author | wenzelm |
Fri, 12 Oct 2012 18:58:20 +0200 | |
changeset 49834 | b27bbb021df1 |
parent 48100 | 0122ba071e1a |
child 51124 | 8fd094d5b7b7 |
permissions | -rw-r--r-- |
48028 | 1 |
(* 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 Cardinality |
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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 ('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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lemma finfun_update_twice [simp]: |
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"f(a $:= b)(a $:= b') = f(a $:= b')" |
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by transfer simp |
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lemma finfun_update_const_same: "(K$ b)(a $:= b) = (K$ b)" |
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by transfer (simp add: fun_eq_iff) |
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subsection {* Code generator setup *} |
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definition finfun_update_code :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow>f 'b" |
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where [simp, code del]: "finfun_update_code = finfun_update" |
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code_datatype finfun_const finfun_update_code |
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lemma finfun_update_const_code [code]: |
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"(K$ b)(a $:= b') = (if b = b' then (K$ b) else finfun_update_code (K$ b) a b')" |
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by(simp add: finfun_update_const_same) |
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lemma finfun_update_update_code [code]: |
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"(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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by(simp add: finfun_update_twist) |
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subsection {* Setup for quickcheck *} |
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quickcheck_generator finfun constructors: finfun_update_code, "finfun_const :: 'b \<Rightarrow> 'a \<Rightarrow>f 'b" |
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subsection {* @{text "finfun_update"} as instance of @{text "comp_fun_commute"} *} |
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parents:
48035
diff
changeset
|
319 |
interpretation finfun_update: comp_fun_commute "\<lambda>a f. f(a :: 'a $:= b')" |
48030 | 320 |
including finfun |
48028 | 321 |
proof |
322 |
fix a a' :: 'a |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
323 |
show "(\<lambda>f. f(a $:= b')) \<circ> (\<lambda>f. f(a' $:= b')) = (\<lambda>f. f(a' $:= b')) \<circ> (\<lambda>f. f(a $:= b'))" |
48028 | 324 |
proof |
325 |
fix b |
|
48029 | 326 |
have "(finfun_apply b)(a := b', a' := b') = (finfun_apply b)(a' := b', a := b')" |
48028 | 327 |
by(cases "a = a'")(auto simp add: fun_upd_twist) |
48036
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FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
328 |
then have "b(a $:= b')(a' $:= b') = b(a' $:= b')(a $:= b')" |
48028 | 329 |
by(auto simp add: finfun_update_def fun_upd_twist) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
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" |
48028 | 331 |
by (simp add: fun_eq_iff) |
332 |
qed |
|
333 |
qed |
|
334 |
||
335 |
lemma fold_finfun_update_finite_univ: |
|
336 |
assumes fin: "finite (UNIV :: 'a set)" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
337 |
shows "Finite_Set.fold (\<lambda>a f. f(a $:= b')) (K$ b) (UNIV :: 'a set) = (K$ b')" |
48028 | 338 |
proof - |
339 |
{ fix A :: "'a set" |
|
340 |
from fin have "finite A" by(auto intro: finite_subset) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
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)" |
48028 | 342 |
proof(induct) |
343 |
case (insert x F) |
|
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)" |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
345 |
by(auto) |
48028 | 346 |
with insert show ?case |
347 |
by(simp add: finfun_const_def fun_upd_def)(simp add: finfun_update_def Abs_finfun_inverse_finite[OF fin] fun_upd_def) |
|
348 |
qed(simp add: finfun_const_def) } |
|
349 |
thus ?thesis by(simp add: finfun_const_def) |
|
350 |
qed |
|
351 |
||
352 |
||
353 |
subsection {* Default value for FinFuns *} |
|
354 |
||
355 |
definition finfun_default_aux :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" |
|
356 |
where [code del]: "finfun_default_aux f = (if finite (UNIV :: 'a set) then undefined else THE b. finite {a. f a \<noteq> b})" |
|
357 |
||
358 |
lemma finfun_default_aux_infinite: |
|
359 |
fixes f :: "'a \<Rightarrow> 'b" |
|
360 |
assumes infin: "\<not> finite (UNIV :: 'a set)" |
|
361 |
and fin: "finite {a. f a \<noteq> b}" |
|
362 |
shows "finfun_default_aux f = b" |
|
363 |
proof - |
|
364 |
let ?B = "{a. f a \<noteq> b}" |
|
365 |
from fin have "(THE b. finite {a. f a \<noteq> b}) = b" |
|
366 |
proof(rule the_equality) |
|
367 |
fix b' |
|
368 |
assume "finite {a. f a \<noteq> b'}" (is "finite ?B'") |
|
369 |
with infin fin have "UNIV - (?B' \<union> ?B) \<noteq> {}" by(auto dest: finite_subset) |
|
370 |
then obtain a where a: "a \<notin> ?B' \<union> ?B" by auto |
|
371 |
thus "b' = b" by auto |
|
372 |
qed |
|
373 |
thus ?thesis using infin by(simp add: finfun_default_aux_def) |
|
374 |
qed |
|
375 |
||
376 |
||
377 |
lemma finite_finfun_default_aux: |
|
378 |
fixes f :: "'a \<Rightarrow> 'b" |
|
379 |
assumes fin: "f \<in> finfun" |
|
380 |
shows "finite {a. f a \<noteq> finfun_default_aux f}" |
|
381 |
proof(cases "finite (UNIV :: 'a set)") |
|
382 |
case True thus ?thesis using fin |
|
383 |
by(auto simp add: finfun_def finfun_default_aux_def intro: finite_subset) |
|
384 |
next |
|
385 |
case False |
|
386 |
from fin obtain b where b: "finite {a. f a \<noteq> b}" (is "finite ?B") |
|
387 |
unfolding finfun_def by blast |
|
388 |
with False show ?thesis by(simp add: finfun_default_aux_infinite) |
|
389 |
qed |
|
390 |
||
391 |
lemma finfun_default_aux_update_const: |
|
392 |
fixes f :: "'a \<Rightarrow> 'b" |
|
393 |
assumes fin: "f \<in> finfun" |
|
394 |
shows "finfun_default_aux (f(a := b)) = finfun_default_aux f" |
|
395 |
proof(cases "finite (UNIV :: 'a set)") |
|
396 |
case False |
|
397 |
from fin obtain b' where b': "finite {a. f a \<noteq> b'}" unfolding finfun_def by blast |
|
398 |
hence "finite {a'. (f(a := b)) a' \<noteq> b'}" |
|
399 |
proof(cases "b = b' \<and> f a \<noteq> b'") |
|
400 |
case True |
|
401 |
hence "{a. f a \<noteq> b'} = insert a {a'. (f(a := b)) a' \<noteq> b'}" by auto |
|
402 |
thus ?thesis using b' by simp |
|
403 |
next |
|
404 |
case False |
|
405 |
moreover |
|
406 |
{ assume "b \<noteq> b'" |
|
407 |
hence "{a'. (f(a := b)) a' \<noteq> b'} = insert a {a. f a \<noteq> b'}" by auto |
|
408 |
hence ?thesis using b' by simp } |
|
409 |
moreover |
|
410 |
{ assume "b = b'" "f a = b'" |
|
411 |
hence "{a'. (f(a := b)) a' \<noteq> b'} = {a. f a \<noteq> b'}" by auto |
|
412 |
hence ?thesis using b' by simp } |
|
413 |
ultimately show ?thesis by blast |
|
414 |
qed |
|
415 |
with False b' show ?thesis by(auto simp del: fun_upd_apply simp add: finfun_default_aux_infinite) |
|
416 |
next |
|
417 |
case True thus ?thesis by(simp add: finfun_default_aux_def) |
|
418 |
qed |
|
419 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
420 |
lift_definition finfun_default :: "'a \<Rightarrow>f 'b \<Rightarrow> 'b" |
48028 | 421 |
is "finfun_default_aux" .. |
422 |
||
48031 | 423 |
lemma finfun_apply_transfer [transfer_rule]: |
424 |
"(fun_rel cr_finfun (fun_rel op = op =)) (\<lambda>f. f) finfun_apply" |
|
425 |
unfolding Rel_def fun_rel_def cr_finfun_def by simp |
|
426 |
||
48029 | 427 |
lemma finite_finfun_default: "finite {a. finfun_apply f a \<noteq> finfun_default f}" |
48031 | 428 |
by transfer (erule finite_finfun_default_aux) |
48028 | 429 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
430 |
lemma finfun_default_const: "finfun_default ((K$ b) :: 'a \<Rightarrow>f 'b) = (if finite (UNIV :: 'a set) then undefined else b)" |
48031 | 431 |
by(transfer)(auto simp add: finfun_default_aux_infinite finfun_default_aux_def) |
48028 | 432 |
|
433 |
lemma finfun_default_update_const: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
434 |
"finfun_default (f(a $:= b)) = finfun_default f" |
48028 | 435 |
by transfer (simp add: finfun_default_aux_update_const) |
436 |
||
437 |
lemma finfun_default_const_code [code]: |
|
48070
02d64fd40852
more sort constraints for FinFun code generation
Andreas Lochbihler
parents:
48059
diff
changeset
|
438 |
"finfun_default ((K$ c) :: 'a :: card_UNIV \<Rightarrow>f 'b) = (if CARD('a) = 0 then c else undefined)" |
48059 | 439 |
by(simp add: finfun_default_const) |
48028 | 440 |
|
441 |
lemma finfun_default_update_code [code]: |
|
442 |
"finfun_default (finfun_update_code f a b) = finfun_default f" |
|
443 |
by(simp add: finfun_default_update_const) |
|
444 |
||
445 |
subsection {* Recursion combinator and well-formedness conditions *} |
|
446 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
447 |
definition finfun_rec :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow>f 'b) \<Rightarrow> 'c" |
48028 | 448 |
where [code del]: |
449 |
"finfun_rec cnst upd f \<equiv> |
|
450 |
let b = finfun_default f; |
|
451 |
g = THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g |
|
452 |
in Finite_Set.fold (\<lambda>a. upd a (map_default b g a)) (cnst b) (dom g)" |
|
453 |
||
454 |
locale finfun_rec_wf_aux = |
|
455 |
fixes cnst :: "'b \<Rightarrow> 'c" |
|
456 |
and upd :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" |
|
457 |
assumes upd_const_same: "upd a b (cnst b) = cnst b" |
|
458 |
and upd_commute: "a \<noteq> a' \<Longrightarrow> upd a b (upd a' b' c) = upd a' b' (upd a b c)" |
|
459 |
and upd_idemp: "b \<noteq> b' \<Longrightarrow> upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)" |
|
460 |
begin |
|
461 |
||
462 |
||
463 |
lemma upd_left_comm: "comp_fun_commute (\<lambda>a. upd a (f a))" |
|
464 |
by(unfold_locales)(auto intro: upd_commute simp add: fun_eq_iff) |
|
465 |
||
466 |
lemma upd_upd_twice: "upd a b'' (upd a b' (cnst b)) = upd a b'' (cnst b)" |
|
467 |
by(cases "b \<noteq> b'")(auto simp add: fun_upd_def upd_const_same upd_idemp) |
|
468 |
||
469 |
lemma map_default_update_const: |
|
470 |
assumes fin: "finite (dom f)" |
|
471 |
and anf: "a \<notin> dom f" |
|
472 |
and fg: "f \<subseteq>\<^sub>m g" |
|
473 |
shows "upd a d (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)) = |
|
474 |
Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f)" |
|
475 |
proof - |
|
476 |
let ?upd = "\<lambda>a. upd a (map_default d g a)" |
|
477 |
let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A" |
|
478 |
interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm) |
|
479 |
||
480 |
from fin anf fg show ?thesis |
|
481 |
proof(induct "dom f" arbitrary: f) |
|
482 |
case empty |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
483 |
from `{} = dom f` have "f = empty" by(auto simp add: dom_def) |
48028 | 484 |
thus ?case by(simp add: finfun_const_def upd_const_same) |
485 |
next |
|
486 |
case (insert a' A) |
|
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)` |
|
488 |
note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A` |
|
489 |
note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g` |
|
490 |
||
491 |
from domf obtain b where b: "f a' = Some b" by auto |
|
492 |
let ?f' = "f(a' := None)" |
|
493 |
have "upd a d (?fr (insert a' A)) = upd a d (upd a' (map_default d g a') (?fr A))" |
|
494 |
by(subst gwf.fold_insert[OF fin a'nA]) rule |
|
495 |
also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec) |
|
496 |
hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def) |
|
497 |
also from anf domf have "a \<noteq> a'" by auto note upd_commute[OF this] |
|
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) |
|
499 |
note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`] |
|
500 |
also have "upd a' (map_default d f a') (?fr (dom (f(a' := None)))) = ?fr (dom f)" |
|
501 |
unfolding domf[symmetric] gwf.fold_insert[OF fin a'nA] ga' unfolding A .. |
|
502 |
also have "insert a' (dom ?f') = dom f" using domf by auto |
|
503 |
finally show ?case . |
|
504 |
qed |
|
505 |
qed |
|
506 |
||
507 |
lemma map_default_update_twice: |
|
508 |
assumes fin: "finite (dom f)" |
|
509 |
and anf: "a \<notin> dom f" |
|
510 |
and fg: "f \<subseteq>\<^sub>m g" |
|
511 |
shows "upd a d'' (upd a d' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))) = |
|
512 |
upd a d'' (Finite_Set.fold (\<lambda>a. upd a (map_default d g a)) (cnst d) (dom f))" |
|
513 |
proof - |
|
514 |
let ?upd = "\<lambda>a. upd a (map_default d g a)" |
|
515 |
let ?fr = "\<lambda>A. Finite_Set.fold ?upd (cnst d) A" |
|
516 |
interpret gwf: comp_fun_commute "?upd" by(rule upd_left_comm) |
|
517 |
||
518 |
from fin anf fg show ?thesis |
|
519 |
proof(induct "dom f" arbitrary: f) |
|
520 |
case empty |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
521 |
from `{} = dom f` have "f = empty" by(auto simp add: dom_def) |
48028 | 522 |
thus ?case by(auto simp add: finfun_const_def finfun_update_def upd_upd_twice) |
523 |
next |
|
524 |
case (insert a' A) |
|
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))` |
|
526 |
note fin = `finite A` note anf = `a \<notin> dom f` note a'nA = `a' \<notin> A` |
|
527 |
note domf = `insert a' A = dom f` note fg = `f \<subseteq>\<^sub>m g` |
|
528 |
||
529 |
from domf obtain b where b: "f a' = Some b" by auto |
|
530 |
let ?f' = "f(a' := None)" |
|
531 |
let ?b' = "case f a' of None \<Rightarrow> d | Some b \<Rightarrow> b" |
|
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 |
|
533 |
also note gwf.fold_insert[OF fin a'nA] |
|
534 |
also from b fg have "g a' = f a'" by(auto simp add: map_le_def intro: domI dest: bspec) |
|
535 |
hence ga': "map_default d g a' = map_default d f a'" by(simp add: map_default_def) |
|
536 |
also from anf domf have ana': "a \<noteq> a'" by auto note upd_commute[OF this] |
|
537 |
also note upd_commute[OF ana'] |
|
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) |
|
539 |
note A also note IH[OF A `a \<notin> dom ?f'` `?f' \<subseteq>\<^sub>m g`] |
|
540 |
also note upd_commute[OF ana'[symmetric]] also note ga'[symmetric] also note A[symmetric] |
|
541 |
also note gwf.fold_insert[symmetric, OF fin a'nA] also note domf |
|
542 |
finally show ?case . |
|
543 |
qed |
|
544 |
qed |
|
545 |
||
546 |
lemma map_default_eq_id [simp]: "map_default d ((\<lambda>a. Some (f a)) |` {a. f a \<noteq> d}) = f" |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
547 |
by(auto simp add: map_default_def restrict_map_def) |
48028 | 548 |
|
549 |
lemma finite_rec_cong1: |
|
550 |
assumes f: "comp_fun_commute f" and g: "comp_fun_commute g" |
|
551 |
and fin: "finite A" |
|
552 |
and eq: "\<And>a. a \<in> A \<Longrightarrow> f a = g a" |
|
553 |
shows "Finite_Set.fold f z A = Finite_Set.fold g z A" |
|
554 |
proof - |
|
555 |
interpret f: comp_fun_commute f by(rule f) |
|
556 |
interpret g: comp_fun_commute g by(rule g) |
|
557 |
{ fix B |
|
558 |
assume BsubA: "B \<subseteq> A" |
|
559 |
with fin have "finite B" by(blast intro: finite_subset) |
|
560 |
hence "B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B" |
|
561 |
proof(induct) |
|
562 |
case empty thus ?case by simp |
|
563 |
next |
|
564 |
case (insert a B) |
|
565 |
note finB = `finite B` note anB = `a \<notin> B` note sub = `insert a B \<subseteq> A` |
|
566 |
note IH = `B \<subseteq> A \<Longrightarrow> Finite_Set.fold f z B = Finite_Set.fold g z B` |
|
567 |
from sub anB have BpsubA: "B \<subset> A" and BsubA: "B \<subseteq> A" and aA: "a \<in> A" by auto |
|
568 |
from IH[OF BsubA] eq[OF aA] finB anB |
|
569 |
show ?case by(auto) |
|
570 |
qed |
|
571 |
with BsubA have "Finite_Set.fold f z B = Finite_Set.fold g z B" by blast } |
|
572 |
thus ?thesis by blast |
|
573 |
qed |
|
574 |
||
575 |
lemma finfun_rec_upd [simp]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
576 |
"finfun_rec cnst upd (f(a' $:= b')) = upd a' b' (finfun_rec cnst upd f)" |
48030 | 577 |
including finfun |
48028 | 578 |
proof - |
579 |
obtain b where b: "b = finfun_default f" by auto |
|
580 |
let ?the = "\<lambda>f g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g" |
|
581 |
obtain g where g: "g = The (?the f)" by blast |
|
582 |
obtain y where f: "f = Abs_finfun y" and y: "y \<in> finfun" by (cases f) |
|
583 |
from f y b have bfin: "finite {a. y a \<noteq> b}" by(simp add: finfun_default_def finite_finfun_default_aux) |
|
584 |
||
585 |
let ?g = "(\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}" |
|
586 |
from bfin have fing: "finite (dom ?g)" by auto |
|
587 |
have bran: "b \<notin> ran ?g" by(auto simp add: ran_def restrict_map_def) |
|
588 |
have yg: "y = map_default b ?g" by simp |
|
589 |
have gg: "g = ?g" unfolding g |
|
590 |
proof(rule the_equality) |
|
591 |
from f y bfin show "?the f ?g" |
|
592 |
by(auto)(simp add: restrict_map_def ran_def split: split_if_asm) |
|
593 |
next |
|
594 |
fix g' |
|
595 |
assume "?the f g'" |
|
596 |
hence fin': "finite (dom g')" and ran': "b \<notin> ran g'" |
|
597 |
and eq: "Abs_finfun (map_default b ?g) = Abs_finfun (map_default b g')" using f yg by auto |
|
598 |
from fin' fing have "map_default b ?g \<in> finfun" "map_default b g' \<in> finfun" by(blast intro: map_default_in_finfun)+ |
|
599 |
with eq have "map_default b ?g = map_default b g'" by simp |
|
600 |
with fing bran fin' ran' show "g' = ?g" by(rule map_default_inject[OF disjI2[OF refl], THEN sym]) |
|
601 |
qed |
|
602 |
||
603 |
show ?thesis |
|
604 |
proof(cases "b' = b") |
|
605 |
case True |
|
606 |
note b'b = True |
|
607 |
||
608 |
let ?g' = "(\<lambda>a. Some ((y(a' := b)) a)) |` {a. (y(a' := b)) a \<noteq> b}" |
|
609 |
from bfin b'b have fing': "finite (dom ?g')" |
|
610 |
by(auto simp add: Collect_conj_eq Collect_imp_eq intro: finite_subset) |
|
611 |
have brang': "b \<notin> ran ?g'" by(auto simp add: ran_def restrict_map_def) |
|
612 |
||
613 |
let ?b' = "\<lambda>a. case ?g' a of None \<Rightarrow> b | Some b \<Rightarrow> b" |
|
614 |
let ?b = "map_default b ?g" |
|
615 |
from upd_left_comm upd_left_comm fing' |
|
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')" |
|
617 |
by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b b map_default_def) |
|
618 |
also interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm) |
|
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))" |
|
620 |
proof(cases "y a' = b") |
|
621 |
case True |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
622 |
with b'b have g': "?g' = ?g" by(auto simp add: restrict_map_def) |
48028 | 623 |
from True have a'ndomg: "a' \<notin> dom ?g" by auto |
624 |
from f b'b b show ?thesis unfolding g' |
|
625 |
by(subst map_default_update_const[OF fing a'ndomg map_le_refl, symmetric]) simp |
|
626 |
next |
|
627 |
case False |
|
628 |
hence domg: "dom ?g = insert a' (dom ?g')" by auto |
|
629 |
from False b'b have a'ndomg': "a' \<notin> dom ?g'" by auto |
|
630 |
have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g')) = |
|
631 |
upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'))" |
|
632 |
using fing' a'ndomg' unfolding b'b by(rule gwf.fold_insert) |
|
633 |
hence "upd a' b (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g'))) = |
|
634 |
upd a' b (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g')))" by simp |
|
635 |
also from b'b have g'leg: "?g' \<subseteq>\<^sub>m ?g" by(auto simp add: restrict_map_def map_le_def) |
|
636 |
note map_default_update_twice[OF fing' a'ndomg' this, of b "?b a'" b] |
|
637 |
also note map_default_update_const[OF fing' a'ndomg' g'leg, of b] |
|
638 |
finally show ?thesis unfolding b'b domg[unfolded b'b] by(rule sym) |
|
639 |
qed |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
640 |
also have "The (?the (f(a' $:= b'))) = ?g'" |
48028 | 641 |
proof(rule the_equality) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
642 |
from f y b b'b brang' fing' show "?the (f(a' $:= b')) ?g'" |
48028 | 643 |
by(auto simp del: fun_upd_apply simp add: finfun_update_def) |
644 |
next |
|
645 |
fix g' |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
646 |
assume "?the (f(a' $:= b')) g'" |
48028 | 647 |
hence fin': "finite (dom g')" and ran': "b \<notin> ran g'" |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
648 |
and eq: "f(a' $:= b') = Abs_finfun (map_default b g')" |
48028 | 649 |
by(auto simp del: fun_upd_apply) |
650 |
from fin' fing' have "map_default b g' \<in> finfun" "map_default b ?g' \<in> finfun" |
|
651 |
by(blast intro: map_default_in_finfun)+ |
|
652 |
with eq f b'b b have "map_default b ?g' = map_default b g'" |
|
653 |
by(simp del: fun_upd_apply add: finfun_update_def) |
|
654 |
with fing' brang' fin' ran' show "g' = ?g'" |
|
655 |
by(rule map_default_inject[OF disjI2[OF refl], THEN sym]) |
|
656 |
qed |
|
657 |
ultimately show ?thesis unfolding finfun_rec_def Let_def b gg[unfolded g b] using bfin b'b b |
|
658 |
by(simp only: finfun_default_update_const map_default_def) |
|
659 |
next |
|
660 |
case False |
|
661 |
note b'b = this |
|
662 |
let ?g' = "?g(a' \<mapsto> b')" |
|
663 |
let ?b' = "map_default b ?g'" |
|
664 |
let ?b = "map_default b ?g" |
|
665 |
from fing have fing': "finite (dom ?g')" by auto |
|
666 |
from bran b'b have bnrang': "b \<notin> ran ?g'" by(auto simp add: ran_def) |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
667 |
have ffmg': "map_default b ?g' = y(a' := b')" by(auto simp add: map_default_def restrict_map_def) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
668 |
with f y have f_Abs: "f(a' $:= b') = Abs_finfun (map_default b ?g')" by(auto simp add: finfun_update_def) |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
669 |
have g': "The (?the (f(a' $:= b'))) = ?g'" |
48028 | 670 |
proof (rule the_equality) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
671 |
from fing' bnrang' f_Abs show "?the (f(a' $:= b')) ?g'" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
672 |
by(auto simp add: finfun_update_def restrict_map_def) |
48028 | 673 |
next |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
674 |
fix g' assume "?the (f(a' $:= b')) g'" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
675 |
hence f': "f(a' $:= b') = Abs_finfun (map_default b g')" |
48028 | 676 |
and fin': "finite (dom g')" and brang': "b \<notin> ran g'" by auto |
677 |
from fing' fin' have "map_default b ?g' \<in> finfun" "map_default b g' \<in> finfun" |
|
678 |
by(auto intro: map_default_in_finfun) |
|
679 |
with f' f_Abs have "map_default b g' = map_default b ?g'" by simp |
|
680 |
with fin' brang' fing' bnrang' show "g' = ?g'" |
|
681 |
by(rule map_default_inject[OF disjI2[OF refl]]) |
|
682 |
qed |
|
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}))" |
|
684 |
by auto |
|
685 |
show ?thesis |
|
686 |
proof(cases "y a' = b") |
|
687 |
case True |
|
688 |
hence a'ndomg: "a' \<notin> dom ?g" by auto |
|
689 |
from f y b'b True have yff: "y = map_default b (?g' |` dom ?g)" |
|
690 |
by(auto simp add: restrict_map_def map_default_def intro!: ext) |
|
691 |
hence f': "f = Abs_finfun (map_default b (?g' |` dom ?g))" using f by simp |
|
692 |
interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm) |
|
693 |
from upd_left_comm upd_left_comm fing |
|
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)" |
|
695 |
by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b True map_default_def) |
|
696 |
thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] |
|
697 |
unfolding g' g[symmetric] gg g'wf.fold_insert[OF fing a'ndomg, of "cnst b", folded dom] |
|
698 |
by -(rule arg_cong2[where f="upd a'"], simp_all add: map_default_def) |
|
699 |
next |
|
700 |
case False |
|
701 |
hence "insert a' (dom ?g) = dom ?g" by auto |
|
702 |
moreover { |
|
703 |
let ?g'' = "?g(a' := None)" |
|
704 |
let ?b'' = "map_default b ?g''" |
|
705 |
from False have domg: "dom ?g = insert a' (dom ?g'')" by auto |
|
706 |
from False have a'ndomg'': "a' \<notin> dom ?g''" by auto |
|
707 |
have fing'': "finite (dom ?g'')" by(rule finite_subset[OF _ fing]) auto |
|
708 |
have bnrang'': "b \<notin> ran ?g''" by(auto simp add: ran_def restrict_map_def) |
|
709 |
interpret gwf: comp_fun_commute "\<lambda>a. upd a (?b a)" by(rule upd_left_comm) |
|
710 |
interpret g'wf: comp_fun_commute "\<lambda>a. upd a (?b' a)" by(rule upd_left_comm) |
|
711 |
have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (insert a' (dom ?g''))) = |
|
712 |
upd a' b' (upd a' (?b a') (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')))" |
|
713 |
unfolding gwf.fold_insert[OF fing'' a'ndomg''] f .. |
|
714 |
also have g''leg: "?g |` dom ?g'' \<subseteq>\<^sub>m ?g" by(auto simp add: map_le_def) |
|
715 |
have "dom (?g |` dom ?g'') = dom ?g''" by auto |
|
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", |
|
717 |
unfolded this, OF fing'' a'ndomg'' g''leg] |
|
718 |
also have b': "b' = ?b' a'" by(auto simp add: map_default_def) |
|
719 |
from upd_left_comm upd_left_comm fing'' |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
720 |
have "Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'') = |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
721 |
Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g'')" |
48028 | 722 |
by(rule finite_rec_cong1)(auto simp add: restrict_map_def b'b map_default_def) |
723 |
with b' have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g'')) = |
|
724 |
upd a' (?b' a') (Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g''))" by simp |
|
725 |
also note g'wf.fold_insert[OF fing'' a'ndomg'', symmetric] |
|
726 |
finally have "upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g)) = |
|
727 |
Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (dom ?g)" |
|
728 |
unfolding domg . } |
|
729 |
ultimately have "Finite_Set.fold (\<lambda>a. upd a (?b' a)) (cnst b) (insert a' (dom ?g)) = |
|
730 |
upd a' b' (Finite_Set.fold (\<lambda>a. upd a (?b a)) (cnst b) (dom ?g))" by simp |
|
731 |
thus ?thesis unfolding finfun_rec_def Let_def finfun_default_update_const b[symmetric] g[symmetric] g' dom[symmetric] |
|
732 |
using b'b gg by(simp add: map_default_insert) |
|
733 |
qed |
|
734 |
qed |
|
735 |
qed |
|
736 |
||
737 |
end |
|
738 |
||
739 |
locale finfun_rec_wf = finfun_rec_wf_aux + |
|
740 |
assumes const_update_all: |
|
741 |
"finite (UNIV :: 'a set) \<Longrightarrow> Finite_Set.fold (\<lambda>a. upd a b') (cnst b) (UNIV :: 'a set) = cnst b'" |
|
742 |
begin |
|
743 |
||
48030 | 744 |
lemma finfun_rec_const [simp]: includes finfun shows |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
745 |
"finfun_rec cnst upd (K$ c) = cnst c" |
48028 | 746 |
proof(cases "finite (UNIV :: 'a set)") |
747 |
case False |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
748 |
hence "finfun_default ((K$ c) :: 'a \<Rightarrow>f 'b) = c" by(simp add: finfun_default_const) |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
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" |
48028 | 750 |
proof (rule the_equality) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
751 |
show "(K$ c) = Abs_finfun (map_default c empty) \<and> finite (dom empty) \<and> c \<notin> ran empty" |
48028 | 752 |
by(auto simp add: finfun_const_def) |
753 |
next |
|
754 |
fix g :: "'a \<rightharpoonup> 'b" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
755 |
assume "(K$ c) = Abs_finfun (map_default c g) \<and> finite (dom g) \<and> c \<notin> ran g" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
756 |
hence g: "(K$ c) = Abs_finfun (map_default c g)" and fin: "finite (dom g)" and ran: "c \<notin> ran g" by blast+ |
48028 | 757 |
from g map_default_in_finfun[OF fin, of c] have "map_default c g = (\<lambda>a. c)" |
758 |
by(simp add: finfun_const_def) |
|
759 |
moreover have "map_default c empty = (\<lambda>a. c)" by simp |
|
760 |
ultimately show "g = empty" by-(rule map_default_inject[OF disjI2[OF refl] fin ran], auto) |
|
761 |
qed |
|
762 |
ultimately show ?thesis by(simp add: finfun_rec_def) |
|
763 |
next |
|
764 |
case True |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
765 |
hence default: "finfun_default ((K$ c) :: 'a \<Rightarrow>f 'b) = undefined" by(simp add: finfun_default_const) |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
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" |
48028 | 767 |
show ?thesis |
768 |
proof(cases "c = undefined") |
|
769 |
case True |
|
770 |
have the: "The ?the = empty" |
|
771 |
proof (rule the_equality) |
|
772 |
from True show "?the empty" by(auto simp add: finfun_const_def) |
|
773 |
next |
|
774 |
fix g' |
|
775 |
assume "?the g'" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
776 |
hence fg: "(K$ c) = Abs_finfun (map_default undefined g')" |
48028 | 777 |
and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all |
778 |
from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun) |
|
779 |
with fg have "map_default undefined g' = (\<lambda>a. c)" |
|
48030 | 780 |
by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1, symmetric]) |
48028 | 781 |
with True show "g' = empty" |
782 |
by -(rule map_default_inject(2)[OF _ fin g], auto) |
|
783 |
qed |
|
784 |
show ?thesis unfolding finfun_rec_def using `finite UNIV` True |
|
785 |
unfolding Let_def the default by(simp) |
|
786 |
next |
|
787 |
case False |
|
788 |
have the: "The ?the = (\<lambda>a :: 'a. Some c)" |
|
789 |
proof (rule the_equality) |
|
790 |
from False True show "?the (\<lambda>a :: 'a. Some c)" |
|
791 |
by(auto simp add: map_default_def [abs_def] finfun_const_def dom_def ran_def) |
|
792 |
next |
|
793 |
fix g' :: "'a \<rightharpoonup> 'b" |
|
794 |
assume "?the g'" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
795 |
hence fg: "(K$ c) = Abs_finfun (map_default undefined g')" |
48028 | 796 |
and fin: "finite (dom g')" and g: "undefined \<notin> ran g'" by simp_all |
797 |
from fin have "map_default undefined g' \<in> finfun" by(rule map_default_in_finfun) |
|
798 |
with fg have "map_default undefined g' = (\<lambda>a. c)" |
|
799 |
by(auto simp add: finfun_const_def intro: Abs_finfun_inject[THEN iffD1]) |
|
800 |
with True False show "g' = (\<lambda>a::'a. Some c)" |
|
801 |
by - (rule map_default_inject(2)[OF _ fin g], |
|
802 |
auto simp add: dom_def ran_def map_default_def [abs_def]) |
|
803 |
qed |
|
804 |
show ?thesis unfolding finfun_rec_def using True False |
|
805 |
unfolding Let_def the default by(simp add: dom_def map_default_def const_update_all) |
|
806 |
qed |
|
807 |
qed |
|
808 |
||
809 |
end |
|
810 |
||
811 |
subsection {* Weak induction rule and case analysis for FinFuns *} |
|
812 |
||
813 |
lemma finfun_weak_induct [consumes 0, case_names const update]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
814 |
assumes const: "\<And>b. P (K$ b)" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
815 |
and update: "\<And>f a b. P f \<Longrightarrow> P (f(a $:= b))" |
48028 | 816 |
shows "P x" |
48030 | 817 |
including finfun |
48028 | 818 |
proof(induct x rule: Abs_finfun_induct) |
819 |
case (Abs_finfun y) |
|
820 |
then obtain b where "finite {a. y a \<noteq> b}" unfolding finfun_def by blast |
|
821 |
thus ?case using `y \<in> finfun` |
|
822 |
proof(induct "{a. y a \<noteq> b}" arbitrary: y rule: finite_induct) |
|
823 |
case empty |
|
824 |
hence "\<And>a. y a = b" by blast |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
825 |
hence "y = (\<lambda>a. b)" by(auto) |
48028 | 826 |
hence "Abs_finfun y = finfun_const b" unfolding finfun_const_def by simp |
827 |
thus ?case by(simp add: const) |
|
828 |
next |
|
829 |
case (insert a A) |
|
830 |
note IH = `\<And>y. \<lbrakk> A = {a. y a \<noteq> b}; y \<in> finfun \<rbrakk> \<Longrightarrow> P (Abs_finfun y)` |
|
831 |
note y = `y \<in> finfun` |
|
832 |
with `insert a A = {a. y a \<noteq> b}` `a \<notin> A` |
|
833 |
have "A = {a'. (y(a := b)) a' \<noteq> b}" "y(a := b) \<in> finfun" by auto |
|
834 |
from IH[OF this] have "P (finfun_update (Abs_finfun (y(a := b))) a (y a))" by(rule update) |
|
835 |
thus ?case using y unfolding finfun_update_def by simp |
|
836 |
qed |
|
837 |
qed |
|
838 |
||
839 |
lemma finfun_exhaust_disj: "(\<exists>b. x = finfun_const b) \<or> (\<exists>f a b. x = finfun_update f a b)" |
|
840 |
by(induct x rule: finfun_weak_induct) blast+ |
|
841 |
||
842 |
lemma finfun_exhaust: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
843 |
obtains b where "x = (K$ b)" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
844 |
| f a b where "x = f(a $:= b)" |
48028 | 845 |
by(atomize_elim)(rule finfun_exhaust_disj) |
846 |
||
847 |
lemma finfun_rec_unique: |
|
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
848 |
fixes f :: "'a \<Rightarrow>f 'b \<Rightarrow> 'c" |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
849 |
assumes c: "\<And>c. f (K$ c) = cnst c" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
850 |
and u: "\<And>g a b. f (g(a $:= b)) = upd g a b (f g)" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
851 |
and c': "\<And>c. f' (K$ c) = cnst c" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
852 |
and u': "\<And>g a b. f' (g(a $:= b)) = upd g a b (f' g)" |
48028 | 853 |
shows "f = f'" |
854 |
proof |
|
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
855 |
fix g :: "'a \<Rightarrow>f 'b" |
48028 | 856 |
show "f g = f' g" |
857 |
by(induct g rule: finfun_weak_induct)(auto simp add: c u c' u') |
|
858 |
qed |
|
859 |
||
860 |
||
861 |
subsection {* Function application *} |
|
862 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
863 |
notation finfun_apply (infixl "$" 999) |
48028 | 864 |
|
865 |
interpretation finfun_apply_aux: finfun_rec_wf_aux "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c" |
|
866 |
by(unfold_locales) auto |
|
867 |
||
868 |
interpretation finfun_apply: finfun_rec_wf "\<lambda>b. b" "\<lambda>a' b c. if (a = a') then b else c" |
|
869 |
proof(unfold_locales) |
|
870 |
fix b' b :: 'a |
|
871 |
assume fin: "finite (UNIV :: 'b set)" |
|
872 |
{ fix A :: "'b set" |
|
873 |
interpret comp_fun_commute "\<lambda>a'. If (a = a') b'" by(rule finfun_apply_aux.upd_left_comm) |
|
874 |
from fin have "finite A" by(auto intro: finite_subset) |
|
875 |
hence "Finite_Set.fold (\<lambda>a'. If (a = a') b') b A = (if a \<in> A then b' else b)" |
|
876 |
by induct auto } |
|
877 |
from this[of UNIV] show "Finite_Set.fold (\<lambda>a'. If (a = a') b') b UNIV = b'" by simp |
|
878 |
qed |
|
879 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
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)" |
48029 | 881 |
proof(rule finfun_rec_unique) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
882 |
fix c show "op $ (K$ c) = (\<lambda>a. c)" by(simp add: finfun_const.rep_eq) |
48029 | 883 |
next |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
884 |
fix g a b show "op $ g(a $:= b) = (\<lambda>c. if c = a then b else g $ c)" |
48029 | 885 |
by(auto simp add: finfun_update_def fun_upd_finfun Abs_finfun_inverse finfun_apply) |
886 |
qed auto |
|
48028 | 887 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
888 |
lemma finfun_upd_apply: "f(a $:= b) $ a' = (if a = a' then b else f $ a')" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
889 |
and finfun_upd_apply_code [code]: "(finfun_update_code f a b) $ a' = (if a = a' then b else f $ a')" |
48028 | 890 |
by(simp_all add: finfun_apply_def) |
891 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
892 |
lemma finfun_const_apply [simp, code]: "(K$ b) $ a = b" |
48029 | 893 |
by(simp add: finfun_apply_def) |
894 |
||
48028 | 895 |
lemma finfun_upd_apply_same [simp]: |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
896 |
"f(a $:= b) $ a = b" |
48028 | 897 |
by(simp add: finfun_upd_apply) |
898 |
||
899 |
lemma finfun_upd_apply_other [simp]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
900 |
"a \<noteq> a' \<Longrightarrow> f(a $:= b) $ a' = f $ a'" |
48028 | 901 |
by(simp add: finfun_upd_apply) |
902 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
903 |
lemma finfun_ext: "(\<And>a. f $ a = g $ a) \<Longrightarrow> f = g" |
48029 | 904 |
by(auto simp add: finfun_apply_inject[symmetric] simp del: finfun_apply_inject) |
48028 | 905 |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
906 |
lemma expand_finfun_eq: "(f = g) = (op $ f = op $ g)" |
48028 | 907 |
by(auto intro: finfun_ext) |
908 |
||
48100 | 909 |
lemma finfun_upd_triv [simp]: "f(x $:= f $ x) = f" |
910 |
by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
|
911 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
912 |
lemma finfun_const_inject [simp]: "(K$ b) = (K$ b') \<equiv> b = b'" |
48028 | 913 |
by(simp add: expand_finfun_eq fun_eq_iff) |
914 |
||
915 |
lemma finfun_const_eq_update: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
916 |
"((K$ b) = f(a $:= b')) = (b = b' \<and> (\<forall>a'. a \<noteq> a' \<longrightarrow> f $ a' = b))" |
48028 | 917 |
by(auto simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
918 |
||
919 |
subsection {* Function composition *} |
|
920 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
921 |
definition finfun_comp :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow>f 'a \<Rightarrow> 'c \<Rightarrow>f 'b" (infixr "o$" 55) |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
922 |
where [code del]: "g o$ f = finfun_rec (\<lambda>b. (K$ g b)) (\<lambda>a b c. c(a $:= g b)) f" |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
923 |
|
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
924 |
notation (xsymbols) finfun_comp (infixr "\<circ>$" 55) |
48028 | 925 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
926 |
interpretation finfun_comp_aux: finfun_rec_wf_aux "(\<lambda>b. (K$ g b))" "(\<lambda>a b c. c(a $:= g b))" |
48028 | 927 |
by(unfold_locales)(auto simp add: finfun_upd_apply intro: finfun_ext) |
928 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
929 |
interpretation finfun_comp: finfun_rec_wf "(\<lambda>b. (K$ g b))" "(\<lambda>a b c. c(a $:= g b))" |
48028 | 930 |
proof |
931 |
fix b' b :: 'a |
|
932 |
assume fin: "finite (UNIV :: 'c set)" |
|
933 |
{ fix A :: "'c set" |
|
934 |
from fin have "finite A" by(auto intro: finite_subset) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
935 |
hence "Finite_Set.fold (\<lambda>(a :: 'c) c. c(a $:= g b')) (K$ g b) A = |
48028 | 936 |
Abs_finfun (\<lambda>a. if a \<in> A then g b' else g b)" |
937 |
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) } |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
938 |
from this[of UNIV] show "Finite_Set.fold (\<lambda>(a :: 'c) c. c(a $:= g b')) (K$ g b) UNIV = (K$ g b')" |
48028 | 939 |
by(simp add: finfun_const_def) |
940 |
qed |
|
941 |
||
942 |
lemma finfun_comp_const [simp, code]: |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
943 |
"g \<circ>$ (K$ c) = (K$ g c)" |
48028 | 944 |
by(simp add: finfun_comp_def) |
945 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
946 |
lemma finfun_comp_update [simp]: "g \<circ>$ (f(a $:= b)) = (g \<circ>$ f)(a $:= g b)" |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
947 |
and finfun_comp_update_code [code]: |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
948 |
"g \<circ>$ (finfun_update_code f a b) = finfun_update_code (g \<circ>$ f) a (g b)" |
48028 | 949 |
by(simp_all add: finfun_comp_def) |
950 |
||
951 |
lemma finfun_comp_apply [simp]: |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
952 |
"op $ (g \<circ>$ f) = g \<circ> op $ f" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
953 |
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_upd_apply) |
48028 | 954 |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
955 |
lemma finfun_comp_comp_collapse [simp]: "f \<circ>$ g \<circ>$ h = (f \<circ> g) \<circ>$ h" |
48028 | 956 |
by(induct h rule: finfun_weak_induct) simp_all |
957 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
958 |
lemma finfun_comp_const1 [simp]: "(\<lambda>x. c) \<circ>$ f = (K$ c)" |
48028 | 959 |
by(induct f rule: finfun_weak_induct)(auto intro: finfun_ext simp add: finfun_upd_apply) |
960 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
961 |
lemma finfun_comp_id1 [simp]: "(\<lambda>x. x) \<circ>$ f = f" "id \<circ>$ f = f" |
48028 | 962 |
by(induct f rule: finfun_weak_induct) auto |
963 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
964 |
lemma finfun_comp_conv_comp: "g \<circ>$ f = Abs_finfun (g \<circ> op $ f)" |
48030 | 965 |
including finfun |
48028 | 966 |
proof - |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
967 |
have "(\<lambda>f. g \<circ>$ f) = (\<lambda>f. Abs_finfun (g \<circ> op $ f))" |
48028 | 968 |
proof(rule finfun_rec_unique) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
969 |
{ fix c show "Abs_finfun (g \<circ> op $ (K$ c)) = (K$ g c)" |
48028 | 970 |
by(simp add: finfun_comp_def o_def)(simp add: finfun_const_def) } |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
971 |
{ fix g' a b show "Abs_finfun (g \<circ> op $ g'(a $:= b)) = (Abs_finfun (g \<circ> op $ g'))(a $:= g b)" |
48028 | 972 |
proof - |
973 |
obtain y where y: "y \<in> finfun" and g': "g' = Abs_finfun y" by(cases g') |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
974 |
moreover hence "(g \<circ> op $ g') \<in> finfun" by(simp add: finfun_left_compose) |
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
975 |
moreover have "g \<circ> y(a := b) = (g \<circ> y)(a := g b)" by(auto) |
48029 | 976 |
ultimately show ?thesis by(simp add: finfun_comp_def finfun_update_def) |
48028 | 977 |
qed } |
978 |
qed auto |
|
979 |
thus ?thesis by(auto simp add: fun_eq_iff) |
|
980 |
qed |
|
981 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
982 |
definition finfun_comp2 :: "'b \<Rightarrow>f 'c \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow>f 'c" (infixr "$o" 55) |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
983 |
where [code del]: "g $o f = Abs_finfun (op $ g \<circ> f)" |
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
984 |
|
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
985 |
notation (xsymbol) finfun_comp2 (infixr "$\<circ>" 55) |
48028 | 986 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
987 |
lemma finfun_comp2_const [code, simp]: "finfun_comp2 (K$ c) f = (K$ c)" |
48030 | 988 |
including finfun |
48028 | 989 |
by(simp add: finfun_comp2_def finfun_const_def comp_def) |
990 |
||
991 |
lemma finfun_comp2_update: |
|
48030 | 992 |
includes finfun |
48028 | 993 |
assumes inj: "inj f" |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
994 |
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)" |
48028 | 995 |
proof(cases "b \<in> range f") |
996 |
case True |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
997 |
from inj have "\<And>x. (op $ g)(f x := c) \<circ> f = (op $ g \<circ> f)(x := c)" by(auto intro!: ext dest: injD) |
48028 | 998 |
with inj True show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def finfun_right_compose) |
999 |
next |
|
1000 |
case False |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1001 |
hence "(op $ g)(b := c) \<circ> f = op $ g \<circ> f" by(auto simp add: fun_eq_iff) |
48028 | 1002 |
with False show ?thesis by(auto simp add: finfun_comp2_def finfun_update_def) |
1003 |
qed |
|
1004 |
||
1005 |
subsection {* Universal quantification *} |
|
1006 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1007 |
definition finfun_All_except :: "'a list \<Rightarrow> 'a \<Rightarrow>f bool \<Rightarrow> bool" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1008 |
where [code del]: "finfun_All_except A P \<equiv> \<forall>a. a \<in> set A \<or> P $ a" |
48028 | 1009 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1010 |
lemma finfun_All_except_const: "finfun_All_except A (K$ b) \<longleftrightarrow> b \<or> set A = UNIV" |
48028 | 1011 |
by(auto simp add: finfun_All_except_def) |
1012 |
||
1013 |
lemma finfun_All_except_const_finfun_UNIV_code [code]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1014 |
"finfun_All_except A (K$ b) = (b \<or> is_list_UNIV A)" |
48028 | 1015 |
by(simp add: finfun_All_except_const is_list_UNIV_iff) |
1016 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1017 |
lemma finfun_All_except_update: |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1018 |
"finfun_All_except A f(a $:= b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)" |
48028 | 1019 |
by(fastforce simp add: finfun_All_except_def finfun_upd_apply) |
1020 |
||
1021 |
lemma finfun_All_except_update_code [code]: |
|
1022 |
fixes a :: "'a :: card_UNIV" |
|
1023 |
shows "finfun_All_except A (finfun_update_code f a b) = ((a \<in> set A \<or> b) \<and> finfun_All_except (a # A) f)" |
|
1024 |
by(simp add: finfun_All_except_update) |
|
1025 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1026 |
definition finfun_All :: "'a \<Rightarrow>f bool \<Rightarrow> bool" |
48028 | 1027 |
where "finfun_All = finfun_All_except []" |
1028 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1029 |
lemma finfun_All_const [simp]: "finfun_All (K$ b) = b" |
48028 | 1030 |
by(simp add: finfun_All_def finfun_All_except_def) |
1031 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1032 |
lemma finfun_All_update: "finfun_All f(a $:= b) = (b \<and> finfun_All_except [a] f)" |
48028 | 1033 |
by(simp add: finfun_All_def finfun_All_except_update) |
1034 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1035 |
lemma finfun_All_All: "finfun_All P = All (op $ P)" |
48028 | 1036 |
by(simp add: finfun_All_def finfun_All_except_def) |
1037 |
||
1038 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1039 |
definition finfun_Ex :: "'a \<Rightarrow>f bool \<Rightarrow> bool" |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1040 |
where "finfun_Ex P = Not (finfun_All (Not \<circ>$ P))" |
48028 | 1041 |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1042 |
lemma finfun_Ex_Ex: "finfun_Ex P = Ex (op $ P)" |
48028 | 1043 |
unfolding finfun_Ex_def finfun_All_All by simp |
1044 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1045 |
lemma finfun_Ex_const [simp]: "finfun_Ex (K$ b) = b" |
48028 | 1046 |
by(simp add: finfun_Ex_def) |
1047 |
||
1048 |
||
1049 |
subsection {* A diagonal operator for FinFuns *} |
|
1050 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1051 |
definition finfun_Diag :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow>f 'c \<Rightarrow> 'a \<Rightarrow>f ('b \<times> 'c)" ("(1'($_,/ _$'))" [0, 0] 1000) |
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1052 |
where [code del]: "($f, g$) = finfun_rec (\<lambda>b. Pair b \<circ>$ g) (\<lambda>a b c. c(a $:= (b, g $ a))) f" |
48028 | 1053 |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1054 |
interpretation finfun_Diag_aux: finfun_rec_wf_aux "\<lambda>b. Pair b \<circ>$ g" "\<lambda>a b c. c(a $:= (b, g $ a))" |
48028 | 1055 |
by(unfold_locales)(simp_all add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
1056 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1057 |
interpretation finfun_Diag: finfun_rec_wf "\<lambda>b. Pair b \<circ>$ g" "\<lambda>a b c. c(a $:= (b, g $ a))" |
48028 | 1058 |
proof |
1059 |
fix b' b :: 'a |
|
1060 |
assume fin: "finite (UNIV :: 'c set)" |
|
1061 |
{ fix A :: "'c set" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1062 |
interpret comp_fun_commute "\<lambda>a c. c(a $:= (b', g $ a))" by(rule finfun_Diag_aux.upd_left_comm) |
48028 | 1063 |
from fin have "finite A" by(auto intro: finite_subset) |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1064 |
hence "Finite_Set.fold (\<lambda>a c. c(a $:= (b', g $ a))) (Pair b \<circ>$ g) A = |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1065 |
Abs_finfun (\<lambda>a. (if a \<in> A then b' else b, g $ a))" |
48028 | 1066 |
by(induct)(simp_all add: finfun_const_def finfun_comp_conv_comp o_def, |
1067 |
auto simp add: finfun_update_def Abs_finfun_inverse_finite fun_upd_def Abs_finfun_inject_finite fun_eq_iff fin) } |
|
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1068 |
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" |
48028 | 1069 |
by(simp add: finfun_const_def finfun_comp_conv_comp o_def) |
1070 |
qed |
|
1071 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1072 |
lemma finfun_Diag_const1: "($K$ b, g$) = Pair b \<circ>$ g" |
48028 | 1073 |
by(simp add: finfun_Diag_def) |
1074 |
||
1075 |
text {* |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1076 |
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>$"}. |
48028 | 1077 |
*} |
1078 |
||
1079 |
lemma finfun_Diag_const_code [code]: |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1080 |
"($K$ b, K$ c$) = (K$ (b, c))" |
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1081 |
"($K$ b, finfun_update_code g a c$) = finfun_update_code ($K$ b, g$) a (b, c)" |
48028 | 1082 |
by(simp_all add: finfun_Diag_const1) |
1083 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1084 |
lemma finfun_Diag_update1: "($f(a $:= b), g$) = ($f, g$)(a $:= (b, g $ a))" |
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1085 |
and finfun_Diag_update1_code [code]: "($finfun_update_code f a b, g$) = ($f, g$)(a $:= (b, g $ a))" |
48028 | 1086 |
by(simp_all add: finfun_Diag_def) |
1087 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1088 |
lemma finfun_Diag_const2: "($f, K$ c$) = (\<lambda>b. (b, c)) \<circ>$ f" |
48028 | 1089 |
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1) |
1090 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1091 |
lemma finfun_Diag_update2: "($f, g(a $:= c)$) = ($f, g$)(a $:= (f $ a, c))" |
48028 | 1092 |
by(induct f rule: finfun_weak_induct)(auto intro!: finfun_ext simp add: finfun_upd_apply finfun_Diag_const1 finfun_Diag_update1) |
1093 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1094 |
lemma finfun_Diag_const_const [simp]: "($K$ b, K$ c$) = (K$ (b, c))" |
48028 | 1095 |
by(simp add: finfun_Diag_const1) |
1096 |
||
1097 |
lemma finfun_Diag_const_update: |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1098 |
"($K$ b, g(a $:= c)$) = ($K$ b, g$)(a $:= (b, c))" |
48028 | 1099 |
by(simp add: finfun_Diag_const1) |
1100 |
||
1101 |
lemma finfun_Diag_update_const: |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1102 |
"($f(a $:= b), K$ c$) = ($f, K$ c$)(a $:= (b, c))" |
48028 | 1103 |
by(simp add: finfun_Diag_def) |
1104 |
||
1105 |
lemma finfun_Diag_update_update: |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1106 |
"($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)))" |
48028 | 1107 |
by(auto simp add: finfun_Diag_update1 finfun_Diag_update2) |
1108 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1109 |
lemma finfun_Diag_apply [simp]: "op $ ($f, g$) = (\<lambda>x. (f $ x, g $ x))" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1110 |
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_Diag_const1 finfun_Diag_update1 finfun_upd_apply) |
48028 | 1111 |
|
1112 |
lemma finfun_Diag_conv_Abs_finfun: |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1113 |
"($f, g$) = Abs_finfun ((\<lambda>x. (f $ x, g $ x)))" |
48030 | 1114 |
including finfun |
48028 | 1115 |
proof - |
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1116 |
have "(\<lambda>f :: 'a \<Rightarrow>f 'b. ($f, g$)) = (\<lambda>f. Abs_finfun ((\<lambda>x. (f $ x, g $ x))))" |
48028 | 1117 |
proof(rule finfun_rec_unique) |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1118 |
{ fix c show "Abs_finfun (\<lambda>x. ((K$ c) $ x, g $ x)) = Pair c \<circ>$ g" |
48029 | 1119 |
by(simp add: finfun_comp_conv_comp o_def finfun_const_def) } |
48028 | 1120 |
{ fix g' a b |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1121 |
show "Abs_finfun (\<lambda>x. (g'(a $:= b) $ x, g $ x)) = |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1122 |
(Abs_finfun (\<lambda>x. (g' $ x, g $ x)))(a $:= (b, g $ a))" |
48029 | 1123 |
by(auto simp add: finfun_update_def fun_eq_iff simp del: fun_upd_apply) simp } |
48028 | 1124 |
qed(simp_all add: finfun_Diag_const1 finfun_Diag_update1) |
1125 |
thus ?thesis by(auto simp add: fun_eq_iff) |
|
1126 |
qed |
|
1127 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1128 |
lemma finfun_Diag_eq: "($f, g$) = ($f', g'$) \<longleftrightarrow> f = f' \<and> g = g'" |
48028 | 1129 |
by(auto simp add: expand_finfun_eq fun_eq_iff) |
1130 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1131 |
definition finfun_fst :: "'a \<Rightarrow>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>f 'b" |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1132 |
where [code]: "finfun_fst f = fst \<circ>$ f" |
48028 | 1133 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1134 |
lemma finfun_fst_const: "finfun_fst (K$ bc) = (K$ fst bc)" |
48028 | 1135 |
by(simp add: finfun_fst_def) |
1136 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1137 |
lemma finfun_fst_update: "finfun_fst (f(a $:= bc)) = (finfun_fst f)(a $:= fst bc)" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1138 |
and finfun_fst_update_code: "finfun_fst (finfun_update_code f a bc) = (finfun_fst f)(a $:= fst bc)" |
48028 | 1139 |
by(simp_all add: finfun_fst_def) |
1140 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1141 |
lemma finfun_fst_comp_conv: "finfun_fst (f \<circ>$ g) = (fst \<circ> f) \<circ>$ g" |
48028 | 1142 |
by(simp add: finfun_fst_def) |
1143 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1144 |
lemma finfun_fst_conv [simp]: "finfun_fst ($f, g$) = f" |
48028 | 1145 |
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) |
1146 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1147 |
lemma finfun_fst_conv_Abs_finfun: "finfun_fst = (\<lambda>f. Abs_finfun (fst \<circ> op $ f))" |
48029 | 1148 |
by(simp add: finfun_fst_def [abs_def] finfun_comp_conv_comp) |
48028 | 1149 |
|
1150 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1151 |
definition finfun_snd :: "'a \<Rightarrow>f ('b \<times> 'c) \<Rightarrow> 'a \<Rightarrow>f 'c" |
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1152 |
where [code]: "finfun_snd f = snd \<circ>$ f" |
48028 | 1153 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1154 |
lemma finfun_snd_const: "finfun_snd (K$ bc) = (K$ snd bc)" |
48028 | 1155 |
by(simp add: finfun_snd_def) |
1156 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1157 |
lemma finfun_snd_update: "finfun_snd (f(a $:= bc)) = (finfun_snd f)(a $:= snd bc)" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1158 |
and finfun_snd_update_code [code]: "finfun_snd (finfun_update_code f a bc) = (finfun_snd f)(a $:= snd bc)" |
48028 | 1159 |
by(simp_all add: finfun_snd_def) |
1160 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1161 |
lemma finfun_snd_comp_conv: "finfun_snd (f \<circ>$ g) = (snd \<circ> f) \<circ>$ g" |
48028 | 1162 |
by(simp add: finfun_snd_def) |
1163 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1164 |
lemma finfun_snd_conv [simp]: "finfun_snd ($f, g$) = g" |
48028 | 1165 |
apply(induct f rule: finfun_weak_induct) |
1166 |
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) |
|
1167 |
done |
|
1168 |
||
48037
6c4b3e78f03e
syntax for FinFun composition without subscripts
Andreas Lochbihler
parents:
48036
diff
changeset
|
1169 |
lemma finfun_snd_conv_Abs_finfun: "finfun_snd = (\<lambda>f. Abs_finfun (snd \<circ> op $ f))" |
48029 | 1170 |
by(simp add: finfun_snd_def [abs_def] finfun_comp_conv_comp) |
48028 | 1171 |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1172 |
lemma finfun_Diag_collapse [simp]: "($finfun_fst f, finfun_snd f$) = f" |
48028 | 1173 |
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) |
1174 |
||
1175 |
subsection {* Currying for FinFuns *} |
|
1176 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1177 |
definition finfun_curry :: "('a \<times> 'b) \<Rightarrow>f 'c \<Rightarrow> 'a \<Rightarrow>f 'b \<Rightarrow>f 'c" |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1178 |
where [code del]: "finfun_curry = finfun_rec (finfun_const \<circ> finfun_const) (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c)))" |
48028 | 1179 |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1180 |
interpretation finfun_curry_aux: finfun_rec_wf_aux "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))" |
48028 | 1181 |
apply(unfold_locales) |
1182 |
apply(auto simp add: split_def finfun_update_twist finfun_upd_apply split_paired_all finfun_update_const_same) |
|
1183 |
done |
|
1184 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1185 |
interpretation finfun_curry: finfun_rec_wf "finfun_const \<circ> finfun_const" "\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))" |
48028 | 1186 |
proof(unfold_locales) |
1187 |
fix b' b :: 'b |
|
1188 |
assume fin: "finite (UNIV :: ('c \<times> 'a) set)" |
|
1189 |
hence fin1: "finite (UNIV :: 'c set)" and fin2: "finite (UNIV :: 'a set)" |
|
1190 |
unfolding UNIV_Times_UNIV[symmetric] |
|
1191 |
by(fastforce dest: finite_cartesian_productD1 finite_cartesian_productD2)+ |
|
1192 |
note [simp] = Abs_finfun_inverse_finite[OF fin] Abs_finfun_inverse_finite[OF fin1] Abs_finfun_inverse_finite[OF fin2] |
|
1193 |
{ fix A :: "('c \<times> 'a) set" |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1194 |
interpret comp_fun_commute "\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b'" |
48028 | 1195 |
by(rule finfun_curry_aux.upd_left_comm) |
1196 |
from fin have "finite A" by(auto intro: finite_subset) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1197 |
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))" |
48029 | 1198 |
by induct (simp_all, auto simp add: finfun_update_def finfun_const_def split_def intro!: arg_cong[where f="Abs_finfun"] ext) } |
48028 | 1199 |
from this[of UNIV] |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1200 |
show "Finite_Set.fold (\<lambda>a :: 'c \<times> 'a. (\<lambda>(a, b) c f. f(a $:= (f $ a)(b $:= c))) a b') ((finfun_const \<circ> finfun_const) b) UNIV = (finfun_const \<circ> finfun_const) b'" |
48028 | 1201 |
by(simp add: finfun_const_def) |
1202 |
qed |
|
1203 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1204 |
lemma finfun_curry_const [simp, code]: "finfun_curry (K$ c) = (K$ K$ c)" |
48028 | 1205 |
by(simp add: finfun_curry_def) |
1206 |
||
1207 |
lemma finfun_curry_update [simp]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1208 |
"finfun_curry (f((a, b) $:= c)) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))" |
48028 | 1209 |
and finfun_curry_update_code [code]: |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1210 |
"finfun_curry (finfun_update_code f (a, b) c) = (finfun_curry f)(a $:= (finfun_curry f $ a)(b $:= c))" |
48028 | 1211 |
by(simp_all add: finfun_curry_def) |
1212 |
||
1213 |
lemma finfun_Abs_finfun_curry: assumes fin: "f \<in> finfun" |
|
1214 |
shows "(\<lambda>a. Abs_finfun (curry f a)) \<in> finfun" |
|
48030 | 1215 |
including finfun |
48028 | 1216 |
proof - |
1217 |
from fin obtain c where c: "finite {ab. f ab \<noteq> c}" unfolding finfun_def by blast |
|
1218 |
have "{a. \<exists>b. f (a, b) \<noteq> c} = fst ` {ab. f ab \<noteq> c}" by(force) |
|
1219 |
hence "{a. curry f a \<noteq> (\<lambda>x. c)} = fst ` {ab. f ab \<noteq> c}" |
|
1220 |
by(auto simp add: curry_def fun_eq_iff) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1221 |
with fin c have "finite {a. Abs_finfun (curry f a) \<noteq> (K$ c)}" |
48028 | 1222 |
by(simp add: finfun_const_def finfun_curry) |
1223 |
thus ?thesis unfolding finfun_def by auto |
|
1224 |
qed |
|
1225 |
||
1226 |
lemma finfun_curry_conv_curry: |
|
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1227 |
fixes f :: "('a \<times> 'b) \<Rightarrow>f 'c" |
48029 | 1228 |
shows "finfun_curry f = Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply f) a))" |
48030 | 1229 |
including finfun |
48028 | 1230 |
proof - |
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1231 |
have "finfun_curry = (\<lambda>f :: ('a \<times> 'b) \<Rightarrow>f 'c. Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply f) a)))" |
48028 | 1232 |
proof(rule finfun_rec_unique) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1233 |
fix c show "finfun_curry (K$ c) = (K$ K$ c)" by simp |
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1234 |
fix f a |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1235 |
show "finfun_curry (f(a $:= c)) = (finfun_curry f)(fst a $:= (finfun_curry f $ (fst a))(snd a $:= c))" |
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1236 |
by(cases a) simp |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1237 |
show "Abs_finfun (\<lambda>a. Abs_finfun (curry (finfun_apply (K$ c)) a)) = (K$ K$ c)" |
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1238 |
by(simp add: finfun_curry_def finfun_const_def curry_def) |
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1239 |
fix g b |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1240 |
show "Abs_finfun (\<lambda>aa. Abs_finfun (curry (op $ g(a $:= b)) aa)) = |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1241 |
(Abs_finfun (\<lambda>a. Abs_finfun (curry (op $ g) a)))( |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1242 |
fst a $:= ((Abs_finfun (\<lambda>a. Abs_finfun (curry (op $ g) a))) $ (fst a))(snd a $:= b))" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1243 |
by(cases a)(auto intro!: ext arg_cong[where f=Abs_finfun] simp add: finfun_curry_def finfun_update_def finfun_Abs_finfun_curry) |
48028 | 1244 |
qed |
1245 |
thus ?thesis by(auto simp add: fun_eq_iff) |
|
1246 |
qed |
|
1247 |
||
1248 |
subsection {* Executable equality for FinFuns *} |
|
1249 |
||
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1250 |
lemma eq_finfun_All_ext: "(f = g) \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))" |
48028 | 1251 |
by(simp add: expand_finfun_eq fun_eq_iff finfun_All_All o_def) |
1252 |
||
1253 |
instantiation finfun :: ("{card_UNIV,equal}",equal) equal begin |
|
48038
72a8506dd59b
eliminated remaining sub- and superscripts in FinFun syntax
Andreas Lochbihler
parents:
48037
diff
changeset
|
1254 |
definition eq_finfun_def [code]: "HOL.equal f g \<longleftrightarrow> finfun_All ((\<lambda>(x, y). x = y) \<circ>$ ($f, g$))" |
48028 | 1255 |
instance by(intro_classes)(simp add: eq_finfun_All_ext eq_finfun_def) |
1256 |
end |
|
1257 |
||
1258 |
lemma [code nbe]: |
|
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1259 |
"HOL.equal (f :: _ \<Rightarrow>f _) f \<longleftrightarrow> True" |
48028 | 1260 |
by (fact equal_refl) |
1261 |
||
1262 |
subsection {* An operator that explicitly removes all redundant updates in the generated representations *} |
|
1263 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1264 |
definition finfun_clearjunk :: "'a \<Rightarrow>f 'b \<Rightarrow> 'a \<Rightarrow>f 'b" |
48028 | 1265 |
where [simp, code del]: "finfun_clearjunk = id" |
1266 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1267 |
lemma finfun_clearjunk_const [code]: "finfun_clearjunk (K$ b) = (K$ b)" |
48028 | 1268 |
by simp |
1269 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1270 |
lemma finfun_clearjunk_update [code]: |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1271 |
"finfun_clearjunk (finfun_update_code f a b) = f(a $:= b)" |
48028 | 1272 |
by simp |
1273 |
||
1274 |
subsection {* The domain of a FinFun as a FinFun *} |
|
1275 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1276 |
definition finfun_dom :: "('a \<Rightarrow>f 'b) \<Rightarrow> ('a \<Rightarrow>f bool)" |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1277 |
where [code del]: "finfun_dom f = Abs_finfun (\<lambda>a. f $ a \<noteq> finfun_default f)" |
48028 | 1278 |
|
1279 |
lemma finfun_dom_const: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1280 |
"finfun_dom ((K$ c) :: 'a \<Rightarrow>f 'b) = (K$ finite (UNIV :: 'a set) \<and> c \<noteq> undefined)" |
48028 | 1281 |
unfolding finfun_dom_def finfun_default_const |
1282 |
by(auto)(simp_all add: finfun_const_def) |
|
1283 |
||
1284 |
text {* |
|
1285 |
@{term "finfun_dom" } raises an exception when called on a FinFun whose domain is a finite type. |
|
1286 |
For such FinFuns, the default value (and as such the domain) is undefined. |
|
1287 |
*} |
|
1288 |
||
1289 |
lemma finfun_dom_const_code [code]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1290 |
"finfun_dom ((K$ c) :: ('a :: card_UNIV) \<Rightarrow>f 'b) = |
48059 | 1291 |
(if CARD('a) = 0 then (K$ False) else FinFun.code_abort (\<lambda>_. finfun_dom (K$ c)))" |
1292 |
by(simp add: finfun_dom_const card_UNIV card_eq_0_iff) |
|
48028 | 1293 |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1294 |
lemma finfun_dom_finfunI: "(\<lambda>a. f $ a \<noteq> finfun_default f) \<in> finfun" |
48028 | 1295 |
using finite_finfun_default[of f] |
48029 | 1296 |
by(simp add: finfun_def exI[where x=False]) |
48028 | 1297 |
|
1298 |
lemma finfun_dom_update [simp]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1299 |
"finfun_dom (f(a $:= b)) = (finfun_dom f)(a $:= (b \<noteq> finfun_default f))" |
48030 | 1300 |
including finfun unfolding finfun_dom_def finfun_update_def |
48029 | 1301 |
apply(simp add: finfun_default_update_const fun_upd_apply finfun_dom_finfunI) |
48028 | 1302 |
apply(fold finfun_update.rep_eq) |
48029 | 1303 |
apply(simp add: finfun_upd_apply fun_eq_iff fun_upd_def finfun_default_update_const) |
48028 | 1304 |
done |
1305 |
||
1306 |
lemma finfun_dom_update_code [code]: |
|
1307 |
"finfun_dom (finfun_update_code f a b) = finfun_update_code (finfun_dom f) a (b \<noteq> finfun_default f)" |
|
1308 |
by(simp) |
|
1309 |
||
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1310 |
lemma finite_finfun_dom: "finite {x. finfun_dom f $ x}" |
48028 | 1311 |
proof(induct f rule: finfun_weak_induct) |
1312 |
case (const b) |
|
1313 |
thus ?case |
|
1314 |
by (cases "finite (UNIV :: 'a set) \<and> b \<noteq> undefined") |
|
1315 |
(auto simp add: finfun_dom_const UNIV_def [symmetric] Set.empty_def [symmetric]) |
|
1316 |
next |
|
1317 |
case (update f a b) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1318 |
have "{x. finfun_dom f(a $:= b) $ x} = |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1319 |
(if b = finfun_default f then {x. finfun_dom f $ x} - {a} else insert a {x. finfun_dom f $ x})" |
48028 | 1320 |
by (auto simp add: finfun_upd_apply split: split_if_asm) |
1321 |
thus ?case using update by simp |
|
1322 |
qed |
|
1323 |
||
1324 |
||
1325 |
subsection {* The domain of a FinFun as a sorted list *} |
|
1326 |
||
48034
1c5171abe5cc
removed subscripts from FinFun type syntax
Andreas Lochbihler
parents:
48031
diff
changeset
|
1327 |
definition finfun_to_list :: "('a :: linorder) \<Rightarrow>f 'b \<Rightarrow> 'a list" |
48028 | 1328 |
where |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1329 |
"finfun_to_list f = (THE xs. set xs = {x. finfun_dom f $ x} \<and> sorted xs \<and> distinct xs)" |
48028 | 1330 |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1331 |
lemma set_finfun_to_list [simp]: "set (finfun_to_list f) = {x. finfun_dom f $ x}" (is ?thesis1) |
48028 | 1332 |
and sorted_finfun_to_list: "sorted (finfun_to_list f)" (is ?thesis2) |
1333 |
and distinct_finfun_to_list: "distinct (finfun_to_list f)" (is ?thesis3) |
|
1334 |
proof - |
|
1335 |
have "?thesis1 \<and> ?thesis2 \<and> ?thesis3" |
|
1336 |
unfolding finfun_to_list_def |
|
1337 |
by(rule theI')(rule finite_sorted_distinct_unique finite_finfun_dom)+ |
|
1338 |
thus ?thesis1 ?thesis2 ?thesis3 by simp_all |
|
1339 |
qed |
|
1340 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1341 |
lemma finfun_const_False_conv_bot: "op $ (K$ False) = bot" |
48028 | 1342 |
by auto |
1343 |
||
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1344 |
lemma finfun_const_True_conv_top: "op $ (K$ True) = top" |
48028 | 1345 |
by auto |
1346 |
||
1347 |
lemma finfun_to_list_const: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1348 |
"finfun_to_list ((K$ c) :: ('a :: {linorder} \<Rightarrow>f 'b)) = |
48028 | 1349 |
(if \<not> finite (UNIV :: 'a set) \<or> c = undefined then [] else THE xs. set xs = UNIV \<and> sorted xs \<and> distinct xs)" |
1350 |
by(auto simp add: finfun_to_list_def finfun_const_False_conv_bot finfun_const_True_conv_top finfun_dom_const) |
|
1351 |
||
1352 |
lemma finfun_to_list_const_code [code]: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1353 |
"finfun_to_list ((K$ c) :: ('a :: {linorder, card_UNIV} \<Rightarrow>f 'b)) = |
48059 | 1354 |
(if CARD('a) = 0 then [] else FinFun.code_abort (\<lambda>_. finfun_to_list ((K$ c) :: ('a \<Rightarrow>f 'b))))" |
1355 |
by(auto simp add: finfun_to_list_const card_UNIV card_eq_0_iff) |
|
48028 | 1356 |
|
1357 |
lemma remove1_insort_insert_same: |
|
1358 |
"x \<notin> set xs \<Longrightarrow> remove1 x (insort_insert x xs) = xs" |
|
1359 |
by (metis insort_insert_insort remove1_insort) |
|
1360 |
||
1361 |
lemma finfun_dom_conv: |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1362 |
"finfun_dom f $ x \<longleftrightarrow> f $ x \<noteq> finfun_default f" |
48028 | 1363 |
by(induct f rule: finfun_weak_induct)(auto simp add: finfun_dom_const finfun_default_const finfun_default_update_const finfun_upd_apply) |
1364 |
||
1365 |
lemma finfun_to_list_update: |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1366 |
"finfun_to_list (f(a $:= b)) = |
48028 | 1367 |
(if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))" |
1368 |
proof(subst finfun_to_list_def, rule the_equality) |
|
1369 |
fix xs |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1370 |
assume "set xs = {x. finfun_dom f(a $:= b) $ x} \<and> sorted xs \<and> distinct xs" |
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1371 |
hence eq: "set xs = {x. finfun_dom f(a $:= b) $ x}" |
48028 | 1372 |
and [simp]: "sorted xs" "distinct xs" by simp_all |
1373 |
show "xs = (if b = finfun_default f then remove1 a (finfun_to_list f) else insort_insert a (finfun_to_list f))" |
|
1374 |
proof(cases "b = finfun_default f") |
|
1375 |
case True [simp] |
|
1376 |
show ?thesis |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1377 |
proof(cases "finfun_dom f $ a") |
48028 | 1378 |
case True |
1379 |
have "finfun_to_list f = insort_insert a xs" |
|
1380 |
unfolding finfun_to_list_def |
|
1381 |
proof(rule the_equality) |
|
1382 |
have "set (insort_insert a xs) = insert a (set xs)" by(simp add: set_insort_insert) |
|
1383 |
also note eq also |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1384 |
have "insert a {x. finfun_dom f(a $:= b) $ x} = {x. finfun_dom f $ x}" using True |
48028 | 1385 |
by(auto simp add: finfun_upd_apply split: split_if_asm) |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1386 |
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)" |
48028 | 1387 |
by(simp add: sorted_insort_insert distinct_insort_insert) |
1388 |
||
1389 |
fix xs' |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1390 |
assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'" |
48028 | 1391 |
thus "xs' = insort_insert a xs" using 1 by(auto dest: sorted_distinct_set_unique) |
1392 |
qed |
|
1393 |
with eq True show ?thesis by(simp add: remove1_insort_insert_same) |
|
1394 |
next |
|
1395 |
case False |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1396 |
hence "f $ a = b" by(auto simp add: finfun_dom_conv) |
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1397 |
hence f: "f(a $:= b) = f" by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
48028 | 1398 |
from eq have "finfun_to_list f = xs" unfolding f finfun_to_list_def |
1399 |
by(auto elim: sorted_distinct_set_unique intro!: the_equality) |
|
1400 |
with eq False show ?thesis unfolding f by(simp add: remove1_idem) |
|
1401 |
qed |
|
1402 |
next |
|
1403 |
case False |
|
1404 |
show ?thesis |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1405 |
proof(cases "finfun_dom f $ a") |
48028 | 1406 |
case True |
1407 |
have "finfun_to_list f = xs" |
|
1408 |
unfolding finfun_to_list_def |
|
1409 |
proof(rule the_equality) |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1410 |
have "finfun_dom f = finfun_dom f(a $:= b)" using False True |
48028 | 1411 |
by(simp add: expand_finfun_eq fun_eq_iff finfun_upd_apply) |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1412 |
with eq show 1: "set xs = {x. finfun_dom f $ x} \<and> sorted xs \<and> distinct xs" |
48028 | 1413 |
by(simp del: finfun_dom_update) |
1414 |
||
1415 |
fix xs' |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1416 |
assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'" |
48028 | 1417 |
thus "xs' = xs" using 1 by(auto elim: sorted_distinct_set_unique) |
1418 |
qed |
|
1419 |
thus ?thesis using False True eq by(simp add: insort_insert_triv) |
|
1420 |
next |
|
1421 |
case False |
|
1422 |
have "finfun_to_list f = remove1 a xs" |
|
1423 |
unfolding finfun_to_list_def |
|
1424 |
proof(rule the_equality) |
|
1425 |
have "set (remove1 a xs) = set xs - {a}" by simp |
|
1426 |
also note eq also |
|
48036
1edcd5f73505
FinFun pseudo-constructor syntax without superscripts
Andreas Lochbihler
parents:
48035
diff
changeset
|
1427 |
have "{x. finfun_dom f(a $:= b) $ x} - {a} = {x. finfun_dom f $ x}" using False |
48028 | 1428 |
by(auto simp add: finfun_upd_apply split: split_if_asm) |
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1429 |
finally show 1: "set (remove1 a xs) = {x. finfun_dom f $ x} \<and> sorted (remove1 a xs) \<and> distinct (remove1 a xs)" |
48028 | 1430 |
by(simp add: sorted_remove1) |
1431 |
||
1432 |
fix xs' |
|
48035
2f9584581cf2
replace FinFun application syntax with $
Andreas Lochbihler
parents:
48034
diff
changeset
|
1433 |
assume "set xs' = {x. finfun_dom f $ x} \<and> sorted xs' \<and> distinct xs'" |
48028 | 1434 |
thus "xs' = remove1 a xs" using 1 by(blast intro: sorted_distinct_set_unique) |
1435 |
qed |
|
1436 |
thus ?thesis using False eq `b \<noteq> finfun_default f` |
|
1437 |
by (simp add: insort_insert_insort insort_remove1) |
|
1438 |
qed |
|
1439 |
qed |
|
1440 |
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) |
|
1441 |
||
1442 |
lemma finfun_to_list_update_code [code]: |
|
1443 |
"finfun_to_list (finfun_update_code f a b) = |
|
1444 |
(if b = finfun_default f then List.remove1 a (finfun_to_list f) else List.insort_insert a (finfun_to_list f))" |
|
1445 |
by(simp add: finfun_to_list_update) |
|
1446 |
||
48041
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1447 |
text {* Deactivate syntax again. Import theory @{text FinFun_Syntax} to reactivate it again *} |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1448 |
|
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1449 |
no_type_notation |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1450 |
finfun ("(_ =>f /_)" [22, 21] 21) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1451 |
|
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1452 |
no_type_notation (xsymbols) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1453 |
finfun ("(_ \<Rightarrow>f /_)" [22, 21] 21) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1454 |
|
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1455 |
no_notation |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1456 |
finfun_const ("K$/ _" [0] 1) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1457 |
finfun_update ("_'(_ $:= _')" [1000,0,0] 1000) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1458 |
finfun_apply (infixl "$" 999) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1459 |
finfun_comp (infixr "o$" 55) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1460 |
finfun_comp2 (infixr "$o" 55) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1461 |
finfun_Diag ("(1'($_,/ _$'))" [0, 0] 1000) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1462 |
|
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1463 |
no_notation (xsymbols) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1464 |
finfun_comp (infixr "\<circ>$" 55) and |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1465 |
finfun_comp2 (infixr "$\<circ>" 55) |
d60f6b41bf2d
remove pretty syntax for FinFuns at the end and provide separate syntax theory
Andreas Lochbihler
parents:
48038
diff
changeset
|
1466 |
|
48028 | 1467 |
end |