src/HOL/Fun.thy
author paulson
Wed, 09 Feb 2005 18:32:28 +0100
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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
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imports Typedef
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begin
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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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lemma inj_on_id: "inj_on id A"
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by (simp add: inj_on_def) 
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lemma surj_id: "surj id"
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by (simp add: surj_def) 
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lemma bij_id: "bij id"
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by (simp add: bij_def inj_on_id surj_id) 
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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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db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
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   164
lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   165
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   166
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
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   167
lemma comp_inj_on:
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parents: 12460
diff changeset
   168
     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   169
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   170
15303
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   171
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
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   172
apply(simp add:inj_on_def image_def)
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   173
apply blast
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parents: 15140
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   174
done
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parents: 15140
diff changeset
   175
15439
71c0f98e31f1 made diff_less a simp rule
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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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parents: 15303
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   177
  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
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   178
apply(unfold inj_on_def)
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apply blast
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parents: 15303
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   180
done
71c0f98e31f1 made diff_less a simp rule
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parents: 15303
diff changeset
   181
13585
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parents: 12460
diff changeset
   182
lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   183
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   184
13585
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parents: 12460
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   185
lemma inj_singleton: "inj (%s. {s})"
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parents: 12460
diff changeset
   186
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   187
15111
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parents: 14565
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   188
lemma inj_on_empty[iff]: "inj_on f {}"
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   189
by(simp add: inj_on_def)
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parents: 14565
diff changeset
   190
15303
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   191
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   192
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   193
15111
c108189645f8 added some inj_on thms
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parents: 14565
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   194
lemma inj_on_Un:
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   195
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
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   196
  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
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diff changeset
   197
apply(unfold inj_on_def)
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parents: 14565
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   198
apply (blast intro:sym)
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parents: 14565
diff changeset
   199
done
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   200
c108189645f8 added some inj_on thms
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   201
lemma inj_on_insert[iff]:
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   202
  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
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parents: 14565
diff changeset
   203
apply(unfold inj_on_def)
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parents: 14565
diff changeset
   204
apply (blast intro:sym)
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parents: 14565
diff changeset
   205
done
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   206
c108189645f8 added some inj_on thms
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   207
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
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parents: 14565
diff changeset
   208
apply(unfold inj_on_def)
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parents: 14565
diff changeset
   209
apply (blast)
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parents: 14565
diff changeset
   210
done
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   211
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   212
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   213
subsection{*The Predicate @{term surj}: Surjectivity*}
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
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   214
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   215
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   216
apply (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   217
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   218
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   219
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   220
lemma surj_range: "surj f ==> range f = UNIV"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   221
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   222
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
lemma surjD: "surj f ==> EX x. y = f x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
by (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   226
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
by (simp add: surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   228
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   229
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   230
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   231
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   232
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   233
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   234
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   235
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   236
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   237
subsection{*The Predicate @{term bij}: Bijectivity*}
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   238
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   240
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   241
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   242
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   243
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   245
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   246
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   247
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   248
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   249
subsection{*Facts About the Identity Function*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   250
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   251
text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   252
forms. The latter can arise by rewriting, while @{term id} may be used
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   253
explicitly.*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   254
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   255
lemma image_ident [simp]: "(%x. x) ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   256
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   257
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   258
lemma image_id [simp]: "id ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   259
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   260
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   261
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   262
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   263
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   264
lemma vimage_id [simp]: "id -` A = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   265
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   266
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   267
lemma vimage_image_eq: "f -` (f ` A) = {y. EX x:A. f x = f y}"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   268
by (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   269
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   270
lemma image_vimage_subset: "f ` (f -` A) <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   271
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   272
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   273
lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   274
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   275
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   276
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   277
by (simp add: surj_range)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   278
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   279
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   280
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   281
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   282
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   283
apply (unfold surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   284
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   285
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   286
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   287
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   288
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   289
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   290
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   291
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   292
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   293
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   294
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   295
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   296
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   297
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   298
lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   299
by blast
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   300
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   301
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   302
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   303
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   304
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   305
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   306
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   307
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   308
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   309
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   310
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   311
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   312
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   313
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   314
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   316
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   317
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   318
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   319
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   320
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   321
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   322
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   324
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   325
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   326
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   327
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   328
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   329
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   332
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   333
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   334
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   335
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   336
(*Compare with image_INT: no use of inj_on, and if f is surjective then
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   337
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   338
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   339
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   340
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   341
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   342
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   343
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   344
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   345
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   346
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   347
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   348
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   349
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   350
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   351
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   352
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   353
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   354
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   355
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   356
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   357
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   358
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   359
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   360
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   361
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   362
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   363
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   364
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   365
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   366
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   367
(* f(x := f x) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   368
declare refl [THEN fun_upd_idem, iff]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   369
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   370
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   371
apply (simp (no_asm) add: fun_upd_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   372
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   373
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   374
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   375
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   376
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   377
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   378
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   379
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   380
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   381
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   382
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   383
by (simp add: expand_fun_eq)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   384
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   385
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   386
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   387
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   388
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   389
by(fastsimp simp:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   390
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   391
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   392
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   393
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   394
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   395
subsection{* overwrite *}
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   396
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   397
lemma overwrite_emptyset[simp]: "f(g|{}) = f"
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   398
by(simp add:overwrite_def)
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   399
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   400
lemma overwrite_apply_notin[simp]: "a ~: A ==> (f(g|A)) a = f a"
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   401
by(simp add:overwrite_def)
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   402
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   403
lemma overwrite_apply_in[simp]: "a : A ==> (f(g|A)) a = g a"
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   404
by(simp add:overwrite_def)
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   405
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   406
subsection{* swap *}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   407
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   408
constdefs
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   409
  swap :: "['a, 'a, 'a => 'b] => ('a => 'b)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   410
   "swap a b f == f(a := f b, b:= f a)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   411
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   412
lemma swap_self: "swap a a f = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   413
by (simp add: swap_def) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   414
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   415
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   416
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   417
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   418
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   419
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   420
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   421
lemma inj_on_imp_inj_on_swap:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   422
     "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   423
by (simp add: inj_on_def swap_def, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   424
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   425
lemma inj_on_swap_iff [simp]:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   426
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   427
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   428
  assume "inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   429
  with A have "inj_on (swap a b (swap a b f)) A" 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   430
    by (rules intro: inj_on_imp_inj_on_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   431
  thus "inj_on f A" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   432
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   433
  assume "inj_on f A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   434
  with A show "inj_on (swap a b f) A" by (rules intro: inj_on_imp_inj_on_swap)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   435
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   436
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   437
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   438
apply (simp add: surj_def swap_def, clarify)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   439
apply (rule_tac P = "y = f b" in case_split_thm, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   440
apply (rule_tac P = "y = f a" in case_split_thm, auto)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   441
  --{*We don't yet have @{text case_tac}*}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   442
done
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   443
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   444
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   445
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   446
  assume "surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   447
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   448
  thus "surj f" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   449
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   450
  assume "surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   451
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   452
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   453
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   454
lemma bij_swap_iff: "bij (swap a b f) = bij f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   455
by (simp add: bij_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   456
 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   457
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   458
text{*The ML section includes some compatibility bindings and a simproc
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   459
for function updates, in addition to the usual ML-bindings of theorems.*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   460
ML
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   461
{*
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   462
val id_def = thm "id_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   463
val inj_on_def = thm "inj_on_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   464
val surj_def = thm "surj_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   465
val bij_def = thm "bij_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   466
val fun_upd_def = thm "fun_upd_def";
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents: 11123
diff changeset
   467
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   468
val o_def = thm "comp_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   469
val injI = thm "inj_onI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   470
val inj_inverseI = thm "inj_on_inverseI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   471
val set_cs = claset() delrules [equalityI];
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   472
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   473
val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   474
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   475
(* simplifies terms of the form f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   476
local
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   477
  fun gen_fun_upd None T _ _ = None
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   478
    | gen_fun_upd (Some f) T x y = Some (Const ("Fun.fun_upd",T) $ f $ x $ y)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   479
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   480
  fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   481
    let
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   482
      fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   483
            if v aconv x then Some g else gen_fun_upd (find g) T v w
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   484
        | find t = None
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   485
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   486
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   487
  val ss = simpset ()
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   488
  val fun_upd_prover = K (rtac eq_reflection 1 THEN rtac ext 1 THEN simp_tac ss 1)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   489
in
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   490
  val fun_upd2_simproc =
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   491
    Simplifier.simproc (Theory.sign_of (the_context ()))
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   492
      "fun_upd2" ["f(v := w, x := y)"]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   493
      (fn sg => fn _ => fn t =>
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   494
        case find_double t of (T, None) => None
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   495
        | (T, Some rhs) => Some (Tactic.prove sg [] [] (Term.equals T $ t $ rhs) fun_upd_prover))
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   496
end;
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   497
Addsimprocs[fun_upd2_simproc];
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   498
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   499
val expand_fun_eq = thm "expand_fun_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   500
val apply_inverse = thm "apply_inverse";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   501
val id_apply = thm "id_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   502
val o_apply = thm "o_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   503
val o_assoc = thm "o_assoc";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   504
val id_o = thm "id_o";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   505
val o_id = thm "o_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   506
val image_compose = thm "image_compose";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   507
val image_eq_UN = thm "image_eq_UN";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   508
val UN_o = thm "UN_o";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   509
val datatype_injI = thm "datatype_injI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   510
val injD = thm "injD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   511
val inj_eq = thm "inj_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   512
val inj_onI = thm "inj_onI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   513
val inj_on_inverseI = thm "inj_on_inverseI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   514
val inj_onD = thm "inj_onD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   515
val inj_on_iff = thm "inj_on_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   516
val comp_inj_on = thm "comp_inj_on";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   517
val inj_on_contraD = thm "inj_on_contraD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   518
val inj_singleton = thm "inj_singleton";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   519
val subset_inj_on = thm "subset_inj_on";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   520
val surjI = thm "surjI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   521
val surj_range = thm "surj_range";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   522
val surjD = thm "surjD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   523
val surjE = thm "surjE";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   524
val comp_surj = thm "comp_surj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   525
val bijI = thm "bijI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   526
val bij_is_inj = thm "bij_is_inj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   527
val bij_is_surj = thm "bij_is_surj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   528
val image_ident = thm "image_ident";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   529
val image_id = thm "image_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   530
val vimage_ident = thm "vimage_ident";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   531
val vimage_id = thm "vimage_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   532
val vimage_image_eq = thm "vimage_image_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   533
val image_vimage_subset = thm "image_vimage_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   534
val image_vimage_eq = thm "image_vimage_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   535
val surj_image_vimage_eq = thm "surj_image_vimage_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   536
val inj_vimage_image_eq = thm "inj_vimage_image_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   537
val vimage_subsetD = thm "vimage_subsetD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   538
val vimage_subsetI = thm "vimage_subsetI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   539
val vimage_subset_eq = thm "vimage_subset_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   540
val image_Int_subset = thm "image_Int_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   541
val image_diff_subset = thm "image_diff_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   542
val inj_on_image_Int = thm "inj_on_image_Int";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   543
val inj_on_image_set_diff = thm "inj_on_image_set_diff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   544
val image_Int = thm "image_Int";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   545
val image_set_diff = thm "image_set_diff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   546
val inj_image_mem_iff = thm "inj_image_mem_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   547
val inj_image_subset_iff = thm "inj_image_subset_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   548
val inj_image_eq_iff = thm "inj_image_eq_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   549
val image_UN = thm "image_UN";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   550
val image_INT = thm "image_INT";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   551
val bij_image_INT = thm "bij_image_INT";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   552
val surj_Compl_image_subset = thm "surj_Compl_image_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   553
val inj_image_Compl_subset = thm "inj_image_Compl_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   554
val bij_image_Compl_eq = thm "bij_image_Compl_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   555
val fun_upd_idem_iff = thm "fun_upd_idem_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   556
val fun_upd_idem = thm "fun_upd_idem";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   557
val fun_upd_apply = thm "fun_upd_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   558
val fun_upd_same = thm "fun_upd_same";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   559
val fun_upd_other = thm "fun_upd_other";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   560
val fun_upd_upd = thm "fun_upd_upd";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   561
val fun_upd_twist = thm "fun_upd_twist";
13637
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   562
val range_ex1_eq = thm "range_ex1_eq";
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   563
*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   564
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   565
end