src/HOL/Library/FinFun.thy
author Andreas Lochbihler
Wed May 30 16:05:06 2012 +0200 (2012-05-30)
changeset 48041 d60f6b41bf2d
parent 48038 72a8506dd59b
child 48051 53a0df441e20
permissions -rw-r--r--
remove pretty syntax for FinFuns at the end and provide separate syntax theory
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(* Author: Andreas Lochbihler, Uni Karlsruhe *)
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header {* Almost everywhere constant functions *}
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theory FinFun
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imports Card_Univ
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begin
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text {*
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  This theory defines functions which are constant except for finitely
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  many points (FinFun) and introduces a type finfin along with a
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  number of operators for them. The code generator is set up such that
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  such functions can be represented as data in the generated code and
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  all operators are executable.
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  For details, see Formalising FinFuns - Generating Code for Functions as Data by A. Lochbihler in TPHOLs 2009.
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*}
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definition "code_abort" :: "(unit \<Rightarrow> 'a) \<Rightarrow> 'a"
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where [simp, code del]: "code_abort f = f ()"
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code_abort "code_abort"
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hide_const (open) "code_abort"
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subsection {* The @{text "map_default"} operation *}
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definition map_default :: "'b \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
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where "map_default b f a \<equiv> case f a of None \<Rightarrow> b | Some b' \<Rightarrow> b'"
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lemma map_default_delete [simp]:
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  "map_default b (f(a := None)) = (map_default b f)(a := b)"
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by(simp add: map_default_def fun_eq_iff)
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lemma map_default_insert:
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  "map_default b (f(a \<mapsto> b')) = (map_default b f)(a := b')"
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by(simp add: map_default_def fun_eq_iff)
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lemma map_default_empty [simp]: "map_default b empty = (\<lambda>a. b)"
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by(simp add: fun_eq_iff map_default_def)
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lemma map_default_inject:
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  fixes g g' :: "'a \<rightharpoonup> 'b"
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  assumes infin_eq: "\<not> finite (UNIV :: 'a set) \<or> b = b'"
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  and fin: "finite (dom g)" and b: "b \<notin> ran g"
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  and fin': "finite (dom g')" and b': "b' \<notin> ran g'"
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  and eq': "map_default b g = map_default b' g'"
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  shows "b = b'" "g = g'"
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proof -
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  from infin_eq show bb': "b = b'"
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  proof
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    assume infin: "\<not> finite (UNIV :: 'a set)"
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    from fin fin' have "finite (dom g \<union> dom g')" by auto
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    with infin have "UNIV - (dom g \<union> dom g') \<noteq> {}" by(auto dest: finite_subset)
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    then obtain a where a: "a \<notin> dom g \<union> dom g'" by auto
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    hence "map_default b g a = b" "map_default b' g' a = b'" by(auto simp add: map_default_def)
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    with eq' show "b = b'" by simp
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  qed
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  show "g = g'"
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  proof
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    fix x
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    show "g x = g' x"
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    proof(cases "g x")
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      case None
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      hence "map_default b g x = b" by(simp add: map_default_def)
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      with bb' eq' have "map_default b' g' x = b'" by simp
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      with b' have "g' x = None" by(simp add: map_default_def ran_def split: option.split_asm)
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      with None show ?thesis by simp
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    next
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      case (Some c)
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      with b have cb: "c \<noteq> b" by(auto simp add: ran_def)
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      moreover from Some have "map_default b g x = c" by(simp add: map_default_def)
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      with eq' have "map_default b' g' x = c" by simp
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      ultimately have "g' x = Some c" using b' bb' by(auto simp add: map_default_def split: option.splits)
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      with Some show ?thesis by simp
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    qed
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  qed
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qed
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subsection {* The finfun type *}
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definition "finfun = {f::'a\<Rightarrow>'b. \<exists>b. finite {a. f a \<noteq> b}}"
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typedef (open) ('a,'b) finfun  ("(_ =>f /_)" [22, 21] 21) = "finfun :: ('a => 'b) set"
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  morphisms finfun_apply Abs_finfun
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proof -
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  have "\<exists>f. finite {x. f x \<noteq> undefined}"
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  proof
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    show "finite {x. (\<lambda>y. undefined) x \<noteq> undefined}" by auto
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  qed
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  then show ?thesis unfolding finfun_def by auto
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qed
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type_notation finfun ("(_ \<Rightarrow>f /_)" [22, 21] 21)
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setup_lifting type_definition_finfun
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lemma fun_upd_finfun: "y(a := b) \<in> finfun \<longleftrightarrow> y \<in> finfun"
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proof -
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  { fix b'
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    have "finite {a'. (y(a := b)) a' \<noteq> b'} = finite {a'. y a' \<noteq> b'}"
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    proof(cases "b = b'")
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      case True
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      hence "{a'. (y(a := b)) a' \<noteq> b'} = {a'. y a' \<noteq> b'} - {a}" by auto
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      thus ?thesis by simp
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    next
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      case False
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      hence "{a'. (y(a := b)) a' \<noteq> b'} = insert a {a'. y a' \<noteq> b'}" by auto
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      thus ?thesis by simp
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    qed }
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  thus ?thesis unfolding finfun_def by blast
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qed
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lemma const_finfun: "(\<lambda>x. a) \<in> finfun"
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by(auto simp add: finfun_def)
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lemma finfun_left_compose:
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  assumes "y \<in> finfun"
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  shows "g \<circ> y \<in> finfun"
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proof -
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  from assms obtain b where "finite {a. y a \<noteq> b}"
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    unfolding finfun_def by blast
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  hence "finite {c. g (y c) \<noteq> g b}"
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  proof(induct "{a. y a \<noteq> b}" arbitrary: y)
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    case empty
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    hence "y = (\<lambda>a. b)" by(auto)
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    thus ?case by(simp)
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  next
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    case (insert x F)
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    note IH = `\<And>y. F = {a. y a \<noteq> b} \<Longrightarrow> finite {c. g (y c) \<noteq> g b}`
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    from `insert x F = {a. y a \<noteq> b}` `x \<notin> F`
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    have F: "F = {a. (y(x := b)) a \<noteq> b}" by(auto)
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    show ?case
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    proof(cases "g (y x) = g b")
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      case True
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      hence "{c. g ((y(x := b)) c) \<noteq> g b} = {c. g (y c) \<noteq> g b}" by auto
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      with IH[OF F] show ?thesis by simp
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    next
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      case False
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      hence "{c. g (y c) \<noteq> g b} = insert x {c. g ((y(x := b)) c) \<noteq> g b}" by auto
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      with IH[OF F] show ?thesis by(simp)
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    qed
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  qed
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  thus ?thesis unfolding finfun_def by auto
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qed
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lemma assumes "y \<in> finfun"
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  shows fst_finfun: "fst \<circ> y \<in> finfun"
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  and snd_finfun: "snd \<circ> y \<in> finfun"
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proof -
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  from assms obtain b c where bc: "finite {a. y a \<noteq> (b, c)}"
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    unfolding finfun_def by auto
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  have "{a. fst (y a) \<noteq> b} \<subseteq> {a. y a \<noteq> (b, c)}"
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    and "{a. snd (y a) \<noteq> c} \<subseteq> {a. y a \<noteq> (b, c)}" by auto
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  hence "finite {a. fst (y a) \<noteq> b}" 
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    and "finite {a. snd (y a) \<noteq> c}" using bc by(auto intro: finite_subset)
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  thus "fst \<circ> y \<in> finfun" "snd \<circ> y \<in> finfun"
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    unfolding finfun_def by auto
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qed
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lemma map_of_finfun: "map_of xs \<in> finfun"
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unfolding finfun_def
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by(induct xs)(auto simp add: Collect_neg_eq Collect_conj_eq Collect_imp_eq intro: finite_subset)
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lemma Diag_finfun: "(\<lambda>x. (f x, g x)) \<in> finfun \<longleftrightarrow> f \<in> finfun \<and> g \<in> finfun"
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by(auto intro: finite_subset simp add: Collect_neg_eq Collect_imp_eq Collect_conj_eq finfun_def)
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lemma finfun_right_compose:
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  assumes g: "g \<in> finfun" and inj: "inj f"
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  shows "g o f \<in> finfun"
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proof -
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  from g obtain b where b: "finite {a. g a \<noteq> b}" unfolding finfun_def by blast
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  moreover have "f ` {a. g (f a) \<noteq> b} \<subseteq> {a. g a \<noteq> b}" by auto
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  moreover from inj have "inj_on f {a.  g (f a) \<noteq> b}" by(rule subset_inj_on) blast
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  ultimately have "finite {a. g (f a) \<noteq> b}"
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    by(blast intro: finite_imageD[where f=f] finite_subset)
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  thus ?thesis unfolding finfun_def by auto
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qed
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lemma finfun_curry:
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  assumes fin: "f \<in> finfun"
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  shows "curry f \<in> finfun" "curry f a \<in> finfun"
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proof -
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  from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast
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  moreover have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force)
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  hence "{a. curry f a \<noteq> (\<lambda>b. c)} = fst ` {ab. f ab \<noteq> c}"
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    by(auto simp add: curry_def fun_eq_iff)
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  ultimately have "finite {a. curry f a \<noteq> (\<lambda>b. c)}" by simp
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  thus "curry f \<in> finfun" unfolding finfun_def by blast
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  have "snd ` {ab. f ab \<noteq> c} = {b. \<exists>a. f (a, b) \<noteq> c}" by(force)
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  hence "{b. f (a, b) \<noteq> c} \<subseteq> snd ` {ab. f ab \<noteq> c}" by auto
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  hence "finite {b. f (a, b) \<noteq> c}" by(rule finite_subset)(rule finite_imageI[OF c])
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  thus "curry f a \<in> finfun" unfolding finfun_def by auto
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qed
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bundle finfun =
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  fst_finfun[simp] snd_finfun[simp] Abs_finfun_inverse[simp] 
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  finfun_apply_inverse[simp] Abs_finfun_inject[simp] finfun_apply_inject[simp]
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  Diag_finfun[simp] finfun_curry[simp]
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  const_finfun[iff] fun_upd_finfun[iff] finfun_apply[iff] map_of_finfun[iff]
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  finfun_left_compose[intro] fst_finfun[intro] snd_finfun[intro]
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lemma Abs_finfun_inject_finite:
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  fixes x y :: "'a \<Rightarrow> 'b"
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  assumes fin: "finite (UNIV :: 'a set)"
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  shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y"
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proof
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  assume "Abs_finfun x = Abs_finfun y"
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  moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def
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    by(auto intro: finite_subset[OF _ fin])
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  ultimately show "x = y" by(simp add: Abs_finfun_inject)
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qed simp
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lemma Abs_finfun_inject_finite_class:
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  fixes x y :: "('a :: finite) \<Rightarrow> 'b"
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  shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y"
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using finite_UNIV
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by(simp add: Abs_finfun_inject_finite)
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lemma Abs_finfun_inj_finite:
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  assumes fin: "finite (UNIV :: 'a set)"
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  shows "inj (Abs_finfun :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>f 'b)"
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proof(rule inj_onI)
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  fix x y :: "'a \<Rightarrow> 'b"
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  assume "Abs_finfun x = Abs_finfun y"
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  moreover have "x \<in> finfun" "y \<in> finfun" unfolding finfun_def
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    by(auto intro: finite_subset[OF _ fin])
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  ultimately show "x = y" by(simp add: Abs_finfun_inject)
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qed
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lemma Abs_finfun_inverse_finite:
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  fixes x :: "'a \<Rightarrow> 'b"
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  assumes fin: "finite (UNIV :: 'a set)"
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  shows "finfun_apply (Abs_finfun x) = x"
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  including finfun
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proof -
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  from fin have "x \<in> finfun"
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    by(auto simp add: finfun_def intro: finite_subset)
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  thus ?thesis by simp
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qed
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lemma Abs_finfun_inverse_finite_class:
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  fixes x :: "('a :: finite) \<Rightarrow> 'b"
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  shows "finfun_apply (Abs_finfun x) = x"
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using finite_UNIV by(simp add: Abs_finfun_inverse_finite)
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lemma finfun_eq_finite_UNIV: "finite (UNIV :: 'a set) \<Longrightarrow> (finfun :: ('a \<Rightarrow> 'b) set) = UNIV"
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unfolding finfun_def by(auto intro: finite_subset)
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lemma finfun_finite_UNIV_class: "finfun = (UNIV :: ('a :: finite \<Rightarrow> 'b) set)"
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by(simp add: finfun_eq_finite_UNIV)
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lemma map_default_in_finfun:
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  assumes fin: "finite (dom f)"
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  shows "map_default b f \<in> finfun"
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unfolding finfun_def
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proof(intro CollectI exI)
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  from fin show "finite {a. map_default b f a \<noteq> b}"
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    by(auto simp add: map_default_def dom_def Collect_conj_eq split: option.splits)
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qed
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lemma finfun_cases_map_default:
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  obtains b g where "f = Abs_finfun (map_default b g)" "finite (dom g)" "b \<notin> ran g"
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proof -
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  obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by(cases f)
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  from y obtain b where b: "finite {a. y a \<noteq> b}" unfolding finfun_def by auto
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  let ?g = "(\<lambda>a. if y a = b then None else Some (y a))"
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  have "map_default b ?g = y" by(simp add: fun_eq_iff map_default_def)
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  with f have "f = Abs_finfun (map_default b ?g)" by simp
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  moreover from b have "finite (dom ?g)" by(auto simp add: dom_def)
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  moreover have "b \<notin> ran ?g" by(auto simp add: ran_def)
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  ultimately show ?thesis by(rule that)
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qed
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subsection {* Kernel functions for type @{typ "'a \<Rightarrow>f 'b"} *}
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lift_definition finfun_const :: "'b \<Rightarrow> 'a \<Rightarrow>f 'b" ("K$/ _" [0] 1)
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is "\<lambda> b x. b" by (rule const_finfun)
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lift_definition finfun_update :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>f 'b" ("_'(_ $:= _')" [1000,0,0] 1000) is "fun_upd"
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by (simp add: fun_upd_finfun)
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lemma finfun_update_twist: "a \<noteq> a' \<Longrightarrow> f(a $:= b)(a' $:= b') = f(a' $:= b')(a $:= b)"
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by transfer (simp add: fun_upd_twist)
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   289
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   290
lemma finfun_update_twice [simp]:
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   291
  "f(a $:= b)(a $:= b') = f(a $:= b')"
Andreas@48028
   292
by transfer simp
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   293
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   294
lemma finfun_update_const_same: "(K$ b)(a $:= b) = (K$ b)"
Andreas@48028
   295
by transfer (simp add: fun_eq_iff)
Andreas@48028
   296
Andreas@48028
   297
subsection {* Code generator setup *}
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   298
Andreas@48036
   299
definition finfun_update_code :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>f 'b"
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   300
where [simp, code del]: "finfun_update_code = finfun_update"
Andreas@48028
   301
Andreas@48028
   302
code_datatype finfun_const finfun_update_code
Andreas@48028
   303
Andreas@48028
   304
lemma finfun_update_const_code [code]:
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   305
  "(K$ b)(a $:= b') = (if b = b' then (K$ b) else finfun_update_code (K$ b) a b')"
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   306
by(simp add: finfun_update_const_same)
Andreas@48028
   307
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   308
lemma finfun_update_update_code [code]:
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   309
  "(finfun_update_code f a b)(a' $:= b') = (if a = a' then f(a $:= b') else finfun_update_code (f(a' $:= b')) a b)"
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   310
by(simp add: finfun_update_twist)
Andreas@48028
   311
Andreas@48028
   312
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   313
subsection {* Setup for quickcheck *}
Andreas@48028
   314
Andreas@48034
   315
quickcheck_generator finfun constructors: finfun_update_code, "finfun_const :: 'b \<Rightarrow> 'a \<Rightarrow>f 'b"
Andreas@48028
   316
Andreas@48028
   317
subsection {* @{text "finfun_update"} as instance of @{text "comp_fun_commute"} *}
Andreas@48028
   318
Andreas@48036
   319
interpretation finfun_update: comp_fun_commute "\<lambda>a f. f(a :: 'a $:= b')"
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   320
  including finfun
Andreas@48028
   321
proof
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   322
  fix a a' :: 'a
Andreas@48036
   323
  show "(\<lambda>f. f(a $:= b')) \<circ> (\<lambda>f. f(a' $:= b')) = (\<lambda>f. f(a' $:= b')) \<circ> (\<lambda>f. f(a $:= b'))"
Andreas@48028
   324
  proof
Andreas@48028
   325
    fix b
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   326
    have "(finfun_apply b)(a := b', a' := b') = (finfun_apply b)(a' := b', a := b')"
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   327
      by(cases "a = a'")(auto simp add: fun_upd_twist)
Andreas@48036
   328
    then have "b(a $:= b')(a' $:= b') = b(a' $:= b')(a $:= b')"
Andreas@48028
   329
      by(auto simp add: finfun_update_def fun_upd_twist)
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   330
    then show "((\<lambda>f. f(a $:= b')) \<circ> (\<lambda>f. f(a' $:= b'))) b = ((\<lambda>f. f(a' $:= b')) \<circ> (\<lambda>f. f(a $:= b'))) b"
Andreas@48028
   331
      by (simp add: fun_eq_iff)
Andreas@48028
   332
  qed
Andreas@48028
   333
qed
Andreas@48028
   334
Andreas@48028
   335
lemma fold_finfun_update_finite_univ:
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   336
  assumes fin: "finite (UNIV :: 'a set)"
Andreas@48036
   337
  shows "Finite_Set.fold (\<lambda>a f. f(a $:= b')) (K$ b) (UNIV :: 'a set) = (K$ b')"
Andreas@48028
   338
proof -
Andreas@48028
   339
  { fix A :: "'a set"
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   340
    from fin have "finite A" by(auto intro: finite_subset)
Andreas@48036
   341
    hence "Finite_Set.fold (\<lambda>a f. f(a $:= b')) (K$ b) A = Abs_finfun (\<lambda>a. if a \<in> A then b' else b)"
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   342
    proof(induct)
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   343
      case (insert x F)
Andreas@48028
   344
      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)"
Andreas@48037
   345
        by(auto)
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   346
      with insert show ?case
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   347
        by(simp add: finfun_const_def fun_upd_def)(simp add: finfun_update_def Abs_finfun_inverse_finite[OF fin] fun_upd_def)
Andreas@48028
   348
    qed(simp add: finfun_const_def) }
Andreas@48028
   349
  thus ?thesis by(simp add: finfun_const_def)
Andreas@48028
   350
qed
Andreas@48028
   351
Andreas@48028
   352
Andreas@48028
   353
subsection {* Default value for FinFuns *}
Andreas@48028
   354
Andreas@48028
   355
definition finfun_default_aux :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b"
Andreas@48028
   356
where [code del]: "finfun_default_aux f = (if finite (UNIV :: 'a set) then undefined else THE b. finite {a. f a \<noteq> b})"
Andreas@48028
   357
Andreas@48028
   358
lemma finfun_default_aux_infinite:
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   359
  fixes f :: "'a \<Rightarrow> 'b"
Andreas@48028
   360
  assumes infin: "\<not> finite (UNIV :: 'a set)"
Andreas@48028
   361
  and fin: "finite {a. f a \<noteq> b}"
Andreas@48028
   362
  shows "finfun_default_aux f = b"
Andreas@48028
   363
proof -
Andreas@48028
   364
  let ?B = "{a. f a \<noteq> b}"
Andreas@48028
   365
  from fin have "(THE b. finite {a. f a \<noteq> b}) = b"
Andreas@48028
   366
  proof(rule the_equality)
Andreas@48028
   367
    fix b'
Andreas@48028
   368
    assume "finite {a. f a \<noteq> b'}" (is "finite ?B'")
Andreas@48028
   369
    with infin fin have "UNIV - (?B' \<union> ?B) \<noteq> {}" by(auto dest: finite_subset)
Andreas@48028
   370
    then obtain a where a: "a \<notin> ?B' \<union> ?B" by auto
Andreas@48028
   371
    thus "b' = b" by auto
Andreas@48028
   372
  qed
Andreas@48028
   373
  thus ?thesis using infin by(simp add: finfun_default_aux_def)
Andreas@48028
   374
qed
Andreas@48028
   375
Andreas@48028
   376
Andreas@48028
   377
lemma finite_finfun_default_aux:
Andreas@48028
   378
  fixes f :: "'a \<Rightarrow> 'b"
Andreas@48028
   379
  assumes fin: "f \<in> finfun"
Andreas@48028
   380
  shows "finite {a. f a \<noteq> finfun_default_aux f}"
Andreas@48028
   381
proof(cases "finite (UNIV :: 'a set)")
Andreas@48028
   382
  case True thus ?thesis using fin
Andreas@48028
   383
    by(auto simp add: finfun_def finfun_default_aux_def intro: finite_subset)
Andreas@48028
   384
next
Andreas@48028
   385
  case False
Andreas@48028
   386
  from fin obtain b where b: "finite {a. f a \<noteq> b}" (is "finite ?B")
Andreas@48028
   387
    unfolding finfun_def by blast
Andreas@48028
   388
  with False show ?thesis by(simp add: finfun_default_aux_infinite)
Andreas@48028
   389
qed
Andreas@48028
   390
Andreas@48028
   391
lemma finfun_default_aux_update_const:
Andreas@48028
   392
  fixes f :: "'a \<Rightarrow> 'b"
Andreas@48028
   393
  assumes fin: "f \<in> finfun"
Andreas@48028
   394
  shows "finfun_default_aux (f(a := b)) = finfun_default_aux f"
Andreas@48028
   395
proof(cases "finite (UNIV :: 'a set)")
Andreas@48028
   396
  case False
Andreas@48028
   397
  from fin obtain b' where b': "finite {a. f a \<noteq> b'}" unfolding finfun_def by blast
Andreas@48028
   398
  hence "finite {a'. (f(a := b)) a' \<noteq> b'}"
Andreas@48028
   399
  proof(cases "b = b' \<and> f a \<noteq> b'") 
Andreas@48028
   400
    case True
Andreas@48028
   401
    hence "{a. f a \<noteq> b'} = insert a {a'. (f(a := b)) a' \<noteq> b'}" by auto
Andreas@48028
   402
    thus ?thesis using b' by simp
Andreas@48028
   403
  next
Andreas@48028
   404
    case False
Andreas@48028
   405
    moreover
Andreas@48028
   406
    { assume "b \<noteq> b'"
Andreas@48028
   407
      hence "{a'. (f(a := b)) a' \<noteq> b'} = insert a {a. f a \<noteq> b'}" by auto
Andreas@48028
   408
      hence ?thesis using b' by simp }
Andreas@48028
   409
    moreover
Andreas@48028
   410
    { assume "b = b'" "f a = b'"
Andreas@48028
   411
      hence "{a'. (f(a := b)) a' \<noteq> b'} = {a. f a \<noteq> b'}" by auto
Andreas@48028
   412
      hence ?thesis using b' by simp }
Andreas@48028
   413
    ultimately show ?thesis by blast
Andreas@48028
   414
  qed
Andreas@48028
   415
  with False b' show ?thesis by(auto simp del: fun_upd_apply simp add: finfun_default_aux_infinite)
Andreas@48028
   416
next
Andreas@48028
   417
  case True thus ?thesis by(simp add: finfun_default_aux_def)
Andreas@48028
   418
qed
Andreas@48028
   419
Andreas@48034
   420
lift_definition finfun_default :: "'a \<Rightarrow>f 'b \<Rightarrow> 'b"
Andreas@48028
   421
is "finfun_default_aux" ..
Andreas@48028
   422
Andreas@48031
   423
lemma finfun_apply_transfer [transfer_rule]: 
Andreas@48031
   424
  "(fun_rel cr_finfun (fun_rel op = op =)) (\<lambda>f. f) finfun_apply"
Andreas@48031
   425
unfolding Rel_def fun_rel_def cr_finfun_def by simp
Andreas@48031
   426
Andreas@48029
   427
lemma finite_finfun_default: "finite {a. finfun_apply f a \<noteq> finfun_default f}"
Andreas@48031
   428
by transfer (erule finite_finfun_default_aux)
Andreas@48028
   429
Andreas@48036
   430
lemma finfun_default_const: "finfun_default ((K$ b) :: 'a \<Rightarrow>f 'b) = (if finite (UNIV :: 'a set) then undefined else b)"
Andreas@48031
   431
by(transfer)(auto simp add: finfun_default_aux_infinite finfun_default_aux_def)
Andreas@48028
   432
Andreas@48028
   433
lemma finfun_default_update_const:
Andreas@48036
   434
  "finfun_default (f(a $:= b)) = finfun_default f"
Andreas@48028
   435
by transfer (simp add: finfun_default_aux_update_const)
Andreas@48028
   436
Andreas@48028
   437
lemma finfun_default_const_code [code]:
Andreas@48036
   438
  "finfun_default ((K$ c) :: ('a :: card_UNIV) \<Rightarrow>f 'b) = (if card_UNIV (TYPE('a)) = 0 then c else undefined)"
Andreas@48028
   439
by(simp add: finfun_default_const card_UNIV_eq_0_infinite_UNIV)
Andreas@48028
   440
Andreas@48028
   441
lemma finfun_default_update_code [code]:
Andreas@48028
   442
  "finfun_default (finfun_update_code f a b) = finfun_default f"
Andreas@48028
   443
by(simp add: finfun_default_update_const)
Andreas@48028
   444
Andreas@48028
   445
subsection {* Recursion combinator and well-formedness conditions *}
Andreas@48028
   446
Andreas@48034
   447
definition finfun_rec :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow>f 'b) \<Rightarrow> 'c"
Andreas@48028
   448
where [code del]:
Andreas@48028
   449
  "finfun_rec cnst upd f \<equiv>
Andreas@48028
   450
   let b = finfun_default f;
Andreas@48028
   451
       g = THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g
Andreas@48028
   452
   in Finite_Set.fold (\<lambda>a. upd a (map_default b g a)) (cnst b) (dom g)"
Andreas@48028
   453
Andreas@48028
   454
locale finfun_rec_wf_aux =
Andreas@48028
   455
  fixes cnst :: "'b \<Rightarrow> 'c"
Andreas@48028
   456
  and upd :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"
Andreas@48028
   457
  assumes upd_const_same: "upd a b (cnst b) = cnst b"
Andreas@48028
   458
  and upd_commute: "a \<noteq> a' \<Longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)"
Andreas@48028
   459
  and upd_idemp: "b \<noteq> b' \<Longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
Andreas@48028
   460
begin
Andreas@48028
   461
Andreas@48028
   462
Andreas@48028
   463
lemma upd_left_comm: "comp_fun_commute (\<lambda>a. upd a (f a))"
Andreas@48028
   464
by(unfold_locales)(auto intro: upd_commute simp add: fun_eq_iff)
Andreas@48028
   465
Andreas@48028
   466
lemma upd_upd_twice: "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)"
Andreas@48028
   467
by(cases "b \<noteq> b'")(auto simp add: fun_upd_def upd_const_same upd_idemp)
Andreas@48028
   468
Andreas@48028
   469
lemma map_default_update_const:
Andreas@48028
   470
  assumes fin: "finite (dom f)"
Andreas@48028
   471
  and anf: "a \<notin> dom f"
Andreas@48028
   472
  and fg: "f \<subseteq>\<^sub>m g"
Andreas@48028
   473
  shows "upd a d  (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)) =
Andreas@48028
   474
         Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)"
Andreas@48028
   475
proof -
Andreas@48028
   476
  let ?upd = "\<lambda>a. upd a (map_default d g a)"
Andreas@48028
   477
  let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A"
Andreas@48028
   478
  interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm)
Andreas@48028
   479
  
Andreas@48028
   480
  from fin anf fg show ?thesis
Andreas@48028
   481
  proof(induct "dom f" arbitrary: f)
Andreas@48028
   482
    case empty
Andreas@48037
   483
    from `{} = dom f` have "f = empty" by(auto simp add: dom_def)
Andreas@48028
   484
    thus ?case by(simp add: finfun_const_def upd_const_same)
Andreas@48028
   485
  next
Andreas@48028
   486
    case (insert a' A)
Andreas@48028
   487
    note IH = `\<And>f.  \<lbrakk> A = dom f; a \<notin> dom f; f \<subseteq>\<^sub>m g \<rbrakk> \<Longrightarrow> upd a d (?fr (dom f)) = ?fr (dom f)`
Andreas@48028
   488
    note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A`
Andreas@48028
   489
    note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g`
Andreas@48028
   490
    
Andreas@48028
   491
    from domf obtain b where b: "f a' = Some b" by auto
Andreas@48028
   492
    let ?f' = "f(a' := None)"
Andreas@48028
   493
    have "upd a d (?fr (insert a' A)) = upd a d (upd a' (map_default d g a') (?fr A))"
Andreas@48028
   494
      by(subst gwf.fold_insert[OF fin a'nA]) rule
Andreas@48028
   495
    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
Andreas@48028
   496
    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
Andreas@48028
   497
    also from anf domf have "a \<noteq> a'" by auto note upd_commute[OF this]
Andreas@48028
   498
    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)
Andreas@48028
   499
    note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`]
Andreas@48028
   500
    also have "upd a' (map_default d f a') (?fr (dom (f(a' := None)))) = ?fr (dom f)"
Andreas@48028
   501
      unfolding domf[symmetric] gwf.fold_insert[OF fin a'nA] ga' unfolding A ..
Andreas@48028
   502
    also have "insert a' (dom ?f') = dom f" using domf by auto
Andreas@48028
   503
    finally show ?case .
Andreas@48028
   504
  qed
Andreas@48028
   505
qed
Andreas@48028
   506
Andreas@48028
   507
lemma map_default_update_twice:
Andreas@48028
   508
  assumes fin: "finite (dom f)"
Andreas@48028
   509
  and anf: "a \<notin> dom f"
Andreas@48028
   510
  and fg: "f \<subseteq>\<^sub>m g"
Andreas@48028
   511
  shows "upd a d'' (upd a d' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))) =
Andreas@48028
   512
         upd a d'' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))"
Andreas@48028
   513
proof -
Andreas@48028
   514
  let ?upd = "\<lambda>a. upd a (map_default d g a)"
Andreas@48028
   515
  let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A"
Andreas@48028
   516
  interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm)
Andreas@48028
   517
  
Andreas@48028
   518
  from fin anf fg show ?thesis
Andreas@48028
   519
  proof(induct "dom f" arbitrary: f)
Andreas@48028
   520
    case empty
Andreas@48037
   521
    from `{} = dom f` have "f = empty" by(auto simp add: dom_def)
Andreas@48028
   522
    thus ?case by(auto simp add: finfun_const_def finfun_update_def upd_upd_twice)
Andreas@48028
   523
  next
Andreas@48028
   524
    case (insert a' A)
Andreas@48028
   525
    note IH = `\<And>f. \<lbrakk>A = dom f; a \<notin> dom f; f \<subseteq>\<^sub>m g\<rbrakk> \<Longrightarrow> upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (?fr (dom f))`
Andreas@48028
   526
    note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A`
Andreas@48028
   527
    note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g`
Andreas@48028
   528
    
Andreas@48028
   529
    from domf obtain b where b: "f a' = Some b" by auto
Andreas@48028
   530
    let ?f' = "f(a' := None)"
Andreas@48028
   531
    let ?b' = "case f a' of None \<Rightarrow> d | Some b \<Rightarrow> b"
Andreas@48028
   532
    from domf have "upd a d'' (upd a d' (?fr (dom f))) = upd a d'' (upd a d' (?fr (insert a' A)))" by simp
Andreas@48028
   533
    also note gwf.fold_insert[OF fin a'nA]
Andreas@48028
   534
    also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec)
Andreas@48028
   535
    hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def)
Andreas@48028
   536
    also from anf domf have ana': "a \<noteq> a'" by auto note upd_commute[OF this]
Andreas@48028
   537
    also note upd_commute[OF ana']
Andreas@48028
   538
    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)
Andreas@48028
   539
    note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`]
Andreas@48028
   540
    also note upd_commute[OF ana'[symmetric]] also note ga'[symmetric] also note A[symmetric]
Andreas@48028
   541
    also note gwf.fold_insert[symmetric, OF fin a'nA] also note domf
Andreas@48028
   542
    finally show ?case .
Andreas@48028
   543
  qed
Andreas@48028
   544
qed
Andreas@48028
   545
Andreas@48028
   546
lemma map_default_eq_id [simp]: "map_default d ((\<lambda>a. Some (f a)) |` {a. f a \<noteq> d}) = f"
Andreas@48037
   547
by(auto simp add: map_default_def restrict_map_def)
Andreas@48028
   548
Andreas@48028
   549
lemma finite_rec_cong1:
Andreas@48028
   550
  assumes f: "comp_fun_commute f" and g: "comp_fun_commute g"
Andreas@48028
   551
  and fin: "finite A"
Andreas@48028
   552
  and eq: "\<And>a. a \<in> A \<Longrightarrow> f a = g a"
Andreas@48028
   553
  shows "Finite_Set.fold f z A = Finite_Set.fold g z A"
Andreas@48028
   554
proof -
Andreas@48028
   555
  interpret f: comp_fun_commute f by(rule f)
Andreas@48028
   556
  interpret g: comp_fun_commute g by(rule g)
Andreas@48028
   557
  { fix B
Andreas@48028
   558
    assume BsubA: "B \<subseteq> A"
Andreas@48028
   559
    with fin have "finite B" by(blast intro: finite_subset)
Andreas@48028
   560
    hence "B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B"
Andreas@48028
   561
    proof(induct)
Andreas@48028
   562
      case empty thus ?case by simp
Andreas@48028
   563
    next
Andreas@48028
   564
      case (insert a B)
Andreas@48028
   565
      note finB = `finite B` note anB = `a \<notin> B` note sub = `insert a B \<subseteq> A`
Andreas@48028
   566
      note IH = `B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B`
Andreas@48028
   567
      from sub anB have BpsubA: "B \<subset> A" and BsubA: "B \<subseteq> A" and aA: "a \<in> A" by auto
Andreas@48028
   568
      from IH[OF BsubA] eq[OF aA] finB anB
Andreas@48028
   569
      show ?case by(auto)
Andreas@48028
   570
    qed
Andreas@48028
   571
    with BsubA have "Finite_Set.fold f z B = Finite_Set.fold g z B" by blast }
Andreas@48028
   572
  thus ?thesis by blast
Andreas@48028
   573
qed
Andreas@48028
   574
Andreas@48028
   575
lemma finfun_rec_upd [simp]:
Andreas@48036
   576
  "finfun_rec cnst upd (f(a' $:= b')) = upd a' b' (finfun_rec cnst upd f)"
Andreas@48030
   577
  including finfun
Andreas@48028
   578
proof -
Andreas@48028
   579
  obtain b where b: "b = finfun_default f" by auto
Andreas@48028
   580
  let ?the = "\<lambda>f g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g"
Andreas@48028
   581
  obtain g where g: "g = The (?the f)" by blast
Andreas@48028
   582
  obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by (cases f)
Andreas@48028
   583
  from f y b have bfin: "finite {a. y a \<noteq> b}" by(simp add: finfun_default_def finite_finfun_default_aux)
Andreas@48028
   584
Andreas@48028
   585
  let ?g = "(\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}"
Andreas@48028
   586
  from bfin have fing: "finite (dom ?g)" by auto
Andreas@48028
   587
  have bran: "b \<notin> ran ?g" by(auto simp add: ran_def restrict_map_def)
Andreas@48028
   588
  have yg: "y = map_default b ?g" by simp
Andreas@48028
   589
  have gg: "g = ?g" unfolding g
Andreas@48028
   590
  proof(rule the_equality)
Andreas@48028
   591
    from f y bfin show "?the f ?g"
Andreas@48028
   592
      by(auto)(simp add: restrict_map_def ran_def split: split_if_asm)
Andreas@48028
   593
  next
Andreas@48028
   594
    fix g'
Andreas@48028
   595
    assume "?the f g'"
Andreas@48028
   596
    hence fin': "finite (dom g')" and ran': "b \<notin> ran g'"
Andreas@48028
   597
      and eq: "Abs_finfun (map_default b ?g) = Abs_finfun (map_default b g')" using f yg by auto
Andreas@48028
   598
    from fin' fing have "map_default b ?g \<in> finfun" "map_default b g' \<in> finfun" by(blast intro: map_default_in_finfun)+
Andreas@48028
   599
    with eq have "map_default b ?g = map_default b g'" by simp
Andreas@48028
   600
    with fing bran fin' ran' show "g' = ?g" by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
Andreas@48028
   601
  qed
Andreas@48028
   602
Andreas@48028
   603
  show ?thesis
Andreas@48028
   604
  proof(cases "b' = b")
Andreas@48028
   605
    case True
Andreas@48028
   606
    note b'b = True
Andreas@48028
   607
Andreas@48028
   608
    let ?g' = "(\<lambda>a. Some ((y(a' := b)) a)) |` {a. (y(a' := b)) a \<noteq> b}"
Andreas@48028
   609
    from bfin b'b have fing': "finite (dom ?g')"
Andreas@48028
   610
      by(auto simp add: Collect_conj_eq Collect_imp_eq intro: finite_subset)
Andreas@48028
   611
    have brang': "b \<notin> ran ?g'" by(auto simp add: ran_def restrict_map_def)
Andreas@48028
   612
Andreas@48028
   613
    let ?b' = "\<lambda>a. case ?g' a of None \<Rightarrow> b | Some b \<Rightarrow> b"
Andreas@48028
   614
    let ?b = "map_default b ?g"
Andreas@48028
   615
    from upd_left_comm upd_left_comm fing'
Andreas@48028
   616
    have "Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g') = Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')"
Andreas@48028
   617
      by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b b map_default_def)
Andreas@48028
   618
    also interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm)
Andreas@48028
   619
    have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g') = upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))"
Andreas@48028
   620
    proof(cases "y a' = b")
Andreas@48028
   621
      case True
Andreas@48037
   622
      with b'b have g': "?g' = ?g" by(auto simp add: restrict_map_def)
Andreas@48028
   623
      from True have a'ndomg: "a' \<notin> dom ?g" by auto
Andreas@48028
   624
      from f b'b b show ?thesis unfolding g'
Andreas@48028
   625
        by(subst map_default_update_const[OF fing a'ndomg map_le_refl, symmetric]) simp
Andreas@48028
   626
    next
Andreas@48028
   627
      case False
Andreas@48028
   628
      hence domg: "dom ?g = insert a' (dom ?g')" by auto
Andreas@48028
   629
      from False b'b have a'ndomg': "a' \<notin> dom ?g'" by auto
Andreas@48028
   630
      have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g')) = 
Andreas@48028
   631
            upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'))"
Andreas@48028
   632
        using fing' a'ndomg' unfolding b'b by(rule gwf.fold_insert)
Andreas@48028
   633
      hence "upd a' b (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g'))) =
Andreas@48028
   634
             upd a' b (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')))" by simp
Andreas@48028
   635
      also from b'b have g'leg: "?g' \<subseteq>\<^sub>m ?g" by(auto simp add: restrict_map_def map_le_def)
Andreas@48028
   636
      note map_default_update_twice[OF fing' a'ndomg' this, of b "?b a'" b]
Andreas@48028
   637
      also note map_default_update_const[OF fing' a'ndomg' g'leg, of b]
Andreas@48028
   638
      finally show ?thesis unfolding b'b domg[unfolded b'b] by(rule sym)
Andreas@48028
   639
    qed
Andreas@48036
   640
    also have "The (?the (f(a' $:= b'))) = ?g'"
Andreas@48028
   641
    proof(rule the_equality)
Andreas@48036
   642
      from f y b b'b brang' fing' show "?the (f(a' $:= b')) ?g'"
Andreas@48028
   643
        by(auto simp del: fun_upd_apply simp add: finfun_update_def)
Andreas@48028
   644
    next
Andreas@48028
   645
      fix g'
Andreas@48036
   646
      assume "?the (f(a' $:= b')) g'"
Andreas@48028
   647
      hence fin': "finite (dom g')" and ran': "b \<notin> ran g'"
Andreas@48036
   648
        and eq: "f(a' $:= b') = Abs_finfun (map_default b g')" 
Andreas@48028
   649
        by(auto simp del: fun_upd_apply)
Andreas@48028
   650
      from fin' fing' have "map_default b g' \<in> finfun" "map_default b ?g' \<in> finfun"
Andreas@48028
   651
        by(blast intro: map_default_in_finfun)+
Andreas@48028
   652
      with eq f b'b b have "map_default b ?g' = map_default b g'"
Andreas@48028
   653
        by(simp del: fun_upd_apply add: finfun_update_def)
Andreas@48028
   654
      with fing' brang' fin' ran' show "g' = ?g'"
Andreas@48028
   655
        by(rule map_default_inject[OF disjI2[OF refl], THEN sym])
Andreas@48028
   656
    qed
Andreas@48028
   657
    ultimately show ?thesis unfolding finfun_rec_def Let_def b gg[unfolded g b] using bfin b'b b
Andreas@48028
   658
      by(simp only: finfun_default_update_const map_default_def)
Andreas@48028
   659
  next
Andreas@48028
   660
    case False
Andreas@48028
   661
    note b'b = this
Andreas@48028
   662
    let ?g' = "?g(a' \<mapsto> b')"
Andreas@48028
   663
    let ?b' = "map_default b ?g'"
Andreas@48028
   664
    let ?b = "map_default b ?g"
Andreas@48028
   665
    from fing have fing': "finite (dom ?g')" by auto
Andreas@48028
   666
    from bran b'b have bnrang': "b \<notin> ran ?g'" by(auto simp add: ran_def)
Andreas@48037
   667
    have ffmg': "map_default b ?g' = y(a' := b')" by(auto simp add: map_default_def restrict_map_def)
Andreas@48036
   668
    with f y have f_Abs: "f(a' $:= b') = Abs_finfun (map_default b ?g')" by(auto simp add: finfun_update_def)
Andreas@48036
   669
    have g': "The (?the (f(a' $:= b'))) = ?g'"
Andreas@48028
   670
    proof (rule the_equality)
Andreas@48036
   671
      from fing' bnrang' f_Abs show "?the (f(a' $:= b')) ?g'"
Andreas@48036
   672
        by(auto simp add: finfun_update_def restrict_map_def)
Andreas@48028
   673
    next
Andreas@48036
   674
      fix g' assume "?the (f(a' $:= b')) g'"
Andreas@48036
   675
      hence f': "f(a' $:= b') = Abs_finfun (map_default b g')"
Andreas@48028
   676
        and fin': "finite (dom g')" and brang': "b \<notin> ran g'" by auto
Andreas@48028
   677
      from fing' fin' have "map_default b ?g' \<in> finfun" "map_default b g' \<in> finfun"
Andreas@48028
   678
        by(auto intro: map_default_in_finfun)
Andreas@48028
   679
      with f' f_Abs have "map_default b g' = map_default b ?g'" by simp
Andreas@48028
   680
      with fin' brang' fing' bnrang' show "g' = ?g'"
Andreas@48028
   681
        by(rule map_default_inject[OF disjI2[OF refl]])
Andreas@48028
   682
    qed
Andreas@48028
   683
    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}))"
Andreas@48028
   684
      by auto
Andreas@48028
   685
    show ?thesis
Andreas@48028
   686
    proof(cases "y a' = b")
Andreas@48028
   687
      case True
Andreas@48028
   688
      hence a'ndomg: "a' \<notin> dom ?g" by auto
Andreas@48028
   689
      from f y b'b True have yff: "y = map_default b (?g' |` dom ?g)"
Andreas@48028
   690
        by(auto simp add: restrict_map_def map_default_def intro!: ext)
Andreas@48028
   691
      hence f': "f = Abs_finfun (map_default b (?g' |` dom ?g))" using f by simp
Andreas@48028
   692
      interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm)
Andreas@48028
   693
      from upd_left_comm upd_left_comm fing
Andreas@48028
   694
      have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g) = Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)"
Andreas@48028
   695
        by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b True map_default_def)
Andreas@48028
   696
      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric]
Andreas@48028
   697
        unfolding g' g[symmetric] gg g'wf.fold_insert[OF fing a'ndomg, of "cnst b", folded dom]
Andreas@48028
   698
        by -(rule arg_cong2[where f="upd a'"], simp_all add: map_default_def)
Andreas@48028
   699
    next
Andreas@48028
   700
      case False
Andreas@48028
   701
      hence "insert a' (dom ?g) = dom ?g" by auto
Andreas@48028
   702
      moreover {
Andreas@48028
   703
        let ?g'' = "?g(a' := None)"
Andreas@48028
   704
        let ?b'' = "map_default b ?g''"
Andreas@48028
   705
        from False have domg: "dom ?g = insert a' (dom ?g'')" by auto
Andreas@48028
   706
        from False have a'ndomg'': "a' \<notin> dom ?g''" by auto
Andreas@48028
   707
        have fing'': "finite (dom ?g'')" by(rule finite_subset[OF _ fing]) auto
Andreas@48028
   708
        have bnrang'': "b \<notin> ran ?g''" by(auto simp add: ran_def restrict_map_def)
Andreas@48028
   709
        interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm)
Andreas@48028
   710
        interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm)
Andreas@48028
   711
        have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g''))) =
Andreas@48028
   712
              upd a' b' (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')))"
Andreas@48028
   713
          unfolding gwf.fold_insert[OF fing'' a'ndomg''] f ..
Andreas@48028
   714
        also have g''leg: "?g |` dom ?g'' \<subseteq>\<^sub>m ?g" by(auto simp add: map_le_def)
Andreas@48028
   715
        have "dom (?g |` dom ?g'') = dom ?g''" by auto
Andreas@48028
   716
        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",
Andreas@48028
   717
                                     unfolded this, OF fing'' a'ndomg'' g''leg]
Andreas@48028
   718
        also have b': "b' = ?b' a'" by(auto simp add: map_default_def)
Andreas@48028
   719
        from upd_left_comm upd_left_comm fing''
Andreas@48036
   720
        have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'') =
Andreas@48036
   721
          Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g'')"
Andreas@48028
   722
          by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b map_default_def)
Andreas@48028
   723
        with b' have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')) =
Andreas@48028
   724
                     upd a' (?b' a') (Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g''))" by simp
Andreas@48028
   725
        also note g'wf.fold_insert[OF fing'' a'ndomg'', symmetric]
Andreas@48028
   726
        finally have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g)) =
Andreas@48028
   727
                   Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)"
Andreas@48028
   728
          unfolding domg . }
Andreas@48028
   729
      ultimately have "Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (insert a' (dom ?g)) =
Andreas@48028
   730
                    upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))" by simp
Andreas@48028
   731
      thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] g[symmetric] g' dom[symmetric]
Andreas@48028
   732
        using b'b gg by(simp add: map_default_insert)
Andreas@48028
   733
    qed
Andreas@48028
   734
  qed
Andreas@48028
   735
qed
Andreas@48028
   736
Andreas@48028
   737
end
Andreas@48028
   738
Andreas@48028
   739
locale finfun_rec_wf = finfun_rec_wf_aux + 
Andreas@48028
   740
  assumes const_update_all:
Andreas@48028
   741
  "finite (UNIV :: 'a set) \<Longrightarrow> Finite_Set.fold (\<lambda>a. upd a b') (cnst b) (UNIV :: 'a set) = cnst b'"
Andreas@48028
   742
begin
Andreas@48028
   743
Andreas@48030
   744
lemma finfun_rec_const [simp]: includes finfun shows
Andreas@48036
   745
  "finfun_rec cnst upd (K$ c) = cnst c"
Andreas@48028
   746
proof(cases "finite (UNIV :: 'a set)")
Andreas@48028
   747
  case False
Andreas@48036
   748
  hence "finfun_default ((K$ c) :: 'a \<Rightarrow>f 'b) = c" by(simp add: finfun_default_const)
Andreas@48036
   749
  moreover have "(THE g :: 'a \<rightharpoonup> 'b. (K$ c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g) = empty"
Andreas@48028
   750
  proof (rule the_equality)
Andreas@48036
   751
    show "(K$ c) = Abs_finfun (map_default c empty) \<and> finite (dom empty) \<and> c \<notin> ran empty"
Andreas@48028
   752
      by(auto simp add: finfun_const_def)
Andreas@48028
   753
  next
Andreas@48028
   754
    fix g :: "'a \<rightharpoonup> 'b"
Andreas@48036
   755
    assume "(K$ c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g"
Andreas@48036
   756
    hence g: "(K$ c) = Abs_finfun (map_default c g)" and fin: "finite (dom g)" and ran: "c \<notin> ran g" by blast+
Andreas@48028
   757
    from g map_default_in_finfun[OF fin, of c] have "map_default c g = (\<lambda>a. c)"
Andreas@48028
   758
      by(simp add: finfun_const_def)
Andreas@48028
   759
    moreover have "map_default c empty = (\<lambda>a. c)" by simp
Andreas@48028
   760
    ultimately show "g = empty" by-(rule map_default_inject[OF disjI2[OF refl] fin ran], auto)
Andreas@48028
   761
  qed
Andreas@48028
   762
  ultimately show ?thesis by(simp add: finfun_rec_def)
Andreas@48028
   763
next
Andreas@48028
   764
  case True
Andreas@48036
   765
  hence default: "finfun_default ((K$ c) :: 'a \<Rightarrow>f 'b) = undefined" by(simp add: finfun_default_const)
Andreas@48036
   766
  let ?the = "\<lambda>g :: 'a \<rightharpoonup> 'b. (K$ c) = Abs_finfun (map_default undefined g) \<and> finite (dom g) \<and> undefined \<notin> ran g"
Andreas@48028
   767
  show ?thesis
Andreas@48028
   768
  proof(cases "c = undefined")
Andreas@48028
   769
    case True
Andreas@48028
   770
    have the: "The ?the = empty"
Andreas@48028
   771
    proof (rule the_equality)
Andreas@48028
   772
      from True show "?the empty" by(auto simp add: finfun_const_def)
Andreas@48028
   773
    next
Andreas@48028
   774
      fix g'
Andreas@48028
   775
      assume "?the g'"
Andreas@48036
   776
      hence fg: "(K$ c) = Abs_finfun (map_default undefined g')"
Andreas@48028
   777
        and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all
Andreas@48028
   778
      from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun)
Andreas@48028
   779
      with fg have "map_default undefined g' = (\<lambda>a. c)"
Andreas@48030
   780
        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1, symmetric])
Andreas@48028
   781
      with True show "g' = empty"
Andreas@48028
   782
        by -(rule map_default_inject(2)[OF _ fin g], auto)
Andreas@48028
   783
    qed
Andreas@48028
   784
    show ?thesis unfolding finfun_rec_def using `finite UNIV` True
Andreas@48028
   785
      unfolding Let_def the default by(simp)
Andreas@48028
   786
  next
Andreas@48028
   787
    case False
Andreas@48028
   788
    have the: "The ?the = (\<lambda>a :: 'a. Some c)"
Andreas@48028
   789
    proof (rule the_equality)
Andreas@48028
   790
      from False True show "?the (\<lambda>a :: 'a. Some c)"
Andreas@48028
   791
        by(auto simp add: map_default_def [abs_def] finfun_const_def dom_def ran_def)
Andreas@48028
   792
    next
Andreas@48028
   793
      fix g' :: "'a \<rightharpoonup> 'b"
Andreas@48028
   794
      assume "?the g'"
Andreas@48036
   795
      hence fg: "(K$ c) = Abs_finfun (map_default undefined g')"
Andreas@48028
   796
        and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all
Andreas@48028
   797
      from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun)
Andreas@48028
   798
      with fg have "map_default undefined g' = (\<lambda>a. c)"
Andreas@48028
   799
        by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1])
Andreas@48028
   800
      with True False show "g' = (\<lambda>a::'a. Some c)"
Andreas@48028
   801
        by - (rule map_default_inject(2)[OF _ fin g],
Andreas@48028
   802
          auto simp add: dom_def ran_def map_default_def [abs_def])
Andreas@48028
   803
    qed
Andreas@48028
   804
    show ?thesis unfolding finfun_rec_def using True False
Andreas@48028
   805
      unfolding Let_def the default by(simp add: dom_def map_default_def const_update_all)
Andreas@48028
   806
  qed
Andreas@48028
   807
qed
Andreas@48028
   808
Andreas@48028
   809
end
Andreas@48028
   810
Andreas@48028
   811
subsection {* Weak induction rule and case analysis for FinFuns *}
Andreas@48028
   812
Andreas@48028
   813
lemma finfun_weak_induct [consumes 0, case_names const update]:
Andreas@48036
   814
  assumes const: "\<And>b. P (K$ b)"
Andreas@48036
   815
  and update: "\<And>f a b. P f \<Longrightarrow> P (f(a $:= b))"
Andreas@48028
   816
  shows "P x"
Andreas@48030
   817
  including finfun
Andreas@48028
   818
proof(induct x rule: Abs_finfun_induct)
Andreas@48028
   819
  case (Abs_finfun y)
Andreas@48028
   820
  then obtain b where "finite {a. y a \<noteq> b}" unfolding finfun_def by blast
Andreas@48028
   821
  thus ?case using `y \<in> finfun`
Andreas@48028
   822
  proof(induct "{a. y a \<noteq> b}" arbitrary: y rule: finite_induct)
Andreas@48028
   823
    case empty
Andreas@48028
   824
    hence "\<And>a. y a = b" by blast
Andreas@48037
   825
    hence "y = (\<lambda>a. b)" by(auto)
Andreas@48028
   826
    hence "Abs_finfun y = finfun_const b" unfolding finfun_const_def by simp
Andreas@48028
   827
    thus ?case by(simp add: const)
Andreas@48028
   828
  next
Andreas@48028
   829
    case (insert a A)
Andreas@48028
   830
    note IH = `\<And>y. \<lbrakk> A = {a. y a \<noteq> b}; y \<in> finfun  \<rbrakk> \<Longrightarrow> P (Abs_finfun y)`
Andreas@48028
   831
    note y = `y \<in> finfun`
Andreas@48028
   832
    with `insert a A = {a. y a \<noteq> b}` `a \<notin> A`
Andreas@48028
   833
    have "A = {a'. (y(a := b)) a' \<noteq> b}" "y(a := b) \<in> finfun" by auto
Andreas@48028
   834
    from IH[OF this] have "P (finfun_update (Abs_finfun (y(a := b))) a (y a))" by(rule update)
Andreas@48028
   835
    thus ?case using y unfolding finfun_update_def by simp
Andreas@48028
   836
  qed
Andreas@48028
   837
qed
Andreas@48028
   838
Andreas@48028
   839
lemma finfun_exhaust_disj: "(\<exists>b. x = finfun_const b) \<or> (\<exists>f a b. x = finfun_update f a b)"
Andreas@48028
   840
by(induct x rule: finfun_weak_induct) blast+
Andreas@48028
   841
Andreas@48028
   842
lemma finfun_exhaust:
Andreas@48036
   843
  obtains b where "x = (K$ b)"
Andreas@48036
   844
        | f a b where "x = f(a $:= b)"
Andreas@48028
   845
by(atomize_elim)(rule finfun_exhaust_disj)
Andreas@48028
   846
Andreas@48028
   847
lemma finfun_rec_unique:
Andreas@48034
   848
  fixes f :: "'a \<Rightarrow>f 'b \<Rightarrow> 'c"
Andreas@48036
   849
  assumes c: "\<And>c. f (K$ c) = cnst c"
Andreas@48036
   850
  and u: "\<And>g a b. f (g(a $:= b)) = upd g a b (f g)"
Andreas@48036
   851
  and c': "\<And>c. f' (K$ c) = cnst c"
Andreas@48036
   852
  and u': "\<And>g a b. f' (g(a $:= b)) = upd g a b (f' g)"
Andreas@48028
   853
  shows "f = f'"
Andreas@48028
   854
proof
Andreas@48034
   855
  fix g :: "'a \<Rightarrow>f 'b"
Andreas@48028
   856
  show "f g = f' g"
Andreas@48028
   857
    by(induct g rule: finfun_weak_induct)(auto simp add: c u c' u')
Andreas@48028
   858
qed
Andreas@48028
   859
Andreas@48028
   860
Andreas@48028
   861
subsection {* Function application *}
Andreas@48028
   862
Andreas@48035
   863
notation finfun_apply (infixl "$" 999)
Andreas@48028
   864
Andreas@48028
   865
interpretation finfun_apply_aux: finfun_rec_wf_aux "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c"
Andreas@48028
   866
by(unfold_locales) auto
Andreas@48028
   867
Andreas@48028
   868
interpretation finfun_apply: finfun_rec_wf "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c"
Andreas@48028
   869
proof(unfold_locales)
Andreas@48028
   870
  fix b' b :: 'a
Andreas@48028
   871
  assume fin: "finite (UNIV :: 'b set)"
Andreas@48028
   872
  { fix A :: "'b set"
Andreas@48028
   873
    interpret comp_fun_commute "\<lambda>a'. If (a = a') b'" by(rule finfun_apply_aux.upd_left_comm)
Andreas@48028
   874
    from fin have "finite A" by(auto intro: finite_subset)
Andreas@48028
   875
    hence "Finite_Set.fold (\<lambda>a'. If (a = a') b') b A = (if a \<in> A then b' else b)"
Andreas@48028
   876
      by induct auto }
Andreas@48028
   877
  from this[of UNIV] show "Finite_Set.fold (\<lambda>a'. If (a = a') b') b UNIV = b'" by simp
Andreas@48028
   878
qed
Andreas@48028
   879
Andreas@48035
   880
lemma finfun_apply_def: "op $ = (\<lambda>f a. finfun_rec (\<lambda>b. b) (\<lambda>a' b c. if (a = a') then b else c) f)"
Andreas@48029
   881
proof(rule finfun_rec_unique)
Andreas@48036
   882
  fix c show "op $ (K$ c) = (\<lambda>a. c)" by(simp add: finfun_const.rep_eq)
Andreas@48029
   883
next
Andreas@48036
   884
  fix g a b show "op $ g(a $:= b) = (\<lambda>c. if c = a then b else g $ c)"
Andreas@48029
   885
    by(auto simp add: finfun_update_def fun_upd_finfun Abs_finfun_inverse finfun_apply)
Andreas@48029
   886
qed auto
Andreas@48028
   887
Andreas@48036
   888
lemma finfun_upd_apply: "f(a $:= b) $ a' = (if a = a' then b else f $ a')"
Andreas@48035
   889
  and finfun_upd_apply_code [code]: "(finfun_update_code f a b) $ a' = (if a = a' then b else f $ a')"
Andreas@48028
   890
by(simp_all add: finfun_apply_def)
Andreas@48028
   891
Andreas@48036
   892
lemma finfun_const_apply [simp, code]: "(K$ b) $ a = b"
Andreas@48029
   893
by(simp add: finfun_apply_def)
Andreas@48029
   894
Andreas@48028
   895
lemma finfun_upd_apply_same [simp]:
Andreas@48036
   896
  "f(a $:= b) $ a = b"
Andreas@48028
   897
by(simp add: finfun_upd_apply)
Andreas@48028
   898
Andreas@48028
   899
lemma finfun_upd_apply_other [simp]:
Andreas@48036
   900
  "a \<noteq> a' \<Longrightarrow> f(a $:= b) $ a' = f $ a'"
Andreas@48028
   901
by(simp add: finfun_upd_apply)
Andreas@48028
   902
Andreas@48035
   903
lemma finfun_ext: "(\<And>a. f $ a = g $ a) \<Longrightarrow> f = g"
Andreas@48029
   904
by(auto simp add: finfun_apply_inject[symmetric] simp del: finfun_apply_inject)
Andreas@48028
   905
Andreas@48035
   906
lemma expand_finfun_eq: "(f = g) = (op $ f = op $ g)"
Andreas@48028
   907
by(auto intro: finfun_ext)
Andreas@48028
   908
Andreas@48036
   909
lemma finfun_const_inject [simp]: "(K$ b) = (K$ b') \<equiv> b = b'"
Andreas@48028
   910
by(simp add: expand_finfun_eq fun_eq_iff)
Andreas@48028
   911
Andreas@48028
   912
lemma finfun_const_eq_update:
Andreas@48036
   913
  "((K$ b) = f(a $:= b')) = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> f $ a' = b))"
Andreas@48028
   914
by(auto simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
Andreas@48028
   915
Andreas@48028
   916
subsection {* Function composition *}
Andreas@48028
   917
Andreas@48037
   918
definition finfun_comp :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow>f 'a \<Rightarrow> 'c \<Rightarrow>f 'b" (infixr "o$" 55)
Andreas@48037
   919
where [code del]: "g o$ f  = finfun_rec (\<lambda>b. (K$ g b)) (\<lambda>a b c. c(a $:= g b)) f"
Andreas@48037
   920
Andreas@48037
   921
notation (xsymbols) finfun_comp (infixr "\<circ>$" 55)
Andreas@48028
   922
Andreas@48036
   923
interpretation finfun_comp_aux: finfun_rec_wf_aux "(\<lambda>b. (K$ g b))" "(\<lambda>a b c. c(a $:= g b))"
Andreas@48028
   924
by(unfold_locales)(auto simp add: finfun_upd_apply intro: finfun_ext)
Andreas@48028
   925
Andreas@48036
   926
interpretation finfun_comp: finfun_rec_wf "(\<lambda>b. (K$ g b))" "(\<lambda>a b c. c(a $:= g b))"
Andreas@48028
   927
proof
Andreas@48028
   928
  fix b' b :: 'a
Andreas@48028
   929
  assume fin: "finite (UNIV :: 'c set)"
Andreas@48028
   930
  { fix A :: "'c set"
Andreas@48028
   931
    from fin have "finite A" by(auto intro: finite_subset)
Andreas@48036
   932
    hence "Finite_Set.fold (\<lambda>(a :: 'c) c. c(a $:= g b')) (K$ g b) A =
Andreas@48028
   933
      Abs_finfun (\<lambda>a. if a \<in> A then g b' else g b)"
Andreas@48028
   934
      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 fun_eq_iff fin) }
Andreas@48036
   935
  from this[of UNIV] show "Finite_Set.fold (\<lambda>(a :: 'c) c. c(a $:= g b')) (K$ g b) UNIV = (K$ g b')"
Andreas@48028
   936
    by(simp add: finfun_const_def)
Andreas@48028
   937
qed
Andreas@48028
   938
Andreas@48028
   939
lemma finfun_comp_const [simp, code]:
Andreas@48037
   940
  "g \<circ>$ (K$ c) = (K$ g c)"
Andreas@48028
   941
by(simp add: finfun_comp_def)
Andreas@48028
   942
Andreas@48037
   943
lemma finfun_comp_update [simp]: "g \<circ>$ (f(a $:= b)) = (g \<circ>$ f)(a $:= g b)"
Andreas@48037
   944
  and finfun_comp_update_code [code]: 
Andreas@48037
   945
  "g \<circ>$ (finfun_update_code f a b) = finfun_update_code (g \<circ>$ f) a (g b)"
Andreas@48028
   946
by(simp_all add: finfun_comp_def)
Andreas@48028
   947
Andreas@48028
   948
lemma finfun_comp_apply [simp]:
Andreas@48037
   949
  "op $ (g \<circ>$ f) = g \<circ> op $ f"
Andreas@48035
   950
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_upd_apply)
Andreas@48028
   951
Andreas@48037
   952
lemma finfun_comp_comp_collapse [simp]: "f \<circ>$ g \<circ>$ h = (f \<circ> g) \<circ>$ h"
Andreas@48028
   953
by(induct h rule: finfun_weak_induct) simp_all
Andreas@48028
   954
Andreas@48037
   955
lemma finfun_comp_const1 [simp]: "(\<lambda>x. c) \<circ>$ f = (K$ c)"
Andreas@48028
   956
by(induct f rule: finfun_weak_induct)(auto intro: finfun_ext simp add: finfun_upd_apply)
Andreas@48028
   957
Andreas@48037
   958
lemma finfun_comp_id1 [simp]: "(\<lambda>x. x) \<circ>$ f = f" "id \<circ>$ f = f"
Andreas@48028
   959
by(induct f rule: finfun_weak_induct) auto
Andreas@48028
   960
Andreas@48037
   961
lemma finfun_comp_conv_comp: "g \<circ>$ f = Abs_finfun (g \<circ> op $ f)"
Andreas@48030
   962
  including finfun
Andreas@48028
   963
proof -
Andreas@48037
   964
  have "(\<lambda>f. g \<circ>$ f) = (\<lambda>f. Abs_finfun (g \<circ> op $ f))"
Andreas@48028
   965
  proof(rule finfun_rec_unique)
Andreas@48036
   966
    { fix c show "Abs_finfun (g \<circ> op $ (K$ c)) = (K$ g c)"
Andreas@48028
   967
        by(simp add: finfun_comp_def o_def)(simp add: finfun_const_def) }
Andreas@48036
   968
    { fix g' a b show "Abs_finfun (g \<circ> op $ g'(a $:= b)) = (Abs_finfun (g \<circ> op $ g'))(a $:= g b)"
Andreas@48028
   969
      proof -
Andreas@48028
   970
        obtain y where y: "y \<in> finfun" and g': "g' = Abs_finfun y" by(cases g')
Andreas@48035
   971
        moreover hence "(g \<circ> op $ g') \<in> finfun" by(simp add: finfun_left_compose)
Andreas@48035
   972
        moreover have "g \<circ> y(a := b) = (g \<circ> y)(a := g b)" by(auto)
Andreas@48029
   973
        ultimately show ?thesis by(simp add: finfun_comp_def finfun_update_def)
Andreas@48028
   974
      qed }
Andreas@48028
   975
  qed auto
Andreas@48028
   976
  thus ?thesis by(auto simp add: fun_eq_iff)
Andreas@48028
   977
qed
Andreas@48028
   978
Andreas@48037
   979
definition finfun_comp2 :: "'b \<Rightarrow>f 'c \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>f 'c" (infixr "$o" 55)
Andreas@48037
   980
where [code del]: "g $o f = Abs_finfun (op $ g \<circ> f)"
Andreas@48037
   981
Andreas@48037
   982
notation (xsymbol) finfun_comp2 (infixr "$\<circ>" 55)
Andreas@48028
   983
Andreas@48036
   984
lemma finfun_comp2_const [code, simp]: "finfun_comp2 (K$ c) f = (K$ c)"
Andreas@48030
   985
  including finfun
Andreas@48028
   986
by(simp add: finfun_comp2_def finfun_const_def comp_def)
Andreas@48028
   987
Andreas@48028
   988
lemma finfun_comp2_update:
Andreas@48030
   989
  includes finfun
Andreas@48028
   990
  assumes inj: "inj f"
Andreas@48036
   991
  shows "finfun_comp2 (g(b $:= c)) f = (if b \<in> range f then (finfun_comp2 g f)(inv f b $:= c) else finfun_comp2 g f)"
Andreas@48028
   992
proof(cases "b \<in> range f")
Andreas@48028
   993
  case True
Andreas@48035
   994
  from inj have "\<And>x. (op $ g)(f x := c) \<circ> f = (op $ g \<circ> f)(x := c)" by(auto intro!: ext dest: injD)
Andreas@48028
   995
  with inj True show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def finfun_right_compose)
Andreas@48028
   996
next
Andreas@48028
   997
  case False
Andreas@48035
   998
  hence "(op $ g)(b := c) \<circ> f = op $ g \<circ> f" by(auto simp add: fun_eq_iff)
Andreas@48028
   999
  with False show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def)
Andreas@48028
  1000
qed
Andreas@48028
  1001
Andreas@48028
  1002
subsection {* Universal quantification *}
Andreas@48028
  1003
Andreas@48034
  1004
definition finfun_All_except :: "'a list \<Rightarrow> 'a \<Rightarrow>f bool \<Rightarrow> bool"
Andreas@48035
  1005
where [code del]: "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> P $ a"
Andreas@48028
  1006
Andreas@48036
  1007
lemma finfun_All_except_const: "finfun_All_except A (K$ b) \<longleftrightarrow> b \<or> set A = UNIV"
Andreas@48028
  1008
by(auto simp add: finfun_All_except_def)
Andreas@48028
  1009
Andreas@48028
  1010
lemma finfun_All_except_const_finfun_UNIV_code [code]:
Andreas@48036
  1011
  "finfun_All_except A (K$ b) = (b \<or> is_list_UNIV A)"
Andreas@48028
  1012
by(simp add: finfun_All_except_const is_list_UNIV_iff)
Andreas@48028
  1013
Andreas@48035
  1014
lemma finfun_All_except_update:
Andreas@48036
  1015
  "finfun_All_except A f(a $:= b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)"
Andreas@48028
  1016
by(fastforce simp add: finfun_All_except_def finfun_upd_apply)
Andreas@48028
  1017
Andreas@48028
  1018
lemma finfun_All_except_update_code [code]:
Andreas@48028
  1019
  fixes a :: "'a :: card_UNIV"
Andreas@48028
  1020
  shows "finfun_All_except A (finfun_update_code f a b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)"
Andreas@48028
  1021
by(simp add: finfun_All_except_update)
Andreas@48028
  1022
Andreas@48034
  1023
definition finfun_All :: "'a \<Rightarrow>f bool \<Rightarrow> bool"
Andreas@48028
  1024
where "finfun_All = finfun_All_except []"
Andreas@48028
  1025
Andreas@48036
  1026
lemma finfun_All_const [simp]: "finfun_All (K$ b) = b"
Andreas@48028
  1027
by(simp add: finfun_All_def finfun_All_except_def)
Andreas@48028
  1028
Andreas@48036
  1029
lemma finfun_All_update: "finfun_All f(a $:= b) = (b \<and> finfun_All_except [a] f)"
Andreas@48028
  1030
by(simp add: finfun_All_def finfun_All_except_update)
Andreas@48028
  1031
Andreas@48035
  1032
lemma finfun_All_All: "finfun_All P = All (op $ P)"
Andreas@48028
  1033
by(simp add: finfun_All_def finfun_All_except_def)
Andreas@48028
  1034
Andreas@48028
  1035
Andreas@48034
  1036
definition finfun_Ex :: "'a \<Rightarrow>f bool \<Rightarrow> bool"
Andreas@48037
  1037
where "finfun_Ex P = Not (finfun_All (Not \<circ>$ P))"
Andreas@48028
  1038
Andreas@48035
  1039
lemma finfun_Ex_Ex: "finfun_Ex P = Ex (op $ P)"
Andreas@48028
  1040
unfolding finfun_Ex_def finfun_All_All by simp
Andreas@48028
  1041
Andreas@48036
  1042
lemma finfun_Ex_const [simp]: "finfun_Ex (K$ b) = b"
Andreas@48028
  1043
by(simp add: finfun_Ex_def)
Andreas@48028
  1044
Andreas@48028
  1045
Andreas@48028
  1046
subsection {* A diagonal operator for FinFuns *}
Andreas@48028
  1047
Andreas@48038
  1048
definition finfun_Diag :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow>f 'c \<Rightarrow> 'a \<Rightarrow>f ('b \<times> 'c)" ("(1'($_,/ _$'))" [0, 0] 1000)
Andreas@48038
  1049
where [code del]: "($f, g$) = finfun_rec (\<lambda>b. Pair b \<circ>$ g) (\<lambda>a b c. c(a $:= (b, g $ a))) f"
Andreas@48028
  1050
Andreas@48037
  1051
interpretation finfun_Diag_aux: finfun_rec_wf_aux "\<lambda>b. Pair b \<circ>$ g" "\<lambda>a b c. c(a $:= (b, g $ a))"
Andreas@48028
  1052
by(unfold_locales)(simp_all add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
Andreas@48028
  1053
Andreas@48037
  1054
interpretation finfun_Diag: finfun_rec_wf "\<lambda>b. Pair b \<circ>$ g" "\<lambda>a b c. c(a $:= (b, g $ a))"
Andreas@48028
  1055
proof
Andreas@48028
  1056
  fix b' b :: 'a
Andreas@48028
  1057
  assume fin: "finite (UNIV :: 'c set)"
Andreas@48028
  1058
  { fix A :: "'c set"
Andreas@48036
  1059
    interpret comp_fun_commute "\<lambda>a c. c(a $:= (b', g $ a))" by(rule finfun_Diag_aux.upd_left_comm)
Andreas@48028
  1060
    from fin have "finite A" by(auto intro: finite_subset)
Andreas@48037
  1061
    hence "Finite_Set.fold (\<lambda>a c. c(a $:= (b', g $ a))) (Pair b \<circ>$ g) A =
Andreas@48035
  1062
      Abs_finfun (\<lambda>a. (if a \<in> A then b' else b, g $ a))"
Andreas@48028
  1063
      by(induct)(simp_all add: finfun_const_def finfun_comp_conv_comp o_def,
Andreas@48028
  1064
                 auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) }
Andreas@48037
  1065
  from this[of UNIV] show "Finite_Set.fold (\<lambda>a c. c(a $:= (b', g $ a))) (Pair b \<circ>$ g) UNIV = Pair b' \<circ>$ g"
Andreas@48028
  1066
    by(simp add: finfun_const_def finfun_comp_conv_comp o_def)
Andreas@48028
  1067
qed
Andreas@48028
  1068
Andreas@48038
  1069
lemma finfun_Diag_const1: "($K$ b, g$) = Pair b \<circ>$ g"
Andreas@48028
  1070
by(simp add: finfun_Diag_def)
Andreas@48028
  1071
Andreas@48028
  1072
text {*
Andreas@48038
  1073
  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 @{term "op \<circ>$"}.
Andreas@48028
  1074
*}
Andreas@48028
  1075
Andreas@48028
  1076
lemma finfun_Diag_const_code [code]:
Andreas@48038
  1077
  "($K$ b, K$ c$) = (K$ (b, c))"
Andreas@48038
  1078
  "($K$ b, finfun_update_code g a c$) = finfun_update_code ($K$ b, g$) a (b, c)"
Andreas@48028
  1079
by(simp_all add: finfun_Diag_const1)
Andreas@48028
  1080
Andreas@48038
  1081
lemma finfun_Diag_update1: "($f(a $:= b), g$) = ($f, g$)(a $:= (b, g $ a))"
Andreas@48038
  1082
  and finfun_Diag_update1_code [code]: "($finfun_update_code f a b, g$) = ($f, g$)(a $:= (b, g $ a))"
Andreas@48028
  1083
by(simp_all add: finfun_Diag_def)
Andreas@48028
  1084
Andreas@48038
  1085
lemma finfun_Diag_const2: "($f, K$ c$) = (\<lambda>b. (b, c)) \<circ>$ f"
Andreas@48028
  1086
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)
Andreas@48028
  1087
Andreas@48038
  1088
lemma finfun_Diag_update2: "($f, g(a $:= c)$) = ($f, g$)(a $:= (f $ a, c))"
Andreas@48028
  1089
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1)
Andreas@48028
  1090
Andreas@48038
  1091
lemma finfun_Diag_const_const [simp]: "($K$ b, K$ c$) = (K$ (b, c))"
Andreas@48028
  1092
by(simp add: finfun_Diag_const1)
Andreas@48028
  1093
Andreas@48028
  1094
lemma finfun_Diag_const_update:
Andreas@48038
  1095
  "($K$ b, g(a $:= c)$) = ($K$ b, g$)(a $:= (b, c))"
Andreas@48028
  1096
by(simp add: finfun_Diag_const1)
Andreas@48028
  1097
Andreas@48028
  1098
lemma finfun_Diag_update_const:
Andreas@48038
  1099
  "($f(a $:= b), K$ c$) = ($f, K$ c$)(a $:= (b, c))"
Andreas@48028
  1100
by(simp add: finfun_Diag_def)
Andreas@48028
  1101
Andreas@48028
  1102
lemma finfun_Diag_update_update:
Andreas@48038
  1103
  "($f(a $:= b), g(a' $:= c)$) = (if a = a' then ($f, g$)(a $:= (b, c)) else ($f, g$)(a $:= (b, g $ a))(a' $:= (f $ a', c)))"
Andreas@48028
  1104
by(auto simp add: finfun_Diag_update1 finfun_Diag_update2)
Andreas@48028
  1105
Andreas@48038
  1106
lemma finfun_Diag_apply [simp]: "op $ ($f, g$) = (\<lambda>x. (f $ x, g $ x))"
Andreas@48035
  1107
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_Diag_const1 finfun_Diag_update1 finfun_upd_apply)
Andreas@48028
  1108
Andreas@48028
  1109
lemma finfun_Diag_conv_Abs_finfun:
Andreas@48038
  1110
  "($f, g$) = Abs_finfun ((\<lambda>x. (f $ x, g $ x)))"
Andreas@48030
  1111
  including finfun
Andreas@48028
  1112
proof -
Andreas@48038
  1113
  have "(\<lambda>f :: 'a \<Rightarrow>f 'b. ($f, g$)) = (\<lambda>f. Abs_finfun ((\<lambda>x. (f $ x, g $ x))))"
Andreas@48028
  1114
  proof(rule finfun_rec_unique)
Andreas@48037
  1115
    { fix c show "Abs_finfun (\<lambda>x. ((K$ c) $ x, g $ x)) = Pair c \<circ>$ g"
Andreas@48029
  1116
        by(simp add: finfun_comp_conv_comp o_def finfun_const_def) }
Andreas@48028
  1117
    { fix g' a b
Andreas@48036
  1118
      show "Abs_finfun (\<lambda>x. (g'(a $:= b) $ x, g $ x)) =
Andreas@48036
  1119
            (Abs_finfun (\<lambda>x. (g' $ x, g $ x)))(a $:= (b, g $ a))"
Andreas@48029
  1120
        by(auto simp add: finfun_update_def fun_eq_iff simp del: fun_upd_apply) simp }
Andreas@48028
  1121
  qed(simp_all add: finfun_Diag_const1 finfun_Diag_update1)
Andreas@48028
  1122
  thus ?thesis by(auto simp add: fun_eq_iff)
Andreas@48028
  1123
qed
Andreas@48028
  1124
Andreas@48038
  1125
lemma finfun_Diag_eq: "($f, g$) = ($f', g'$) \<longleftrightarrow> f = f' \<and> g = g'"
Andreas@48028
  1126
by(auto simp add: expand_finfun_eq fun_eq_iff)
Andreas@48028
  1127
Andreas@48034
  1128
definition finfun_fst :: "'a \<Rightarrow>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>f 'b"
Andreas@48037
  1129
where [code]: "finfun_fst f = fst \<circ>$ f"
Andreas@48028
  1130
Andreas@48036
  1131
lemma finfun_fst_const: "finfun_fst (K$ bc) = (K$ fst bc)"
Andreas@48028
  1132
by(simp add: finfun_fst_def)
Andreas@48028
  1133
Andreas@48036
  1134
lemma finfun_fst_update: "finfun_fst (f(a $:= bc)) = (finfun_fst f)(a $:= fst bc)"
Andreas@48036
  1135
  and finfun_fst_update_code: "finfun_fst (finfun_update_code f a bc) = (finfun_fst f)(a $:= fst bc)"
Andreas@48028
  1136
by(simp_all add: finfun_fst_def)
Andreas@48028
  1137
Andreas@48037
  1138
lemma finfun_fst_comp_conv: "finfun_fst (f \<circ>$ g) = (fst \<circ> f) \<circ>$ g"
Andreas@48028
  1139
by(simp add: finfun_fst_def)
Andreas@48028
  1140
Andreas@48038
  1141
lemma finfun_fst_conv [simp]: "finfun_fst ($f, g$) = f"
Andreas@48028
  1142
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)
Andreas@48028
  1143
Andreas@48037
  1144
lemma finfun_fst_conv_Abs_finfun: "finfun_fst = (\<lambda>f. Abs_finfun (fst \<circ> op $ f))"
Andreas@48029
  1145
by(simp add: finfun_fst_def [abs_def] finfun_comp_conv_comp)
Andreas@48028
  1146
Andreas@48028
  1147
Andreas@48034
  1148
definition finfun_snd :: "'a \<Rightarrow>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>f 'c"
Andreas@48037
  1149
where [code]: "finfun_snd f = snd \<circ>$ f"
Andreas@48028
  1150
Andreas@48036
  1151
lemma finfun_snd_const: "finfun_snd (K$ bc) = (K$ snd bc)"
Andreas@48028
  1152
by(simp add: finfun_snd_def)
Andreas@48028
  1153
Andreas@48036
  1154
lemma finfun_snd_update: "finfun_snd (f(a $:= bc)) = (finfun_snd f)(a $:= snd bc)"
Andreas@48036
  1155
  and finfun_snd_update_code [code]: "finfun_snd (finfun_update_code f a bc) = (finfun_snd f)(a $:= snd bc)"
Andreas@48028
  1156
by(simp_all add: finfun_snd_def)
Andreas@48028
  1157
Andreas@48037
  1158
lemma finfun_snd_comp_conv: "finfun_snd (f \<circ>$ g) = (snd \<circ> f) \<circ>$ g"
Andreas@48028
  1159
by(simp add: finfun_snd_def)
Andreas@48028
  1160
Andreas@48038
  1161
lemma finfun_snd_conv [simp]: "finfun_snd ($f, g$) = g"
Andreas@48028
  1162
apply(induct f rule: finfun_weak_induct)
Andreas@48028
  1163
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)
Andreas@48028
  1164
done
Andreas@48028
  1165
Andreas@48037
  1166
lemma finfun_snd_conv_Abs_finfun: "finfun_snd = (\<lambda>f. Abs_finfun (snd \<circ> op $ f))"
Andreas@48029
  1167
by(simp add: finfun_snd_def [abs_def] finfun_comp_conv_comp)
Andreas@48028
  1168
Andreas@48038
  1169
lemma finfun_Diag_collapse [simp]: "($finfun_fst f, finfun_snd f$) = f"
Andreas@48028
  1170
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)
Andreas@48028
  1171
Andreas@48028
  1172
subsection {* Currying for FinFuns *}
Andreas@48028
  1173
Andreas@48034
  1174
definition finfun_curry :: "('a \<times> 'b) \<Rightarrow>f 'c \<Rightarrow> 'a \<Rightarrow>f 'b \<Rightarrow>f 'c"
Andreas@48036
  1175
where [code del]: "finfun_curry = finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c)))"
Andreas@48028
  1176
Andreas@48036
  1177
interpretation finfun_curry_aux: finfun_rec_wf_aux "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))"
Andreas@48028
  1178
apply(unfold_locales)
Andreas@48028
  1179
apply(auto simp add: split_def finfun_update_twist finfun_upd_apply split_paired_all finfun_update_const_same)
Andreas@48028
  1180
done
Andreas@48028
  1181
Andreas@48036
  1182
interpretation finfun_curry: finfun_rec_wf "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))"
Andreas@48028
  1183
proof(unfold_locales)
Andreas@48028
  1184
  fix b' b :: 'b
Andreas@48028
  1185
  assume fin: "finite (UNIV :: ('c \<times> 'a) set)"
Andreas@48028
  1186
  hence fin1: "finite (UNIV :: 'c set)" and fin2: "finite (UNIV :: 'a set)"
Andreas@48028
  1187
    unfolding UNIV_Times_UNIV[symmetric]
Andreas@48028
  1188
    by(fastforce dest: finite_cartesian_productD1 finite_cartesian_productD2)+
Andreas@48028
  1189
  note [simp] = Abs_finfun_inverse_finite[OF fin] Abs_finfun_inverse_finite[OF fin1] Abs_finfun_inverse_finite[OF fin2]
Andreas@48028
  1190
  { fix A :: "('c \<times> 'a) set"
Andreas@48036
  1191
    interpret comp_fun_commute "\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b'"
Andreas@48028
  1192
      by(rule finfun_curry_aux.upd_left_comm)
Andreas@48028
  1193
    from fin have "finite A" by(auto intro: finite_subset)
Andreas@48036
  1194
    hence "Finite_Set.fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(a $:= (f $ a)(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))"
Andreas@48029
  1195
      by induct (simp_all, auto simp add: finfun_update_def finfun_const_def split_def intro!: arg_cong[where f="Abs_finfun"] ext) }
Andreas@48028
  1196
  from this[of UNIV]
Andreas@48036
  1197
  show "Finite_Set.fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b') ((finfun_const \<circ> finfun_const) b) UNIV = (finfun_const \<circ> finfun_const) b'"
Andreas@48028
  1198
    by(simp add: finfun_const_def)
Andreas@48028
  1199
qed
Andreas@48028
  1200
Andreas@48036
  1201
lemma finfun_curry_const [simp, code]: "finfun_curry (K$ c) = (K$ K$ c)"
Andreas@48028
  1202
by(simp add: finfun_curry_def)
Andreas@48028
  1203
Andreas@48028
  1204
lemma finfun_curry_update [simp]:
Andreas@48036
  1205
  "finfun_curry (f((a, b) $:= c)) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))"
Andreas@48028
  1206
  and finfun_curry_update_code [code]:
Andreas@48036
  1207
  "finfun_curry (finfun_update_code f (a, b) c) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))"
Andreas@48028
  1208
by(simp_all add: finfun_curry_def)
Andreas@48028
  1209
Andreas@48028
  1210
lemma finfun_Abs_finfun_curry: assumes fin: "f \<in> finfun"
Andreas@48028
  1211
  shows "(\<lambda>a. Abs_finfun (curry f a)) \<in> finfun"
Andreas@48030
  1212
  including finfun
Andreas@48028
  1213
proof -
Andreas@48028
  1214
  from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast
Andreas@48028
  1215
  have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force)
Andreas@48028
  1216
  hence "{a. curry f a \<noteq> (\<lambda>x. c)} = fst ` {ab. f ab \<noteq> c}"
Andreas@48028
  1217
    by(auto simp add: curry_def fun_eq_iff)
Andreas@48036
  1218
  with fin c have "finite {a.  Abs_finfun (curry f a) \<noteq> (K$ c)}"
Andreas@48028
  1219
    by(simp add: finfun_const_def finfun_curry)
Andreas@48028
  1220
  thus ?thesis unfolding finfun_def by auto
Andreas@48028
  1221
qed
Andreas@48028
  1222
Andreas@48028
  1223
lemma finfun_curry_conv_curry:
Andreas@48034
  1224
  fixes f :: "('a \<times> 'b) \<Rightarrow>f 'c"
Andreas@48029
  1225
  shows "finfun_curry f = Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply f) a))"
Andreas@48030
  1226
  including finfun
Andreas@48028
  1227
proof -
Andreas@48034
  1228
  have "finfun_curry = (\<lambda>f :: ('a \<times> 'b) \<Rightarrow>f 'c. Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply f) a)))"
Andreas@48028
  1229
  proof(rule finfun_rec_unique)
Andreas@48036
  1230
    fix c show "finfun_curry (K$ c) = (K$ K$ c)" by simp
Andreas@48034
  1231
    fix f a
Andreas@48036
  1232
    show "finfun_curry (f(a $:= c)) = (finfun_curry f)(fst a $:= (finfun_curry f $ (fst a))(snd a $:= c))"
Andreas@48034
  1233
      by(cases a) simp
Andreas@48036
  1234
    show "Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply (K$ c)) a)) = (K$ K$ c)"
Andreas@48034
  1235
      by(simp add: finfun_curry_def finfun_const_def curry_def)
Andreas@48034
  1236
    fix g b
Andreas@48036
  1237
    show "Abs_finfun (\<lambda>aa. Abs_finfun (curry (op $ g(a $:= b)) aa)) =
Andreas@48036
  1238
      (Abs_finfun (\<lambda>a. Abs_finfun (curry (op $ g) a)))(
Andreas@48036
  1239
      fst a $:= ((Abs_finfun (\<lambda>a. Abs_finfun (curry (op $ g) a))) $ (fst a))(snd a $:= b))"
Andreas@48035
  1240
      by(cases a)(auto intro!: ext arg_cong[where f=Abs_finfun] simp add: finfun_curry_def finfun_update_def finfun_Abs_finfun_curry)
Andreas@48028
  1241
  qed
Andreas@48028
  1242
  thus ?thesis by(auto simp add: fun_eq_iff)
Andreas@48028
  1243
qed
Andreas@48028
  1244
Andreas@48028
  1245
subsection {* Executable equality for FinFuns *}
Andreas@48028
  1246
Andreas@48038
  1247
lemma eq_finfun_All_ext: "(f = g) \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))"
Andreas@48028
  1248
by(simp add: expand_finfun_eq fun_eq_iff finfun_All_All o_def)
Andreas@48028
  1249
Andreas@48028
  1250
instantiation finfun :: ("{card_UNIV,equal}",equal) equal begin
Andreas@48038
  1251
definition eq_finfun_def [code]: "HOL.equal f g \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))"
Andreas@48028
  1252
instance by(intro_classes)(simp add: eq_finfun_All_ext eq_finfun_def)
Andreas@48028
  1253
end
Andreas@48028
  1254
Andreas@48028
  1255
lemma [code nbe]:
Andreas@48034
  1256
  "HOL.equal (f :: _ \<Rightarrow>f _) f \<longleftrightarrow> True"
Andreas@48028
  1257
  by (fact equal_refl)
Andreas@48028
  1258
Andreas@48028
  1259
subsection {* An operator that explicitly removes all redundant updates in the generated representations *}
Andreas@48028
  1260
Andreas@48034
  1261
definition finfun_clearjunk :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow>f 'b"
Andreas@48028
  1262
where [simp, code del]: "finfun_clearjunk = id"
Andreas@48028
  1263
Andreas@48036
  1264
lemma finfun_clearjunk_const [code]: "finfun_clearjunk (K$ b) = (K$ b)"
Andreas@48028
  1265
by simp
Andreas@48028
  1266
Andreas@48036
  1267
lemma finfun_clearjunk_update [code]: 
Andreas@48036
  1268
  "finfun_clearjunk (finfun_update_code f a b) = f(a $:= b)"
Andreas@48028
  1269
by simp
Andreas@48028
  1270
Andreas@48028
  1271
subsection {* The domain of a FinFun as a FinFun *}
Andreas@48028
  1272
Andreas@48034
  1273
definition finfun_dom :: "('a \<Rightarrow>f 'b) \<Rightarrow> ('a \<Rightarrow>f bool)"
Andreas@48035
  1274
where [code del]: "finfun_dom f = Abs_finfun (\<lambda>a. f $ a \<noteq> finfun_default f)"
Andreas@48028
  1275
Andreas@48028
  1276
lemma finfun_dom_const:
Andreas@48036
  1277
  "finfun_dom ((K$ c) :: 'a \<Rightarrow>f 'b) = (K$ finite (UNIV :: 'a set) \<and> c \<noteq> undefined)"
Andreas@48028
  1278
unfolding finfun_dom_def finfun_default_const
Andreas@48028
  1279
by(auto)(simp_all add: finfun_const_def)
Andreas@48028
  1280
Andreas@48028
  1281
text {*
Andreas@48028
  1282
  @{term "finfun_dom" } raises an exception when called on a FinFun whose domain is a finite type. 
Andreas@48028
  1283
  For such FinFuns, the default value (and as such the domain) is undefined.
Andreas@48028
  1284
*}
Andreas@48028
  1285
Andreas@48028
  1286
lemma finfun_dom_const_code [code]:
Andreas@48036
  1287
  "finfun_dom ((K$ c) :: ('a :: card_UNIV) \<Rightarrow>f 'b) = 
Andreas@48036
  1288
   (if card_UNIV (TYPE('a)) = 0 then (K$ False) else FinFun.code_abort (\<lambda>_. finfun_dom (K$ c)))"
Andreas@48028
  1289
unfolding card_UNIV_eq_0_infinite_UNIV
Andreas@48028
  1290
by(simp add: finfun_dom_const)
Andreas@48028
  1291
Andreas@48035
  1292
lemma finfun_dom_finfunI: "(\<lambda>a. f $ a \<noteq> finfun_default f) \<in> finfun"
Andreas@48028
  1293
using finite_finfun_default[of f]
Andreas@48029
  1294
by(simp add: finfun_def exI[where x=False])
Andreas@48028
  1295
Andreas@48028
  1296
lemma finfun_dom_update [simp]:
Andreas@48036
  1297
  "finfun_dom (f(a $:= b)) = (finfun_dom f)(a $:= (b \<noteq> finfun_default f))"
Andreas@48030
  1298
including finfun unfolding finfun_dom_def finfun_update_def
Andreas@48029
  1299
apply(simp add: finfun_default_update_const fun_upd_apply finfun_dom_finfunI)
Andreas@48028
  1300
apply(fold finfun_update.rep_eq)
Andreas@48029
  1301
apply(simp add: finfun_upd_apply fun_eq_iff fun_upd_def finfun_default_update_const)
Andreas@48028
  1302
done
Andreas@48028
  1303
Andreas@48028
  1304
lemma finfun_dom_update_code [code]:
Andreas@48028
  1305
  "finfun_dom (finfun_update_code f a b) = finfun_update_code (finfun_dom f) a (b \<noteq> finfun_default f)"
Andreas@48028
  1306
by(simp)
Andreas@48028
  1307
Andreas@48035
  1308
lemma finite_finfun_dom: "finite {x. finfun_dom f $ x}"
Andreas@48028
  1309
proof(induct f rule: finfun_weak_induct)
Andreas@48028
  1310
  case (const b)
Andreas@48028
  1311
  thus ?case
Andreas@48028
  1312
    by (cases "finite (UNIV :: 'a set) \<and> b \<noteq> undefined")
Andreas@48028
  1313
      (auto simp add: finfun_dom_const UNIV_def [symmetric] Set.empty_def [symmetric])
Andreas@48028
  1314
next
Andreas@48028
  1315
  case (update f a b)
Andreas@48036
  1316
  have "{x. finfun_dom f(a $:= b) $ x} =
Andreas@48035
  1317
    (if b = finfun_default f then {x. finfun_dom f $ x} - {a} else insert a {x. finfun_dom f $ x})"
Andreas@48028
  1318
    by (auto simp add: finfun_upd_apply split: split_if_asm)
Andreas@48028
  1319
  thus ?case using update by simp
Andreas@48028
  1320
qed
Andreas@48028
  1321
Andreas@48028
  1322
Andreas@48028
  1323
subsection {* The domain of a FinFun as a sorted list *}
Andreas@48028
  1324
Andreas@48034
  1325
definition finfun_to_list :: "('a :: linorder) \<Rightarrow>f 'b \<Rightarrow> 'a list"
Andreas@48028
  1326
where
Andreas@48035
  1327
  "finfun_to_list f = (THE xs. set xs = {x. finfun_dom f $ x} \<and> sorted xs \<and> distinct xs)"
Andreas@48028
  1328
Andreas@48035
  1329
lemma set_finfun_to_list [simp]: "set (finfun_to_list f) = {x. finfun_dom f $ x}" (is ?thesis1)
Andreas@48028
  1330
  and sorted_finfun_to_list: "sorted (finfun_to_list f)" (is ?thesis2)
Andreas@48028
  1331
  and distinct_finfun_to_list: "distinct (finfun_to_list f)" (is ?thesis3)
Andreas@48028
  1332
proof -
Andreas@48028
  1333
  have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"
Andreas@48028
  1334
    unfolding finfun_to_list_def
Andreas@48028
  1335
    by(rule theI')(rule finite_sorted_distinct_unique finite_finfun_dom)+
Andreas@48028
  1336
  thus ?thesis1 ?thesis2 ?thesis3 by simp_all
Andreas@48028
  1337
qed
Andreas@48028
  1338
Andreas@48036
  1339
lemma finfun_const_False_conv_bot: "op $ (K$ False) = bot"
Andreas@48028
  1340
by auto
Andreas@48028
  1341
Andreas@48036
  1342
lemma finfun_const_True_conv_top: "op $ (K$ True) = top"
Andreas@48028
  1343
by auto
Andreas@48028
  1344
Andreas@48028
  1345
lemma finfun_to_list_const:
Andreas@48036
  1346
  "finfun_to_list ((K$ c) :: ('a :: {linorder} \<Rightarrow>f 'b)) = 
Andreas@48028
  1347
  (if \<not> finite (UNIV :: 'a set) \<or> c = undefined then [] else THE xs. set xs = UNIV \<and> sorted xs \<and> distinct xs)"
Andreas@48028
  1348
by(auto simp add: finfun_to_list_def finfun_const_False_conv_bot finfun_const_True_conv_top finfun_dom_const)
Andreas@48028
  1349
Andreas@48028
  1350
lemma finfun_to_list_const_code [code]:
Andreas@48036
  1351
  "finfun_to_list ((K$ c) :: ('a :: {linorder, card_UNIV} \<Rightarrow>f 'b)) =
Andreas@48036
  1352
   (if card_UNIV (TYPE('a)) = 0 then [] else FinFun.code_abort (\<lambda>_. finfun_to_list ((K$ c) :: ('a \<Rightarrow>f 'b))))"
Andreas@48028
  1353
unfolding card_UNIV_eq_0_infinite_UNIV
Andreas@48028
  1354
by(auto simp add: finfun_to_list_const)
Andreas@48028
  1355
Andreas@48028
  1356
lemma remove1_insort_insert_same:
Andreas@48028
  1357
  "x \<notin> set xs \<Longrightarrow> remove1 x (insort_insert x xs) = xs"
Andreas@48028
  1358
by (metis insort_insert_insort remove1_insort)
Andreas@48028
  1359
Andreas@48028
  1360
lemma finfun_dom_conv:
Andreas@48035
  1361
  "finfun_dom f $ x \<longleftrightarrow> f $ x \<noteq> finfun_default f"
Andreas@48028
  1362
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_dom_const finfun_default_const finfun_default_update_const finfun_upd_apply)
Andreas@48028
  1363
Andreas@48028
  1364
lemma finfun_to_list_update:
Andreas@48036
  1365
  "finfun_to_list (f(a $:= b)) = 
Andreas@48028
  1366
  (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))"
Andreas@48028
  1367
proof(subst finfun_to_list_def, rule the_equality)
Andreas@48028
  1368
  fix xs
Andreas@48036
  1369
  assume "set xs = {x. finfun_dom f(a $:= b) $ x} \<and> sorted xs \<and> distinct xs"
Andreas@48036
  1370
  hence eq: "set xs = {x. finfun_dom f(a $:= b) $ x}"
Andreas@48028
  1371
    and [simp]: "sorted xs" "distinct xs" by simp_all
Andreas@48028
  1372
  show "xs = (if b = finfun_default f then remove1 a (finfun_to_list f) else insort_insert a (finfun_to_list f))"
Andreas@48028
  1373
  proof(cases "b = finfun_default f")
Andreas@48028
  1374
    case True [simp]
Andreas@48028
  1375
    show ?thesis
Andreas@48035
  1376
    proof(cases "finfun_dom f $ a")
Andreas@48028
  1377
      case True
Andreas@48028
  1378
      have "finfun_to_list f = insort_insert a xs"
Andreas@48028
  1379
        unfolding finfun_to_list_def
Andreas@48028
  1380
      proof(rule the_equality)
Andreas@48028
  1381
        have "set (insort_insert a xs) = insert a (set xs)" by(simp add: set_insort_insert)
Andreas@48028
  1382
        also note eq also
Andreas@48036
  1383
        have "insert a {x. finfun_dom f(a $:= b) $ x} = {x. finfun_dom f $ x}" using True
Andreas@48028
  1384
          by(auto simp add: finfun_upd_apply split: split_if_asm)
Andreas@48035
  1385
        finally show 1: "set (insort_insert a xs) = {x. finfun_dom f $ x} \<and> sorted (insort_insert a xs) \<and> distinct (insort_insert a xs)"
Andreas@48028
  1386
          by(simp add: sorted_insort_insert distinct_insort_insert)
Andreas@48028
  1387
Andreas@48028
  1388
        fix xs'
Andreas@48035
  1389
        assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'"
Andreas@48028
  1390
        thus "xs' = insort_insert a xs" using 1 by(auto dest: sorted_distinct_set_unique)
Andreas@48028
  1391
      qed
Andreas@48028
  1392
      with eq True show ?thesis by(simp add: remove1_insort_insert_same)
Andreas@48028
  1393
    next
Andreas@48028
  1394
      case False
Andreas@48035
  1395
      hence "f $ a = b" by(auto simp add: finfun_dom_conv)
Andreas@48036
  1396
      hence f: "f(a $:= b) = f" by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
Andreas@48028
  1397
      from eq have "finfun_to_list f = xs" unfolding f finfun_to_list_def
Andreas@48028
  1398
        by(auto elim: sorted_distinct_set_unique intro!: the_equality)
Andreas@48028
  1399
      with eq False show ?thesis unfolding f by(simp add: remove1_idem)
Andreas@48028
  1400
    qed
Andreas@48028
  1401
  next
Andreas@48028
  1402
    case False
Andreas@48028
  1403
    show ?thesis
Andreas@48035
  1404
    proof(cases "finfun_dom f $ a")
Andreas@48028
  1405
      case True
Andreas@48028
  1406
      have "finfun_to_list f = xs"
Andreas@48028
  1407
        unfolding finfun_to_list_def
Andreas@48028
  1408
      proof(rule the_equality)
Andreas@48036
  1409
        have "finfun_dom f = finfun_dom f(a $:= b)" using False True
Andreas@48028
  1410
          by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply)
Andreas@48035
  1411
        with eq show 1: "set xs = {x. finfun_dom f $ x} \<and> sorted xs \<and> distinct xs"
Andreas@48028
  1412
          by(simp del: finfun_dom_update)
Andreas@48028
  1413
        
Andreas@48028
  1414
        fix xs'
Andreas@48035
  1415
        assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'"
Andreas@48028
  1416
        thus "xs' = xs" using 1 by(auto elim: sorted_distinct_set_unique)
Andreas@48028
  1417
      qed
Andreas@48028
  1418
      thus ?thesis using False True eq by(simp add: insort_insert_triv)
Andreas@48028
  1419
    next
Andreas@48028
  1420
      case False
Andreas@48028
  1421
      have "finfun_to_list f = remove1 a xs"
Andreas@48028
  1422
        unfolding finfun_to_list_def
Andreas@48028
  1423
      proof(rule the_equality)
Andreas@48028
  1424
        have "set (remove1 a xs) = set xs - {a}" by simp
Andreas@48028
  1425
        also note eq also
Andreas@48036
  1426
        have "{x. finfun_dom f(a $:= b) $ x} - {a} = {x. finfun_dom f $ x}" using False
Andreas@48028
  1427
          by(auto simp add: finfun_upd_apply split: split_if_asm)
Andreas@48035
  1428
        finally show 1: "set (remove1 a xs) = {x. finfun_dom f $ x} \<and> sorted (remove1 a xs) \<and> distinct (remove1 a xs)"
Andreas@48028
  1429
          by(simp add: sorted_remove1)
Andreas@48028
  1430
        
Andreas@48028
  1431
        fix xs'
Andreas@48035
  1432
        assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'"
Andreas@48028
  1433
        thus "xs' = remove1 a xs" using 1 by(blast intro: sorted_distinct_set_unique)
Andreas@48028
  1434
      qed
Andreas@48028
  1435
      thus ?thesis using False eq `b \<noteq> finfun_default f` 
Andreas@48028
  1436
        by (simp add: insort_insert_insort insort_remove1)
Andreas@48028
  1437
    qed
Andreas@48028
  1438
  qed
Andreas@48028
  1439
qed (auto simp add: distinct_finfun_to_list sorted_finfun_to_list sorted_remove1 set_insort_insert sorted_insort_insert distinct_insort_insert finfun_upd_apply split: split_if_asm)
Andreas@48028
  1440
Andreas@48028
  1441
lemma finfun_to_list_update_code [code]:
Andreas@48028
  1442
  "finfun_to_list (finfun_update_code f a b) = 
Andreas@48028
  1443
  (if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))"
Andreas@48028
  1444
by(simp add: finfun_to_list_update)
Andreas@48028
  1445
Andreas@48041
  1446
text {* Deactivate syntax again. Import theory @{text FinFun_Syntax} to reactivate it again *}
Andreas@48041
  1447
Andreas@48041
  1448
no_type_notation
Andreas@48041
  1449
  finfun ("(_ =>f /_)" [22, 21] 21)
Andreas@48041
  1450
Andreas@48041
  1451
no_type_notation (xsymbols)
Andreas@48041
  1452
  finfun ("(_ \<Rightarrow>f /_)" [22, 21] 21)
Andreas@48041
  1453
Andreas@48041
  1454
no_notation
Andreas@48041
  1455
  finfun_const ("K$/ _" [0] 1) and
Andreas@48041
  1456
  finfun_update ("_'(_ $:= _')" [1000,0,0] 1000) and
Andreas@48041
  1457
  finfun_apply (infixl "$" 999) and
Andreas@48041
  1458
  finfun_comp (infixr "o$" 55) and
Andreas@48041
  1459
  finfun_comp2 (infixr "$o" 55) and
Andreas@48041
  1460
  finfun_Diag ("(1'($_,/ _$'))" [0, 0] 1000)
Andreas@48041
  1461
Andreas@48041
  1462
no_notation (xsymbols) 
Andreas@48041
  1463
  finfun_comp (infixr "\<circ>$" 55) and
Andreas@48041
  1464
  finfun_comp2 (infixr "$\<circ>" 55)
Andreas@48041
  1465
Andreas@48028
  1466
end