src/HOL/Fun.thy
author nipkow
Wed Aug 04 19:11:02 2004 +0200 (2004-08-04)
changeset 15111 c108189645f8
parent 14565 c6dc17aab88a
child 15131 c69542757a4d
permissions -rw-r--r--
added some inj_on thms
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(*  Title:      HOL/Fun.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Notions about functions.
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*)
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theory Fun = Typedef:
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instance set :: (type) order
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  by (intro_classes,
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      (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+)
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constdefs
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  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
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   "fun_upd f a b == % x. if x=a then b else f x"
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nonterminals
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  updbinds updbind
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syntax
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  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
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  ""         :: "updbind => updbinds"             ("_")
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  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
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  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
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translations
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  "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
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  "f(x:=y)"                     == "fun_upd f x y"
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(* Hint: to define the sum of two functions (or maps), use sum_case.
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         A nice infix syntax could be defined (in Datatype.thy or below) by
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consts
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  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
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translations
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 "fun_sum" == sum_case
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*)
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constdefs
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 overwrite :: "('a => 'b) => ('a => 'b) => 'a set => ('a => 'b)"
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              ("_/'(_|/_')"  [900,0,0]900)
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"f(g|A) == %a. if a : A then g a else f a"
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 id :: "'a => 'a"
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"id == %x. x"
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 comp :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixl "o" 55)
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"f o g == %x. f(g(x))"
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text{*compatibility*}
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lemmas o_def = comp_def
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syntax (xsymbols)
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  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "\<circ>" 55)
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syntax (HTML output)
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  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"        (infixl "\<circ>" 55)
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constdefs
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  inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
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    "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
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text{*A common special case: functions injective over the entire domain type.*}
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syntax inj   :: "('a => 'b) => bool"
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translations
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  "inj f" == "inj_on f UNIV"
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constdefs
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  surj :: "('a => 'b) => bool"                   (*surjective*)
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    "surj f == ! y. ? x. y=f(x)"
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  bij :: "('a => 'b) => bool"                    (*bijective*)
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    "bij f == inj f & surj f"
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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 expand_fun_eq: "(f = g) = (! 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, simp (no_asm_simp))
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done
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lemma apply_inverse:
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    "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
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by auto
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text{*The Identity Function: @{term id}*}
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lemma id_apply [simp]: "id x = x"
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by (simp add: id_def)
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subsection{*The Composition Operator: @{term "f \<circ> g"}*}
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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 image_compose: "(f o g) ` r = f`(g`r)"
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by (simp add: comp_def, blast)
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lemma image_eq_UN: "f`A = (UN x:A. {f x})"
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by blast
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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 Injectivity Predicate, @{term inj}*}
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text{*NB: @{term inj} now just translates to @{term inj_on}*}
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text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
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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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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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(*Useful with the simplifier*)
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lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
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by (force simp add: inj_on_def)
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subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}
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lemma inj_onI:
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    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
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by (simp add: inj_on_def)
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lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
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by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
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lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
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by (unfold inj_on_def, blast)
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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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by (blast dest!: inj_onD)
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lemma comp_inj_on:
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     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
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by (simp add: comp_def inj_on_def)
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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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by (unfold inj_on_def, blast)
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lemma inj_singleton: "inj (%s. {s})"
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by (simp add: inj_on_def)
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lemma inj_on_empty[iff]: "inj_on f {}"
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by(simp add: inj_on_def)
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lemma subset_inj_on: "[| A<=B; inj_on f B |] ==> inj_on f A"
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by (unfold inj_on_def, blast)
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lemma inj_on_Un:
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 "inj_on f (A Un B) =
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  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_insert[iff]:
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  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
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apply(unfold inj_on_def)
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apply (blast intro:sym)
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done
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lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
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apply(unfold inj_on_def)
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apply (blast)
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done
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subsection{*The Predicate @{term surj}: Surjectivity*}
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lemma surjI: "(!! x. g(f x) = x) ==> surj g"
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apply (simp add: surj_def)
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apply (blast intro: sym)
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done
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lemma surj_range: "surj f ==> range f = UNIV"
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by (auto simp add: surj_def)
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lemma surjD: "surj f ==> EX x. y = f x"
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by (simp add: surj_def)
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lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
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by (simp add: surj_def, blast)
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lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
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apply (simp add: comp_def surj_def, clarify)
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apply (drule_tac x = y in spec, clarify)
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apply (drule_tac x = x in spec, blast)
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done
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subsection{*The Predicate @{term bij}: Bijectivity*}
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lemma bijI: "[| inj f; surj f |] ==> bij f"
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by (simp add: bij_def)
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lemma bij_is_inj: "bij f ==> inj f"
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by (simp add: bij_def)
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lemma bij_is_surj: "bij f ==> surj f"
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by (simp add: bij_def)
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subsection{*Facts About the Identity Function*}
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text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"}
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forms. The latter can arise by rewriting, while @{term id} may be used
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explicitly.*}
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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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lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"
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by (blast intro: sym)
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lemma image_vimage_subset: "f ` (f -` A) <= A"
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by blast
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lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
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by blast
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lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
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by (simp add: surj_range)
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lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
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by (simp add: inj_on_def, blast)
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lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
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apply (unfold surj_def)
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apply (blast intro: sym)
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done
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lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
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by (unfold inj_on_def, blast)
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lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
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apply (unfold bij_def)
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apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
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done
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lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
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by blast
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lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
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by blast
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lemma inj_on_image_Int:
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   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
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apply (simp add: inj_on_def, blast)
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done
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lemma inj_on_image_set_diff:
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   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
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apply (simp add: inj_on_def, blast)
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done
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lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
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by (simp add: inj_on_def, blast)
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lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
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by (simp add: inj_on_def, blast)
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lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
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by (blast dest: injD)
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lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
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by (simp add: inj_on_def, blast)
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lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
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by (blast dest: injD)
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lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
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by blast
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(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
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lemma image_INT:
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   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
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    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
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apply (simp add: inj_on_def, blast)
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done
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(*Compare with image_INT: no use of inj_on, and if f is surjective then
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  it doesn't matter whether A is empty*)
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lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
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apply (simp add: bij_def)
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apply (simp add: inj_on_def surj_def, blast)
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done
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lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
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by (auto simp add: surj_def)
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lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
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by (auto simp add: inj_on_def)
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lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
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apply (simp add: bij_def)
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apply (rule equalityI)
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apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
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done
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subsection{*Function Updating*}
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lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
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apply (simp add: fun_upd_def, safe)
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apply (erule subst)
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apply (rule_tac [2] ext, auto)
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done
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(* f x = y ==> f(x:=y) = f *)
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lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
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(* f(x := f x) = f *)
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declare refl [THEN fun_upd_idem, iff]
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lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
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apply (simp (no_asm) add: fun_upd_def)
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done
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(* fun_upd_apply supersedes these two,   but they are useful
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   if fun_upd_apply is intentionally removed from the simpset *)
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lemma fun_upd_same: "(f(x:=y)) x = y"
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by simp
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lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
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by simp
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lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
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by (simp add: expand_fun_eq)
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lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
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by (rule ext, auto)
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subsection{* overwrite *}
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lemma overwrite_emptyset[simp]: "f(g|{}) = f"
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by(simp add:overwrite_def)
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lemma overwrite_apply_notin[simp]: "a ~: A ==> (f(g|A)) a = f a"
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by(simp add:overwrite_def)
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lemma overwrite_apply_in[simp]: "a : A ==> (f(g|A)) a = g a"
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by(simp add:overwrite_def)
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text{*The ML section includes some compatibility bindings and a simproc
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for function updates, in addition to the usual ML-bindings of theorems.*}
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ML
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{*
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val id_def = thm "id_def";
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val inj_on_def = thm "inj_on_def";
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val surj_def = thm "surj_def";
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val bij_def = thm "bij_def";
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val fun_upd_def = thm "fun_upd_def";
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val o_def = thm "comp_def";
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val injI = thm "inj_onI";
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val inj_inverseI = thm "inj_on_inverseI";
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val set_cs = claset() delrules [equalityI];
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val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
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(* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *)
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local
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  fun gen_fun_upd None T _ _ = None
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    | gen_fun_upd (Some f) T x y = Some (Const ("Fun.fun_upd",T) $ f $ x $ y)
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  fun dest_fun_T1 (Type (_, T :: Ts)) = T
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  fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
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    let
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      fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
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            if v aconv x then Some g else gen_fun_upd (find g) T v w
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   402
        | find t = None
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   403
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
paulson@13585
   404
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   405
  val ss = simpset ()
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  val fun_upd_prover = K (rtac eq_reflection 1 THEN rtac ext 1 THEN simp_tac ss 1)
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in
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  val fun_upd2_simproc =
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    Simplifier.simproc (Theory.sign_of (the_context ()))
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      "fun_upd2" ["f(v := w, x := y)"]
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      (fn sg => fn _ => fn t =>
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   412
        case find_double t of (T, None) => None
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        | (T, Some rhs) => Some (Tactic.prove sg [] [] (Term.equals T $ t $ rhs) fun_upd_prover))
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   414
end;
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   415
Addsimprocs[fun_upd2_simproc];
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   416
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   417
val expand_fun_eq = thm "expand_fun_eq";
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   418
val apply_inverse = thm "apply_inverse";
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   419
val id_apply = thm "id_apply";
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   420
val o_apply = thm "o_apply";
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   421
val o_assoc = thm "o_assoc";
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   422
val id_o = thm "id_o";
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   423
val o_id = thm "o_id";
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   424
val image_compose = thm "image_compose";
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   425
val image_eq_UN = thm "image_eq_UN";
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   426
val UN_o = thm "UN_o";
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   427
val datatype_injI = thm "datatype_injI";
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   428
val injD = thm "injD";
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   429
val inj_eq = thm "inj_eq";
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   430
val inj_onI = thm "inj_onI";
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   431
val inj_on_inverseI = thm "inj_on_inverseI";
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   432
val inj_onD = thm "inj_onD";
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   433
val inj_on_iff = thm "inj_on_iff";
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   434
val comp_inj_on = thm "comp_inj_on";
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   435
val inj_on_contraD = thm "inj_on_contraD";
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   436
val inj_singleton = thm "inj_singleton";
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   437
val subset_inj_on = thm "subset_inj_on";
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   438
val surjI = thm "surjI";
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   439
val surj_range = thm "surj_range";
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   440
val surjD = thm "surjD";
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   441
val surjE = thm "surjE";
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   442
val comp_surj = thm "comp_surj";
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   443
val bijI = thm "bijI";
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   444
val bij_is_inj = thm "bij_is_inj";
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   445
val bij_is_surj = thm "bij_is_surj";
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   446
val image_ident = thm "image_ident";
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   447
val image_id = thm "image_id";
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   448
val vimage_ident = thm "vimage_ident";
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   449
val vimage_id = thm "vimage_id";
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   450
val vimage_image_eq = thm "vimage_image_eq";
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   451
val image_vimage_subset = thm "image_vimage_subset";
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   452
val image_vimage_eq = thm "image_vimage_eq";
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   453
val surj_image_vimage_eq = thm "surj_image_vimage_eq";
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   454
val inj_vimage_image_eq = thm "inj_vimage_image_eq";
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   455
val vimage_subsetD = thm "vimage_subsetD";
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   456
val vimage_subsetI = thm "vimage_subsetI";
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   457
val vimage_subset_eq = thm "vimage_subset_eq";
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   458
val image_Int_subset = thm "image_Int_subset";
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   459
val image_diff_subset = thm "image_diff_subset";
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   460
val inj_on_image_Int = thm "inj_on_image_Int";
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   461
val inj_on_image_set_diff = thm "inj_on_image_set_diff";
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   462
val image_Int = thm "image_Int";
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   463
val image_set_diff = thm "image_set_diff";
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   464
val inj_image_mem_iff = thm "inj_image_mem_iff";
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   465
val inj_image_subset_iff = thm "inj_image_subset_iff";
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   466
val inj_image_eq_iff = thm "inj_image_eq_iff";
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   467
val image_UN = thm "image_UN";
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   468
val image_INT = thm "image_INT";
paulson@13585
   469
val bij_image_INT = thm "bij_image_INT";
paulson@13585
   470
val surj_Compl_image_subset = thm "surj_Compl_image_subset";
paulson@13585
   471
val inj_image_Compl_subset = thm "inj_image_Compl_subset";
paulson@13585
   472
val bij_image_Compl_eq = thm "bij_image_Compl_eq";
paulson@13585
   473
val fun_upd_idem_iff = thm "fun_upd_idem_iff";
paulson@13585
   474
val fun_upd_idem = thm "fun_upd_idem";
paulson@13585
   475
val fun_upd_apply = thm "fun_upd_apply";
paulson@13585
   476
val fun_upd_same = thm "fun_upd_same";
paulson@13585
   477
val fun_upd_other = thm "fun_upd_other";
paulson@13585
   478
val fun_upd_upd = thm "fun_upd_upd";
paulson@13585
   479
val fun_upd_twist = thm "fun_upd_twist";
berghofe@13637
   480
val range_ex1_eq = thm "range_ex1_eq";
paulson@13585
   481
*}
paulson@5852
   482
nipkow@2912
   483
end