clasohm@1475
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(* Title: HOL/Fun.thy
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clasohm@1475
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Author: Tobias Nipkow, Cambridge University Computer Laboratory
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clasohm@923
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Copyright 1994 University of Cambridge
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huffman@18154
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*)
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clasohm@923
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huffman@18154
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header {* Notions about functions *}
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clasohm@923
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paulson@15510
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theory Fun
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haftmann@32139
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imports Complete_Lattice
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nipkow@15131
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begin
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nipkow@2912
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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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nipkow@39535
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lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
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haftmann@26147
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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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paulson@6171
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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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oheimb@5305
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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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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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wenzelm@21210
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notation (xsymbols)
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comp (infixl "\<circ>" 55)
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wenzelm@19656
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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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paulson@33044
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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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haftmann@31202
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code_const fcomp
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haftmann@31202
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(Eval infixl 1 "#>")
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haftmann@31202
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haftmann@37750
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no_notation fcomp (infixl "\<circ>>" 60)
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haftmann@26357
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haftmann@40850
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subsection {* Mapping functions *}
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haftmann@40850
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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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haftmann@40850
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"map_fun f g h = g \<circ> h \<circ> f"
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haftmann@40850
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haftmann@40850
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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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haftmann@40850
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by (simp add: map_fun_def)
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haftmann@40850
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haftmann@40850
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type_mapper map_fun
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by (simp_all add: fun_eq_iff)
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haftmann@40850
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haftmann@40850
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hoelzl@40950
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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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hoelzl@39310
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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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hoelzl@39310
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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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haftmann@26147
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assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
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haftmann@26147
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shows "inj f"
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haftmann@26147
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using assms unfolding inj_on_def by auto
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paulson@13585
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haftmann@31771
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text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
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paulson@13585
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lemma datatype_injI:
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"(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
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by (simp add: inj_on_def)
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berghofe@13637
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theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
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berghofe@13637
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by (unfold inj_on_def, blast)
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berghofe@13637
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paulson@13585
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lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
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paulson@13585
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by (simp add: inj_on_def)
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paulson@13585
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nipkow@32988
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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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paulson@13585
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by (force simp add: inj_on_def)
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paulson@13585
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hoelzl@40951
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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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haftmann@38843
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lemma inj_comp:
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haftmann@38843
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"inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
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haftmann@38843
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by (simp add: inj_on_def)
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haftmann@38843
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haftmann@38843
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lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
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nipkow@39535
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by (simp add: inj_on_def fun_eq_iff)
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haftmann@38843
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lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
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nipkow@32988
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by (simp add: inj_on_eq_iff)
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nipkow@32988
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haftmann@26147
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lemma inj_on_id[simp]: "inj_on id A"
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hoelzl@39310
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by (simp add: inj_on_def)
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paulson@13585
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haftmann@26147
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lemma inj_on_id2[simp]: "inj_on (%x. x) A"
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hoelzl@39310
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by (simp add: inj_on_def)
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haftmann@26147
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hoelzl@40951
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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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hoelzl@40951
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unfolding inj_on_def by blast
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hoelzl@40951
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hoelzl@40951
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lemma inj_on_INTER:
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hoelzl@40951
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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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hoelzl@40951
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unfolding inj_on_def by blast
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hoelzl@40951
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hoelzl@40951
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lemma inj_on_Inter:
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hoelzl@40951
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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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hoelzl@40951
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unfolding inj_on_def by blast
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hoelzl@40951
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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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hoelzl@40951
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INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
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hoelzl@40951
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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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hoelzl@40951
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assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
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hoelzl@40951
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and ***: "f x = f y"
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hoelzl@40951
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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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hoelzl@40951
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with ** have "x \<in> A j" by auto
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hoelzl@40951
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with INJ * ** *** have ?thesis
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hoelzl@40951
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by(auto simp add: inj_on_def)
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hoelzl@40951
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}
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hoelzl@40951
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moreover
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hoelzl@40951
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{assume "A j \<le> A i"
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hoelzl@40951
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with ** have "y \<in> A i" by auto
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hoelzl@40951
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with INJ * ** *** have ?thesis
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hoelzl@40951
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by(auto simp add: inj_on_def)
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hoelzl@40951
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}
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hoelzl@40951
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ultimately show ?thesis using CH * by blast
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hoelzl@40951
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qed
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hoelzl@40951
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qed
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hoelzl@40951
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hoelzl@40950
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lemma surj_id: "surj id"
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hoelzl@40950
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by simp
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haftmann@26147
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hoelzl@39335
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lemma bij_id[simp]: "bij id"
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hoelzl@39310
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by (simp add: bij_betw_def)
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paulson@13585
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paulson@13585
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lemma inj_onI:
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paulson@13585
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"(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A"
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paulson@13585
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by (simp add: inj_on_def)
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paulson@13585
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paulson@13585
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lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
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paulson@13585
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by (auto dest: arg_cong [of concl: g] simp add: inj_on_def)
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paulson@13585
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paulson@13585
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lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y"
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paulson@13585
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by (unfold inj_on_def, blast)
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paulson@13585
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paulson@13585
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lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)"
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paulson@13585
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by (blast dest!: inj_onD)
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paulson@13585
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paulson@13585
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lemma comp_inj_on:
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paulson@13585
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"[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A"
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paulson@13585
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by (simp add: comp_def inj_on_def)
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paulson@13585
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nipkow@15303
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lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
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nipkow@15303
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apply(simp add:inj_on_def image_def)
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nipkow@15303
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apply blast
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nipkow@15303
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done
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nipkow@15303
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nipkow@15439
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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);
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nipkow@15439
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inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
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nipkow@15439
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apply(unfold inj_on_def)
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nipkow@15439
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apply blast
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nipkow@15439
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done
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nipkow@15439
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paulson@13585
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lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)"
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paulson@13585
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by (unfold inj_on_def, blast)
|
paulson@13585
|
268 |
|
paulson@13585
|
269 |
lemma inj_singleton: "inj (%s. {s})"
|
paulson@13585
|
270 |
by (simp add: inj_on_def)
|
paulson@13585
|
271 |
|
nipkow@15111
|
272 |
lemma inj_on_empty[iff]: "inj_on f {}"
|
nipkow@15111
|
273 |
by(simp add: inj_on_def)
|
nipkow@15111
|
274 |
|
nipkow@15303
|
275 |
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
|
paulson@13585
|
276 |
by (unfold inj_on_def, blast)
|
paulson@13585
|
277 |
|
nipkow@15111
|
278 |
lemma inj_on_Un:
|
nipkow@15111
|
279 |
"inj_on f (A Un B) =
|
nipkow@15111
|
280 |
(inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
|
nipkow@15111
|
281 |
apply(unfold inj_on_def)
|
nipkow@15111
|
282 |
apply (blast intro:sym)
|
nipkow@15111
|
283 |
done
|
nipkow@15111
|
284 |
|
nipkow@15111
|
285 |
lemma inj_on_insert[iff]:
|
nipkow@15111
|
286 |
"inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
|
nipkow@15111
|
287 |
apply(unfold inj_on_def)
|
nipkow@15111
|
288 |
apply (blast intro:sym)
|
nipkow@15111
|
289 |
done
|
nipkow@15111
|
290 |
|
nipkow@15111
|
291 |
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
|
nipkow@15111
|
292 |
apply(unfold inj_on_def)
|
nipkow@15111
|
293 |
apply (blast)
|
nipkow@15111
|
294 |
done
|
nipkow@15111
|
295 |
|
hoelzl@40951
|
296 |
lemma comp_inj_on_iff:
|
hoelzl@40951
|
297 |
"inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
|
hoelzl@40951
|
298 |
by(auto simp add: comp_inj_on inj_on_def)
|
hoelzl@40951
|
299 |
|
hoelzl@40951
|
300 |
lemma inj_on_imageI2:
|
hoelzl@40951
|
301 |
"inj_on (f' o f) A \<Longrightarrow> inj_on f A"
|
hoelzl@40951
|
302 |
by(auto simp add: comp_inj_on inj_on_def)
|
hoelzl@40951
|
303 |
|
hoelzl@40950
|
304 |
lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
|
hoelzl@40950
|
305 |
by auto
|
paulson@13585
|
306 |
|
hoelzl@40950
|
307 |
lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
|
hoelzl@40950
|
308 |
using *[symmetric] by auto
|
hoelzl@39310
|
309 |
|
hoelzl@39310
|
310 |
lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
|
hoelzl@39310
|
311 |
by (simp add: surj_def)
|
hoelzl@39310
|
312 |
|
hoelzl@39310
|
313 |
lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
|
hoelzl@39310
|
314 |
by (simp add: surj_def, blast)
|
paulson@13585
|
315 |
|
paulson@13585
|
316 |
lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)"
|
paulson@13585
|
317 |
apply (simp add: comp_def surj_def, clarify)
|
paulson@13585
|
318 |
apply (drule_tac x = y in spec, clarify)
|
paulson@13585
|
319 |
apply (drule_tac x = x in spec, blast)
|
paulson@13585
|
320 |
done
|
paulson@13585
|
321 |
|
hoelzl@39308
|
322 |
lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
|
hoelzl@40950
|
323 |
unfolding bij_betw_def by auto
|
hoelzl@39308
|
324 |
|
hoelzl@40951
|
325 |
lemma bij_betw_empty1:
|
hoelzl@40951
|
326 |
assumes "bij_betw f {} A"
|
hoelzl@40951
|
327 |
shows "A = {}"
|
hoelzl@40951
|
328 |
using assms unfolding bij_betw_def by blast
|
hoelzl@40951
|
329 |
|
hoelzl@40951
|
330 |
lemma bij_betw_empty2:
|
hoelzl@40951
|
331 |
assumes "bij_betw f A {}"
|
hoelzl@40951
|
332 |
shows "A = {}"
|
hoelzl@40951
|
333 |
using assms unfolding bij_betw_def by blast
|
hoelzl@40951
|
334 |
|
hoelzl@40951
|
335 |
lemma inj_on_imp_bij_betw:
|
hoelzl@40951
|
336 |
"inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
|
hoelzl@40951
|
337 |
unfolding bij_betw_def by simp
|
hoelzl@40951
|
338 |
|
hoelzl@39310
|
339 |
lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
|
hoelzl@40950
|
340 |
unfolding bij_betw_def ..
|
hoelzl@39308
|
341 |
|
paulson@13585
|
342 |
lemma bijI: "[| inj f; surj f |] ==> bij f"
|
paulson@13585
|
343 |
by (simp add: bij_def)
|
paulson@13585
|
344 |
|
paulson@13585
|
345 |
lemma bij_is_inj: "bij f ==> inj f"
|
paulson@13585
|
346 |
by (simp add: bij_def)
|
paulson@13585
|
347 |
|
paulson@13585
|
348 |
lemma bij_is_surj: "bij f ==> surj f"
|
paulson@13585
|
349 |
by (simp add: bij_def)
|
paulson@13585
|
350 |
|
nipkow@26105
|
351 |
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
|
nipkow@26105
|
352 |
by (simp add: bij_betw_def)
|
nipkow@26105
|
353 |
|
nipkow@31424
|
354 |
lemma bij_betw_trans:
|
nipkow@31424
|
355 |
"bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
|
nipkow@31424
|
356 |
by(auto simp add:bij_betw_def comp_inj_on)
|
nipkow@31424
|
357 |
|
hoelzl@40950
|
358 |
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
|
hoelzl@40950
|
359 |
by (rule bij_betw_trans)
|
hoelzl@40950
|
360 |
|
hoelzl@40951
|
361 |
lemma bij_betw_comp_iff:
|
hoelzl@40951
|
362 |
"bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
|
hoelzl@40951
|
363 |
by(auto simp add: bij_betw_def inj_on_def)
|
hoelzl@40951
|
364 |
|
hoelzl@40951
|
365 |
lemma bij_betw_comp_iff2:
|
hoelzl@40951
|
366 |
assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
|
hoelzl@40951
|
367 |
shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
|
hoelzl@40951
|
368 |
using assms
|
hoelzl@40951
|
369 |
proof(auto simp add: bij_betw_comp_iff)
|
hoelzl@40951
|
370 |
assume *: "bij_betw (f' \<circ> f) A A''"
|
hoelzl@40951
|
371 |
thus "bij_betw f A A'"
|
hoelzl@40951
|
372 |
using IM
|
hoelzl@40951
|
373 |
proof(auto simp add: bij_betw_def)
|
hoelzl@40951
|
374 |
assume "inj_on (f' \<circ> f) A"
|
hoelzl@40951
|
375 |
thus "inj_on f A" using inj_on_imageI2 by blast
|
hoelzl@40951
|
376 |
next
|
hoelzl@40951
|
377 |
fix a' assume **: "a' \<in> A'"
|
hoelzl@40951
|
378 |
hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
|
hoelzl@40951
|
379 |
then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
|
hoelzl@40951
|
380 |
unfolding bij_betw_def by force
|
hoelzl@40951
|
381 |
hence "f a \<in> A'" using IM by auto
|
hoelzl@40951
|
382 |
hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
|
hoelzl@40951
|
383 |
thus "a' \<in> f ` A" using 1 by auto
|
hoelzl@40951
|
384 |
qed
|
hoelzl@40951
|
385 |
qed
|
hoelzl@40951
|
386 |
|
nipkow@26105
|
387 |
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
|
nipkow@26105
|
388 |
proof -
|
nipkow@26105
|
389 |
have i: "inj_on f A" and s: "f ` A = B"
|
nipkow@26105
|
390 |
using assms by(auto simp:bij_betw_def)
|
nipkow@26105
|
391 |
let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
|
nipkow@26105
|
392 |
{ fix a b assume P: "?P b a"
|
nipkow@26105
|
393 |
hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
|
nipkow@26105
|
394 |
hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
|
nipkow@26105
|
395 |
hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
|
nipkow@26105
|
396 |
} note g = this
|
nipkow@26105
|
397 |
have "inj_on ?g B"
|
nipkow@26105
|
398 |
proof(rule inj_onI)
|
nipkow@26105
|
399 |
fix x y assume "x:B" "y:B" "?g x = ?g y"
|
nipkow@26105
|
400 |
from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
|
nipkow@26105
|
401 |
from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
|
nipkow@26105
|
402 |
from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
|
nipkow@26105
|
403 |
qed
|
nipkow@26105
|
404 |
moreover have "?g ` B = A"
|
nipkow@26105
|
405 |
proof(auto simp:image_def)
|
nipkow@26105
|
406 |
fix b assume "b:B"
|
nipkow@26105
|
407 |
with s obtain a where P: "?P b a" unfolding image_def by blast
|
nipkow@26105
|
408 |
thus "?g b \<in> A" using g[OF P] by auto
|
nipkow@26105
|
409 |
next
|
nipkow@26105
|
410 |
fix a assume "a:A"
|
nipkow@26105
|
411 |
then obtain b where P: "?P b a" using s unfolding image_def by blast
|
nipkow@26105
|
412 |
then have "b:B" using s unfolding image_def by blast
|
nipkow@26105
|
413 |
with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
|
nipkow@26105
|
414 |
qed
|
nipkow@26105
|
415 |
ultimately show ?thesis by(auto simp:bij_betw_def)
|
nipkow@26105
|
416 |
qed
|
nipkow@26105
|
417 |
|
hoelzl@40951
|
418 |
lemma bij_betw_cong:
|
hoelzl@40951
|
419 |
"(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
|
hoelzl@40951
|
420 |
unfolding bij_betw_def inj_on_def by force
|
hoelzl@40951
|
421 |
|
hoelzl@40951
|
422 |
lemma bij_betw_id[intro, simp]:
|
hoelzl@40951
|
423 |
"bij_betw id A A"
|
hoelzl@40951
|
424 |
unfolding bij_betw_def id_def by auto
|
hoelzl@40951
|
425 |
|
hoelzl@40951
|
426 |
lemma bij_betw_id_iff:
|
hoelzl@40951
|
427 |
"bij_betw id A B \<longleftrightarrow> A = B"
|
hoelzl@40951
|
428 |
by(auto simp add: bij_betw_def)
|
hoelzl@40951
|
429 |
|
hoelzl@39309
|
430 |
lemma bij_betw_combine:
|
hoelzl@39309
|
431 |
assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
|
hoelzl@39309
|
432 |
shows "bij_betw f (A \<union> C) (B \<union> D)"
|
hoelzl@39309
|
433 |
using assms unfolding bij_betw_def inj_on_Un image_Un by auto
|
hoelzl@39309
|
434 |
|
hoelzl@40951
|
435 |
lemma bij_betw_UNION_chain:
|
hoelzl@40951
|
436 |
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
|
hoelzl@40951
|
437 |
BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
|
hoelzl@40951
|
438 |
shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
|
hoelzl@40951
|
439 |
proof(unfold bij_betw_def, auto simp add: image_def)
|
hoelzl@40951
|
440 |
have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
|
hoelzl@40951
|
441 |
using BIJ bij_betw_def[of f] by auto
|
hoelzl@40951
|
442 |
thus "inj_on f (\<Union> i \<in> I. A i)"
|
hoelzl@40951
|
443 |
using CH inj_on_UNION_chain[of I A f] by auto
|
hoelzl@40951
|
444 |
next
|
hoelzl@40951
|
445 |
fix i x
|
hoelzl@40951
|
446 |
assume *: "i \<in> I" "x \<in> A i"
|
hoelzl@40951
|
447 |
hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
|
hoelzl@40951
|
448 |
thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
|
hoelzl@40951
|
449 |
next
|
hoelzl@40951
|
450 |
fix i x'
|
hoelzl@40951
|
451 |
assume *: "i \<in> I" "x' \<in> A' i"
|
hoelzl@40951
|
452 |
hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
|
hoelzl@40951
|
453 |
thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
|
hoelzl@40951
|
454 |
using * by blast
|
hoelzl@40951
|
455 |
qed
|
hoelzl@40951
|
456 |
|
hoelzl@40951
|
457 |
lemma bij_betw_Disj_Un:
|
hoelzl@40951
|
458 |
assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and
|
hoelzl@40951
|
459 |
B1: "bij_betw f A A'" and B2: "bij_betw f B B'"
|
hoelzl@40951
|
460 |
shows "bij_betw f (A \<union> B) (A' \<union> B')"
|
hoelzl@40951
|
461 |
proof-
|
hoelzl@40951
|
462 |
have 1: "inj_on f A \<and> inj_on f B"
|
hoelzl@40951
|
463 |
using B1 B2 by (auto simp add: bij_betw_def)
|
hoelzl@40951
|
464 |
have 2: "f`A = A' \<and> f`B = B'"
|
hoelzl@40951
|
465 |
using B1 B2 by (auto simp add: bij_betw_def)
|
hoelzl@40951
|
466 |
hence "f`(A - B) \<inter> f`(B - A) = {}"
|
hoelzl@40951
|
467 |
using DISJ DISJ' by blast
|
hoelzl@40951
|
468 |
hence "inj_on f (A \<union> B)"
|
hoelzl@40951
|
469 |
using 1 by (auto simp add: inj_on_Un)
|
hoelzl@40951
|
470 |
(* *)
|
hoelzl@40951
|
471 |
moreover
|
hoelzl@40951
|
472 |
have "f`(A \<union> B) = A' \<union> B'"
|
hoelzl@40951
|
473 |
using 2 by auto
|
hoelzl@40951
|
474 |
ultimately show ?thesis
|
hoelzl@40951
|
475 |
unfolding bij_betw_def by auto
|
hoelzl@40951
|
476 |
qed
|
hoelzl@40951
|
477 |
|
hoelzl@40951
|
478 |
lemma bij_betw_subset:
|
hoelzl@40951
|
479 |
assumes BIJ: "bij_betw f A A'" and
|
hoelzl@40951
|
480 |
SUB: "B \<le> A" and IM: "f ` B = B'"
|
hoelzl@40951
|
481 |
shows "bij_betw f B B'"
|
hoelzl@40951
|
482 |
using assms
|
hoelzl@40951
|
483 |
by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
|
hoelzl@40951
|
484 |
|
paulson@13585
|
485 |
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
|
hoelzl@40950
|
486 |
by simp
|
paulson@13585
|
487 |
|
paulson@13585
|
488 |
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
|
paulson@13585
|
489 |
by (simp add: inj_on_def, blast)
|
paulson@13585
|
490 |
|
paulson@13585
|
491 |
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
|
hoelzl@40950
|
492 |
by (blast intro: sym)
|
paulson@13585
|
493 |
|
paulson@13585
|
494 |
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
|
paulson@13585
|
495 |
by (unfold inj_on_def, blast)
|
paulson@13585
|
496 |
|
paulson@13585
|
497 |
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
|
paulson@13585
|
498 |
apply (unfold bij_def)
|
paulson@13585
|
499 |
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
|
paulson@13585
|
500 |
done
|
paulson@13585
|
501 |
|
nipkow@31424
|
502 |
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
|
nipkow@31424
|
503 |
by(blast dest: inj_onD)
|
nipkow@31424
|
504 |
|
paulson@13585
|
505 |
lemma inj_on_image_Int:
|
paulson@13585
|
506 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B"
|
paulson@13585
|
507 |
apply (simp add: inj_on_def, blast)
|
paulson@13585
|
508 |
done
|
paulson@13585
|
509 |
|
paulson@13585
|
510 |
lemma inj_on_image_set_diff:
|
paulson@13585
|
511 |
"[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B"
|
paulson@13585
|
512 |
apply (simp add: inj_on_def, blast)
|
paulson@13585
|
513 |
done
|
paulson@13585
|
514 |
|
paulson@13585
|
515 |
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
|
paulson@13585
|
516 |
by (simp add: inj_on_def, blast)
|
paulson@13585
|
517 |
|
paulson@13585
|
518 |
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
|
paulson@13585
|
519 |
by (simp add: inj_on_def, blast)
|
paulson@13585
|
520 |
|
paulson@13585
|
521 |
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
|
paulson@13585
|
522 |
by (blast dest: injD)
|
paulson@13585
|
523 |
|
paulson@13585
|
524 |
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
|
paulson@13585
|
525 |
by (simp add: inj_on_def, blast)
|
paulson@13585
|
526 |
|
paulson@13585
|
527 |
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
|
paulson@13585
|
528 |
by (blast dest: injD)
|
paulson@13585
|
529 |
|
paulson@13585
|
530 |
(*injectivity's required. Left-to-right inclusion holds even if A is empty*)
|
paulson@13585
|
531 |
lemma image_INT:
|
paulson@13585
|
532 |
"[| inj_on f C; ALL x:A. B x <= C; j:A |]
|
paulson@13585
|
533 |
==> f ` (INTER A B) = (INT x:A. f ` B x)"
|
paulson@13585
|
534 |
apply (simp add: inj_on_def, blast)
|
paulson@13585
|
535 |
done
|
paulson@13585
|
536 |
|
paulson@13585
|
537 |
(*Compare with image_INT: no use of inj_on, and if f is surjective then
|
paulson@13585
|
538 |
it doesn't matter whether A is empty*)
|
paulson@13585
|
539 |
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
|
paulson@13585
|
540 |
apply (simp add: bij_def)
|
paulson@13585
|
541 |
apply (simp add: inj_on_def surj_def, blast)
|
paulson@13585
|
542 |
done
|
paulson@13585
|
543 |
|
paulson@13585
|
544 |
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
|
hoelzl@40950
|
545 |
by auto
|
paulson@13585
|
546 |
|
paulson@13585
|
547 |
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
|
paulson@13585
|
548 |
by (auto simp add: inj_on_def)
|
paulson@13585
|
549 |
|
paulson@13585
|
550 |
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
|
paulson@13585
|
551 |
apply (simp add: bij_def)
|
paulson@13585
|
552 |
apply (rule equalityI)
|
paulson@13585
|
553 |
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
|
paulson@13585
|
554 |
done
|
paulson@13585
|
555 |
|
hoelzl@35584
|
556 |
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
|
hoelzl@35580
|
557 |
by (auto intro!: inj_onI)
|
paulson@13585
|
558 |
|
hoelzl@35584
|
559 |
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
|
hoelzl@35584
|
560 |
by (auto intro!: inj_onI dest: strict_mono_eq)
|
hoelzl@35584
|
561 |
|
paulson@13585
|
562 |
subsection{*Function Updating*}
|
paulson@13585
|
563 |
|
haftmann@35413
|
564 |
definition
|
haftmann@35413
|
565 |
fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
|
haftmann@26147
|
566 |
"fun_upd f a b == % x. if x=a then b else f x"
|
haftmann@26147
|
567 |
|
haftmann@26147
|
568 |
nonterminals
|
haftmann@26147
|
569 |
updbinds updbind
|
haftmann@26147
|
570 |
syntax
|
haftmann@26147
|
571 |
"_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)")
|
haftmann@26147
|
572 |
"" :: "updbind => updbinds" ("_")
|
haftmann@26147
|
573 |
"_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
|
wenzelm@35118
|
574 |
"_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900)
|
haftmann@26147
|
575 |
|
haftmann@26147
|
576 |
translations
|
wenzelm@35118
|
577 |
"_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
|
wenzelm@35118
|
578 |
"f(x:=y)" == "CONST fun_upd f x y"
|
haftmann@26147
|
579 |
|
haftmann@26147
|
580 |
(* Hint: to define the sum of two functions (or maps), use sum_case.
|
haftmann@26147
|
581 |
A nice infix syntax could be defined (in Datatype.thy or below) by
|
wenzelm@35118
|
582 |
notation
|
wenzelm@35118
|
583 |
sum_case (infixr "'(+')"80)
|
haftmann@26147
|
584 |
*)
|
haftmann@26147
|
585 |
|
paulson@13585
|
586 |
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
|
paulson@13585
|
587 |
apply (simp add: fun_upd_def, safe)
|
paulson@13585
|
588 |
apply (erule subst)
|
paulson@13585
|
589 |
apply (rule_tac [2] ext, auto)
|
paulson@13585
|
590 |
done
|
paulson@13585
|
591 |
|
paulson@13585
|
592 |
(* f x = y ==> f(x:=y) = f *)
|
paulson@13585
|
593 |
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
|
paulson@13585
|
594 |
|
paulson@13585
|
595 |
(* f(x := f x) = f *)
|
paulson@17084
|
596 |
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
|
paulson@17084
|
597 |
declare fun_upd_triv [iff]
|
paulson@13585
|
598 |
|
paulson@13585
|
599 |
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
|
paulson@17084
|
600 |
by (simp add: fun_upd_def)
|
paulson@13585
|
601 |
|
paulson@13585
|
602 |
(* fun_upd_apply supersedes these two, but they are useful
|
paulson@13585
|
603 |
if fun_upd_apply is intentionally removed from the simpset *)
|
paulson@13585
|
604 |
lemma fun_upd_same: "(f(x:=y)) x = y"
|
paulson@13585
|
605 |
by simp
|
paulson@13585
|
606 |
|
paulson@13585
|
607 |
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
|
paulson@13585
|
608 |
by simp
|
paulson@13585
|
609 |
|
paulson@13585
|
610 |
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
|
nipkow@39535
|
611 |
by (simp add: fun_eq_iff)
|
paulson@13585
|
612 |
|
paulson@13585
|
613 |
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
|
paulson@13585
|
614 |
by (rule ext, auto)
|
paulson@13585
|
615 |
|
nipkow@15303
|
616 |
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
|
krauss@34209
|
617 |
by (fastsimp simp:inj_on_def image_def)
|
nipkow@15303
|
618 |
|
paulson@15510
|
619 |
lemma fun_upd_image:
|
paulson@15510
|
620 |
"f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
|
paulson@15510
|
621 |
by auto
|
paulson@15510
|
622 |
|
nipkow@31080
|
623 |
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
|
krauss@34209
|
624 |
by (auto intro: ext)
|
nipkow@31080
|
625 |
|
haftmann@26147
|
626 |
|
haftmann@26147
|
627 |
subsection {* @{text override_on} *}
|
haftmann@26147
|
628 |
|
haftmann@26147
|
629 |
definition
|
haftmann@26147
|
630 |
override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
|
haftmann@26147
|
631 |
where
|
haftmann@26147
|
632 |
"override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
|
nipkow@13910
|
633 |
|
nipkow@15691
|
634 |
lemma override_on_emptyset[simp]: "override_on f g {} = f"
|
nipkow@15691
|
635 |
by(simp add:override_on_def)
|
nipkow@13910
|
636 |
|
nipkow@15691
|
637 |
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
|
nipkow@15691
|
638 |
by(simp add:override_on_def)
|
nipkow@13910
|
639 |
|
nipkow@15691
|
640 |
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
|
nipkow@15691
|
641 |
by(simp add:override_on_def)
|
nipkow@13910
|
642 |
|
haftmann@26147
|
643 |
|
haftmann@26147
|
644 |
subsection {* @{text swap} *}
|
paulson@15510
|
645 |
|
haftmann@22744
|
646 |
definition
|
haftmann@22744
|
647 |
swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
|
haftmann@22744
|
648 |
where
|
haftmann@22744
|
649 |
"swap a b f = f (a := f b, b:= f a)"
|
paulson@15510
|
650 |
|
huffman@34095
|
651 |
lemma swap_self [simp]: "swap a a f = f"
|
nipkow@15691
|
652 |
by (simp add: swap_def)
|
paulson@15510
|
653 |
|
paulson@15510
|
654 |
lemma swap_commute: "swap a b f = swap b a f"
|
paulson@15510
|
655 |
by (rule ext, simp add: fun_upd_def swap_def)
|
paulson@15510
|
656 |
|
paulson@15510
|
657 |
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
|
paulson@15510
|
658 |
by (rule ext, simp add: fun_upd_def swap_def)
|
paulson@15510
|
659 |
|
huffman@34134
|
660 |
lemma swap_triple:
|
huffman@34134
|
661 |
assumes "a \<noteq> c" and "b \<noteq> c"
|
huffman@34134
|
662 |
shows "swap a b (swap b c (swap a b f)) = swap a c f"
|
nipkow@39535
|
663 |
using assms by (simp add: fun_eq_iff swap_def)
|
huffman@34134
|
664 |
|
huffman@34095
|
665 |
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
|
huffman@34095
|
666 |
by (rule ext, simp add: fun_upd_def swap_def)
|
huffman@34095
|
667 |
|
hoelzl@39310
|
668 |
lemma swap_image_eq [simp]:
|
hoelzl@39310
|
669 |
assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
|
hoelzl@39310
|
670 |
proof -
|
hoelzl@39310
|
671 |
have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
|
hoelzl@39310
|
672 |
using assms by (auto simp: image_iff swap_def)
|
hoelzl@39310
|
673 |
then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
|
hoelzl@39310
|
674 |
with subset[of f] show ?thesis by auto
|
hoelzl@39310
|
675 |
qed
|
hoelzl@39310
|
676 |
|
paulson@15510
|
677 |
lemma inj_on_imp_inj_on_swap:
|
hoelzl@39310
|
678 |
"\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
|
hoelzl@39310
|
679 |
by (simp add: inj_on_def swap_def, blast)
|
paulson@15510
|
680 |
|
paulson@15510
|
681 |
lemma inj_on_swap_iff [simp]:
|
hoelzl@39310
|
682 |
assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
|
hoelzl@39309
|
683 |
proof
|
paulson@15510
|
684 |
assume "inj_on (swap a b f) A"
|
hoelzl@39309
|
685 |
with A have "inj_on (swap a b (swap a b f)) A"
|
hoelzl@39309
|
686 |
by (iprover intro: inj_on_imp_inj_on_swap)
|
hoelzl@39309
|
687 |
thus "inj_on f A" by simp
|
paulson@15510
|
688 |
next
|
paulson@15510
|
689 |
assume "inj_on f A"
|
krauss@34209
|
690 |
with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
|
paulson@15510
|
691 |
qed
|
paulson@15510
|
692 |
|
hoelzl@39310
|
693 |
lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
|
hoelzl@40950
|
694 |
by simp
|
paulson@15510
|
695 |
|
hoelzl@39310
|
696 |
lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
|
hoelzl@40950
|
697 |
by simp
|
paulson@15510
|
698 |
|
hoelzl@39310
|
699 |
lemma bij_betw_swap_iff [simp]:
|
hoelzl@39310
|
700 |
"\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
|
hoelzl@39310
|
701 |
by (auto simp: bij_betw_def)
|
haftmann@21547
|
702 |
|
hoelzl@39310
|
703 |
lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
|
hoelzl@39310
|
704 |
by simp
|
hoelzl@39309
|
705 |
|
wenzelm@36176
|
706 |
hide_const (open) swap
|
haftmann@21547
|
707 |
|
haftmann@31949
|
708 |
subsection {* Inversion of injective functions *}
|
haftmann@31949
|
709 |
|
nipkow@33057
|
710 |
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
|
nipkow@33057
|
711 |
"the_inv_into A f == %x. THE y. y : A & f y = x"
|
nipkow@32957
|
712 |
|
nipkow@33057
|
713 |
lemma the_inv_into_f_f:
|
nipkow@33057
|
714 |
"[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x"
|
nipkow@33057
|
715 |
apply (simp add: the_inv_into_def inj_on_def)
|
krauss@34209
|
716 |
apply blast
|
nipkow@32957
|
717 |
done
|
nipkow@32957
|
718 |
|
nipkow@33057
|
719 |
lemma f_the_inv_into_f:
|
nipkow@33057
|
720 |
"inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y"
|
nipkow@33057
|
721 |
apply (simp add: the_inv_into_def)
|
nipkow@32957
|
722 |
apply (rule the1I2)
|
nipkow@32957
|
723 |
apply(blast dest: inj_onD)
|
nipkow@32957
|
724 |
apply blast
|
nipkow@32957
|
725 |
done
|
nipkow@32957
|
726 |
|
nipkow@33057
|
727 |
lemma the_inv_into_into:
|
nipkow@33057
|
728 |
"[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
|
nipkow@33057
|
729 |
apply (simp add: the_inv_into_def)
|
nipkow@32957
|
730 |
apply (rule the1I2)
|
nipkow@32957
|
731 |
apply(blast dest: inj_onD)
|
nipkow@32957
|
732 |
apply blast
|
nipkow@32957
|
733 |
done
|
nipkow@32957
|
734 |
|
nipkow@33057
|
735 |
lemma the_inv_into_onto[simp]:
|
nipkow@33057
|
736 |
"inj_on f A ==> the_inv_into A f ` (f ` A) = A"
|
nipkow@33057
|
737 |
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
|
nipkow@32957
|
738 |
|
nipkow@33057
|
739 |
lemma the_inv_into_f_eq:
|
nipkow@33057
|
740 |
"[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
|
nipkow@32957
|
741 |
apply (erule subst)
|
nipkow@33057
|
742 |
apply (erule the_inv_into_f_f, assumption)
|
nipkow@32957
|
743 |
done
|
nipkow@32957
|
744 |
|
nipkow@33057
|
745 |
lemma the_inv_into_comp:
|
nipkow@32957
|
746 |
"[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
|
nipkow@33057
|
747 |
the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
|
nipkow@33057
|
748 |
apply (rule the_inv_into_f_eq)
|
nipkow@32957
|
749 |
apply (fast intro: comp_inj_on)
|
nipkow@33057
|
750 |
apply (simp add: f_the_inv_into_f the_inv_into_into)
|
nipkow@33057
|
751 |
apply (simp add: the_inv_into_into)
|
nipkow@32957
|
752 |
done
|
nipkow@32957
|
753 |
|
nipkow@33057
|
754 |
lemma inj_on_the_inv_into:
|
nipkow@33057
|
755 |
"inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
|
nipkow@33057
|
756 |
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
|
nipkow@32957
|
757 |
|
nipkow@33057
|
758 |
lemma bij_betw_the_inv_into:
|
nipkow@33057
|
759 |
"bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
|
nipkow@33057
|
760 |
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
|
nipkow@32957
|
761 |
|
berghofe@32998
|
762 |
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
|
nipkow@33057
|
763 |
"the_inv f \<equiv> the_inv_into UNIV f"
|
berghofe@32998
|
764 |
|
berghofe@32998
|
765 |
lemma the_inv_f_f:
|
berghofe@32998
|
766 |
assumes "inj f"
|
berghofe@32998
|
767 |
shows "the_inv f (f x) = x" using assms UNIV_I
|
nipkow@33057
|
768 |
by (rule the_inv_into_f_f)
|
berghofe@32998
|
769 |
|
hoelzl@40951
|
770 |
subsection {* Cantor's Paradox *}
|
hoelzl@40951
|
771 |
|
hoelzl@40951
|
772 |
lemma Cantors_paradox:
|
hoelzl@40951
|
773 |
"\<not>(\<exists>f. f ` A = Pow A)"
|
hoelzl@40951
|
774 |
proof clarify
|
hoelzl@40951
|
775 |
fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
|
hoelzl@40951
|
776 |
let ?X = "{a \<in> A. a \<notin> f a}"
|
hoelzl@40951
|
777 |
have "?X \<in> Pow A" unfolding Pow_def by auto
|
hoelzl@40951
|
778 |
with * obtain x where "x \<in> A \<and> f x = ?X" by blast
|
hoelzl@40951
|
779 |
thus False by best
|
hoelzl@40951
|
780 |
qed
|
haftmann@31949
|
781 |
|
haftmann@22845
|
782 |
subsection {* Proof tool setup *}
|
haftmann@22845
|
783 |
|
haftmann@22845
|
784 |
text {* simplifies terms of the form
|
haftmann@22845
|
785 |
f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
|
haftmann@22845
|
786 |
|
wenzelm@24017
|
787 |
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
|
haftmann@22845
|
788 |
let
|
haftmann@22845
|
789 |
fun gen_fun_upd NONE T _ _ = NONE
|
wenzelm@24017
|
790 |
| gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
|
haftmann@22845
|
791 |
fun dest_fun_T1 (Type (_, T :: Ts)) = T
|
haftmann@22845
|
792 |
fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
|
haftmann@22845
|
793 |
let
|
haftmann@22845
|
794 |
fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
|
haftmann@22845
|
795 |
if v aconv x then SOME g else gen_fun_upd (find g) T v w
|
haftmann@22845
|
796 |
| find t = NONE
|
haftmann@22845
|
797 |
in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
|
wenzelm@24017
|
798 |
|
wenzelm@24017
|
799 |
fun proc ss ct =
|
wenzelm@24017
|
800 |
let
|
wenzelm@24017
|
801 |
val ctxt = Simplifier.the_context ss
|
wenzelm@24017
|
802 |
val t = Thm.term_of ct
|
wenzelm@24017
|
803 |
in
|
wenzelm@24017
|
804 |
case find_double t of
|
wenzelm@24017
|
805 |
(T, NONE) => NONE
|
wenzelm@24017
|
806 |
| (T, SOME rhs) =>
|
wenzelm@27330
|
807 |
SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
|
wenzelm@24017
|
808 |
(fn _ =>
|
wenzelm@24017
|
809 |
rtac eq_reflection 1 THEN
|
wenzelm@24017
|
810 |
rtac ext 1 THEN
|
wenzelm@24017
|
811 |
simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
|
wenzelm@24017
|
812 |
end
|
wenzelm@24017
|
813 |
in proc end
|
haftmann@22845
|
814 |
*}
|
haftmann@22845
|
815 |
|
haftmann@22845
|
816 |
|
haftmann@21870
|
817 |
subsection {* Code generator setup *}
|
haftmann@21870
|
818 |
|
berghofe@25886
|
819 |
types_code
|
berghofe@25886
|
820 |
"fun" ("(_ ->/ _)")
|
berghofe@25886
|
821 |
attach (term_of) {*
|
berghofe@25886
|
822 |
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
|
berghofe@25886
|
823 |
*}
|
berghofe@25886
|
824 |
attach (test) {*
|
berghofe@25886
|
825 |
fun gen_fun_type aF aT bG bT i =
|
berghofe@25886
|
826 |
let
|
wenzelm@32740
|
827 |
val tab = Unsynchronized.ref [];
|
berghofe@25886
|
828 |
fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
|
berghofe@25886
|
829 |
(aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
|
berghofe@25886
|
830 |
in
|
berghofe@25886
|
831 |
(fn x =>
|
berghofe@25886
|
832 |
case AList.lookup op = (!tab) x of
|
berghofe@25886
|
833 |
NONE =>
|
berghofe@25886
|
834 |
let val p as (y, _) = bG i
|
berghofe@25886
|
835 |
in (tab := (x, p) :: !tab; y) end
|
berghofe@25886
|
836 |
| SOME (y, _) => y,
|
berghofe@28711
|
837 |
fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
|
berghofe@25886
|
838 |
end;
|
berghofe@25886
|
839 |
*}
|
berghofe@25886
|
840 |
|
haftmann@21870
|
841 |
code_const "op \<circ>"
|
haftmann@21870
|
842 |
(SML infixl 5 "o")
|
haftmann@21870
|
843 |
(Haskell infixr 9 ".")
|
haftmann@21870
|
844 |
|
haftmann@21906
|
845 |
code_const "id"
|
haftmann@21906
|
846 |
(Haskell "id")
|
haftmann@21906
|
847 |
|
nipkow@2912
|
848 |
end
|