author | haftmann |
Mon, 06 Dec 2010 09:19:10 +0100 | |
changeset 40968 | a6fcd305f7dc |
parent 40719 | acb830207103 |
child 40969 | fb2d3ccda5a7 |
permissions | -rw-r--r-- |
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(* Title: HOL/Fun.thy |
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Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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*) |
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|
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header {* Notions about functions *} |
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|
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theory Fun |
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imports Complete_Lattice |
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begin |
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|
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text{*As a simplification rule, it replaces all function equalities by |
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first-order equalities.*} |
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lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" |
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apply (rule iffI) |
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apply (simp (no_asm_simp)) |
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apply (rule ext) |
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apply (simp (no_asm_simp)) |
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done |
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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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subsection {* The Identity Function @{text id} *} |
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definition |
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id :: "'a \<Rightarrow> 'a" |
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where |
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"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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lemma image_ident [simp]: "(%x. x) ` Y = Y" |
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by blast |
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lemma image_id [simp]: "id ` Y = Y" |
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by (simp add: id_def) |
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lemma vimage_ident [simp]: "(%x. x) -` Y = Y" |
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by blast |
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lemma vimage_id [simp]: "id -` A = A" |
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by (simp add: id_def) |
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subsection {* The Composition Operator @{text "f \<circ> g"} *} |
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||
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definition |
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comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) |
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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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notation (HTML output) |
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comp (infixl "\<circ>" 55) |
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text{*compatibility*} |
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lemmas o_def = comp_def |
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lemma o_apply [simp]: "(f o g) x = f (g x)" |
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by (simp add: comp_def) |
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lemma o_assoc: "f o (g o h) = f o g o h" |
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by (simp add: comp_def) |
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lemma id_o [simp]: "id o g = g" |
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by (simp add: comp_def) |
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lemma o_id [simp]: "f o id = f" |
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by (simp add: comp_def) |
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lemma o_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 only: o_def) (fact fun_cong) |
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lemma o_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 (erule meta_mp) (fact o_eq_dest) |
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lemma image_compose: "(f o g) ` r = f`(g`r)" |
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by (simp add: comp_def, blast) |
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lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)" |
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by auto |
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lemma UN_o: "UNION A (g o f) = UNION (f`A) g" |
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by (unfold comp_def, blast) |
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subsection {* The Forward Composition Operator @{text fcomp} *} |
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definition |
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fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) |
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where |
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"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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code_const fcomp |
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(Eval infixl 1 "#>") |
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no_notation fcomp (infixl "\<circ>>" 60) |
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subsection {* Mapping functions *} |
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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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type_lifting map_fun |
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by (simp_all add: fun_eq_iff) |
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subsection {* Injectivity and Bijectivity *} |
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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{*A common special case: functions injective, surjective or bijective over |
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the entire domain type.*} |
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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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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)" |
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by (unfold inj_on_def, blast) |
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lemma injD: "[| inj(f); f(x) = f(y) |] ==> 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 ==> 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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"\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B" |
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unfolding inj_on_def unfolding image_def by blast |
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lemma inj_comp: |
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"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 ==> (f(x) = f(y)) = (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 (%x. x) A" |
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by (simp add: inj_on_def) |
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lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)" |
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unfolding inj_on_def by blast |
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lemma inj_on_INTER: |
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"\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)\<rbrakk> \<Longrightarrow> inj_on f (\<Inter> i \<in> I. A i)" |
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unfolding inj_on_def by blast |
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lemma inj_on_Inter: |
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"\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)" |
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lemma inj_on_UNION_chain: |
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assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and |
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INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
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shows "inj_on f (\<Union> i \<in> I. A i)" |
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proof(unfold inj_on_def UNION_def, auto) |
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fix i j x y |
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assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j" |
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and ***: "f x = f y" |
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show "x = y" |
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proof- |
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{assume "A i \<le> A j" |
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with ** have "x \<in> A j" by auto |
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with INJ * ** *** have ?thesis |
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by(auto simp add: inj_on_def) |
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} |
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moreover |
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{assume "A j \<le> A i" |
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with ** have "y \<in> A i" by auto |
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diff
changeset
|
220 |
with INJ * ** *** have ?thesis |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
221 |
by(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
|
222 |
} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
223 |
ultimately show ?thesis using CH * by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
224 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
225 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
226 |
|
40702 | 227 |
lemma surj_id: "surj id" |
228 |
by simp |
|
26147 | 229 |
|
39101
606432dd1896
Revert bij_betw changes to simp set (Problem in afp/Ordinals_and_Cardinals)
hoelzl
parents:
39076
diff
changeset
|
230 |
lemma bij_id[simp]: "bij id" |
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
231 |
by (simp add: bij_betw_def) |
13585 | 232 |
|
233 |
lemma inj_onI: |
|
234 |
"(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" |
|
235 |
by (simp add: inj_on_def) |
|
236 |
||
237 |
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" |
|
238 |
by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) |
|
239 |
||
240 |
lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" |
|
241 |
by (unfold inj_on_def, blast) |
|
242 |
||
243 |
lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" |
|
244 |
by (blast dest!: inj_onD) |
|
245 |
||
246 |
lemma comp_inj_on: |
|
247 |
"[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" |
|
248 |
by (simp add: comp_def inj_on_def) |
|
249 |
||
15303 | 250 |
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)" |
251 |
apply(simp add:inj_on_def image_def) |
|
252 |
apply blast |
|
253 |
done |
|
254 |
||
15439 | 255 |
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); |
256 |
inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A" |
|
257 |
apply(unfold inj_on_def) |
|
258 |
apply blast |
|
259 |
done |
|
260 |
||
13585 | 261 |
lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" |
262 |
by (unfold inj_on_def, blast) |
|
12258 | 263 |
|
13585 | 264 |
lemma inj_singleton: "inj (%s. {s})" |
265 |
by (simp add: inj_on_def) |
|
266 |
||
15111 | 267 |
lemma inj_on_empty[iff]: "inj_on f {}" |
268 |
by(simp add: inj_on_def) |
|
269 |
||
15303 | 270 |
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" |
13585 | 271 |
by (unfold inj_on_def, blast) |
272 |
||
15111 | 273 |
lemma inj_on_Un: |
274 |
"inj_on f (A Un B) = |
|
275 |
(inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" |
|
276 |
apply(unfold inj_on_def) |
|
277 |
apply (blast intro:sym) |
|
278 |
done |
|
279 |
||
280 |
lemma inj_on_insert[iff]: |
|
281 |
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" |
|
282 |
apply(unfold inj_on_def) |
|
283 |
apply (blast intro:sym) |
|
284 |
done |
|
285 |
||
286 |
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" |
|
287 |
apply(unfold inj_on_def) |
|
288 |
apply (blast) |
|
289 |
done |
|
290 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
291 |
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
|
292 |
"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
|
293 |
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
|
294 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
295 |
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
|
296 |
"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
|
297 |
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
|
298 |
|
40702 | 299 |
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)" |
300 |
by auto |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
301 |
|
40702 | 302 |
lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g" |
303 |
using *[symmetric] by auto |
|
13585 | 304 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
305 |
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
|
306 |
by (simp add: surj_def) |
13585 | 307 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
308 |
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
|
309 |
by (simp add: surj_def, blast) |
13585 | 310 |
|
311 |
lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" |
|
312 |
apply (simp add: comp_def surj_def, clarify) |
|
313 |
apply (drule_tac x = y in spec, clarify) |
|
314 |
apply (drule_tac x = x in spec, blast) |
|
315 |
done |
|
316 |
||
39074 | 317 |
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f" |
40702 | 318 |
unfolding bij_betw_def by auto |
39074 | 319 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
320 |
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
|
321 |
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
|
322 |
shows "A = {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
323 |
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
|
324 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
325 |
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
|
326 |
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
|
327 |
shows "A = {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
328 |
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
|
329 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
330 |
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
|
331 |
"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
|
332 |
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
|
333 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
334 |
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f" |
40702 | 335 |
unfolding bij_betw_def .. |
39074 | 336 |
|
13585 | 337 |
lemma bijI: "[| inj f; surj f |] ==> bij f" |
338 |
by (simp add: bij_def) |
|
339 |
||
340 |
lemma bij_is_inj: "bij f ==> inj f" |
|
341 |
by (simp add: bij_def) |
|
342 |
||
343 |
lemma bij_is_surj: "bij f ==> surj f" |
|
344 |
by (simp add: bij_def) |
|
345 |
||
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
346 |
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
|
347 |
by (simp add: bij_betw_def) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
348 |
|
31438 | 349 |
lemma bij_betw_trans: |
350 |
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C" |
|
351 |
by(auto simp add:bij_betw_def comp_inj_on) |
|
352 |
||
40702 | 353 |
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)" |
354 |
by (rule bij_betw_trans) |
|
355 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
356 |
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
|
357 |
"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
|
358 |
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
|
359 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
360 |
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
|
361 |
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
|
362 |
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
|
363 |
using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
364 |
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
|
365 |
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
|
366 |
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
|
367 |
using IM |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
368 |
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
|
369 |
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
|
370 |
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
|
371 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
372 |
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
|
373 |
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
|
374 |
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
|
375 |
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
|
376 |
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
|
377 |
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
|
378 |
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
|
379 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
380 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
381 |
|
26105
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
382 |
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
|
383 |
proof - |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
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
|
387 |
{ fix a b assume P: "?P b a" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
388 |
hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
389 |
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
|
390 |
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
|
391 |
} note g = this |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
392 |
have "inj_on ?g B" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
393 |
proof(rule inj_onI) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
394 |
fix x y assume "x:B" "y:B" "?g x = ?g y" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
395 |
from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
396 |
from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
397 |
from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
398 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
399 |
moreover have "?g ` B = A" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
400 |
proof(auto simp:image_def) |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
401 |
fix b assume "b:B" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
402 |
with s obtain a where P: "?P b a" unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
403 |
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
|
404 |
next |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
405 |
fix a assume "a:A" |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
406 |
then obtain b where P: "?P b a" using s unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
407 |
then have "b:B" using s unfolding image_def by blast |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
408 |
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
|
409 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
410 |
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
|
411 |
qed |
ae06618225ec
moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents:
25886
diff
changeset
|
412 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
413 |
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
|
414 |
"(\<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
|
415 |
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
|
416 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
417 |
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
|
418 |
"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
|
419 |
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
|
420 |
|
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_id_iff: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
422 |
"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
|
423 |
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
|
424 |
|
39075 | 425 |
lemma bij_betw_combine: |
426 |
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}" |
|
427 |
shows "bij_betw f (A \<union> C) (B \<union> D)" |
|
428 |
using assms unfolding bij_betw_def inj_on_Un image_Un by auto |
|
429 |
||
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
430 |
lemma bij_betw_UNION_chain: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
431 |
assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
432 |
BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
433 |
shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
434 |
proof(unfold bij_betw_def, auto simp add: image_def) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
435 |
have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
436 |
using BIJ bij_betw_def[of f] by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
437 |
thus "inj_on f (\<Union> i \<in> I. A i)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
438 |
using CH inj_on_UNION_chain[of I A f] by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
439 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
440 |
fix i x |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
441 |
assume *: "i \<in> I" "x \<in> A i" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
442 |
hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
443 |
thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
444 |
next |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
445 |
fix i x' |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
446 |
assume *: "i \<in> I" "x' \<in> A' i" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
447 |
hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
448 |
thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
449 |
using * by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
450 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
451 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
452 |
lemma bij_betw_Disj_Un: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
453 |
assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
454 |
B1: "bij_betw f A A'" and B2: "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
|
455 |
shows "bij_betw f (A \<union> B) (A' \<union> B')" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
456 |
proof- |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
457 |
have 1: "inj_on f A \<and> inj_on f B" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
458 |
using B1 B2 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
|
459 |
have 2: "f`A = A' \<and> 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
|
460 |
using B1 B2 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
|
461 |
hence "f`(A - B) \<inter> f`(B - A) = {}" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
462 |
using DISJ DISJ' by blast |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
463 |
hence "inj_on f (A \<union> B)" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
464 |
using 1 by (auto simp add: inj_on_Un) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
465 |
(* *) |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
466 |
moreover |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
467 |
have "f`(A \<union> B) = A' \<union> B'" |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
468 |
using 2 by auto |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
469 |
ultimately show ?thesis |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
470 |
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
|
471 |
qed |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
472 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
473 |
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
|
474 |
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
|
475 |
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
|
476 |
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
|
477 |
using assms |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
478 |
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
|
479 |
|
13585 | 480 |
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" |
40702 | 481 |
by simp |
13585 | 482 |
|
483 |
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" |
|
484 |
by (simp add: inj_on_def, blast) |
|
485 |
||
486 |
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" |
|
40702 | 487 |
by (blast intro: sym) |
13585 | 488 |
|
489 |
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" |
|
490 |
by (unfold inj_on_def, blast) |
|
491 |
||
492 |
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" |
|
493 |
apply (unfold bij_def) |
|
494 |
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) |
|
495 |
done |
|
496 |
||
31438 | 497 |
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B" |
498 |
by(blast dest: inj_onD) |
|
499 |
||
13585 | 500 |
lemma inj_on_image_Int: |
501 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" |
|
502 |
apply (simp add: inj_on_def, blast) |
|
503 |
done |
|
504 |
||
505 |
lemma inj_on_image_set_diff: |
|
506 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B" |
|
507 |
apply (simp add: inj_on_def, blast) |
|
508 |
done |
|
509 |
||
510 |
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" |
|
511 |
by (simp add: inj_on_def, blast) |
|
512 |
||
513 |
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" |
|
514 |
by (simp add: inj_on_def, blast) |
|
515 |
||
516 |
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" |
|
517 |
by (blast dest: injD) |
|
518 |
||
519 |
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" |
|
520 |
by (simp add: inj_on_def, blast) |
|
521 |
||
522 |
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" |
|
523 |
by (blast dest: injD) |
|
524 |
||
525 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*) |
|
526 |
lemma image_INT: |
|
527 |
"[| inj_on f C; ALL x:A. B x <= C; j:A |] |
|
528 |
==> f ` (INTER A B) = (INT x:A. f ` B x)" |
|
529 |
apply (simp add: inj_on_def, blast) |
|
530 |
done |
|
531 |
||
532 |
(*Compare with image_INT: no use of inj_on, and if f is surjective then |
|
533 |
it doesn't matter whether A is empty*) |
|
534 |
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" |
|
535 |
apply (simp add: bij_def) |
|
536 |
apply (simp add: inj_on_def surj_def, blast) |
|
537 |
done |
|
538 |
||
539 |
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" |
|
40702 | 540 |
by auto |
13585 | 541 |
|
542 |
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" |
|
543 |
by (auto simp add: inj_on_def) |
|
5852 | 544 |
|
13585 | 545 |
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" |
546 |
apply (simp add: bij_def) |
|
547 |
apply (rule equalityI) |
|
548 |
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) |
|
549 |
done |
|
550 |
||
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
551 |
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" |
35580 | 552 |
by (auto intro!: inj_onI) |
13585 | 553 |
|
35584
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
554 |
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
|
555 |
by (auto intro!: inj_onI dest: strict_mono_eq) |
768f8d92b767
generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents:
35580
diff
changeset
|
556 |
|
13585 | 557 |
subsection{*Function Updating*} |
558 |
||
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset
|
559 |
definition |
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35115
diff
changeset
|
560 |
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where |
26147 | 561 |
"fun_upd f a b == % x. if x=a then b else f x" |
562 |
||
563 |
nonterminals |
|
564 |
updbinds updbind |
|
565 |
syntax |
|
566 |
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") |
|
567 |
"" :: "updbind => updbinds" ("_") |
|
568 |
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") |
|
35115 | 569 |
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) |
26147 | 570 |
|
571 |
translations |
|
35115 | 572 |
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" |
573 |
"f(x:=y)" == "CONST fun_upd f x y" |
|
26147 | 574 |
|
575 |
(* Hint: to define the sum of two functions (or maps), use sum_case. |
|
576 |
A nice infix syntax could be defined (in Datatype.thy or below) by |
|
35115 | 577 |
notation |
578 |
sum_case (infixr "'(+')"80) |
|
26147 | 579 |
*) |
580 |
||
13585 | 581 |
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" |
582 |
apply (simp add: fun_upd_def, safe) |
|
583 |
apply (erule subst) |
|
584 |
apply (rule_tac [2] ext, auto) |
|
585 |
done |
|
586 |
||
587 |
(* f x = y ==> f(x:=y) = f *) |
|
588 |
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] |
|
589 |
||
590 |
(* f(x := f x) = f *) |
|
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
591 |
lemmas fun_upd_triv = refl [THEN fun_upd_idem] |
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16973
diff
changeset
|
592 |
declare fun_upd_triv [iff] |
13585 | 593 |
|
594 |
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
|
595 |
by (simp add: fun_upd_def) |
13585 | 596 |
|
597 |
(* fun_upd_apply supersedes these two, but they are useful |
|
598 |
if fun_upd_apply is intentionally removed from the simpset *) |
|
599 |
lemma fun_upd_same: "(f(x:=y)) x = y" |
|
600 |
by simp |
|
601 |
||
602 |
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" |
|
603 |
by simp |
|
604 |
||
605 |
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
|
606 |
by (simp add: fun_eq_iff) |
13585 | 607 |
|
608 |
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" |
|
609 |
by (rule ext, auto) |
|
610 |
||
15303 | 611 |
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A" |
34209 | 612 |
by (fastsimp simp:inj_on_def image_def) |
15303 | 613 |
|
15510 | 614 |
lemma fun_upd_image: |
615 |
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)" |
|
616 |
by auto |
|
617 |
||
31080 | 618 |
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)" |
34209 | 619 |
by (auto intro: ext) |
31080 | 620 |
|
26147 | 621 |
|
622 |
subsection {* @{text override_on} *} |
|
623 |
||
624 |
definition |
|
625 |
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" |
|
626 |
where |
|
627 |
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)" |
|
13910 | 628 |
|
15691 | 629 |
lemma override_on_emptyset[simp]: "override_on f g {} = f" |
630 |
by(simp add:override_on_def) |
|
13910 | 631 |
|
15691 | 632 |
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" |
633 |
by(simp add:override_on_def) |
|
13910 | 634 |
|
15691 | 635 |
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" |
636 |
by(simp add:override_on_def) |
|
13910 | 637 |
|
26147 | 638 |
|
639 |
subsection {* @{text swap} *} |
|
15510 | 640 |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
641 |
definition |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
642 |
swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
643 |
where |
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22577
diff
changeset
|
644 |
"swap a b f = f (a := f b, b:= f a)" |
15510 | 645 |
|
34101 | 646 |
lemma swap_self [simp]: "swap a a f = f" |
15691 | 647 |
by (simp add: swap_def) |
15510 | 648 |
|
649 |
lemma swap_commute: "swap a b f = swap b a f" |
|
650 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
651 |
||
652 |
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" |
|
653 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
654 |
||
34145 | 655 |
lemma swap_triple: |
656 |
assumes "a \<noteq> c" and "b \<noteq> c" |
|
657 |
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
|
658 |
using assms by (simp add: fun_eq_iff swap_def) |
34145 | 659 |
|
34101 | 660 |
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)" |
661 |
by (rule ext, simp add: fun_upd_def swap_def) |
|
662 |
||
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
663 |
lemma swap_image_eq [simp]: |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
664 |
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
|
665 |
proof - |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
666 |
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
|
667 |
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
|
668 |
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
|
669 |
with subset[of f] show ?thesis by auto |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
670 |
qed |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
671 |
|
15510 | 672 |
lemma inj_on_imp_inj_on_swap: |
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
673 |
"\<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
|
674 |
by (simp add: inj_on_def swap_def, blast) |
15510 | 675 |
|
676 |
lemma inj_on_swap_iff [simp]: |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
677 |
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A" |
39075 | 678 |
proof |
15510 | 679 |
assume "inj_on (swap a b f) A" |
39075 | 680 |
with A have "inj_on (swap a b (swap a b f)) A" |
681 |
by (iprover intro: inj_on_imp_inj_on_swap) |
|
682 |
thus "inj_on f A" by simp |
|
15510 | 683 |
next |
684 |
assume "inj_on f A" |
|
34209 | 685 |
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) |
15510 | 686 |
qed |
687 |
||
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
688 |
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)" |
40702 | 689 |
by simp |
15510 | 690 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
691 |
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f" |
40702 | 692 |
by simp |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
693 |
|
39076
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
694 |
lemma bij_betw_swap_iff [simp]: |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
695 |
"\<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
|
696 |
by (auto simp: bij_betw_def) |
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
697 |
|
b3a9b6734663
Introduce surj_on and replace surj and bij by abbreviations.
hoelzl
parents:
39075
diff
changeset
|
698 |
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
|
699 |
by simp |
39075 | 700 |
|
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
|
701 |
hide_const (open) swap |
21547
9c9fdf4c2949
moved order arities for fun and bool to Fun/Orderings
haftmann
parents:
21327
diff
changeset
|
702 |
|
31949 | 703 |
subsection {* Inversion of injective functions *} |
704 |
||
33057 | 705 |
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where |
706 |
"the_inv_into A f == %x. THE y. y : A & f y = x" |
|
32961 | 707 |
|
33057 | 708 |
lemma the_inv_into_f_f: |
709 |
"[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x" |
|
710 |
apply (simp add: the_inv_into_def inj_on_def) |
|
34209 | 711 |
apply blast |
32961 | 712 |
done |
713 |
||
33057 | 714 |
lemma f_the_inv_into_f: |
715 |
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" |
|
716 |
apply (simp add: the_inv_into_def) |
|
32961 | 717 |
apply (rule the1I2) |
718 |
apply(blast dest: inj_onD) |
|
719 |
apply blast |
|
720 |
done |
|
721 |
||
33057 | 722 |
lemma the_inv_into_into: |
723 |
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B" |
|
724 |
apply (simp add: the_inv_into_def) |
|
32961 | 725 |
apply (rule the1I2) |
726 |
apply(blast dest: inj_onD) |
|
727 |
apply blast |
|
728 |
done |
|
729 |
||
33057 | 730 |
lemma the_inv_into_onto[simp]: |
731 |
"inj_on f A ==> the_inv_into A f ` (f ` A) = A" |
|
732 |
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) |
|
32961 | 733 |
|
33057 | 734 |
lemma the_inv_into_f_eq: |
735 |
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" |
|
32961 | 736 |
apply (erule subst) |
33057 | 737 |
apply (erule the_inv_into_f_f, assumption) |
32961 | 738 |
done |
739 |
||
33057 | 740 |
lemma the_inv_into_comp: |
32961 | 741 |
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> |
33057 | 742 |
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" |
743 |
apply (rule the_inv_into_f_eq) |
|
32961 | 744 |
apply (fast intro: comp_inj_on) |
33057 | 745 |
apply (simp add: f_the_inv_into_f the_inv_into_into) |
746 |
apply (simp add: the_inv_into_into) |
|
32961 | 747 |
done |
748 |
||
33057 | 749 |
lemma inj_on_the_inv_into: |
750 |
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)" |
|
751 |
by (auto intro: inj_onI simp: image_def the_inv_into_f_f) |
|
32961 | 752 |
|
33057 | 753 |
lemma bij_betw_the_inv_into: |
754 |
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A" |
|
755 |
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) |
|
32961 | 756 |
|
32998
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
757 |
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where |
33057 | 758 |
"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
|
759 |
|
31b19fa0de0b
Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents:
32988
diff
changeset
|
760 |
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
|
761 |
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
|
762 |
shows "the_inv f (f x) = x" using assms UNIV_I |
33057 | 763 |
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
|
764 |
|
40703
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
765 |
subsection {* Cantor's Paradox *} |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
766 |
|
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
767 |
lemma Cantors_paradox: |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
768 |
"\<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
|
769 |
proof clarify |
d1fc454d6735
Move some missing lemmas from Andrei Popescus 'Ordinals and Cardinals' AFP entry to the HOL-image.
hoelzl
parents:
40702
diff
changeset
|
770 |
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
|
771 |
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
|
772 |
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
|
773 |
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
|
774 |
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
|
775 |
qed |
31949 | 776 |
|
22845 | 777 |
subsection {* Proof tool setup *} |
778 |
||
779 |
text {* simplifies terms of the form |
|
780 |
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} |
|
781 |
||
24017 | 782 |
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => |
22845 | 783 |
let |
784 |
fun gen_fun_upd NONE T _ _ = NONE |
|
24017 | 785 |
| gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y) |
22845 | 786 |
fun dest_fun_T1 (Type (_, T :: Ts)) = T |
787 |
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) = |
|
788 |
let |
|
789 |
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) = |
|
790 |
if v aconv x then SOME g else gen_fun_upd (find g) T v w |
|
791 |
| find t = NONE |
|
792 |
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end |
|
24017 | 793 |
|
794 |
fun proc ss ct = |
|
795 |
let |
|
796 |
val ctxt = Simplifier.the_context ss |
|
797 |
val t = Thm.term_of ct |
|
798 |
in |
|
799 |
case find_double t of |
|
800 |
(T, NONE) => NONE |
|
801 |
| (T, SOME rhs) => |
|
27330 | 802 |
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) |
24017 | 803 |
(fn _ => |
804 |
rtac eq_reflection 1 THEN |
|
805 |
rtac ext 1 THEN |
|
806 |
simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) |
|
807 |
end |
|
808 |
in proc end |
|
22845 | 809 |
*} |
810 |
||
811 |
||
21870 | 812 |
subsection {* Code generator setup *} |
813 |
||
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
814 |
types_code |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
815 |
"fun" ("(_ ->/ _)") |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
816 |
attach (term_of) {* |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
817 |
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT); |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
818 |
*} |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
819 |
attach (test) {* |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
820 |
fun gen_fun_type aF aT bG bT i = |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
821 |
let |
32740 | 822 |
val tab = Unsynchronized.ref []; |
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
823 |
fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
824 |
(aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y () |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
825 |
in |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
826 |
(fn x => |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
827 |
case AList.lookup op = (!tab) x of |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
828 |
NONE => |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
829 |
let val p as (y, _) = bG i |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
830 |
in (tab := (x, p) :: !tab; y) end |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
831 |
| SOME (y, _) => y, |
28711 | 832 |
fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT))) |
25886
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
833 |
end; |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
834 |
*} |
7753e0d81b7a
Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents:
24286
diff
changeset
|
835 |
|
21870 | 836 |
code_const "op \<circ>" |
837 |
(SML infixl 5 "o") |
|
838 |
(Haskell infixr 9 ".") |
|
839 |
||
21906 | 840 |
code_const "id" |
841 |
(Haskell "id") |
|
842 |
||
2912 | 843 |
end |