src/HOL/Fun.thy
author haftmann
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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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*)
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header {* Notions about functions *}
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theory Fun
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imports Set Code_Generator
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begin
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constdefs
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  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
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  [code func]: "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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definition
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  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
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where
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  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
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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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definition
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  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
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where
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  "f o g = (\<lambda>x. f (g x))"
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notation (xsymbols)
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  comp  (infixl "\<circ>" 55)
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notation (HTML output)
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  comp  (infixl "\<circ>" 55)
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text{*compatibility*}
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lemmas o_def = comp_def
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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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abbreviation
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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 \<longleftrightarrow> (\<forall>x. f x = g x)"
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apply (rule iffI)
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apply (simp (no_asm_simp))
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apply (rule ext)
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apply (simp (no_asm_simp))
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done
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lemma apply_inverse:
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    "[| f(x)=u;  !!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[simp]: "inj_on id A"
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by (simp add: inj_on_def) 
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lemma inj_on_id2[simp]: "inj_on (%x. x) A"
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by (simp add: inj_on_def) 
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lemma surj_id[simp]: "surj id"
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by (simp add: surj_def) 
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lemma bij_id[simp]: "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"
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   163
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
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parents: 12460
diff changeset
   164
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   165
lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   166
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   167
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   168
lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   169
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   170
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   171
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   172
     "[| 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
   173
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   174
15303
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   175
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
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parents: 15140
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   176
apply(simp add:inj_on_def image_def)
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   177
apply blast
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   178
done
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parents: 15140
diff changeset
   179
15439
71c0f98e31f1 made diff_less a simp rule
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   180
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
diff changeset
   181
  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
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   182
apply(unfold inj_on_def)
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parents: 15303
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   183
apply blast
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parents: 15303
diff changeset
   184
done
71c0f98e31f1 made diff_less a simp rule
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parents: 15303
diff changeset
   185
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   186
lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   187
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   188
13585
db4005b40cc6 Converted Fun to Isar style.
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parents: 12460
diff changeset
   189
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   190
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   191
15111
c108189645f8 added some inj_on thms
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parents: 14565
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   192
lemma inj_on_empty[iff]: "inj_on f {}"
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   193
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   194
15303
eedbb8d22ca2 added lemmas
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   195
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
   196
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   197
15111
c108189645f8 added some inj_on thms
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parents: 14565
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   198
lemma inj_on_Un:
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   199
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
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   200
  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   201
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   202
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   203
done
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   204
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   205
lemma inj_on_insert[iff]:
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parents: 14565
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   206
  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   207
apply(unfold inj_on_def)
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parents: 14565
diff changeset
   208
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   209
done
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   210
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   211
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   212
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   213
apply (blast)
c108189645f8 added some inj_on thms
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parents: 14565
diff changeset
   214
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   215
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   216
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   217
subsection{*The Predicate @{term surj}: Surjectivity*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   218
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   219
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   220
apply (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   221
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   222
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
lemma surj_range: "surj f ==> range f = UNIV"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   226
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
lemma surjD: "surj f ==> EX x. y = f x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   228
by (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   229
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   230
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   231
by (simp add: surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   232
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   233
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   234
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   235
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   236
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   237
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   238
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   240
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   241
subsection{*The Predicate @{term bij}: Bijectivity*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   242
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   243
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   245
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   246
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   247
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   248
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   249
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   250
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   251
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   252
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   253
subsection{*Facts About the Identity Function*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   254
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   255
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
   256
forms. The latter can arise by rewriting, while @{term id} may be used
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   257
explicitly.*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   258
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   259
lemma image_ident [simp]: "(%x. x) ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   260
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   261
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   262
lemma image_id [simp]: "id ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   263
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   264
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   265
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   266
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   267
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   268
lemma vimage_id [simp]: "id -` A = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   269
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   270
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   271
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
   272
by (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   273
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   274
lemma image_vimage_subset: "f ` (f -` A) <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   275
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   276
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   277
lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   278
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   279
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   280
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   281
by (simp add: surj_range)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   282
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   283
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   284
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   285
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   286
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   287
apply (unfold surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   288
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   289
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   290
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   291
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   292
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   293
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   294
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   295
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   296
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   297
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   298
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   299
lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   300
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   301
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   302
lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   303
by blast
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   304
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   305
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   306
   "[| 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
   307
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   308
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   309
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   310
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   311
   "[| 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
   312
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   313
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   314
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
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
   316
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   317
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   318
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   319
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   320
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   321
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   322
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   324
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   325
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   326
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   327
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   328
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   329
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   332
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   333
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   334
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   335
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   336
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   337
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   338
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   339
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   340
(*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
   341
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   342
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
   343
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   344
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   345
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   346
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   347
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   348
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   349
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   350
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   351
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   352
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   353
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   354
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   355
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   356
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
   357
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   358
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   359
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   360
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   361
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   362
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   363
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   364
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   365
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   366
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   367
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   368
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   369
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   370
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   371
(* f(x := f x) = f *)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   372
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   373
declare fun_upd_triv [iff]
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   374
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   375
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   376
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   377
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   378
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   379
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   380
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   381
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   382
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   383
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   384
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   385
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   386
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   387
by (simp add: expand_fun_eq)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   388
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   389
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
   390
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   391
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   392
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
   393
by(fastsimp simp:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   394
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   395
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   396
     "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
   397
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   398
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   399
subsection{* @{text override_on} *}
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   400
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   401
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   402
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   403
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   404
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   405
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   406
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   407
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   408
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   409
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   410
subsection{* swap *}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   411
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   412
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   413
  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   414
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   415
  "swap a b f = f (a := f b, b:= f a)"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   416
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   417
lemma swap_self: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   418
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   419
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   420
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   421
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   422
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   423
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   424
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   425
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   426
lemma inj_on_imp_inj_on_swap:
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   427
  "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   428
by (simp add: inj_on_def swap_def, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   429
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   430
lemma inj_on_swap_iff [simp]:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   431
  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
   432
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   433
  assume "inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   434
  with A have "inj_on (swap a b (swap a b f)) A" 
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   435
    by (iprover intro: inj_on_imp_inj_on_swap) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   436
  thus "inj_on f A" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   437
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   438
  assume "inj_on f A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   439
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   440
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   441
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   442
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   443
apply (simp add: surj_def swap_def, clarify)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   444
apply (rule_tac P = "y = f b" in case_split_thm, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   445
apply (rule_tac P = "y = f a" in case_split_thm, auto)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   446
  --{*We don't yet have @{text case_tac}*}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   447
done
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   448
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   449
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   450
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   451
  assume "surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   452
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   453
  thus "surj f" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   454
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   455
  assume "surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   456
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   457
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   458
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   459
lemma bij_swap_iff: "bij (swap a b f) = bij f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   460
by (simp add: bij_def)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   461
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   462
22453
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   463
subsection {* Order and lattice on functions *}
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   464
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   465
instance "fun" :: (type, ord) ord
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   466
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   467
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g" ..
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   468
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   469
lemmas [code func del] = le_fun_def less_fun_def
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   470
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   471
instance "fun" :: (type, order) order
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   472
  by default
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   473
    (auto simp add: le_fun_def less_fun_def expand_fun_eq
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   474
       intro: order_trans order_antisym)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   475
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   476
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   477
  unfolding le_fun_def by simp
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   478
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   479
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   480
  unfolding le_fun_def by simp
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   481
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   482
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   483
  unfolding le_fun_def by simp
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   484
22453
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   485
text {*
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   486
  Handy introduction and elimination rules for @{text "\<le>"}
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   487
  on unary and binary predicates
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   488
*}
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   489
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   490
lemma predicate1I [Pure.intro!, intro!]:
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   491
  assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   492
  shows "P \<le> Q"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   493
  apply (rule le_funI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   494
  apply (rule le_boolI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   495
  apply (rule PQ)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   496
  apply assumption
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   497
  done
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   498
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   499
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   500
  apply (erule le_funE)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   501
  apply (erule le_boolE)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   502
  apply assumption+
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   503
  done
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   504
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   505
lemma predicate2I [Pure.intro!, intro!]:
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   506
  assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   507
  shows "P \<le> Q"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   508
  apply (rule le_funI)+
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   509
  apply (rule le_boolI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   510
  apply (rule PQ)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   511
  apply assumption
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   512
  done
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   513
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   514
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   515
  apply (erule le_funE)+
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   516
  apply (erule le_boolE)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   517
  apply assumption+
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   518
  done
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   519
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   520
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   521
  by (rule predicate1D)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   522
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   523
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   524
  by (rule predicate2D)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   525
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   526
instance "fun" :: (type, lattice) lattice
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   527
  inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   528
  sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))"
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   529
apply intro_classes
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   530
unfolding inf_fun_eq sup_fun_eq
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   531
apply (auto intro: le_funI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   532
apply (rule le_funI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   533
apply (auto dest: le_funD)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   534
apply (rule le_funI)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   535
apply (auto dest: le_funD)
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   536
done
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   537
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   538
lemmas [code func del] = inf_fun_eq sup_fun_eq
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   539
22453
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   540
instance "fun" :: (type, distrib_lattice) distrib_lattice
530db8c36f53 moved lattice instance here
haftmann
parents: 21906
diff changeset
   541
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   542
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   543
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   544
subsection {* Proof tool setup *} 
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   545
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   546
text {* simplifies terms of the form
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   547
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   548
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   549
ML {*
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   550
let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   551
  fun gen_fun_upd NONE T _ _ = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   552
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd},T) $ f $ x $ y)
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   553
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   554
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   555
    let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   556
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   557
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   558
        | find t = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   559
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   560
  fun fun_upd_prover ss =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   561
    rtac eq_reflection 1 THEN rtac ext 1 THEN
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   562
    simp_tac (Simplifier.inherit_context ss @{simpset}) 1
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   563
  val fun_upd2_simproc =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   564
    Simplifier.simproc @{theory}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   565
      "fun_upd2" ["f(v := w, x := y)"]
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   566
      (fn _ => fn ss => fn t =>
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   567
        case find_double t of (T, NONE) => NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   568
        | (T, SOME rhs) =>
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   569
            SOME (Goal.prove (Simplifier.the_context ss) [] []
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   570
              (Term.equals T $ t $ rhs) (K (fun_upd_prover ss))))
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   571
in
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   572
  Addsimprocs [fun_upd2_simproc]
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   573
end;
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   574
*}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   575
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   576
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   577
subsection {* Code generator setup *}
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   578
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   579
code_const "op \<circ>"
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   580
  (SML infixl 5 "o")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   581
  (Haskell infixr 9 ".")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   582
21906
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   583
code_const "id"
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   584
  (Haskell "id")
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   585
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   586
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   587
subsection {* ML legacy bindings *} 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   588
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   589
ML {*
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   590
val set_cs = claset() delrules [equalityI]
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   591
*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   592
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   593
ML {*
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   594
val id_apply = @{thm id_apply}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   595
val id_def = @{thm id_def}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   596
val o_apply = @{thm o_apply}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   597
val o_assoc = @{thm o_assoc}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   598
val o_def = @{thm o_def}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   599
val injD = @{thm injD}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   600
val datatype_injI = @{thm datatype_injI}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   601
val range_ex1_eq = @{thm range_ex1_eq}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   602
val expand_fun_eq = @{thm expand_fun_eq}
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   603
*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   604
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   605
end