author | wenzelm |
Mon, 20 Jun 2016 17:25:08 +0200 | |
changeset 63323 | 814541a57d89 |
parent 63322 | bc1f17d45e91 |
child 63324 | 1e98146f3581 |
permissions | -rw-r--r-- |
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(* Title: HOL/Fun.thy |
2 |
Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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Author: Andrei Popescu, TU Muenchen |
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Copyright 1994, 2012 |
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*) |
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section \<open>Notions about functions\<close> |
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theory Fun |
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imports Set |
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keywords "functor" :: thy_goal |
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begin |
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lemma apply_inverse: "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u" |
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by auto |
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text \<open>Uniqueness, so NOT the axiom of choice.\<close> |
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lemma uniq_choice: "\<forall>x. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)" |
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by (force intro: theI') |
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lemma b_uniq_choice: "\<forall>x\<in>S. \<exists>!y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)" |
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by (force intro: theI') |
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subsection \<open>The Identity Function \<open>id\<close>\<close> |
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definition id :: "'a \<Rightarrow> 'a" |
27 |
where "id = (\<lambda>x. x)" |
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lemma id_apply [simp]: "id x = x" |
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by (simp add: id_def) |
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||
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lemma image_id [simp]: "image id = id" |
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by (simp add: id_def fun_eq_iff) |
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lemma vimage_id [simp]: "vimage id = id" |
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by (simp add: id_def fun_eq_iff) |
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lemma eq_id_iff: "(\<forall>x. f x = x) \<longleftrightarrow> f = id" |
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by auto |
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code_printing |
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constant id \<rightharpoonup> (Haskell) "id" |
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subsection \<open>The Composition Operator \<open>f \<circ> g\<close>\<close> |
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definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>" 55) |
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where "f \<circ> g = (\<lambda>x. f (g x))" |
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notation (ASCII) |
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comp (infixl "o" 55) |
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lemma comp_apply [simp]: "(f \<circ> g) x = f (g x)" |
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by (simp add: comp_def) |
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lemma comp_assoc: "(f \<circ> g) \<circ> h = f \<circ> (g \<circ> h)" |
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by (simp add: fun_eq_iff) |
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|
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lemma id_comp [simp]: "id \<circ> g = g" |
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by (simp add: fun_eq_iff) |
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lemma comp_id [simp]: "f \<circ> id = f" |
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by (simp add: fun_eq_iff) |
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65 |
lemma comp_eq_dest: |
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"a \<circ> b = c \<circ> d \<Longrightarrow> a (b v) = c (d v)" |
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by (simp add: fun_eq_iff) |
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lemma comp_eq_elim: |
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"a \<circ> b = c \<circ> d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R" |
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by (simp add: fun_eq_iff) |
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lemma comp_eq_dest_lhs: "a \<circ> b = c \<Longrightarrow> a (b v) = c v" |
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by clarsimp |
75 |
||
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lemma comp_eq_id_dest: "a \<circ> b = id \<circ> c \<Longrightarrow> a (b v) = c v" |
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by clarsimp |
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78 |
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lemma image_comp: "f ` (g ` r) = (f \<circ> g) ` r" |
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by auto |
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lemma vimage_comp: "f -` (g -` x) = (g \<circ> f) -` x" |
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by auto |
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lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h \<circ> f) ` A = (h \<circ> g) ` B" |
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by (auto simp: comp_def elim!: equalityE) |
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87 |
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lemma image_bind: "f ` (Set.bind A g) = Set.bind A (op ` f \<circ> g)" |
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by (auto simp add: Set.bind_def) |
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lemma bind_image: "Set.bind (f ` A) g = Set.bind A (g \<circ> f)" |
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by (auto simp add: Set.bind_def) |
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lemma (in group_add) minus_comp_minus [simp]: "uminus \<circ> uminus = id" |
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by (simp add: fun_eq_iff) |
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lemma (in boolean_algebra) minus_comp_minus [simp]: "uminus \<circ> uminus = id" |
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by (simp add: fun_eq_iff) |
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code_printing |
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constant comp \<rightharpoonup> (SML) infixl 5 "o" and (Haskell) infixr 9 "." |
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102 |
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subsection \<open>The Forward Composition Operator \<open>fcomp\<close>\<close> |
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definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) |
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where "f \<circ>> g = (\<lambda>x. g (f x))" |
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lemma fcomp_apply [simp]: "(f \<circ>> g) x = g (f x)" |
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by (simp add: fcomp_def) |
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lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)" |
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by (simp add: fcomp_def) |
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lemma id_fcomp [simp]: "id \<circ>> g = g" |
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by (simp add: fcomp_def) |
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lemma fcomp_id [simp]: "f \<circ>> id = f" |
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by (simp add: fcomp_def) |
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lemma fcomp_comp: "fcomp f g = comp g f" |
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by (simp add: ext) |
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code_printing |
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constant fcomp \<rightharpoonup> (Eval) infixl 1 "#>" |
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no_notation fcomp (infixl "\<circ>>" 60) |
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subsection \<open>Mapping functions\<close> |
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definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" |
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where "map_fun f g h = g \<circ> h \<circ> f" |
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lemma map_fun_apply [simp]: "map_fun f g h x = g (h (f x))" |
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by (simp add: map_fun_def) |
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subsection \<open>Injectivity and Bijectivity\<close> |
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definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" \<comment> \<open>injective\<close> |
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where "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)" |
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definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" \<comment> \<open>bijective\<close> |
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where "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B" |
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text \<open>A common special case: functions injective, surjective or bijective over |
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the entire domain type.\<close> |
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abbreviation "inj f \<equiv> inj_on f UNIV" |
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abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" \<comment> "surjective" |
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where "surj f \<equiv> range f = UNIV" |
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abbreviation "bij f \<equiv> bij_betw f UNIV UNIV" |
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text \<open>The negated case:\<close> |
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translations |
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"\<not> CONST surj f" \<leftharpoondown> "CONST range f \<noteq> CONST UNIV" |
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lemma injI: "(\<And>x y. f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj f" |
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unfolding inj_on_def by auto |
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theorem range_ex1_eq: "inj f \<Longrightarrow> b \<in> range f \<longleftrightarrow> (\<exists>!x. b = f x)" |
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lemma injD: "inj f \<Longrightarrow> f x = f y \<Longrightarrow> x = y" |
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by (simp add: inj_on_def) |
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lemma inj_on_eq_iff: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<longleftrightarrow> x = y" |
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by (force simp add: inj_on_def) |
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lemma inj_on_cong: "(\<And>a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A" |
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unfolding inj_on_def by auto |
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lemma inj_on_strict_subset: "inj_on f B \<Longrightarrow> A \<subset> B \<Longrightarrow> f ` A \<subset> f ` B" |
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unfolding inj_on_def by blast |
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lemma inj_comp: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)" |
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by (simp add: inj_on_def) |
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lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)" |
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by (simp add: inj_on_def fun_eq_iff) |
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lemma inj_eq: "inj f \<Longrightarrow> f x = f y \<longleftrightarrow> x = y" |
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by (simp add: inj_on_eq_iff) |
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lemma inj_on_id[simp]: "inj_on id A" |
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by (simp add: inj_on_def) |
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lemma inj_on_id2[simp]: "inj_on (\<lambda>x. x) A" |
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by (simp add: inj_on_def) |
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lemma inj_on_Int: "inj_on f A \<or> inj_on f B \<Longrightarrow> inj_on f (A \<inter> B)" |
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lemma surj_id: "surj id" |
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by simp |
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lemma bij_id[simp]: "bij id" |
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by (simp add: bij_betw_def) |
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lemma bij_uminus: "bij (uminus :: 'a \<Rightarrow> 'a::ab_group_add)" |
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unfolding bij_betw_def inj_on_def |
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by (force intro: minus_minus [symmetric]) |
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lemma inj_onI [intro?]: "(\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> x = y) \<Longrightarrow> inj_on f A" |
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by (simp add: inj_on_def) |
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lemma inj_on_inverseI: "(\<And>x. x \<in> A \<Longrightarrow> g (f x) = x) \<Longrightarrow> inj_on f A" |
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by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) |
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lemma inj_onD: "inj_on f A \<Longrightarrow> f x = f y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x = y" |
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unfolding inj_on_def by blast |
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lemma comp_inj_on: "inj_on f A \<Longrightarrow> inj_on g (f ` A) \<Longrightarrow> inj_on (g \<circ> f) A" |
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by (simp add: comp_def inj_on_def) |
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||
219 |
lemma inj_on_imageI: "inj_on (g \<circ> f) A \<Longrightarrow> inj_on g (f ` A)" |
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by (auto simp add: inj_on_def) |
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lemma inj_on_image_iff: |
223 |
"\<forall>x\<in>A. \<forall>y\<in>A. g (f x) = g (f y) \<longleftrightarrow> g x = g y \<Longrightarrow> inj_on f A \<Longrightarrow> inj_on g (f ` A) = inj_on g A" |
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unfolding inj_on_def by blast |
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lemma inj_on_contraD: "inj_on f A \<Longrightarrow> x \<noteq> y \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x \<noteq> f y" |
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unfolding inj_on_def by blast |
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lemma inj_singleton [simp]: "inj_on (\<lambda>x. {x}) A" |
230 |
by (simp add: inj_on_def) |
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lemma inj_on_empty[iff]: "inj_on f {}" |
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by (simp add: inj_on_def) |
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lemma subset_inj_on: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> inj_on f A" |
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unfolding inj_on_def by blast |
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237 |
||
238 |
lemma inj_on_Un: "inj_on f (A \<union> B) \<longleftrightarrow> inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {}" |
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239 |
unfolding inj_on_def by (blast intro: sym) |
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lemma inj_on_insert [iff]: "inj_on f (insert a A) \<longleftrightarrow> inj_on f A \<and> f a \<notin> f ` (A - {a})" |
242 |
unfolding inj_on_def by (blast intro: sym) |
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243 |
||
244 |
lemma inj_on_diff: "inj_on f A \<Longrightarrow> inj_on f (A - B)" |
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245 |
unfolding inj_on_def by blast |
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lemma comp_inj_on_iff: "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' \<circ> f) A" |
248 |
by (auto simp add: comp_inj_on inj_on_def) |
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lemma inj_on_imageI2: "inj_on (f' \<circ> f) A \<Longrightarrow> inj_on f A" |
251 |
by (auto simp add: comp_inj_on inj_on_def) |
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lemma inj_img_insertE: |
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254 |
assumes "inj_on f A" |
63322 | 255 |
assumes "x \<notin> B" |
256 |
and "insert x B = f ` A" |
|
257 |
obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" and "x = f x'" and "B = f ` A'" |
|
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
258 |
proof - |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
259 |
from assms have "x \<in> f ` A" by auto |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
260 |
then obtain x' where *: "x' \<in> A" "x = f x'" by auto |
63322 | 261 |
then have A: "A = insert x' (A - {x'})" by auto |
262 |
with assms * have B: "B = f ` (A - {x'})" by (auto dest: inj_on_contraD) |
|
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
263 |
have "x' \<notin> A - {x'}" by simp |
63322 | 264 |
from this A \<open>x = f x'\<close> B show ?thesis .. |
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
265 |
qed |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
266 |
|
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
267 |
lemma linorder_injI: |
63322 | 268 |
assumes hyp: "\<And>x y::'a::linorder. x < y \<Longrightarrow> f x \<noteq> f y" |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
269 |
shows "inj f" |
61799 | 270 |
\<comment> \<open>Courtesy of Stephan Merz\<close> |
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
271 |
proof (rule inj_onI) |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
272 |
fix x y |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
273 |
assume f_eq: "f x = f y" |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
274 |
show "x = y" by (rule linorder_cases) (auto dest: hyp simp: f_eq) |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
275 |
qed |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
276 |
|
40702 | 277 |
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)" |
278 |
by auto |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
279 |
|
63322 | 280 |
lemma surjI: |
281 |
assumes *: "\<And> x. g (f x) = x" |
|
282 |
shows "surj g" |
|
40702 | 283 |
using *[symmetric] by auto |
13585 | 284 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
285 |
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
286 |
by (simp add: surj_def) |
13585 | 287 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
288 |
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
289 |
by (simp add: surj_def, blast) |
13585 | 290 |
|
63322 | 291 |
lemma comp_surj: "surj f \<Longrightarrow> surj g \<Longrightarrow> surj (g \<circ> f)" |
292 |
apply (simp add: comp_def surj_def) |
|
293 |
apply clarify |
|
294 |
apply (drule_tac x = y in spec) |
|
295 |
apply clarify |
|
296 |
apply (drule_tac x = x in spec) |
|
297 |
apply blast |
|
298 |
done |
|
13585 | 299 |
|
63322 | 300 |
lemma bij_betw_imageI: "inj_on f A \<Longrightarrow> f ` A = B \<Longrightarrow> bij_betw f A B" |
301 |
unfolding bij_betw_def by clarify |
|
57282 | 302 |
|
303 |
lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B" |
|
304 |
unfolding bij_betw_def by clarify |
|
305 |
||
39074 | 306 |
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f" |
40702 | 307 |
unfolding bij_betw_def by auto |
39074 | 308 |
|
63322 | 309 |
lemma bij_betw_empty1: "bij_betw f {} A \<Longrightarrow> A = {}" |
310 |
unfolding bij_betw_def by blast |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
311 |
|
63322 | 312 |
lemma bij_betw_empty2: "bij_betw f A {} \<Longrightarrow> A = {}" |
313 |
unfolding bij_betw_def by blast |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
314 |
|
63322 | 315 |
lemma inj_on_imp_bij_betw: "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)" |
316 |
unfolding bij_betw_def by simp |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
317 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
318 |
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f" |
40702 | 319 |
unfolding bij_betw_def .. |
39074 | 320 |
|
63322 | 321 |
lemma bijI: "inj f \<Longrightarrow> surj f \<Longrightarrow> bij f" |
322 |
by (simp add: bij_def) |
|
13585 | 323 |
|
63322 | 324 |
lemma bij_is_inj: "bij f \<Longrightarrow> inj f" |
325 |
by (simp add: bij_def) |
|
13585 | 326 |
|
63322 | 327 |
lemma bij_is_surj: "bij f \<Longrightarrow> surj f" |
328 |
by (simp add: bij_def) |
|
13585 | 329 |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
330 |
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" |
63322 | 331 |
by (simp add: bij_betw_def) |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
332 |
|
63322 | 333 |
lemma bij_betw_trans: "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g \<circ> f) A C" |
334 |
by (auto simp add:bij_betw_def comp_inj_on) |
|
31438 | 335 |
|
63322 | 336 |
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g \<circ> f)" |
40702 | 337 |
by (rule bij_betw_trans) |
338 |
||
63322 | 339 |
lemma bij_betw_comp_iff: "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''" |
340 |
by (auto simp add: bij_betw_def inj_on_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
341 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
342 |
lemma bij_betw_comp_iff2: |
63322 | 343 |
assumes bij: "bij_betw f' A' A''" |
344 |
and img: "f ` A \<le> A'" |
|
345 |
shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' \<circ> f) A A''" |
|
346 |
using assms |
|
347 |
proof (auto simp add: bij_betw_comp_iff) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
348 |
assume *: "bij_betw (f' \<circ> f) A A''" |
63322 | 349 |
then show "bij_betw f A A'" |
350 |
using img |
|
351 |
proof (auto simp add: bij_betw_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
352 |
assume "inj_on (f' \<circ> f) A" |
63322 | 353 |
then show "inj_on f A" using inj_on_imageI2 by blast |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
354 |
next |
63322 | 355 |
fix a' |
356 |
assume **: "a' \<in> A'" |
|
357 |
then have "f' a' \<in> A''" using bij unfolding bij_betw_def by auto |
|
358 |
then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" |
|
359 |
using * unfolding bij_betw_def by force |
|
360 |
then have "f a \<in> A'" using img by auto |
|
361 |
then have "f a = a'" |
|
362 |
using bij ** 1 unfolding bij_betw_def inj_on_def by auto |
|
363 |
then show "a' \<in> f ` A" |
|
364 |
using 1 by auto |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
365 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
366 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
367 |
|
63322 | 368 |
lemma bij_betw_inv: |
369 |
assumes "bij_betw f A B" |
|
370 |
shows "\<exists>g. bij_betw g B A" |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
371 |
proof - |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
372 |
have i: "inj_on f A" and s: "f ` A = B" |
63322 | 373 |
using assms by (auto simp: bij_betw_def) |
374 |
let ?P = "\<lambda>b a. a \<in> A \<and> f a = b" |
|
375 |
let ?g = "\<lambda>b. The (?P b)" |
|
376 |
have g: "?g b = a" if P: "?P b a" for a b |
|
377 |
proof - |
|
378 |
from that have ex1: "\<exists>a. ?P b a" using s by blast |
|
379 |
then have uex1: "\<exists>!a. ?P b a" by (blast dest:inj_onD[OF i]) |
|
380 |
then show ?thesis using the1_equality[OF uex1, OF P] P by simp |
|
381 |
qed |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
382 |
have "inj_on ?g B" |
63322 | 383 |
proof (rule inj_onI) |
384 |
fix x y |
|
385 |
assume "x \<in> B" "y \<in> B" "?g x = ?g y" |
|
386 |
from s \<open>x \<in> B\<close> obtain a1 where a1: "?P x a1" by blast |
|
387 |
from s \<open>y \<in> B\<close> obtain a2 where a2: "?P y a2" by blast |
|
388 |
from g [OF a1] a1 g [OF a2] a2 \<open>?g x = ?g y\<close> show "x = y" by simp |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
389 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
390 |
moreover have "?g ` B = A" |
63322 | 391 |
proof (auto simp: image_def) |
392 |
fix b |
|
393 |
assume "b \<in> B" |
|
56077 | 394 |
with s obtain a where P: "?P b a" by blast |
63322 | 395 |
then show "?g b \<in> A" using g[OF P] by auto |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
396 |
next |
63322 | 397 |
fix a |
398 |
assume "a \<in> A" |
|
56077 | 399 |
then obtain b where P: "?P b a" using s by blast |
63322 | 400 |
then have "b \<in> B" using s by blast |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
401 |
with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
402 |
qed |
63322 | 403 |
ultimately show ?thesis by (auto simp: bij_betw_def) |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
404 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
405 |
|
63322 | 406 |
lemma bij_betw_cong: "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'" |
407 |
unfolding bij_betw_def inj_on_def by force |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
408 |
|
63322 | 409 |
lemma bij_betw_id[intro, simp]: "bij_betw id A A" |
410 |
unfolding bij_betw_def id_def by auto |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
411 |
|
63322 | 412 |
lemma bij_betw_id_iff: "bij_betw id A B \<longleftrightarrow> A = B" |
413 |
by (auto simp add: bij_betw_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
414 |
|
39075 | 415 |
lemma bij_betw_combine: |
416 |
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}" |
|
417 |
shows "bij_betw f (A \<union> C) (B \<union> D)" |
|
418 |
using assms unfolding bij_betw_def inj_on_Un image_Un by auto |
|
419 |
||
63322 | 420 |
lemma bij_betw_subset: "bij_betw f A A' \<Longrightarrow> B \<le> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw f B B'" |
421 |
by (auto simp add: bij_betw_def inj_on_def) |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
422 |
|
58195 | 423 |
lemma bij_pointE: |
424 |
assumes "bij f" |
|
425 |
obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x" |
|
426 |
proof - |
|
427 |
from assms have "inj f" by (rule bij_is_inj) |
|
428 |
moreover from assms have "surj f" by (rule bij_is_surj) |
|
429 |
then have "y \<in> range f" by simp |
|
430 |
ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq) |
|
431 |
with that show thesis by blast |
|
432 |
qed |
|
433 |
||
63322 | 434 |
lemma surj_image_vimage_eq: "surj f \<Longrightarrow> f ` (f -` A) = A" |
435 |
by simp |
|
13585 | 436 |
|
42903 | 437 |
lemma surj_vimage_empty: |
63322 | 438 |
assumes "surj f" |
439 |
shows "f -` A = {} \<longleftrightarrow> A = {}" |
|
440 |
using surj_image_vimage_eq [OF \<open>surj f\<close>, of A] |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44860
diff
changeset
|
441 |
by (intro iffI) fastforce+ |
42903 | 442 |
|
63322 | 443 |
lemma inj_vimage_image_eq: "inj f \<Longrightarrow> f -` (f ` A) = A" |
444 |
unfolding inj_on_def by blast |
|
13585 | 445 |
|
63322 | 446 |
lemma vimage_subsetD: "surj f \<Longrightarrow> f -` B \<subseteq> A \<Longrightarrow> B \<subseteq> f ` A" |
447 |
by (blast intro: sym) |
|
13585 | 448 |
|
63322 | 449 |
lemma vimage_subsetI: "inj f \<Longrightarrow> B \<subseteq> f ` A \<Longrightarrow> f -` B \<subseteq> A" |
450 |
unfolding inj_on_def by blast |
|
13585 | 451 |
|
63322 | 452 |
lemma vimage_subset_eq: "bij f \<Longrightarrow> f -` B \<subseteq> A \<longleftrightarrow> B \<subseteq> f ` A" |
453 |
unfolding bij_def by (blast del: subsetI intro: vimage_subsetI vimage_subsetD) |
|
13585 | 454 |
|
63322 | 455 |
lemma inj_on_image_eq_iff: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
456 |
by (fastforce simp add: inj_on_def) |
|
53927 | 457 |
|
31438 | 458 |
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
63322 | 459 |
by (erule inj_on_image_eq_iff) simp_all |
31438 | 460 |
|
63322 | 461 |
lemma inj_on_image_Int: "inj_on f C \<Longrightarrow> A \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B" |
462 |
unfolding inj_on_def by blast |
|
463 |
||
464 |
lemma inj_on_image_set_diff: "inj_on f C \<Longrightarrow> A - B \<subseteq> C \<Longrightarrow> B \<subseteq> C \<Longrightarrow> f ` (A - B) = f ` A - f ` B" |
|
465 |
unfolding inj_on_def by blast |
|
13585 | 466 |
|
63322 | 467 |
lemma image_Int: "inj f \<Longrightarrow> f ` (A \<inter> B) = f ` A \<inter> f ` B" |
468 |
unfolding inj_on_def by blast |
|
13585 | 469 |
|
63322 | 470 |
lemma image_set_diff: "inj f \<Longrightarrow> f ` (A - B) = f ` A - f ` B" |
471 |
unfolding inj_on_def by blast |
|
13585 | 472 |
|
63322 | 473 |
lemma inj_on_image_mem_iff: "inj_on f B \<Longrightarrow> a \<in> B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A" |
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
474 |
by (auto simp: inj_on_def) |
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
475 |
|
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61378
diff
changeset
|
476 |
(*FIXME DELETE*) |
63322 | 477 |
lemma inj_on_image_mem_iff_alt: "inj_on f B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> f a \<in> f ` A \<Longrightarrow> a \<in> B \<Longrightarrow> a \<in> A" |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61378
diff
changeset
|
478 |
by (blast dest: inj_onD) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61378
diff
changeset
|
479 |
|
63322 | 480 |
lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f ` A \<longleftrightarrow> a \<in> A" |
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
481 |
by (blast dest: injD) |
13585 | 482 |
|
63322 | 483 |
lemma inj_image_subset_iff: "inj f \<Longrightarrow> f ` A \<subseteq> f ` B \<longleftrightarrow> A \<subseteq> B" |
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
484 |
by (blast dest: injD) |
13585 | 485 |
|
63322 | 486 |
lemma inj_image_eq_iff: "inj f \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
487 |
by (blast dest: injD) |
13585 | 488 |
|
63322 | 489 |
lemma surj_Compl_image_subset: "surj f \<Longrightarrow> - (f ` A) \<subseteq> f ` (- A)" |
490 |
by auto |
|
5852 | 491 |
|
63322 | 492 |
lemma inj_image_Compl_subset: "inj f \<Longrightarrow> f ` (- A) \<subseteq> - (f ` A)" |
493 |
by (auto simp add: inj_on_def) |
|
494 |
||
495 |
lemma bij_image_Compl_eq: "bij f \<Longrightarrow> f ` (- A) = - (f ` A)" |
|
496 |
by (simp add: bij_def inj_image_Compl_subset surj_Compl_image_subset equalityI) |
|
13585 | 497 |
|
41657 | 498 |
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}" |
63322 | 499 |
\<comment> \<open>The inverse image of a singleton under an injective function is included in a singleton.\<close> |
41657 | 500 |
apply (auto simp add: inj_on_def) |
501 |
apply (blast intro: the_equality [symmetric]) |
|
502 |
done |
|
503 |
||
63322 | 504 |
lemma inj_on_vimage_singleton: "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}" |
43991 | 505 |
by (auto simp add: inj_on_def intro: the_equality [symmetric]) |
506 |
||
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
507 |
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" |
35580 | 508 |
by (auto intro!: inj_onI) |
13585 | 509 |
|
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
510 |
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A" |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
511 |
by (auto intro!: inj_onI dest: strict_mono_eq) |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
512 |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
513 |
lemma bij_betw_byWitness: |
63322 | 514 |
assumes left: "\<forall>a \<in> A. f' (f a) = a" |
515 |
and right: "\<forall>a' \<in> A'. f (f' a') = a'" |
|
516 |
and "f ` A \<le> A'" |
|
517 |
and img2: "f' ` A' \<le> A" |
|
518 |
shows "bij_betw f A A'" |
|
519 |
using assms |
|
520 |
proof (unfold bij_betw_def inj_on_def, safe) |
|
521 |
fix a b |
|
522 |
assume *: "a \<in> A" "b \<in> A" and **: "f a = f b" |
|
523 |
have "a = f' (f a) \<and> b = f'(f b)" using * left by simp |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
524 |
with ** show "a = b" by simp |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
525 |
next |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
526 |
fix a' assume *: "a' \<in> A'" |
63322 | 527 |
hence "f' a' \<in> A" using img2 by blast |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
528 |
moreover |
63322 | 529 |
have "a' = f (f' a')" using * right by simp |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
530 |
ultimately show "a' \<in> f ` A" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
531 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
532 |
|
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
533 |
corollary notIn_Un_bij_betw: |
63322 | 534 |
assumes "b \<notin> A" |
535 |
and "f b \<notin> A'" |
|
536 |
and "bij_betw f A A'" |
|
537 |
shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})" |
|
538 |
proof - |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
539 |
have "bij_betw f {b} {f b}" |
63322 | 540 |
unfolding bij_betw_def inj_on_def by simp |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
541 |
with assms show ?thesis |
63322 | 542 |
using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
543 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
544 |
|
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
545 |
lemma notIn_Un_bij_betw3: |
63322 | 546 |
assumes "b \<notin> A" |
547 |
and "f b \<notin> A'" |
|
548 |
shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})" |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
549 |
proof |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
550 |
assume "bij_betw f A A'" |
63322 | 551 |
then show "bij_betw f (A \<union> {b}) (A' \<union> {f b})" |
552 |
using assms notIn_Un_bij_betw [of b A f A'] by blast |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
553 |
next |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
554 |
assume *: "bij_betw f (A \<union> {b}) (A' \<union> {f b})" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
555 |
have "f ` A = A'" |
63322 | 556 |
proof auto |
557 |
fix a |
|
558 |
assume **: "a \<in> A" |
|
559 |
then have "f a \<in> A' \<union> {f b}" |
|
560 |
using * unfolding bij_betw_def by blast |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
561 |
moreover |
63322 | 562 |
have False if "f a = f b" |
563 |
proof - |
|
564 |
have "a = b" using * ** that unfolding bij_betw_def inj_on_def by blast |
|
565 |
with \<open>b \<notin> A\<close> ** show ?thesis by blast |
|
566 |
qed |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
567 |
ultimately show "f a \<in> A'" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
568 |
next |
63322 | 569 |
fix a' |
570 |
assume **: "a' \<in> A'" |
|
571 |
then have "a' \<in> f ` (A \<union> {b})" |
|
572 |
using * by (auto simp add: bij_betw_def) |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
573 |
then obtain a where 1: "a \<in> A \<union> {b} \<and> f a = a'" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
574 |
moreover |
63322 | 575 |
have False if "a = b" using 1 ** \<open>f b \<notin> A'\<close> that by blast |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
576 |
ultimately have "a \<in> A" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
577 |
with 1 show "a' \<in> f ` A" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
578 |
qed |
63322 | 579 |
then show "bij_betw f A A'" |
580 |
using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
581 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
582 |
|
41657 | 583 |
|
63322 | 584 |
subsection \<open>Function Updating\<close> |
13585 | 585 |
|
63322 | 586 |
definition fun_upd :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> 'b)" |
587 |
where "fun_upd f a b \<equiv> \<lambda>x. if x = a then b else f x" |
|
26147 | 588 |
|
41229
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset
|
589 |
nonterminal updbinds and updbind |
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset
|
590 |
|
26147 | 591 |
syntax |
63322 | 592 |
"_updbind" :: "'a \<Rightarrow> 'a \<Rightarrow> updbind" ("(2_ :=/ _)") |
593 |
"" :: "updbind \<Rightarrow> updbinds" ("_") |
|
594 |
"_updbinds":: "updbind \<Rightarrow> updbinds \<Rightarrow> updbinds" ("_,/ _") |
|
595 |
"_Update" :: "'a \<Rightarrow> updbinds \<Rightarrow> 'a" ("_/'((_)')" [1000, 0] 900) |
|
26147 | 596 |
|
597 |
translations |
|
63322 | 598 |
"_Update f (_updbinds b bs)" \<rightleftharpoons> "_Update (_Update f b) bs" |
599 |
"f(x:=y)" \<rightleftharpoons> "CONST fun_upd f x y" |
|
26147 | 600 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55066
diff
changeset
|
601 |
(* Hint: to define the sum of two functions (or maps), use case_sum. |
58111 | 602 |
A nice infix syntax could be defined by |
35115 | 603 |
notation |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55066
diff
changeset
|
604 |
case_sum (infixr "'(+')"80) |
26147 | 605 |
*) |
606 |
||
63322 | 607 |
lemma fun_upd_idem_iff: "f(x:=y) = f \<longleftrightarrow> f x = y" |
608 |
unfolding fun_upd_def |
|
609 |
apply safe |
|
610 |
apply (erule subst) |
|
611 |
apply (rule_tac [2] ext) |
|
612 |
apply auto |
|
613 |
done |
|
13585 | 614 |
|
63322 | 615 |
lemma fun_upd_idem: "f x = y \<Longrightarrow> f(x := y) = f" |
45603 | 616 |
by (simp only: fun_upd_idem_iff) |
13585 | 617 |
|
45603 | 618 |
lemma fun_upd_triv [iff]: "f(x := f x) = f" |
619 |
by (simp only: fun_upd_idem) |
|
13585 | 620 |
|
63322 | 621 |
lemma fun_upd_apply [simp]: "(f(x := y)) z = (if z = x then y else f z)" |
622 |
by (simp add: fun_upd_def) |
|
13585 | 623 |
|
63322 | 624 |
(* fun_upd_apply supersedes these two, but they are useful |
13585 | 625 |
if fun_upd_apply is intentionally removed from the simpset *) |
63322 | 626 |
lemma fun_upd_same: "(f(x := y)) x = y" |
627 |
by simp |
|
13585 | 628 |
|
63322 | 629 |
lemma fun_upd_other: "z \<noteq> x \<Longrightarrow> (f(x := y)) z = f z" |
630 |
by simp |
|
13585 | 631 |
|
63322 | 632 |
lemma fun_upd_upd [simp]: "f(x := y, x := z) = f(x := z)" |
633 |
by (simp add: fun_eq_iff) |
|
13585 | 634 |
|
63322 | 635 |
lemma fun_upd_twist: "a \<noteq> c \<Longrightarrow> (m(a := b))(c := d) = (m(c := d))(a := b)" |
636 |
by (rule ext) auto |
|
637 |
||
638 |
lemma inj_on_fun_updI: "inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A" |
|
56077 | 639 |
by (fastforce simp: inj_on_def) |
15303 | 640 |
|
63322 | 641 |
lemma fun_upd_image: "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)" |
642 |
by auto |
|
15510 | 643 |
|
31080 | 644 |
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" |
44921 | 645 |
by auto |
31080 | 646 |
|
61630 | 647 |
lemma fun_upd_eqD: "f(x := y) = g(x := z) \<Longrightarrow> y = z" |
63322 | 648 |
by (simp add: fun_eq_iff split: if_split_asm) |
649 |
||
26147 | 650 |
|
61799 | 651 |
subsection \<open>\<open>override_on\<close>\<close> |
26147 | 652 |
|
63322 | 653 |
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" |
654 |
where "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" |
|
13910 | 655 |
|
15691 | 656 |
lemma override_on_emptyset[simp]: "override_on f g {} = f" |
63322 | 657 |
by (simp add:override_on_def) |
13910 | 658 |
|
63322 | 659 |
lemma override_on_apply_notin[simp]: "a \<notin> A \<Longrightarrow> (override_on f g A) a = f a" |
660 |
by (simp add:override_on_def) |
|
13910 | 661 |
|
63322 | 662 |
lemma override_on_apply_in[simp]: "a \<in> A \<Longrightarrow> (override_on f g A) a = g a" |
663 |
by (simp add:override_on_def) |
|
13910 | 664 |
|
26147 | 665 |
|
61799 | 666 |
subsection \<open>\<open>swap\<close>\<close> |
15510 | 667 |
|
56608 | 668 |
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" |
63322 | 669 |
where "swap a b f = f (a := f b, b:= f a)" |
15510 | 670 |
|
56608 | 671 |
lemma swap_apply [simp]: |
672 |
"swap a b f a = f b" |
|
673 |
"swap a b f b = f a" |
|
674 |
"c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c" |
|
675 |
by (simp_all add: swap_def) |
|
676 |
||
63322 | 677 |
lemma swap_self [simp]: "swap a a f = f" |
56608 | 678 |
by (simp add: swap_def) |
15510 | 679 |
|
63322 | 680 |
lemma swap_commute: "swap a b f = swap b a f" |
56608 | 681 |
by (simp add: fun_upd_def swap_def fun_eq_iff) |
15510 | 682 |
|
63322 | 683 |
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" |
56608 | 684 |
by (rule ext, simp add: fun_upd_def swap_def) |
685 |
||
63322 | 686 |
lemma swap_comp_involutory [simp]: "swap a b \<circ> swap a b = id" |
56608 | 687 |
by (rule ext) simp |
15510 | 688 |
|
34145 | 689 |
lemma swap_triple: |
690 |
assumes "a \<noteq> c" and "b \<noteq> c" |
|
691 |
shows "swap a b (swap b c (swap a b f)) = swap a c f" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39213
diff
changeset
|
692 |
using assms by (simp add: fun_eq_iff swap_def) |
34145 | 693 |
|
34101 | 694 |
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" |
63322 | 695 |
by (rule ext) (simp add: fun_upd_def swap_def) |
34101 | 696 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
697 |
lemma swap_image_eq [simp]: |
63322 | 698 |
assumes "a \<in> A" "b \<in> A" |
699 |
shows "swap a b f ` A = f ` A" |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
700 |
proof - |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
701 |
have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
702 |
using assms by (auto simp: image_iff swap_def) |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
703 |
then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" . |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
704 |
with subset[of f] show ?thesis by auto |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
705 |
qed |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
706 |
|
63322 | 707 |
lemma inj_on_imp_inj_on_swap: "inj_on f A \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> inj_on (swap a b f) A" |
708 |
by (auto simp add: inj_on_def swap_def) |
|
15510 | 709 |
|
710 |
lemma inj_on_swap_iff [simp]: |
|
63322 | 711 |
assumes A: "a \<in> A" "b \<in> A" |
712 |
shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A" |
|
39075 | 713 |
proof |
15510 | 714 |
assume "inj_on (swap a b f) A" |
39075 | 715 |
with A have "inj_on (swap a b (swap a b f)) A" |
716 |
by (iprover intro: inj_on_imp_inj_on_swap) |
|
63322 | 717 |
then show "inj_on f A" by simp |
15510 | 718 |
next |
719 |
assume "inj_on f A" |
|
63322 | 720 |
with A show "inj_on (swap a b f) A" |
721 |
by (iprover intro: inj_on_imp_inj_on_swap) |
|
15510 | 722 |
qed |
723 |
||
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
724 |
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)" |
40702 | 725 |
by simp |
15510 | 726 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
727 |
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f" |
40702 | 728 |
by simp |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
729 |
|
63322 | 730 |
lemma bij_betw_swap_iff [simp]: "x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B" |
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
731 |
by (auto simp: bij_betw_def) |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
732 |
|
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
733 |
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
734 |
by simp |
39075 | 735 |
|
36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
wenzelm
parents:
35584
diff
changeset
|
736 |
hide_const (open) swap |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
737 |
|
56608 | 738 |
|
60758 | 739 |
subsection \<open>Inversion of injective functions\<close> |
31949 | 740 |
|
63322 | 741 |
definition the_inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" |
742 |
where "the_inv_into A f \<equiv> \<lambda>x. THE y. y \<in> A \<and> f y = x" |
|
743 |
||
744 |
lemma the_inv_into_f_f: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f (f x) = x" |
|
745 |
unfolding the_inv_into_def inj_on_def by blast |
|
32961 | 746 |
|
63322 | 747 |
lemma f_the_inv_into_f: "inj_on f A \<Longrightarrow> y \<in> f ` A \<Longrightarrow> f (the_inv_into A f y) = y" |
748 |
apply (simp add: the_inv_into_def) |
|
749 |
apply (rule the1I2) |
|
750 |
apply(blast dest: inj_onD) |
|
751 |
apply blast |
|
752 |
done |
|
32961 | 753 |
|
63322 | 754 |
lemma the_inv_into_into: "inj_on f A \<Longrightarrow> x \<in> f ` A \<Longrightarrow> A \<subseteq> B \<Longrightarrow> the_inv_into A f x \<in> B" |
755 |
apply (simp add: the_inv_into_def) |
|
756 |
apply (rule the1I2) |
|
757 |
apply(blast dest: inj_onD) |
|
758 |
apply blast |
|
759 |
done |
|
32961 | 760 |
|
63322 | 761 |
lemma the_inv_into_onto [simp]: "inj_on f A \<Longrightarrow> the_inv_into A f ` (f ` A) = A" |
762 |
by (fast intro: the_inv_into_into the_inv_into_f_f [symmetric]) |
|
32961 | 763 |
|
63322 | 764 |
lemma the_inv_into_f_eq: "inj_on f A \<Longrightarrow> f x = y \<Longrightarrow> x \<in> A \<Longrightarrow> the_inv_into A f y = x" |
32961 | 765 |
apply (erule subst) |
63322 | 766 |
apply (erule the_inv_into_f_f) |
767 |
apply assumption |
|
32961 | 768 |
done |
769 |
||
33057 | 770 |
lemma the_inv_into_comp: |
63322 | 771 |
"inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow> |
772 |
the_inv_into A (f \<circ> g) x = (the_inv_into A g \<circ> the_inv_into (g ` A) f) x" |
|
773 |
apply (rule the_inv_into_f_eq) |
|
774 |
apply (fast intro: comp_inj_on) |
|
775 |
apply (simp add: f_the_inv_into_f the_inv_into_into) |
|
776 |
apply (simp add: the_inv_into_into) |
|
777 |
done |
|
32961 | 778 |
|
63322 | 779 |
lemma inj_on_the_inv_into: "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)" |
780 |
by (auto intro: inj_onI simp: the_inv_into_f_f) |
|
32961 | 781 |
|
63322 | 782 |
lemma bij_betw_the_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A" |
783 |
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) |
|
32961 | 784 |
|
63322 | 785 |
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" |
786 |
where "the_inv f \<equiv> the_inv_into UNIV f" |
|
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
787 |
|
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
788 |
lemma the_inv_f_f: |
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
789 |
assumes "inj f" |
63322 | 790 |
shows "the_inv f (f x) = x" |
791 |
using assms UNIV_I by (rule the_inv_into_f_f) |
|
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
792 |
|
44277
bcb696533579
moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents:
43991
diff
changeset
|
793 |
|
60758 | 794 |
subsection \<open>Cantor's Paradox\<close> |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
795 |
|
63323 | 796 |
theorem Cantors_paradox: "\<nexists>f. f ` A = Pow A" |
797 |
proof |
|
798 |
assume "\<exists>f. f ` A = Pow A" |
|
799 |
then obtain f where f: "f ` A = Pow A" .. |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
800 |
let ?X = "{a \<in> A. a \<notin> f a}" |
63323 | 801 |
have "?X \<in> Pow A" by blast |
802 |
then have "?X \<in> f ` A" by (simp only: f) |
|
803 |
then obtain x where "x \<in> A" and "f x = ?X" by blast |
|
804 |
then show False by blast |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
805 |
qed |
31949 | 806 |
|
63322 | 807 |
|
61204 | 808 |
subsection \<open>Setup\<close> |
40969 | 809 |
|
60758 | 810 |
subsubsection \<open>Proof tools\<close> |
22845 | 811 |
|
63322 | 812 |
text \<open>Simplify terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close> |
22845 | 813 |
|
60758 | 814 |
simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ => |
63322 | 815 |
let |
816 |
fun gen_fun_upd NONE T _ _ = NONE |
|
817 |
| gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) |
|
818 |
fun dest_fun_T1 (Type (_, T :: Ts)) = T |
|
819 |
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = |
|
820 |
let |
|
821 |
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = |
|
822 |
if v aconv x then SOME g else gen_fun_upd (find g) T v w |
|
823 |
| find t = NONE |
|
824 |
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end |
|
24017 | 825 |
|
63322 | 826 |
val ss = simpset_of @{context} |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51598
diff
changeset
|
827 |
|
63322 | 828 |
fun proc ctxt ct = |
829 |
let |
|
830 |
val t = Thm.term_of ct |
|
831 |
in |
|
832 |
case find_double t of |
|
833 |
(T, NONE) => NONE |
|
834 |
| (T, SOME rhs) => |
|
835 |
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) |
|
836 |
(fn _ => |
|
837 |
resolve_tac ctxt [eq_reflection] 1 THEN |
|
838 |
resolve_tac ctxt @{thms ext} 1 THEN |
|
839 |
simp_tac (put_simpset ss ctxt) 1)) |
|
840 |
end |
|
841 |
in proc end |
|
60758 | 842 |
\<close> |
22845 | 843 |
|
844 |
||
60758 | 845 |
subsubsection \<open>Functorial structure of types\<close> |
40969 | 846 |
|
55467
a5c9002bc54d
renamed 'enriched_type' to more informative 'functor' (following the renaming of enriched type constructors to bounded natural functors)
blanchet
parents:
55414
diff
changeset
|
847 |
ML_file "Tools/functor.ML" |
40969 | 848 |
|
55467
a5c9002bc54d
renamed 'enriched_type' to more informative 'functor' (following the renaming of enriched type constructors to bounded natural functors)
blanchet
parents:
55414
diff
changeset
|
849 |
functor map_fun: map_fun |
47488
be6dd389639d
centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents:
46950
diff
changeset
|
850 |
by (simp_all add: fun_eq_iff) |
be6dd389639d
centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents:
46950
diff
changeset
|
851 |
|
55467
a5c9002bc54d
renamed 'enriched_type' to more informative 'functor' (following the renaming of enriched type constructors to bounded natural functors)
blanchet
parents:
55414
diff
changeset
|
852 |
functor vimage |
49739 | 853 |
by (simp_all add: fun_eq_iff vimage_comp) |
854 |
||
63322 | 855 |
|
60758 | 856 |
text \<open>Legacy theorem names\<close> |
49739 | 857 |
|
858 |
lemmas o_def = comp_def |
|
859 |
lemmas o_apply = comp_apply |
|
860 |
lemmas o_assoc = comp_assoc [symmetric] |
|
861 |
lemmas id_o = id_comp |
|
862 |
lemmas o_id = comp_id |
|
863 |
lemmas o_eq_dest = comp_eq_dest |
|
864 |
lemmas o_eq_elim = comp_eq_elim |
|
55066 | 865 |
lemmas o_eq_dest_lhs = comp_eq_dest_lhs |
866 |
lemmas o_eq_id_dest = comp_eq_id_dest |
|
47488
be6dd389639d
centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents:
46950
diff
changeset
|
867 |
|
2912 | 868 |
end |