auto simp add: map_default_def_raw finfun_const_def dom_def ran_def)
   1.823 -    next
   1.824 -      fix g' :: "'a \<rightharpoonup> 'b"
   1.825 -      assume "?the g'"
   1.826 -      hence fg: "(\<lambda>\<^isup>f c) = Abs_finfun (map_default undefined g')"
   1.827 -        and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all
   1.828 -      from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun)
   1.829 -      with fg have "map_default undefined g' = (\<lambda>a. c)"
   1.830 -        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1])
   1.831 -      with True False show "g' = (\<lambda>a::'a. Some c)"
   1.832 -        by -(rule map_default_inject(2)[OF _ fin g], auto simp add: dom_def ran_def map_default_def_raw)
   1.833 -    qed
   1.834 -    show ?thesis unfolding finfun_rec_def using True False
   1.835 -      unfolding Let_def the default by(simp add: dom_def map_default_def const_update_all)
   1.836 -  qed
   1.837 -qed
   1.838 -
   1.839 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.840 -
   1.841 -end
   1.842 -
   1.843 -subsection {* Weak induction rule and case analysis for FinFuns *}
   1.844 -
   1.845 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.846 -
   1.847 -lemma finfun_weak_induct [consumes 0, case_names const update]:
   1.848 -  assumes const: "\<And>b. P (\<lambda>\<^isup>f b)"
   1.849 -  and update: "\<And>f a b. P f \<Longrightarrow> P (f(\<^sup>f a := b))"
   1.850 -  shows "P x"
   1.851 -proof(induct x rule: Abs_finfun_induct)
   1.852 -  case (Abs_finfun y)
   1.853 -  then obtain b where "finite {a. y a \<noteq> b}" unfolding finfun_def by blast
   1.854 -  thus ?case using `y \<in> finfun`
   1.855 -  proof(induct x\<equiv>"{a. y a \<noteq> b}" arbitrary: y rule: finite_induct)
   1.856 -    case empty
   1.857 -    hence "\<And>a. y a = b" by blast
   1.858 -    hence "y = (\<lambda>a. b)" by(auto intro: ext)
   1.859 -    hence "Abs_finfun y = finfun_const b" unfolding finfun_const_def by simp
   1.860 -    thus ?case by(simp add: const)
   1.861 -  next
   1.862 -    case (insert a A)
   1.863 -    note IH = `\<And>y. \<lbrakk> y \<in> finfun; A = {a. y a \<noteq> b} \<rbrakk> \<Longrightarrow> P (Abs_finfun y)`
   1.864 -    note y = `y \<in> finfun`
   1.865 -    with `insert a A = {a. y a \<noteq> b}` `a \<notin> A`
   1.866 -    have "y(a := b) \<in> finfun" "A = {a'. (y(a := b)) a' \<noteq> b}" by auto
   1.867 -    from IH[OF this] have "P (finfun_update (Abs_finfun (y(a := b))) a (y a))" by(rule update)
   1.868 -    thus ?case using y unfolding finfun_update_def by simp
   1.869 -  qed
   1.870 -qed
   1.871 -
   1.872 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.873 -
   1.874 -lemma finfun_exhaust_disj: "(\<exists>b. x = finfun_const b) \<or> (\<exists>f a b. x = finfun_update f a b)"
   1.875 -by(induct x rule: finfun_weak_induct) blast+
   1.876 -
   1.877 -lemma finfun_exhaust:
   1.878 -  obtains b where "x = (\<lambda>\<^isup>f b)"
   1.879 -        | f a b where "x = f(\<^sup>f a := b)"
   1.880 -by(atomize_elim)(rule finfun_exhaust_disj)
   1.881 -
   1.882 -lemma finfun_rec_unique:
   1.883 -  fixes f :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'c"
   1.884 -  assumes c: "\<And>c. f (\<lambda>\<^isup>f c) = cnst c"
   1.885 -  and u: "\<And>g a b. f (g(\<^sup>f a := b)) = upd g a b (f g)"
   1.886 -  and c': "\<And>c. f' (\<lambda>\<^isup>f c) = cnst c"
   1.887 -  and u': "\<And>g a b. f' (g(\<^sup>f a := b)) = upd g a b (f' g)"
   1.888 -  shows "f = f'"
   1.889 -proof
   1.890 -  fix g :: "'a \<Rightarrow>\<^isub>f 'b"
   1.891 -  show "f g = f' g"
   1.892 -    by(induct g rule: finfun_weak_induct)(auto simp add: c u c' u')
   1.893 -qed
   1.894 -
   1.895 -
   1.896 -subsection {* Function application *}
   1.897 -
   1.898 -definition finfun_apply :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow> 'b" ("_\<^sub>f" [1000] 1000)
   1.899 -where [code del]: "finfun_apply = (\<lambda>f a. finfun_rec (\<lambda>b. b) (\<lambda>a' b c. if (a = a') then b else c) f)"
   1.900 -
   1.901 -interpretation finfun_apply_aux: finfun_rec_wf_aux "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c"
   1.902 -by(unfold_locales) auto
   1.903 -
   1.904 -interpretation finfun_apply: finfun_rec_wf "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c"
   1.905 -proof(unfold_locales)
   1.906 -  fix b' b :: 'a
   1.907 -  assume fin: "finite (UNIV :: 'b set)"
   1.908 -  { fix A :: "'b set"
   1.909 -    interpret fun_left_comm "\<lambda>a'. If (a = a') b'" by(rule finfun_apply_aux.upd_left_comm)
   1.910 -    from fin have "finite A" by(auto intro: finite_subset)
   1.911 -    hence "fold (\<lambda>a'. If (a = a') b') b A = (if a \<in> A then b' else b)"
   1.912 -      by induct auto }
   1.913 -  from this[of UNIV] show "fold (\<lambda>a'. If (a = a') b') b UNIV = b'" by simp
   1.914 -qed
   1.915 -
   1.916 -lemma finfun_const_apply [simp, code]: "(\<lambda>\<^isup>f b)\<^sub>f a = b"
   1.917 -by(simp add: finfun_apply_def)
   1.918 -
   1.919 -lemma finfun_upd_apply: "f(\<^sup>fa := b)\<^sub>f a' = (if a = a' then b else f\<^sub>f a')"
   1.920 -  and finfun_upd_apply_code [code]: "(finfun_update_code f a b)\<^sub>f a' = (if a = a' then b else f\<^sub>f a')"
   1.921 -by(simp_all add: finfun_apply_def)
   1.922 -
   1.923 -lemma finfun_upd_apply_same [simp]:
   1.924 -  "f(\<^sup>fa := b)\<^sub>f a = b"
   1.925 -by(simp add: finfun_upd_apply)
   1.926 -
   1.927 -lemma finfun_upd_apply_other [simp]:
   1.928 -  "a \<noteq> a' \<Longrightarrow> f(\<^sup>fa := b)\<^sub>f a' = f\<^sub>f a'"
   1.929 -by(simp add: finfun_upd_apply)
   1.930 -
   1.931 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
   1.932 -
   1.933 -lemma finfun_apply_Rep_finfun:
   1.934 -  "finfun_apply = Rep_finfun"
   1.935 -proof(rule finfun_rec_unique)
   1.936 -  fix c show "Rep_finfun (\<lambda>\<^isup>f c) = (\<lambda>a. c)" by(auto simp add: finfun_const_def)
   1.937 -next
   1.938 -  fix g a b show "Rep_finfun g(\<^sup>f a := b) = (\<lambda>c. if c = a then b else Rep_finfun g c)"
   1.939 -    by(auto simp add: finfun_update_def fun_upd_finfun Abs_finfun_inverse Rep_finfun intro: ext)
   1.940 -qed(auto intro: ext)
   1.941 -
   1.942 -lemma finfun_ext: "(\<And>a. f\<^sub>f a = g\<^sub>f a) \<Longrightarrow> f = g"
   1.943 -by(auto simp add: finfun_apply_Rep_finfun Rep_finfun_inject[symmetric] simp del: Rep_finfun_inject intro: ext)
   1.944 -
   1.945 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
   1.946 -
   1.947 -lemma expand_finfun_eq: "(f = g) = (f\<^sub>f = g\<^sub>f)"
   1.948 -by(auto intro: finfun_ext)
   1.949 -
   1.950 -lemma finfun_const_inject [simp]: "(\<lambda>\<^isup>f b) = (\<lambda>\<^isup>f b') \<equiv> b = b'"
   1.951 -by(simp add: expand_finfun_eq expand_fun_eq)
   1.952 -
   1.953 -lemma finfun_const_eq_update:
   1.954 -  "((\<lambda>\<^isup>f b) = f(\<^sup>f a := b')) = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> f\<^sub>f a' = b))"
   1.955 -by(auto simp add: expand_finfun_eq expand_fun_eq finfun_upd_apply)
   1.956 -
   1.957 -subsection {* Function composition *}
   1.958 -
   1.959 -definition finfun_comp :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow>\<^isub>f 'a \<Rightarrow> 'c \<Rightarrow>\<^isub>f 'b" (infixr "\<circ>\<^isub>f" 55)
   1.960 -where [code del]: "g \<circ>\<^isub>f f  = finfun_rec (\<lambda>b. (\<lambda>\<^isup>f g b)) (\<lambda>a b c. c(\<^sup>f a := g b)) f"
   1.961 -
   1.962 -interpretation finfun_comp_aux: finfun_rec_wf_aux "(\<lambda>b. (\<lambda>\<^isup>f g b))" "(\<lambda>a b c. c(\<^sup>f a := g b))"
   1.963 -by(unfold_locales)(auto simp add: finfun_upd_apply intro: finfun_ext)
   1.964 -
   1.965 -interpretation finfun_comp: finfun_rec_wf "(\<lambda>b. (\<lambda>\<^isup>f g b))" "(\<lambda>a b c. c(\<^sup>f a := g b))"
   1.966 -proof
   1.967 -  fix b' b :: 'a
   1.968 -  assume fin: "finite (UNIV :: 'c set)"
   1.969 -  { fix A :: "'c set"
   1.970 -    from fin have "finite A" by(auto intro: finite_subset)
   1.971 -    hence "fold (\<lambda>(a :: 'c) c. c(\<^sup>f a := g b')) (\<lambda>\<^isup>f g b) A =
   1.972 -      Abs_finfun (\<lambda>a. if a \<in> A then g b' else g b)"
   1.973 -      by induct (simp_all add: finfun_const_def, auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite expand_fun_eq fin) }
   1.974 -  from this[of UNIV] show "fold (\<lambda>(a :: 'c) c. c(\<^sup>f a := g b')) (\<lambda>\<^isup>f g b) UNIV = (\<lambda>\<^isup>f g b')"
   1.975 -    by(simp add: finfun_const_def)
   1.976 -qed
   1.977 -
   1.978 -lemma finfun_comp_const [simp, code]:
   1.979 -  "g \<circ>\<^isub>f (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f g c)"
   1.980 -by(simp add: finfun_comp_def)
   1.981 -
   1.982 -lemma finfun_comp_update [simp]: "g \<circ>\<^isub>f (f(\<^sup>f a := b)) = (g \<circ>\<^isub>f f)(\<^sup>f a := g b)"
   1.983 -  and finfun_comp_update_code [code]: "g \<circ>\<^isub>f (finfun_update_code f a b) = finfun_update_code (g \<circ>\<^isub>f f) a (g b)"
   1.984 -by(simp_all add: finfun_comp_def)
   1.985 -
   1.986 -lemma finfun_comp_apply [simp]:
   1.987 -  "(g \<circ>\<^isub>f f)\<^sub>f = g \<circ> f\<^sub>f"
   1.988 -by(induct f rule: finfun_weak_induct)(auto simp add: finfun_upd_apply intro: ext)
   1.989 -
   1.990 -lemma finfun_comp_comp_collapse [simp]: "f \<circ>\<^isub>f g \<circ>\<^isub>f h = (f o g) \<circ>\<^isub>f h"
   1.991 -by(induct h rule: finfun_weak_induct) simp_all
   1.992 -
   1.993 -lemma finfun_comp_const1 [simp]: "(\<lambda>x. c) \<circ>\<^isub>f f = (\<lambda>\<^isup>f c)"
   1.994 -by(induct f rule: finfun_weak_induct)(auto intro: finfun_ext simp add: finfun_upd_apply)
   1.995 -
   1.996 -lemma finfun_comp_id1 [simp]: "(\<lambda>x. x) \<circ>\<^isub>f f = f" "id \<circ>\<^isub>f f = f"
   1.997 -by(induct f rule: finfun_weak_induct) auto
   1.998 -
   1.999 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
  1.1000 -
  1.1001 -lemma finfun_comp_conv_comp: "g \<circ>\<^isub>f f = Abs_finfun (g \<circ> finfun_apply f)"
  1.1002 -proof -
  1.1003 -  have "(\<lambda>f. g \<circ>\<^isub>f f) = (\<lambda>f. Abs_finfun (g \<circ> finfun_apply f))"
  1.1004 -  proof(rule finfun_rec_unique)
  1.1005 -    { fix c show "Abs_finfun (g \<circ> (\<lambda>\<^isup>f c)\<^sub>f) = (\<lambda>\<^isup>f g c)"
  1.1006 -        by(simp add: finfun_comp_def o_def)(simp add: finfun_const_def) }
  1.1007 -    { fix g' a b show "Abs_finfun (g \<circ> g'(\<^sup>f a := b)\<^sub>f) = (Abs_finfun (g \<circ> g'\<^sub>f))(\<^sup>f a := g b)"
  1.1008 -      proof -
  1.1009 -        obtain y where y: "y \<in> finfun" and g': "g' = Abs_finfun y" by(cases g')
  1.1010 -        moreover hence "(g \<circ> g'\<^sub>f) \<in> finfun" by(simp add: finfun_apply_Rep_finfun finfun_left_compose)
  1.1011 -        moreover have "g \<circ> y(a := b) = (g \<circ> y)(a := g b)" by(auto intro: ext)
  1.1012 -        ultimately show ?thesis by(simp add: finfun_comp_def finfun_update_def finfun_apply_Rep_finfun)
  1.1013 -      qed }
  1.1014 -  qed auto
  1.1015 -  thus ?thesis by(auto simp add: expand_fun_eq)
  1.1016 -qed
  1.1017 -
  1.1018 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
  1.1019 -
  1.1020 -
  1.1021 -
  1.1022 -definition finfun_comp2 :: "'b \<Rightarrow>\<^isub>f 'c \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c" (infixr "\<^sub>f\<circ>" 55)
  1.1023 -where [code del]: "finfun_comp2 g f = Abs_finfun (Rep_finfun g \<circ> f)"
  1.1024 -
  1.1025 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
  1.1026 -
  1.1027 -lemma finfun_comp2_const [code, simp]: "finfun_comp2 (\<lambda>\<^isup>f c) f = (\<lambda>\<^isup>f c)"
  1.1028 -by(simp add: finfun_comp2_def finfun_const_def comp_def)
  1.1029 -
  1.1030 -lemma finfun_comp2_update:
  1.1031 -  assumes inj: "inj f"
  1.1032 -  shows "finfun_comp2 (g(\<^sup>f b := c)) f = (if b \<in> range f then (finfun_comp2 g f)(\<^sup>f inv f b := c) else finfun_comp2 g f)"
  1.1033 -proof(cases "b \<in> range f")
  1.1034 -  case True
  1.1035 -  from inj have "\<And>x. (Rep_finfun g)(f x := c) \<circ> f = (Rep_finfun g \<circ> f)(x := c)" by(auto intro!: ext dest: injD)
  1.1036 -  with inj True show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def finfun_right_compose)
  1.1037 -next
  1.1038 -  case False
  1.1039 -  hence "(Rep_finfun g)(b := c) \<circ> f = Rep_finfun g \<circ> f" by(auto simp add: expand_fun_eq)
  1.1040 -  with False show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def)
  1.1041 -qed
  1.1042 -
  1.1043 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
  1.1044 -
  1.1045 -subsection {* A type class for computing the cardinality of a type's universe *}
  1.1046 -
  1.1047 -class card_UNIV = 
  1.1048 -  fixes card_UNIV :: "'a itself \<Rightarrow> nat"
  1.1049 -  assumes card_UNIV: "card_UNIV x = card (UNIV :: 'a set)"
  1.1050 -begin
  1.1051 -
  1.1052 -lemma card_UNIV_neq_0_finite_UNIV:
  1.1053 -  "card_UNIV x \<noteq> 0 \<longleftrightarrow> finite (UNIV :: 'a set)"
  1.1054 -by(simp add: card_UNIV card_eq_0_iff)
  1.1055 -
  1.1056 -lemma card_UNIV_ge_0_finite_UNIV:
  1.1057 -  "card_UNIV x > 0 \<longleftrightarrow> finite (UNIV :: 'a set)"
  1.1058 -by(auto simp add: card_UNIV intro: card_ge_0_finite finite_UNIV_card_ge_0)
  1.1059 -
  1.1060 -lemma card_UNIV_eq_0_infinite_UNIV:
  1.1061 -  "card_UNIV x = 0 \<longleftrightarrow> infinite (UNIV :: 'a set)"
  1.1062 -by(simp add: card_UNIV card_eq_0_iff)
  1.1063 -
  1.1064 -definition is_list_UNIV :: "'a list \<Rightarrow> bool"
  1.1065 -where "is_list_UNIV xs = (let c = card_UNIV (TYPE('a)) in if c = 0 then False else size (remdups xs) = c)"
  1.1066 -
  1.1067 -lemma is_list_UNIV_iff:
  1.1068 -  fixes xs :: "'a list"
  1.1069 -  shows "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
  1.1070 -proof
  1.1071 -  assume "is_list_UNIV xs"
  1.1072 -  hence c: "card_UNIV (TYPE('a)) > 0" and xs: "size (remdups xs) = card_UNIV (TYPE('a))"
  1.1073 -    unfolding is_list_UNIV_def by(simp_all add: Let_def split: split_if_asm)
  1.1074 -  from c have fin: "finite (UNIV :: 'a set)" by(auto simp add: card_UNIV_ge_0_finite_UNIV)
  1.1075 -  have "card (set (remdups xs)) = size (remdups xs)" by(subst distinct_card) auto
  1.1076 -  also note set_remdups
  1.1077 -  finally show "set xs = UNIV" using fin unfolding xs card_UNIV by-(rule card_eq_UNIV_imp_eq_UNIV)
  1.1078 -next
  1.1079 -  assume xs: "set xs = UNIV"
  1.1080 -  from finite_set[of xs] have fin: "finite (UNIV :: 'a set)" unfolding xs .
  1.1081 -  hence "card_UNIV (TYPE ('a)) \<noteq> 0" unfolding card_UNIV_neq_0_finite_UNIV .
  1.1082 -  moreover have "size (remdups xs) = card (set (remdups xs))"
  1.1083 -    by(subst distinct_card) auto
  1.1084 -  ultimately show "is_list_UNIV xs" using xs by(simp add: is_list_UNIV_def Let_def card_UNIV)
  1.1085 -qed
  1.1086 -
  1.1087 -lemma card_UNIV_eq_0_is_list_UNIV_False:
  1.1088 -  assumes cU0: "card_UNIV x = 0"
  1.1089 -  shows "is_list_UNIV = (\<lambda>xs. False)"
  1.1090 -proof(rule ext)
  1.1091 -  fix xs :: "'a list"
  1.1092 -  from cU0 have "infinite (UNIV :: 'a set)"
  1.1093 -    by(auto simp only: card_UNIV_eq_0_infinite_UNIV)
  1.1094 -  moreover have "finite (set xs)" by(rule finite_set)
  1.1095 -  ultimately have "(UNIV :: 'a set) \<noteq> set xs" by(auto simp del: finite_set)
  1.1096 -  thus "is_list_UNIV xs = False" unfolding is_list_UNIV_iff by simp
  1.1097 -qed
  1.1098 -
  1.1099 -end
  1.1100 -
  1.1101 -subsection {* Instantiations for @{text "card_UNIV"} *}
  1.1102 -
  1.1103 -subsubsection {* @{typ "nat"} *}
  1.1104 -
  1.1105 -instantiation nat :: card_UNIV begin
  1.1106 -
  1.1107 -definition card_UNIV_nat_def:
  1.1108 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: nat itself. 0)"
  1.1109 -
  1.1110 -instance proof
  1.1111 -  fix x :: "nat itself"
  1.1112 -  show "card_UNIV x = card (UNIV :: nat set)"
  1.1113 -    unfolding card_UNIV_nat_def by simp
  1.1114 -qed
  1.1115 -
  1.1116 -end
  1.1117 -
  1.1118 -subsubsection {* @{typ "int"} *}
  1.1119 -
  1.1120 -instantiation int :: card_UNIV begin
  1.1121 -
  1.1122 -definition card_UNIV_int_def:
  1.1123 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: int itself. 0)"
  1.1124 -
  1.1125 -instance proof
  1.1126 -  fix x :: "int itself"
  1.1127 -  show "card_UNIV x = card (UNIV :: int set)"
  1.1128 -    unfolding card_UNIV_int_def by simp
  1.1129 -qed
  1.1130 -
  1.1131 -end
  1.1132 -
  1.1133 -subsubsection {* @{typ "'a list"} *}
  1.1134 -
  1.1135 -instantiation list :: (type) card_UNIV begin
  1.1136 -
  1.1137 -definition card_UNIV_list_def:
  1.1138 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a list itself. 0)"
  1.1139 -
  1.1140 -instance proof
  1.1141 -  fix x :: "'a list itself"
  1.1142 -  show "card_UNIV x = card (UNIV :: 'a list set)"
  1.1143 -    unfolding card_UNIV_list_def by(simp add: infinite_UNIV_listI)
  1.1144 -qed
  1.1145 -
  1.1146 -end
  1.1147 -
  1.1148 -subsubsection {* @{typ "unit"} *}
  1.1149 -
  1.1150 -lemma card_UNIV_unit: "card (UNIV :: unit set) = 1"
  1.1151 -  unfolding UNIV_unit by simp
  1.1152 -
  1.1153 -instantiation unit :: card_UNIV begin
  1.1154 -
  1.1155 -definition card_UNIV_unit_def: 
  1.1156 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: unit itself. 1)"
  1.1157 -
  1.1158 -instance proof
  1.1159 -  fix x :: "unit itself"
  1.1160 -  show "card_UNIV x = card (UNIV :: unit set)"
  1.1161 -    by(simp add: card_UNIV_unit_def card_UNIV_unit)
  1.1162 -qed
  1.1163 -
  1.1164 -end
  1.1165 -
  1.1166 -subsubsection {* @{typ "bool"} *}
  1.1167 -
  1.1168 -lemma card_UNIV_bool: "card (UNIV :: bool set) = 2"
  1.1169 -  unfolding UNIV_bool by simp
  1.1170 -
  1.1171 -instantiation bool :: card_UNIV begin
  1.1172 -
  1.1173 -definition card_UNIV_bool_def: 
  1.1174 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: bool itself. 2)"
  1.1175 -
  1.1176 -instance proof
  1.1177 -  fix x :: "bool itself"
  1.1178 -  show "card_UNIV x = card (UNIV :: bool set)"
  1.1179 -    by(simp add: card_UNIV_bool_def card_UNIV_bool)
  1.1180 -qed
  1.1181 -
  1.1182 -end
  1.1183 -
  1.1184 -subsubsection {* @{typ "char"} *}
  1.1185 -
  1.1186 -lemma card_UNIV_char: "card (UNIV :: char set) = 256"
  1.1187 -proof -
  1.1188 -  from enum_distinct
  1.1189 -  have "card (set (enum :: char list)) = length (enum :: char list)"
  1.1190 -    by - (rule distinct_card)
  1.1191 -  also have "set enum = (UNIV :: char set)" by auto
  1.1192 -  also note enum_chars
  1.1193 -  finally show ?thesis by (simp add: chars_def)
  1.1194 -qed
  1.1195 -
  1.1196 -instantiation char :: card_UNIV begin
  1.1197 -
  1.1198 -definition card_UNIV_char_def: 
  1.1199 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: char itself. 256)"
  1.1200 -
  1.1201 -instance proof
  1.1202 -  fix x :: "char itself"
  1.1203 -  show "card_UNIV x = card (UNIV :: char set)"
  1.1204 -    by(simp add: card_UNIV_char_def card_UNIV_char)
  1.1205 -qed
  1.1206 -
  1.1207 -end
  1.1208 -
  1.1209 -subsubsection {* @{typ "'a \<times> 'b"} *}
  1.1210 -
  1.1211 -instantiation * :: (card_UNIV, card_UNIV) card_UNIV begin
  1.1212 -
  1.1213 -definition card_UNIV_product_def: 
  1.1214 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<times> 'b) itself. card_UNIV (TYPE('a)) * card_UNIV (TYPE('b)))"
  1.1215 -
  1.1216 -instance proof
  1.1217 -  fix x :: "('a \<times> 'b) itself"
  1.1218 -  show "card_UNIV x = card (UNIV :: ('a \<times> 'b) set)"
  1.1219 -    by(simp add: card_UNIV_product_def card_UNIV UNIV_Times_UNIV[symmetric] card_cartesian_product del: UNIV_Times_UNIV)
  1.1220 -qed
  1.1221 -
  1.1222 -end
  1.1223 -
  1.1224 -subsubsection {* @{typ "'a + 'b"} *}
  1.1225 -
  1.1226 -instantiation "+" :: (card_UNIV, card_UNIV) card_UNIV begin
  1.1227 -
  1.1228 -definition card_UNIV_sum_def: 
  1.1229 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a + 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b))
  1.1230 -                           in if ca \<noteq> 0 \<and> cb \<noteq> 0 then ca + cb else 0)"
  1.1231 -
  1.1232 -instance proof
  1.1233 -  fix x :: "('a + 'b) itself"
  1.1234 -  show "card_UNIV x = card (UNIV :: ('a + 'b) set)"
  1.1235 -    by (auto simp add: card_UNIV_sum_def card_UNIV card_eq_0_iff UNIV_Plus_UNIV[symmetric] finite_Plus_iff Let_def card_Plus simp del: UNIV_Plus_UNIV dest!: card_ge_0_finite)
  1.1236 -qed
  1.1237 -
  1.1238 -end
  1.1239 -
  1.1240 -subsubsection {* @{typ "'a \<Rightarrow> 'b"} *}
  1.1241 -
  1.1242 -instantiation "fun" :: (card_UNIV, card_UNIV) card_UNIV begin
  1.1243 -
  1.1244 -definition card_UNIV_fun_def: 
  1.1245 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: ('a \<Rightarrow> 'b) itself. let ca = card_UNIV (TYPE('a)); cb = card_UNIV (TYPE('b))
  1.1246 -                           in if ca \<noteq> 0 \<and> cb \<noteq> 0 \<or> cb = 1 then cb ^ ca else 0)"
  1.1247 -
  1.1248 -instance proof
  1.1249 -  fix x :: "('a \<Rightarrow> 'b) itself"
  1.1250 -
  1.1251 -  { assume "0 < card (UNIV :: 'a set)"
  1.1252 -    and "0 < card (UNIV :: 'b set)"
  1.1253 -    hence fina: "finite (UNIV :: 'a set)" and finb: "finite (UNIV :: 'b set)"
  1.1254 -      by(simp_all only: card_ge_0_finite)
  1.1255 -    from finite_distinct_list[OF finb] obtain bs 
  1.1256 -      where bs: "set bs = (UNIV :: 'b set)" and distb: "distinct bs" by blast
  1.1257 -    from finite_distinct_list[OF fina] obtain as
  1.1258 -      where as: "set as = (UNIV :: 'a set)" and dista: "distinct as" by blast
  1.1259 -    have cb: "card (UNIV :: 'b set) = length bs"
  1.1260 -      unfolding bs[symmetric] distinct_card[OF distb] ..
  1.1261 -    have ca: "card (UNIV :: 'a set) = length as"
  1.1262 -      unfolding as[symmetric] distinct_card[OF dista] ..
  1.1263 -    let ?xs = "map (\<lambda>ys. the o map_of (zip as ys)) (n_lists (length as) bs)"
  1.1264 -    have "UNIV = set ?xs"
  1.1265 -    proof(rule UNIV_eq_I)
  1.1266 -      fix f :: "'a \<Rightarrow> 'b"
  1.1267 -      from as have "f = the \<circ> map_of (zip as (map f as))"
  1.1268 -        by(auto simp add: map_of_zip_map intro: ext)
  1.1269 -      thus "f \<in> set ?xs" using bs by(auto simp add: set_n_lists)
  1.1270 -    qed
  1.1271 -    moreover have "distinct ?xs" unfolding distinct_map
  1.1272 -    proof(intro conjI distinct_n_lists distb inj_onI)
  1.1273 -      fix xs ys :: "'b list"
  1.1274 -      assume xs: "xs \<in> set (n_lists (length as) bs)"
  1.1275 -        and ys: "ys \<in> set (n_lists (length as) bs)"
  1.1276 -        and eq: "the \<circ> map_of (zip as xs) = the \<circ> map_of (zip as ys)"
  1.1277 -      from xs ys have [simp]: "length xs = length as" "length ys = length as"
  1.1278 -        by(simp_all add: length_n_lists_elem)
  1.1279 -      have "map_of (zip as xs) = map_of (zip as ys)"
  1.1280 -      proof
  1.1281 -        fix x
  1.1282 -        from as bs have "\<exists>y. map_of (zip as xs) x = Some y" "\<exists>y. map_of (zip as ys) x = Some y"
  1.1283 -          by(simp_all add: map_of_zip_is_Some[symmetric])
  1.1284 -        with eq show "map_of (zip as xs) x = map_of (zip as ys) x"
  1.1285 -          by(auto dest: fun_cong[where x=x])
  1.1286 -      qed
  1.1287 -      with dista show "xs = ys" by(simp add: map_of_zip_inject)
  1.1288 -    qed
  1.1289 -    hence "card (set ?xs) = length ?xs" by(simp only: distinct_card)
  1.1290 -    moreover have "length ?xs = length bs ^ length as" by(simp add: length_n_lists)
  1.1291 -    ultimately have "card (UNIV :: ('a \<Rightarrow> 'b) set) = card (UNIV :: 'b set) ^ card (UNIV :: 'a set)"
  1.1292 -      using cb ca by simp }
  1.1293 -  moreover {
  1.1294 -    assume cb: "card (UNIV :: 'b set) = Suc 0"
  1.1295 -    then obtain b where b: "UNIV = {b :: 'b}" by(auto simp add: card_Suc_eq)
  1.1296 -    have eq: "UNIV = {\<lambda>x :: 'a. b ::'b}"
  1.1297 -    proof(rule UNIV_eq_I)
  1.1298 -      fix x :: "'a \<Rightarrow> 'b"
  1.1299 -      { fix y
  1.1300 -        have "x y \<in> UNIV" ..
  1.1301 -        hence "x y = b" unfolding b by simp }
  1.1302 -      thus "x \<in> {\<lambda>x. b}" by(auto intro: ext)
  1.1303 -    qed
  1.1304 -    have "card (UNIV :: ('a \<Rightarrow> 'b) set) = Suc 0" unfolding eq by simp }
  1.1305 -  ultimately show "card_UNIV x = card (UNIV :: ('a \<Rightarrow> 'b) set)"
  1.1306 -    unfolding card_UNIV_fun_def card_UNIV Let_def
  1.1307 -    by(auto simp del: One_nat_def)(auto simp add: card_eq_0_iff dest: finite_fun_UNIVD2 finite_fun_UNIVD1)
  1.1308 -qed
  1.1309 -
  1.1310 -end
  1.1311 -
  1.1312 -subsubsection {* @{typ "'a option"} *}
  1.1313 -
  1.1314 -instantiation option :: (card_UNIV) card_UNIV
  1.1315 -begin
  1.1316 -
  1.1317 -definition card_UNIV_option_def: 
  1.1318 -  "card_UNIV_class.card_UNIV = (\<lambda>a :: 'a option itself. let c = card_UNIV (TYPE('a))
  1.1319 -                           in if c \<noteq> 0 then Suc c else 0)"
  1.1320 -
  1.1321 -instance proof
  1.1322 -  fix x :: "'a option itself"
  1.1323 -  show "card_UNIV x = card (UNIV :: 'a option set)"
  1.1324 -    unfolding UNIV_option_conv
  1.1325 -    by(auto simp add: card_UNIV_option_def card_UNIV card_eq_0_iff Let_def intro: inj_Some dest: finite_imageD)
  1.1326 -      (subst card_insert_disjoint, auto simp add: card_eq_0_iff card_image inj_Some intro: finite_imageI card_ge_0_finite)
  1.1327 -qed
  1.1328 -
  1.1329 -end
  1.1330 -
  1.1331 -
  1.1332 -subsection {* Universal quantification *}
  1.1333 -
  1.1334 -definition finfun_All_except :: "'a list \<Rightarrow> 'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool"
  1.1335 -where [code del]: "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> P\<^sub>f a"
  1.1336 -
  1.1337 -lemma finfun_All_except_const: "finfun_All_except A (\<lambda>\<^isup>f b) \<longleftrightarrow> b \<or> set A = UNIV"
  1.1338 -by(auto simp add: finfun_All_except_def)
  1.1339 -
  1.1340 -lemma finfun_All_except_const_finfun_UNIV_code [code]:
  1.1341 -  "finfun_All_except A (\<lambda>\<^isup>f b) = (b \<or> is_list_UNIV A)"
  1.1342 -by(simp add: finfun_All_except_const is_list_UNIV_iff)
  1.1343 -
  1.1344 -lemma finfun_All_except_update: 
  1.1345 -  "finfun_All_except A f(\<^sup>f a := b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)"
  1.1346 -by(fastsimp simp add: finfun_All_except_def finfun_upd_apply)
  1.1347 -
  1.1348 -lemma finfun_All_except_update_code [code]:
  1.1349 -  fixes a :: "'a :: card_UNIV"
  1.1350 -  shows "finfun_All_except A (finfun_update_code f a b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)"
  1.1351 -by(simp add: finfun_All_except_update)
  1.1352 -
  1.1353 -definition finfun_All :: "'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool"
  1.1354 -where "finfun_All = finfun_All_except []"
  1.1355 -
  1.1356 -lemma finfun_All_const [simp]: "finfun_All (\<lambda>\<^isup>f b) = b"
  1.1357 -by(simp add: finfun_All_def finfun_All_except_def)
  1.1358 -
  1.1359 -lemma finfun_All_update: "finfun_All f(\<^sup>f a := b) = (b \<and> finfun_All_except [a] f)"
  1.1360 -by(simp add: finfun_All_def finfun_All_except_update)
  1.1361 -
  1.1362 -lemma finfun_All_All: "finfun_All P = All P\<^sub>f"
  1.1363 -by(simp add: finfun_All_def finfun_All_except_def)
  1.1364 -
  1.1365 -
  1.1366 -definition finfun_Ex :: "'a \<Rightarrow>\<^isub>f bool \<Rightarrow> bool"
  1.1367 -where "finfun_Ex P = Not (finfun_All (Not \<circ>\<^isub>f P))"
  1.1368 -
  1.1369 -lemma finfun_Ex_Ex: "finfun_Ex P = Ex P\<^sub>f"
  1.1370 -unfolding finfun_Ex_def finfun_All_All by simp
  1.1371 -
  1.1372 -lemma finfun_Ex_const [simp]: "finfun_Ex (\<lambda>\<^isup>f b) = b"
  1.1373 -by(simp add: finfun_Ex_def)
  1.1374 -
  1.1375 -
  1.1376 -subsection {* A diagonal operator for FinFuns *}
  1.1377 -
  1.1378 -definition finfun_Diag :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c \<Rightarrow> 'a \<Rightarrow>\<^isub>f ('b \<times> 'c)" ("(1'(_,/ _')\<^sup>f)" [0, 0] 1000)
  1.1379 -where [code del]: "finfun_Diag f g = finfun_rec (\<lambda>b. Pair b \<circ>\<^isub>f g) (\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))) f"
  1.1380 -
  1.1381 -interpretation finfun_Diag_aux: finfun_rec_wf_aux "\<lambda>b. Pair b \<circ>\<^isub>f g" "\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))"
  1.1382 -by(unfold_locales)(simp_all add: expand_finfun_eq expand_fun_eq finfun_upd_apply)
  1.1383 -
  1.1384 -interpretation finfun_Diag: finfun_rec_wf "\<lambda>b. Pair b \<circ>\<^isub>f g" "\<lambda>a b c. c(\<^sup>f a := (b, g\<^sub>f a))"
  1.1385 -proof
  1.1386 -  fix b' b :: 'a
  1.1387 -  assume fin: "finite (UNIV :: 'c set)"
  1.1388 -  { fix A :: "'c set"
  1.1389 -    interpret fun_left_comm "\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))" by(rule finfun_Diag_aux.upd_left_comm)
  1.1390 -    from fin have "finite A" by(auto intro: finite_subset)
  1.1391 -    hence "fold (\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))) (Pair b \<circ>\<^isub>f g) A =
  1.1392 -      Abs_finfun (\<lambda>a. (if a \<in> A then b' else b, g\<^sub>f a))"
  1.1393 -      by(induct)(simp_all add: finfun_const_def finfun_comp_conv_comp o_def,
  1.1394 -                 auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite expand_fun_eq fin) }
  1.1395 -  from this[of UNIV] show "fold (\<lambda>a c. c(\<^sup>f a := (b', g\<^sub>f a))) (Pair b \<circ>\<^isub>f g) UNIV = Pair b' \<circ>\<^isub>f g"
  1.1396 -    by(simp add: finfun_const_def finfun_comp_conv_comp o_def)
  1.1397 -qed
  1.1398 -
  1.1399 -lemma finfun_Diag_const1: "(\<lambda>\<^isup>f b, g)\<^sup>f = Pair b \<circ>\<^isub>f g"
  1.1400 -by(simp add: finfun_Diag_def)
  1.1401 -
  1.1402 -text {*
  1.1403 -  Do not use @{thm finfun_Diag_const1} for the code generator because @{term "Pair b"} is injective, i.e. if @{term g} is free of redundant updates, there is no need to check for redundant updates as is done for @{text "\<circ>\<^isub>f"}.
  1.1404 -*}
  1.1405 -
  1.1406 -lemma finfun_Diag_const_code [code]:
  1.1407 -  "(\<lambda>\<^isup>f b, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>\<^isup>f (b, c))"
  1.1408 -  "(\<lambda>\<^isup>f b, g(\<^sup>f\<^sup>c a := c))\<^sup>f = (\<lambda>\<^isup>f b, g)\<^sup>f(\<^sup>f\<^sup>c a := (b, c))"
  1.1409 -by(simp_all add: finfun_Diag_const1)
  1.1410 -
  1.1411 -lemma finfun_Diag_update1: "(f(\<^sup>f a := b), g)\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))"
  1.1412 -  and finfun_Diag_update1_code [code]: "(finfun_update_code f a b, g)\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))"
  1.1413 -by(simp_all add: finfun_Diag_def)
  1.1414 -
  1.1415 -lemma finfun_Diag_const2: "(f, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>b. (b, c)) \<circ>\<^isub>f f"
  1.1416 -by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)
  1.1417 -
  1.1418 -lemma finfun_Diag_update2: "(f, g(\<^sup>f a := c))\<^sup>f = (f, g)\<^sup>f(\<^sup>f a := (f\<^sub>f a, c))"
  1.1419 -by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)
  1.1420 -
  1.1421 -lemma finfun_Diag_const_const [simp]: "(\<lambda>\<^isup>f b, \<lambda>\<^isup>f c)\<^sup>f = (\<lambda>\<^isup>f (b, c))"
  1.1422 -by(simp add: finfun_Diag_const1)
  1.1423 -
  1.1424 -lemma finfun_Diag_const_update:
  1.1425 -  "(\<lambda>\<^isup>f b, g(\<^sup>f a := c))\<^sup>f = (\<lambda>\<^isup>f b, g)\<^sup>f(\<^sup>f a := (b, c))"
  1.1426 -by(simp add: finfun_Diag_const1)
  1.1427 -
  1.1428 -lemma finfun_Diag_update_const:
  1.1429 -  "(f(\<^sup>f a := b), \<lambda>\<^isup>f c)\<^sup>f = (f, \<lambda>\<^isup>f c)\<^sup>f(\<^sup>f a := (b, c))"
  1.1430 -by(simp add: finfun_Diag_def)
  1.1431 -
  1.1432 -lemma finfun_Diag_update_update:
  1.1433 -  "(f(\<^sup>f a := b), g(\<^sup>f a' := c))\<^sup>f = (if a = a' then (f, g)\<^sup>f(\<^sup>f a := (b, c)) else (f, g)\<^sup>f(\<^sup>f a := (b, g\<^sub>f a))(\<^sup>f a' := (f\<^sub>f a', c)))"
  1.1434 -by(auto simp add: finfun_Diag_update1 finfun_Diag_update2)
  1.1435 -
  1.1436 -lemma finfun_Diag_apply [simp]: "(f, g)\<^sup>f\<^sub>f = (\<lambda>x. (f\<^sub>f x, g\<^sub>f x))"
  1.1437 -by(induct f rule: finfun_weak_induct)(auto simp add: finfun_Diag_const1 finfun_Diag_update1 finfun_upd_apply intro: ext)
  1.1438 -
  1.1439 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
  1.1440 -
  1.1441 -lemma finfun_Diag_conv_Abs_finfun:
  1.1442 -  "(f, g)\<^sup>f = Abs_finfun ((\<lambda>x. (Rep_finfun f x, Rep_finfun g x)))"
  1.1443 -proof -
  1.1444 -  have "(\<lambda>f :: 'a \<Rightarrow>\<^isub>f 'b. (f, g)\<^sup>f) = (\<lambda>f. Abs_finfun ((\<lambda>x. (Rep_finfun f x, Rep_finfun g x))))"
  1.1445 -  proof(rule finfun_rec_unique)
  1.1446 -    { fix c show "Abs_finfun (\<lambda>x. (Rep_finfun (\<lambda>\<^isup>f c) x, Rep_finfun g x)) = Pair c \<circ>\<^isub>f g"
  1.1447 -        by(simp add: finfun_comp_conv_comp finfun_apply_Rep_finfun o_def finfun_const_def) }
  1.1448 -    { fix g' a b
  1.1449 -      show "Abs_finfun (\<lambda>x. (Rep_finfun g'(\<^sup>f a := b) x, Rep_finfun g x)) =
  1.1450 -            (Abs_finfun (\<lambda>x. (Rep_finfun g' x, Rep_finfun g x)))(\<^sup>f a := (b, g\<^sub>f a))"
  1.1451 -        by(auto simp add: finfun_update_def expand_fun_eq finfun_apply_Rep_finfun simp del: fun_upd_apply) simp }
  1.1452 -  qed(simp_all add: finfun_Diag_const1 finfun_Diag_update1)
  1.1453 -  thus ?thesis by(auto simp add: expand_fun_eq)
  1.1454 -qed
  1.1455 -
  1.1456 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
  1.1457 -
  1.1458 -lemma finfun_Diag_eq: "(f, g)\<^sup>f = (f', g')\<^sup>f \<longleftrightarrow> f = f' \<and> g = g'"
  1.1459 -by(auto simp add: expand_finfun_eq expand_fun_eq)
  1.1460 -
  1.1461 -definition finfun_fst :: "'a \<Rightarrow>\<^isub>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b"
  1.1462 -where [code]: "finfun_fst f = fst \<circ>\<^isub>f f"
  1.1463 -
  1.1464 -lemma finfun_fst_const: "finfun_fst (\<lambda>\<^isup>f bc) = (\<lambda>\<^isup>f fst bc)"
  1.1465 -by(simp add: finfun_fst_def)
  1.1466 -
  1.1467 -lemma finfun_fst_update: "finfun_fst (f(\<^sup>f a := bc)) = (finfun_fst f)(\<^sup>f a := fst bc)"
  1.1468 -  and finfun_fst_update_code: "finfun_fst (finfun_update_code f a bc) = (finfun_fst f)(\<^sup>f a := fst bc)"
  1.1469 -by(simp_all add: finfun_fst_def)
  1.1470 -
  1.1471 -lemma finfun_fst_comp_conv: "finfun_fst (f \<circ>\<^isub>f g) = (fst \<circ> f) \<circ>\<^isub>f g"
  1.1472 -by(simp add: finfun_fst_def)
  1.1473 -
  1.1474 -lemma finfun_fst_conv [simp]: "finfun_fst (f, g)\<^sup>f = f"
  1.1475 -by(induct f rule: finfun_weak_induct)(simp_all add: finfun_Diag_const1 finfun_fst_comp_conv o_def finfun_Diag_update1 finfun_fst_update)
  1.1476 -
  1.1477 -lemma finfun_fst_conv_Abs_finfun: "finfun_fst = (\<lambda>f. Abs_finfun (fst o Rep_finfun f))"
  1.1478 -by(simp add: finfun_fst_def_raw finfun_comp_conv_comp finfun_apply_Rep_finfun)
  1.1479 -
  1.1480 -
  1.1481 -definition finfun_snd :: "'a \<Rightarrow>\<^isub>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'c"
  1.1482 -where [code]: "finfun_snd f = snd \<circ>\<^isub>f f"
  1.1483 -
  1.1484 -lemma finfun_snd_const: "finfun_snd (\<lambda>\<^isup>f bc) = (\<lambda>\<^isup>f snd bc)"
  1.1485 -by(simp add: finfun_snd_def)
  1.1486 -
  1.1487 -lemma finfun_snd_update: "finfun_snd (f(\<^sup>f a := bc)) = (finfun_snd f)(\<^sup>f a := snd bc)"
  1.1488 -  and finfun_snd_update_code [code]: "finfun_snd (finfun_update_code f a bc) = (finfun_snd f)(\<^sup>f a := snd bc)"
  1.1489 -by(simp_all add: finfun_snd_def)
  1.1490 -
  1.1491 -lemma finfun_snd_comp_conv: "finfun_snd (f \<circ>\<^isub>f g) = (snd \<circ> f) \<circ>\<^isub>f g"
  1.1492 -by(simp add: finfun_snd_def)
  1.1493 -
  1.1494 -lemma finfun_snd_conv [simp]: "finfun_snd (f, g)\<^sup>f = g"
  1.1495 -apply(induct f rule: finfun_weak_induct)
  1.1496 -apply(auto simp add: finfun_Diag_const1 finfun_snd_comp_conv o_def finfun_Diag_update1 finfun_snd_update finfun_upd_apply intro: finfun_ext)
  1.1497 -done
  1.1498 -
  1.1499 -lemma finfun_snd_conv_Abs_finfun: "finfun_snd = (\<lambda>f. Abs_finfun (snd o Rep_finfun f))"
  1.1500 -by(simp add: finfun_snd_def_raw finfun_comp_conv_comp finfun_apply_Rep_finfun)
  1.1501 -
  1.1502 -lemma finfun_Diag_collapse [simp]: "(finfun_fst f, finfun_snd f)\<^sup>f = f"
  1.1503 -by(induct f rule: finfun_weak_induct)(simp_all add: finfun_fst_const finfun_snd_const finfun_fst_update finfun_snd_update finfun_Diag_update_update)
  1.1504 -
  1.1505 -subsection {* Currying for FinFuns *}
  1.1506 -
  1.1507 -definition finfun_curry :: "('a \<times> 'b) \<Rightarrow>\<^isub>f 'c \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b \<Rightarrow>\<^isub>f 'c"
  1.1508 -where [code del]: "finfun_curry = finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c)))"
  1.1509 -
  1.1510 -interpretation finfun_curry_aux: finfun_rec_wf_aux "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))"
  1.1511 -apply(unfold_locales)
  1.1512 -apply(auto simp add: split_def finfun_update_twist finfun_upd_apply split_paired_all finfun_update_const_same)
  1.1513 -done
  1.1514 -
  1.1515 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
  1.1516 -
  1.1517 -interpretation finfun_curry: finfun_rec_wf "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))"
  1.1518 -proof(unfold_locales)
  1.1519 -  fix b' b :: 'b
  1.1520 -  assume fin: "finite (UNIV :: ('c \<times> 'a) set)"
  1.1521 -  hence fin1: "finite (UNIV :: 'c set)" and fin2: "finite (UNIV :: 'a set)"
  1.1522 -    unfolding UNIV_Times_UNIV[symmetric]
  1.1523 -    by(fastsimp dest: finite_cartesian_productD1 finite_cartesian_productD2)+
  1.1524 -  note [simp] = Abs_finfun_inverse_finite[OF fin] Abs_finfun_inverse_finite[OF fin1] Abs_finfun_inverse_finite[OF fin2]
  1.1525 -  { fix A :: "('c \<times> 'a) set"
  1.1526 -    interpret fun_left_comm "\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b'"
  1.1527 -      by(rule finfun_curry_aux.upd_left_comm)
  1.1528 -    from fin have "finite A" by(auto intro: finite_subset)
  1.1529 -    hence "fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b') ((finfun_const \<circ> finfun_const) b) A = Abs_finfun (\<lambda>a. Abs_finfun (\<lambda>b''. if (a, b'') \<in> A then b' else b))"
  1.1530 -      by induct (simp_all, auto simp add: finfun_update_def finfun_const_def split_def finfun_apply_Rep_finfun intro!: arg_cong[where f="Abs_finfun"] ext) }
  1.1531 -  from this[of UNIV]
  1.1532 -  show "fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(\<^sup>f a := (f\<^sub>f a)(\<^sup>f b := c))) a b') ((finfun_const \<circ> finfun_const) b) UNIV = (finfun_const \<circ> finfun_const) b'"
  1.1533 -    by(simp add: finfun_const_def)
  1.1534 -qed
  1.1535 -
  1.1536 -declare finfun_simp [simp del] finfun_iff [iff del] finfun_intro [rule del]
  1.1537 -
  1.1538 -lemma finfun_curry_const [simp, code]: "finfun_curry (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)"
  1.1539 -by(simp add: finfun_curry_def)
  1.1540 -
  1.1541 -lemma finfun_curry_update [simp]:
  1.1542 -  "finfun_curry (f(\<^sup>f (a, b) := c)) = (finfun_curry f)(\<^sup>f a := ((finfun_curry f)\<^sub>f a)(\<^sup>f b := c))"
  1.1543 -  and finfun_curry_update_code [code]:
  1.1544 -  "finfun_curry (f(\<^sup>f\<^sup>c (a, b) := c)) = (finfun_curry f)(\<^sup>f a := ((finfun_curry f)\<^sub>f a)(\<^sup>f b := c))"
  1.1545 -by(simp_all add: finfun_curry_def)
  1.1546 -
  1.1547 -declare finfun_simp [simp] finfun_iff [iff] finfun_intro [intro]
  1.1548 -
  1.1549 -lemma finfun_Abs_finfun_curry: assumes fin: "f \<in> finfun"
  1.1550 -  shows "(\<lambda>a. Abs_finfun (curry f a)) \<in> finfun"
  1.1551 -proof -
  1.1552 -  from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast
  1.1553 -  have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force)
  1.1554 -  hence "{a. curry f a \<noteq> (\<lambda>x. c)} = fst ` {ab. f ab \<noteq> c}"
  1.1555 -    by(auto simp add: curry_def expand_fun_eq)
  1.1556 -  with fin c have "finite {a.  Abs_finfun (curry f a) \<noteq> (\<lambda>\<^isup>f c)}"
  1.1557 -    by(simp add: finfun_const_def finfun_curry)
  1.1558 -  thus ?thesis unfolding finfun_def by auto
  1.1559 -qed
  1.1560 -
  1.1561 -lemma finfun_curry_conv_curry:
  1.1562 -  fixes f :: "('a \<times> 'b) \<Rightarrow>\<^isub>f 'c"
  1.1563 -  shows "finfun_curry f = Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun f) a))"
  1.1564 -proof -
  1.1565 -  have "finfun_curry = (\<lambda>f :: ('a \<times> 'b) \<Rightarrow>\<^isub>f 'c. Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun f) a)))"
  1.1566 -  proof(rule finfun_rec_unique)
  1.1567 -    { fix c show "finfun_curry (\<lambda>\<^isup>f c) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)" by simp }
  1.1568 -    { fix f a c show "finfun_curry (f(\<^sup>f a := c)) = (finfun_curry f)(\<^sup>f fst a := ((finfun_curry f)\<^sub>f (fst a))(\<^sup>f snd a := c))"
  1.1569 -        by(cases a) simp }
  1.1570 -    { fix c show "Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun (\<lambda>\<^isup>f c)) a)) = (\<lambda>\<^isup>f \<lambda>\<^isup>f c)"
  1.1571 -        by(simp add: finfun_curry_def finfun_const_def curry_def) }
  1.1572 -    { fix g a b
  1.1573 -      show "Abs_finfun (\<lambda>aa. Abs_finfun (curry (Rep_finfun g(\<^sup>f a := b)) aa)) =
  1.1574 -       (Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun g) a)))(\<^sup>f
  1.1575 -       fst a := ((Abs_finfun (\<lambda>a. Abs_finfun (curry (Rep_finfun g) a)))\<^sub>f (fst a))(\<^sup>f snd a := b))"
  1.1576 -        by(cases a)(auto intro!: ext arg_cong[where f=Abs_finfun] simp add: finfun_curry_def finfun_update_def finfun_apply_Rep_finfun finfun_curry finfun_Abs_finfun_curry) }
  1.1577 -  qed
  1.1578 -  thus ?thesis by(auto simp add: expand_fun_eq)
  1.1579 -qed
  1.1580 -
  1.1581 -subsection {* Executable equality for FinFuns *}
  1.1582 -
  1.1583 -lemma eq_finfun_All_ext: "(f = g) \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>\<^isub>f (f, g)\<^sup>f)"
  1.1584 -by(simp add: expand_finfun_eq expand_fun_eq finfun_All_All o_def)
  1.1585 -
  1.1586 -instantiation finfun :: ("{card_UNIV,eq}",eq) eq begin
  1.1587 -definition eq_finfun_def: "eq_class.eq f g \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>\<^isub>f (f, g)\<^sup>f)"
  1.1588 -instance by(intro_classes)(simp add: eq_finfun_All_ext eq_finfun_def)
  1.1589 -end
  1.1590 -
  1.1591 -subsection {* Operator that explicitly removes all redundant updates in the generated representations *}
  1.1592 -
  1.1593 -definition finfun_clearjunk :: "'a \<Rightarrow>\<^isub>f 'b \<Rightarrow> 'a \<Rightarrow>\<^isub>f 'b"
  1.1594 -where [simp, code del]: "finfun_clearjunk = id"
  1.1595 -
  1.1596 -lemma finfun_clearjunk_const [code]: "finfun_clearjunk (\<lambda>\<^isup>f b) = (\<lambda>\<^isup>f b)"
  1.1597 -by simp
  1.1598 -
  1.1599 -lemma finfun_clearjunk_update [code]: "finfun_clearjunk (finfun_update_code f a b) = f(\<^sup>f a := b)"
  1.1600 -by simp
  1.1601 -
  1.1602 -end
  1.1603 \ No newline at end of file
     2.1 --- a/src/HOL/Library/Library.thy	Mon Oct 26 09:03:57 2009 +0100
     2.2 +++ b/src/HOL/Library/Library.thy	Mon Oct 26 11:19:24 2009 +0100
     2.3 @@ -20,7 +20,6 @@
     2.4    Enum
     2.5    Eval_Witness
     2.6    Executable_Set
     2.7 -  Fin_Fun
     2.8    Float
     2.9    Formal_Power_Series
    2.10    Fraction_Field