author | haftmann |
Tue, 13 Oct 2015 09:21:15 +0200 | |
changeset 61424 | c3658c18b7bc |
parent 61378 | 3e04c9ca001a |
child 61520 | 8f85bb443d33 |
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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|
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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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|
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lemma apply_inverse: |
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"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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|
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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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|
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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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|
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subsection \<open>The Identity Function @{text id}\<close> |
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definition id :: "'a \<Rightarrow> 'a" where |
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"id = (\<lambda>x. x)" |
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|
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lemma id_apply [simp]: "id x = x" |
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by (simp add: id_def) |
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32 |
||
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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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code_printing |
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constant id \<rightharpoonup> (Haskell) "id" |
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subsection \<open>The Composition Operator @{text "f \<circ> g"}\<close> |
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definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where |
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"f o g = (\<lambda>x. f (g x))" |
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notation (xsymbols) |
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comp (infixl "\<circ>" 55) |
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lemma comp_apply [simp]: "(f o g) x = f (g x)" |
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by (simp add: comp_def) |
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lemma comp_assoc: "(f o g) o h = f o (g o h)" |
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by (simp add: fun_eq_iff) |
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lemma id_comp [simp]: "id o g = g" |
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by (simp add: fun_eq_iff) |
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lemma comp_id [simp]: "f o id = f" |
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by (simp add: fun_eq_iff) |
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||
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lemma comp_eq_dest: |
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"a o b = c o d \<Longrightarrow> a (b v) = c (d v)" |
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by (simp add: fun_eq_iff) |
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|
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lemma comp_eq_elim: |
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"a o b = c o 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 o b = c \<Longrightarrow> a (b v) = c v" |
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by clarsimp |
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||
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lemma comp_eq_id_dest: "a o b = id o c \<Longrightarrow> a (b v) = c v" |
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by clarsimp |
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||
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lemma image_comp: |
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"f ` (g ` r) = (f o g) ` r" |
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by auto |
80 |
||
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lemma vimage_comp: |
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"f -` (g -` x) = (g \<circ> f) -` x" |
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by auto |
84 |
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lemma image_eq_imp_comp: "f ` A = g ` B \<Longrightarrow> (h o f) ` A = (h o g) ` B" |
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by (auto simp: comp_def elim!: equalityE) |
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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]: |
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"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]: |
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"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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104 |
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subsection \<open>The Forward Composition Operator @{text fcomp}\<close> |
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definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where |
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"f \<circ>> g = (\<lambda>x. g (f x))" |
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|
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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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||
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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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||
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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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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" where |
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"map_fun f g h = g \<circ> h \<circ> f" |
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lemma map_fun_apply [simp]: |
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"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" where -- "injective" |
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"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" where -- "bijective" |
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"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 |
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"inj f \<equiv> inj_on f UNIV" |
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abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective" |
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"surj f \<equiv> (range f = UNIV)" |
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abbreviation |
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"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" <= "CONST range f \<noteq> CONST UNIV" |
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lemma injI: |
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assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" |
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shows "inj f" |
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using assms unfolding inj_on_def by auto |
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theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)" |
169 |
by (unfold inj_on_def, blast) |
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||
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lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y" |
172 |
by (simp add: inj_on_def) |
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||
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lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)" |
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by (force simp add: inj_on_def) |
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lemma inj_on_cong: |
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"(\<And> a. a : 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: |
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"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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|
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lemma inj_comp: |
186 |
"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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|
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lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)" |
193 |
by (simp add: inj_on_eq_iff) |
|
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||
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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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|
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lemma inj_on_id2[simp]: "inj_on (%x. x) A" |
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by (simp add: inj_on_def) |
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|
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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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unfolding inj_on_def by blast |
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|
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lemma surj_id: "surj id" |
205 |
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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|
210 |
lemma inj_onI: |
|
211 |
"(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" |
|
212 |
by (simp add: inj_on_def) |
|
213 |
||
214 |
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" |
|
215 |
by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) |
|
216 |
||
217 |
lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" |
|
218 |
by (unfold inj_on_def, blast) |
|
219 |
||
220 |
lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" |
|
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by (fact inj_on_eq_iff) |
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|
223 |
lemma comp_inj_on: |
|
224 |
"[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" |
|
225 |
by (simp add: comp_def inj_on_def) |
|
226 |
||
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lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" |
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by (simp add: inj_on_def) blast |
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|
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lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); |
231 |
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" |
|
232 |
apply(unfold inj_on_def) |
|
233 |
apply blast |
|
234 |
done |
|
235 |
||
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lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" |
237 |
by (unfold inj_on_def, blast) |
|
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|
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lemma inj_singleton: "inj (%s. {s})" |
240 |
by (simp add: inj_on_def) |
|
241 |
||
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lemma inj_on_empty[iff]: "inj_on f {}" |
243 |
by(simp add: inj_on_def) |
|
244 |
||
15303 | 245 |
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" |
13585 | 246 |
by (unfold inj_on_def, blast) |
247 |
||
15111 | 248 |
lemma inj_on_Un: |
249 |
"inj_on f (A Un B) = |
|
250 |
(inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" |
|
251 |
apply(unfold inj_on_def) |
|
252 |
apply (blast intro:sym) |
|
253 |
done |
|
254 |
||
255 |
lemma inj_on_insert[iff]: |
|
256 |
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" |
|
257 |
apply(unfold inj_on_def) |
|
258 |
apply (blast intro:sym) |
|
259 |
done |
|
260 |
||
261 |
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" |
|
262 |
apply(unfold inj_on_def) |
|
263 |
apply (blast) |
|
264 |
done |
|
265 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
266 |
lemma comp_inj_on_iff: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
267 |
"inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
268 |
by(auto simp add: comp_inj_on inj_on_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
269 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
270 |
lemma inj_on_imageI2: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
271 |
"inj_on (f' o f) A \<Longrightarrow> inj_on f A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
272 |
by(auto simp add: comp_inj_on inj_on_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
273 |
|
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
274 |
lemma inj_img_insertE: |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
275 |
assumes "inj_on f A" |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
276 |
assumes "x \<notin> B" and "insert x B = f ` A" |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
277 |
obtains x' A' where "x' \<notin> A'" and "A = insert x' A'" |
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
278 |
and "x = f x'" and "B = f ` A'" |
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
279 |
proof - |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
280 |
from assms have "x \<in> f ` A" by auto |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
281 |
then obtain x' where *: "x' \<in> A" "x = f x'" by auto |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
282 |
then have "A = insert x' (A - {x'})" by auto |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
283 |
with assms * have "B = f ` (A - {x'})" |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
284 |
by (auto dest: inj_on_contraD) |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
285 |
have "x' \<notin> A - {x'}" by simp |
60758 | 286 |
from \<open>x' \<notin> A - {x'}\<close> \<open>A = insert x' (A - {x'})\<close> \<open>x = f x'\<close> \<open>B = image f (A - {x'})\<close> |
51598
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
287 |
show ?thesis .. |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
288 |
qed |
5dbe537087aa
generalized lemma fold_image thanks to Peter Lammich
haftmann
parents:
49905
diff
changeset
|
289 |
|
54578
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
290 |
lemma linorder_injI: |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
291 |
assumes hyp: "\<And>x y. x < (y::'a::linorder) \<Longrightarrow> f x \<noteq> f y" |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
292 |
shows "inj f" |
60758 | 293 |
-- \<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
|
294 |
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
|
295 |
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
|
296 |
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
|
297 |
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
|
298 |
qed |
9387251b6a46
eliminated dependence of BNF on Infinite_Set by moving 3 theorems from the latter to Main
traytel
parents:
54147
diff
changeset
|
299 |
|
40702 | 300 |
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)" |
301 |
by auto |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
302 |
|
40702 | 303 |
lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g" |
304 |
using *[symmetric] by auto |
|
13585 | 305 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
306 |
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
|
307 |
by (simp add: surj_def) |
13585 | 308 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
309 |
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
|
310 |
by (simp add: surj_def, blast) |
13585 | 311 |
|
312 |
lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" |
|
313 |
apply (simp add: comp_def surj_def, clarify) |
|
314 |
apply (drule_tac x = y in spec, clarify) |
|
315 |
apply (drule_tac x = x in spec, blast) |
|
316 |
done |
|
317 |
||
57282 | 318 |
lemma bij_betw_imageI: |
319 |
"\<lbrakk> inj_on f A; f ` A = B \<rbrakk> \<Longrightarrow> bij_betw f A B" |
|
320 |
unfolding bij_betw_def by clarify |
|
321 |
||
322 |
lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> f ` A = B" |
|
323 |
unfolding bij_betw_def by clarify |
|
324 |
||
39074 | 325 |
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f" |
40702 | 326 |
unfolding bij_betw_def by auto |
39074 | 327 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
328 |
lemma bij_betw_empty1: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
329 |
assumes "bij_betw f {} A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
330 |
shows "A = {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
331 |
using assms unfolding bij_betw_def by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
332 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
333 |
lemma bij_betw_empty2: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
334 |
assumes "bij_betw f A {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
335 |
shows "A = {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
336 |
using assms unfolding bij_betw_def by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
337 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
338 |
lemma inj_on_imp_bij_betw: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
339 |
"inj_on f A \<Longrightarrow> bij_betw f A (f ` A)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
340 |
unfolding bij_betw_def by simp |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
341 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
342 |
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f" |
40702 | 343 |
unfolding bij_betw_def .. |
39074 | 344 |
|
13585 | 345 |
lemma bijI: "[| inj f; surj f |] ==> bij f" |
346 |
by (simp add: bij_def) |
|
347 |
||
348 |
lemma bij_is_inj: "bij f ==> inj f" |
|
349 |
by (simp add: bij_def) |
|
350 |
||
351 |
lemma bij_is_surj: "bij f ==> surj f" |
|
352 |
by (simp add: bij_def) |
|
353 |
||
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
354 |
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
355 |
by (simp add: bij_betw_def) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
356 |
|
31438 | 357 |
lemma bij_betw_trans: |
358 |
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C" |
|
359 |
by(auto simp add:bij_betw_def comp_inj_on) |
|
360 |
||
40702 | 361 |
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" |
362 |
by (rule bij_betw_trans) |
|
363 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
364 |
lemma bij_betw_comp_iff: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
365 |
"bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
366 |
by(auto simp add: bij_betw_def inj_on_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
367 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
368 |
lemma bij_betw_comp_iff2: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
369 |
assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
370 |
shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
371 |
using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
372 |
proof(auto simp add: bij_betw_comp_iff) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
373 |
assume *: "bij_betw (f' \<circ> f) A A''" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
374 |
thus "bij_betw f A A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
375 |
using IM |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
376 |
proof(auto simp add: bij_betw_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
377 |
assume "inj_on (f' \<circ> f) A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
378 |
thus "inj_on f A" using inj_on_imageI2 by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
379 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
380 |
fix a' assume **: "a' \<in> A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
381 |
hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
382 |
then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using * |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
383 |
unfolding bij_betw_def by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
384 |
hence "f a \<in> A'" using IM by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
385 |
hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
386 |
thus "a' \<in> f ` A" using 1 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
387 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
388 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
389 |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
390 |
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
391 |
proof - |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
392 |
have i: "inj_on f A" and s: "f ` A = B" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
393 |
using assms by(auto simp:bij_betw_def) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
394 |
let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
395 |
{ fix a b assume P: "?P b a" |
56077 | 396 |
hence ex1: "\<exists>a. ?P b a" using s by blast |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
397 |
hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i]) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
398 |
hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
399 |
} note g = this |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
400 |
have "inj_on ?g B" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
401 |
proof(rule inj_onI) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
402 |
fix x y assume "x:B" "y:B" "?g x = ?g y" |
60758 | 403 |
from s \<open>x:B\<close> obtain a1 where a1: "?P x a1" by blast |
404 |
from s \<open>y:B\<close> obtain a2 where a2: "?P y a2" by blast |
|
405 |
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
|
406 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
407 |
moreover have "?g ` B = A" |
56077 | 408 |
proof(auto simp: image_def) |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
409 |
fix b assume "b:B" |
56077 | 410 |
with s obtain a where P: "?P b a" by blast |
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
411 |
thus "?g b \<in> A" using g[OF P] by auto |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
412 |
next |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
413 |
fix a assume "a:A" |
56077 | 414 |
then obtain b where P: "?P b a" using s by blast |
415 |
then have "b:B" using s by blast |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
416 |
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
|
417 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
418 |
ultimately show ?thesis by(auto simp:bij_betw_def) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
419 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
420 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
421 |
lemma bij_betw_cong: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
422 |
"(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
423 |
unfolding bij_betw_def inj_on_def by force |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
424 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
425 |
lemma bij_betw_id[intro, simp]: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
426 |
"bij_betw id A A" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
427 |
unfolding bij_betw_def id_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
428 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
429 |
lemma bij_betw_id_iff: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
430 |
"bij_betw id A B \<longleftrightarrow> A = B" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
431 |
by(auto simp add: bij_betw_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
432 |
|
39075 | 433 |
lemma bij_betw_combine: |
434 |
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}" |
|
435 |
shows "bij_betw f (A \<union> C) (B \<union> D)" |
|
436 |
using assms unfolding bij_betw_def inj_on_Un image_Un by auto |
|
437 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
438 |
lemma bij_betw_subset: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
439 |
assumes BIJ: "bij_betw f A A'" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
440 |
SUB: "B \<le> A" and IM: "f ` B = B'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
441 |
shows "bij_betw f B B'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
442 |
using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
443 |
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
444 |
|
58195 | 445 |
lemma bij_pointE: |
446 |
assumes "bij f" |
|
447 |
obtains x where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x" |
|
448 |
proof - |
|
449 |
from assms have "inj f" by (rule bij_is_inj) |
|
450 |
moreover from assms have "surj f" by (rule bij_is_surj) |
|
451 |
then have "y \<in> range f" by simp |
|
452 |
ultimately have "\<exists>!x. y = f x" by (simp add: range_ex1_eq) |
|
453 |
with that show thesis by blast |
|
454 |
qed |
|
455 |
||
13585 | 456 |
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" |
40702 | 457 |
by simp |
13585 | 458 |
|
42903 | 459 |
lemma surj_vimage_empty: |
460 |
assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}" |
|
60758 | 461 |
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
|
462 |
by (intro iffI) fastforce+ |
42903 | 463 |
|
13585 | 464 |
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" |
465 |
by (simp add: inj_on_def, blast) |
|
466 |
||
467 |
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" |
|
40702 | 468 |
by (blast intro: sym) |
13585 | 469 |
|
470 |
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" |
|
471 |
by (unfold inj_on_def, blast) |
|
472 |
||
473 |
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" |
|
474 |
apply (unfold bij_def) |
|
475 |
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) |
|
476 |
done |
|
477 |
||
53927 | 478 |
lemma inj_on_image_eq_iff: "\<lbrakk> inj_on f C; A \<subseteq> C; B \<subseteq> C \<rbrakk> \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
479 |
by(fastforce simp add: inj_on_def) |
|
480 |
||
31438 | 481 |
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
53927 | 482 |
by(erule inj_on_image_eq_iff) simp_all |
31438 | 483 |
|
13585 | 484 |
lemma inj_on_image_Int: |
485 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" |
|
60303 | 486 |
by (simp add: inj_on_def, blast) |
13585 | 487 |
|
488 |
lemma inj_on_image_set_diff: |
|
60303 | 489 |
"[| inj_on f C; A-B \<subseteq> C; B \<subseteq> C |] ==> f`(A-B) = f`A - f`B" |
490 |
by (simp add: inj_on_def, blast) |
|
13585 | 491 |
|
492 |
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" |
|
60303 | 493 |
by (simp add: inj_on_def, blast) |
13585 | 494 |
|
495 |
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" |
|
496 |
by (simp add: inj_on_def, blast) |
|
497 |
||
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
498 |
lemma inj_on_image_mem_iff: "\<lbrakk>inj_on f B; a \<in> B; A \<subseteq> B\<rbrakk> \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A" |
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
499 |
by (auto simp: inj_on_def) |
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
500 |
|
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
501 |
lemma inj_image_mem_iff: "inj f \<Longrightarrow> f a \<in> f`A \<longleftrightarrow> a \<in> A" |
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
502 |
by (blast dest: injD) |
13585 | 503 |
|
504 |
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" |
|
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
505 |
by (blast dest: injD) |
13585 | 506 |
|
507 |
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" |
|
59504
8c6747dba731
New lemmas and a bit of tidying up.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
508 |
by (blast dest: injD) |
13585 | 509 |
|
510 |
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" |
|
40702 | 511 |
by auto |
13585 | 512 |
|
513 |
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" |
|
514 |
by (auto simp add: inj_on_def) |
|
5852 | 515 |
|
13585 | 516 |
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" |
517 |
apply (simp add: bij_def) |
|
518 |
apply (rule equalityI) |
|
519 |
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) |
|
520 |
done |
|
521 |
||
41657 | 522 |
lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}" |
60758 | 523 |
-- \<open>The inverse image of a singleton under an injective function |
524 |
is included in a singleton.\<close> |
|
41657 | 525 |
apply (auto simp add: inj_on_def) |
526 |
apply (blast intro: the_equality [symmetric]) |
|
527 |
done |
|
528 |
||
43991 | 529 |
lemma inj_on_vimage_singleton: |
530 |
"inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}" |
|
531 |
by (auto simp add: inj_on_def intro: the_equality [symmetric]) |
|
532 |
||
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
533 |
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" |
35580 | 534 |
by (auto intro!: inj_onI) |
13585 | 535 |
|
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
536 |
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
|
537 |
by (auto intro!: inj_onI dest: strict_mono_eq) |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
538 |
|
55019
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
539 |
lemma bij_betw_byWitness: |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
540 |
assumes LEFT: "\<forall>a \<in> A. f'(f a) = a" and |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
541 |
RIGHT: "\<forall>a' \<in> A'. f(f' a') = a'" and |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
542 |
IM1: "f ` A \<le> A'" and IM2: "f' ` A' \<le> A" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
543 |
shows "bij_betw f A A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
544 |
using assms |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
545 |
proof(unfold bij_betw_def inj_on_def, safe) |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
546 |
fix a b assume *: "a \<in> A" "b \<in> A" and **: "f a = f b" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
547 |
have "a = f'(f a) \<and> b = f'(f b)" using * LEFT by simp |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
548 |
with ** show "a = b" by simp |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
549 |
next |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
550 |
fix a' assume *: "a' \<in> A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
551 |
hence "f' a' \<in> A" using IM2 by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
552 |
moreover |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
553 |
have "a' = f(f' a')" using * RIGHT by simp |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
554 |
ultimately show "a' \<in> f ` A" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
555 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
556 |
|
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
557 |
corollary notIn_Un_bij_betw: |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
558 |
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" and |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
559 |
BIJ: "bij_betw f A A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
560 |
shows "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
|
561 |
proof- |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
562 |
have "bij_betw f {b} {f b}" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
563 |
unfolding bij_betw_def inj_on_def by simp |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
564 |
with assms show ?thesis |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
565 |
using bij_betw_combine[of f A A' "{b}" "{f b}"] by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
566 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
567 |
|
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
568 |
lemma notIn_Un_bij_betw3: |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
569 |
assumes NIN: "b \<notin> A" and NIN': "f b \<notin> A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
570 |
shows "bij_betw f A A' = 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
|
571 |
proof |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
572 |
assume "bij_betw f A A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
573 |
thus "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
|
574 |
using assms notIn_Un_bij_betw[of b A f A'] by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
575 |
next |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
576 |
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
|
577 |
have "f ` A = A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
578 |
proof(auto) |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
579 |
fix a assume **: "a \<in> A" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
580 |
hence "f a \<in> A' \<union> {f b}" using * unfolding bij_betw_def by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
581 |
moreover |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
582 |
{assume "f a = f b" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
583 |
hence "a = b" using * ** unfolding bij_betw_def inj_on_def by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
584 |
with NIN ** have False by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
585 |
} |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
586 |
ultimately show "f a \<in> A'" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
587 |
next |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
588 |
fix a' assume **: "a' \<in> A'" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
589 |
hence "a' \<in> f`(A \<union> {b})" |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
590 |
using * by (auto simp add: bij_betw_def) |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
591 |
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
|
592 |
moreover |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
593 |
{assume "a = b" with 1 ** NIN' have False by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
594 |
} |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
595 |
ultimately have "a \<in> A" by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
596 |
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
|
597 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
598 |
thus "bij_betw f A A'" using * bij_betw_subset[of f "A \<union> {b}" _ A] by blast |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
599 |
qed |
0d5e831175de
moved lemmas from 'Fun_More_FP' to where they belong
blanchet
parents:
54578
diff
changeset
|
600 |
|
41657 | 601 |
|
60758 | 602 |
subsection\<open>Function Updating\<close> |
13585 | 603 |
|
44277
bcb696533579
moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents:
43991
diff
changeset
|
604 |
definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where |
26147 | 605 |
"fun_upd f a b == % x. if x=a then b else f x" |
606 |
||
41229
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset
|
607 |
nonterminal updbinds and updbind |
d797baa3d57c
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
wenzelm
parents:
40969
diff
changeset
|
608 |
|
26147 | 609 |
syntax |
610 |
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") |
|
611 |
"" :: "updbind => updbinds" ("_") |
|
612 |
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") |
|
35115 | 613 |
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) |
26147 | 614 |
|
615 |
translations |
|
35115 | 616 |
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" |
617 |
"f(x:=y)" == "CONST fun_upd f x y" |
|
26147 | 618 |
|
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55066
diff
changeset
|
619 |
(* Hint: to define the sum of two functions (or maps), use case_sum. |
58111 | 620 |
A nice infix syntax could be defined by |
35115 | 621 |
notation |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55066
diff
changeset
|
622 |
case_sum (infixr "'(+')"80) |
26147 | 623 |
*) |
624 |
||
13585 | 625 |
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" |
626 |
apply (simp add: fun_upd_def, safe) |
|
627 |
apply (erule subst) |
|
628 |
apply (rule_tac [2] ext, auto) |
|
629 |
done |
|
630 |
||
45603 | 631 |
lemma fun_upd_idem: "f x = y ==> f(x:=y) = f" |
632 |
by (simp only: fun_upd_idem_iff) |
|
13585 | 633 |
|
45603 | 634 |
lemma fun_upd_triv [iff]: "f(x := f x) = f" |
635 |
by (simp only: fun_upd_idem) |
|
13585 | 636 |
|
637 |
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" |
|
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
638 |
by (simp add: fun_upd_def) |
13585 | 639 |
|
640 |
(* fun_upd_apply supersedes these two, but they are useful |
|
641 |
if fun_upd_apply is intentionally removed from the simpset *) |
|
642 |
lemma fun_upd_same: "(f(x:=y)) x = y" |
|
643 |
by simp |
|
644 |
||
645 |
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" |
|
646 |
by simp |
|
647 |
||
648 |
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39213
diff
changeset
|
649 |
by (simp add: fun_eq_iff) |
13585 | 650 |
|
651 |
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" |
|
652 |
by (rule ext, auto) |
|
653 |
||
56077 | 654 |
lemma inj_on_fun_updI: |
655 |
"inj_on f A \<Longrightarrow> y \<notin> f ` A \<Longrightarrow> inj_on (f(x := y)) A" |
|
656 |
by (fastforce simp: inj_on_def) |
|
15303 | 657 |
|
15510 | 658 |
lemma fun_upd_image: |
659 |
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)" |
|
660 |
by auto |
|
661 |
||
31080 | 662 |
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" |
44921 | 663 |
by auto |
31080 | 664 |
|
26147 | 665 |
|
60758 | 666 |
subsection \<open>@{text override_on}\<close> |
26147 | 667 |
|
44277
bcb696533579
moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents:
43991
diff
changeset
|
668 |
definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where |
26147 | 669 |
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" |
13910 | 670 |
|
15691 | 671 |
lemma override_on_emptyset[simp]: "override_on f g {} = f" |
672 |
by(simp add:override_on_def) |
|
13910 | 673 |
|
15691 | 674 |
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" |
675 |
by(simp add:override_on_def) |
|
13910 | 676 |
|
15691 | 677 |
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" |
678 |
by(simp add:override_on_def) |
|
13910 | 679 |
|
26147 | 680 |
|
60758 | 681 |
subsection \<open>@{text swap}\<close> |
15510 | 682 |
|
56608 | 683 |
definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" |
684 |
where |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
685 |
"swap a b f = f (a := f b, b:= f a)" |
15510 | 686 |
|
56608 | 687 |
lemma swap_apply [simp]: |
688 |
"swap a b f a = f b" |
|
689 |
"swap a b f b = f a" |
|
690 |
"c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> swap a b f c = f c" |
|
691 |
by (simp_all add: swap_def) |
|
692 |
||
693 |
lemma swap_self [simp]: |
|
694 |
"swap a a f = f" |
|
695 |
by (simp add: swap_def) |
|
15510 | 696 |
|
56608 | 697 |
lemma swap_commute: |
698 |
"swap a b f = swap b a f" |
|
699 |
by (simp add: fun_upd_def swap_def fun_eq_iff) |
|
15510 | 700 |
|
56608 | 701 |
lemma swap_nilpotent [simp]: |
702 |
"swap a b (swap a b f) = f" |
|
703 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
704 |
||
705 |
lemma swap_comp_involutory [simp]: |
|
706 |
"swap a b \<circ> swap a b = id" |
|
707 |
by (rule ext) simp |
|
15510 | 708 |
|
34145 | 709 |
lemma swap_triple: |
710 |
assumes "a \<noteq> c" and "b \<noteq> c" |
|
711 |
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
|
712 |
using assms by (simp add: fun_eq_iff swap_def) |
34145 | 713 |
|
34101 | 714 |
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" |
56608 | 715 |
by (rule ext, simp add: fun_upd_def swap_def) |
34101 | 716 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
717 |
lemma swap_image_eq [simp]: |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
718 |
assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
719 |
proof - |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
720 |
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
|
721 |
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
|
722 |
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
|
723 |
with subset[of f] show ?thesis by auto |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
724 |
qed |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
725 |
|
15510 | 726 |
lemma inj_on_imp_inj_on_swap: |
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
727 |
"\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
728 |
by (simp add: inj_on_def swap_def, blast) |
15510 | 729 |
|
730 |
lemma inj_on_swap_iff [simp]: |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
731 |
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A" |
39075 | 732 |
proof |
15510 | 733 |
assume "inj_on (swap a b f) A" |
39075 | 734 |
with A have "inj_on (swap a b (swap a b f)) A" |
735 |
by (iprover intro: inj_on_imp_inj_on_swap) |
|
736 |
thus "inj_on f A" by simp |
|
15510 | 737 |
next |
738 |
assume "inj_on f A" |
|
34209 | 739 |
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) |
15510 | 740 |
qed |
741 |
||
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
742 |
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)" |
40702 | 743 |
by simp |
15510 | 744 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
745 |
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f" |
40702 | 746 |
by simp |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
747 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
748 |
lemma bij_betw_swap_iff [simp]: |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
749 |
"\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B" |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
750 |
by (auto simp: bij_betw_def) |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
751 |
|
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
752 |
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
|
753 |
by simp |
39075 | 754 |
|
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
|
755 |
hide_const (open) swap |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
756 |
|
56608 | 757 |
|
60758 | 758 |
subsection \<open>Inversion of injective functions\<close> |
31949 | 759 |
|
33057 | 760 |
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where |
44277
bcb696533579
moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents:
43991
diff
changeset
|
761 |
"the_inv_into A f == %x. THE y. y : A & f y = x" |
32961 | 762 |
|
33057 | 763 |
lemma the_inv_into_f_f: |
764 |
"[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x" |
|
765 |
apply (simp add: the_inv_into_def inj_on_def) |
|
34209 | 766 |
apply blast |
32961 | 767 |
done |
768 |
||
33057 | 769 |
lemma f_the_inv_into_f: |
770 |
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" |
|
771 |
apply (simp add: the_inv_into_def) |
|
32961 | 772 |
apply (rule the1I2) |
773 |
apply(blast dest: inj_onD) |
|
774 |
apply blast |
|
775 |
done |
|
776 |
||
33057 | 777 |
lemma the_inv_into_into: |
778 |
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B" |
|
779 |
apply (simp add: the_inv_into_def) |
|
32961 | 780 |
apply (rule the1I2) |
781 |
apply(blast dest: inj_onD) |
|
782 |
apply blast |
|
783 |
done |
|
784 |
||
33057 | 785 |
lemma the_inv_into_onto[simp]: |
786 |
"inj_on f A ==> the_inv_into A f ` (f ` A) = A" |
|
787 |
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) |
|
32961 | 788 |
|
33057 | 789 |
lemma the_inv_into_f_eq: |
790 |
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" |
|
32961 | 791 |
apply (erule subst) |
33057 | 792 |
apply (erule the_inv_into_f_f, assumption) |
32961 | 793 |
done |
794 |
||
33057 | 795 |
lemma the_inv_into_comp: |
32961 | 796 |
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> |
33057 | 797 |
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" |
798 |
apply (rule the_inv_into_f_eq) |
|
32961 | 799 |
apply (fast intro: comp_inj_on) |
33057 | 800 |
apply (simp add: f_the_inv_into_f the_inv_into_into) |
801 |
apply (simp add: the_inv_into_into) |
|
32961 | 802 |
done |
803 |
||
33057 | 804 |
lemma inj_on_the_inv_into: |
805 |
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)" |
|
56077 | 806 |
by (auto intro: inj_onI simp: the_inv_into_f_f) |
32961 | 807 |
|
33057 | 808 |
lemma bij_betw_the_inv_into: |
809 |
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A" |
|
810 |
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) |
|
32961 | 811 |
|
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
812 |
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where |
33057 | 813 |
"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
|
814 |
|
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
815 |
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
|
816 |
assumes "inj f" |
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
817 |
shows "the_inv f (f x) = x" using assms UNIV_I |
33057 | 818 |
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
|
819 |
|
44277
bcb696533579
moved fundamental lemma fun_eq_iff to theory HOL; tuned whitespace
haftmann
parents:
43991
diff
changeset
|
820 |
|
60758 | 821 |
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
|
822 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53927
diff
changeset
|
823 |
lemma Cantors_paradox: |
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
824 |
"\<not>(\<exists>f. f ` A = Pow A)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
825 |
proof clarify |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
826 |
fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
827 |
let ?X = "{a \<in> A. a \<notin> f a}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
828 |
have "?X \<in> Pow A" unfolding Pow_def by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
829 |
with * obtain x where "x \<in> A \<and> f x = ?X" by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
830 |
thus False by best |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
831 |
qed |
31949 | 832 |
|
61204 | 833 |
subsection \<open>Setup\<close> |
40969 | 834 |
|
60758 | 835 |
subsubsection \<open>Proof tools\<close> |
22845 | 836 |
|
60758 | 837 |
text \<open>simplifies terms of the form |
838 |
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...)\<close> |
|
22845 | 839 |
|
60758 | 840 |
simproc_setup fun_upd2 ("f(v := w, x := y)") = \<open>fn _ => |
22845 | 841 |
let |
842 |
fun gen_fun_upd NONE T _ _ = NONE |
|
24017 | 843 |
| gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) |
22845 | 844 |
fun dest_fun_T1 (Type (_, T :: Ts)) = T |
845 |
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = |
|
846 |
let |
|
847 |
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = |
|
848 |
if v aconv x then SOME g else gen_fun_upd (find g) T v w |
|
849 |
| find t = NONE |
|
850 |
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end |
|
24017 | 851 |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51598
diff
changeset
|
852 |
val ss = simpset_of @{context} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51598
diff
changeset
|
853 |
|
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51598
diff
changeset
|
854 |
fun proc ctxt ct = |
24017 | 855 |
let |
856 |
val t = Thm.term_of ct |
|
857 |
in |
|
858 |
case find_double t of |
|
859 |
(T, NONE) => NONE |
|
860 |
| (T, SOME rhs) => |
|
27330 | 861 |
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) |
24017 | 862 |
(fn _ => |
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
58889
diff
changeset
|
863 |
resolve_tac ctxt [eq_reflection] 1 THEN |
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
58889
diff
changeset
|
864 |
resolve_tac ctxt @{thms ext} 1 THEN |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51598
diff
changeset
|
865 |
simp_tac (put_simpset ss ctxt) 1)) |
24017 | 866 |
end |
867 |
in proc end |
|
60758 | 868 |
\<close> |
22845 | 869 |
|
870 |
||
60758 | 871 |
subsubsection \<open>Functorial structure of types\<close> |
40969 | 872 |
|
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
|
873 |
ML_file "Tools/functor.ML" |
40969 | 874 |
|
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
|
875 |
functor map_fun: map_fun |
47488
be6dd389639d
centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents:
46950
diff
changeset
|
876 |
by (simp_all add: fun_eq_iff) |
be6dd389639d
centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents:
46950
diff
changeset
|
877 |
|
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
|
878 |
functor vimage |
49739 | 879 |
by (simp_all add: fun_eq_iff vimage_comp) |
880 |
||
60758 | 881 |
text \<open>Legacy theorem names\<close> |
49739 | 882 |
|
883 |
lemmas o_def = comp_def |
|
884 |
lemmas o_apply = comp_apply |
|
885 |
lemmas o_assoc = comp_assoc [symmetric] |
|
886 |
lemmas id_o = id_comp |
|
887 |
lemmas o_id = comp_id |
|
888 |
lemmas o_eq_dest = comp_eq_dest |
|
889 |
lemmas o_eq_elim = comp_eq_elim |
|
55066 | 890 |
lemmas o_eq_dest_lhs = comp_eq_dest_lhs |
891 |
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
|
892 |
|
2912 | 893 |
end |
56015
57e2cfba9c6e
bootstrap fundamental Fun theory immediately after Set theory, without dependency on complete lattices
haftmann
parents:
55990
diff
changeset
|
894 |