src/HOL/Fun.thy
author haftmann
Mon, 27 Nov 2006 13:42:39 +0100
changeset 21547 9c9fdf4c2949
parent 21327 2b3c41d02e87
child 21870 c701cdacf69b
permissions -rw-r--r--
moved order arities for fun and bool to Fun/Orderings
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
1475
7f5a4cd08209 expanded tabs; renamed subtype to typedef;
clasohm
parents: 923
diff changeset
     1
(*  Title:      HOL/Fun.thy
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     2
    ID:         $Id$
1475
7f5a4cd08209 expanded tabs; renamed subtype to typedef;
clasohm
parents: 923
diff changeset
     3
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     4
    Copyright   1994  University of Cambridge
18154
0c05abaf6244 add header
huffman
parents: 17956
diff changeset
     5
*)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     6
18154
0c05abaf6244 add header
huffman
parents: 17956
diff changeset
     7
header {* Notions about functions *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     8
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
     9
theory Fun
21327
2b3c41d02e87 dropped Typedef dependency
haftmann
parents: 21210
diff changeset
    10
imports Set
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15111
diff changeset
    11
begin
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    12
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    13
constdefs
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    14
  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    15
   "fun_upd f a b == % x. if x=a then b else f x"
6171
cd237a10cbf8 inj is now a translation of inj_on
paulson
parents: 5852
diff changeset
    16
9141
wenzelm
parents: 8960
diff changeset
    17
nonterminals
wenzelm
parents: 8960
diff changeset
    18
  updbinds updbind
5305
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    19
syntax
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    20
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    21
  ""         :: "updbind => updbinds"             ("_")
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    22
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    23
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
5305
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    24
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    25
translations
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    26
  "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    27
  "f(x:=y)"                     == "fun_upd f x y"
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    28
9340
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    29
(* Hint: to define the sum of two functions (or maps), use sum_case.
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    30
         A nice infix syntax could be defined (in Datatype.thy or below) by
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    31
consts
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    32
  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    33
translations
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    34
 "fun_sum" == sum_case
9340
9666f78ecfab added hint on fun_sum
oheimb
parents: 9309
diff changeset
    35
*)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
    36
6171
cd237a10cbf8 inj is now a translation of inj_on
paulson
parents: 5852
diff changeset
    37
constdefs
19536
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    38
  override_on :: "('a => 'b) => ('a => 'b) => 'a set => ('a => 'b)"
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    39
  "override_on f g A == %a. if a : A then g a else f a"
6171
cd237a10cbf8 inj is now a translation of inj_on
paulson
parents: 5852
diff changeset
    40
19536
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    41
  id :: "'a => 'a"
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    42
  "id == %x. x"
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
    43
19536
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    44
  comp :: "['b => 'c, 'a => 'b, 'a] => 'c"   (infixl "o" 55)
1a3a3cf8b4fa replaced syntax/translations by abbreviation;
wenzelm
parents: 19363
diff changeset
    45
  "f o g == %x. f(g(x))"
11123
15ffc08f905e removed whitespace
oheimb
parents: 10826
diff changeset
    46
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20044
diff changeset
    47
notation (xsymbols)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    48
  comp  (infixl "\<circ>" 55)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    49
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20044
diff changeset
    50
notation (HTML output)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    51
  comp  (infixl "\<circ>" 55)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    52
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    53
text{*compatibility*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    54
lemmas o_def = comp_def
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    55
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    56
constdefs
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    57
  inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    58
  "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
6171
cd237a10cbf8 inj is now a translation of inj_on
paulson
parents: 5852
diff changeset
    59
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    60
text{*A common special case: functions injective over the entire domain type.*}
19323
ec5cd5b1804c Converted translations to abbbreviations.
nipkow
parents: 18154
diff changeset
    61
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    62
abbreviation
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    63
  "inj f == inj_on f UNIV"
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
    64
7374
dec7b838f5cb the bij predicate (at last)
paulson
parents: 6171
diff changeset
    65
constdefs
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    66
  surj :: "('a => 'b) => bool"                   (*surjective*)
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    67
  "surj f == ! y. ? x. y=f(x)"
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
    68
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    69
  bij :: "('a => 'b) => bool"                    (*bijective*)
19363
667b5ea637dd refined 'abbreviation';
wenzelm
parents: 19323
diff changeset
    70
  "bij f == inj f & surj f"
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
    71
7374
dec7b838f5cb the bij predicate (at last)
paulson
parents: 6171
diff changeset
    72
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    73
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    74
text{*As a simplification rule, it replaces all function equalities by
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    75
  first-order equalities.*}
21327
2b3c41d02e87 dropped Typedef dependency
haftmann
parents: 21210
diff changeset
    76
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    77
apply (rule iffI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    78
apply (simp (no_asm_simp))
21327
2b3c41d02e87 dropped Typedef dependency
haftmann
parents: 21210
diff changeset
    79
apply (rule ext)
2b3c41d02e87 dropped Typedef dependency
haftmann
parents: 21210
diff changeset
    80
apply (simp (no_asm_simp))
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    81
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    82
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    83
lemma apply_inverse:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    84
    "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    85
by auto
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    86
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    87
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    88
text{*The Identity Function: @{term id}*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    89
lemma id_apply [simp]: "id x = x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    90
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    91
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
    92
lemma inj_on_id[simp]: "inj_on id A"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
    93
by (simp add: inj_on_def) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
    94
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
    95
lemma inj_on_id2[simp]: "inj_on (%x. x) A"
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
    96
by (simp add: inj_on_def) 
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
    97
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
    98
lemma surj_id[simp]: "surj id"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
    99
by (simp add: surj_def) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   100
16733
236dfafbeb63 linear arithmetic now takes "&" in assumptions apart.
nipkow
parents: 15691
diff changeset
   101
lemma bij_id[simp]: "bij id"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   102
by (simp add: bij_def inj_on_id surj_id) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   103
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   104
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   105
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   106
subsection{*The Composition Operator: @{term "f \<circ> g"}*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   107
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   108
lemma o_apply [simp]: "(f o g) x = f (g x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   109
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   110
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   111
lemma o_assoc: "f o (g o h) = f o g o h"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   112
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   113
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   114
lemma id_o [simp]: "id o g = g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   115
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   116
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   117
lemma o_id [simp]: "f o id = f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   118
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   119
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   120
lemma image_compose: "(f o g) ` r = f`(g`r)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   121
by (simp add: comp_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   122
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   123
lemma image_eq_UN: "f`A = (UN x:A. {f x})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   124
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   125
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   126
lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   127
by (unfold comp_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   128
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   129
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   130
subsection{*The Injectivity Predicate, @{term inj}*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   131
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   132
text{*NB: @{term inj} now just translates to @{term inj_on}*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   133
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   134
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   135
text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   136
lemma datatype_injI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   137
    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   138
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   139
13637
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   140
theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   141
  by (unfold inj_on_def, blast)
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   142
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   143
lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   144
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   145
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   146
(*Useful with the simplifier*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   147
lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   148
by (force simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   149
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   150
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   151
subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   152
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   153
lemma inj_onI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   154
    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   155
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   156
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   157
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   158
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   159
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   160
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
   161
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   162
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   163
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
   164
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   165
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   166
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   167
     "[| 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
   168
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   169
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   170
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   171
apply(simp add:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   172
apply blast
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   173
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   174
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   175
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   176
  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   177
apply(unfold inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   178
apply blast
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   179
done
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   180
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   181
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
   182
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   183
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   184
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   185
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   186
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   187
lemma inj_on_empty[iff]: "inj_on f {}"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   188
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   189
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   190
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
   191
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   192
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   193
lemma inj_on_Un:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   194
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   195
  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   196
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   197
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   198
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   199
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   200
lemma inj_on_insert[iff]:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   201
  "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
   202
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   203
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   204
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   205
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   206
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   207
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   208
apply (blast)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   209
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   210
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   211
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   212
subsection{*The Predicate @{term surj}: Surjectivity*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   213
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   214
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   215
apply (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   216
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   217
done
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 surj_range: "surj f ==> range f = UNIV"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   220
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   221
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   222
lemma surjD: "surj f ==> EX x. y = f x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   223
by (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   224
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   225
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   226
by (simp add: surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   227
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   228
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   229
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   230
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   231
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   232
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   233
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
subsection{*The Predicate @{term bij}: Bijectivity*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   237
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   238
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   240
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   241
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   242
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   243
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   245
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   246
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   247
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   248
subsection{*Facts About the Identity Function*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   249
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   250
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
   251
forms. The latter can arise by rewriting, while @{term id} may be used
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   252
explicitly.*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   253
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   254
lemma image_ident [simp]: "(%x. x) ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   255
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   256
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   257
lemma image_id [simp]: "id ` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   258
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   259
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   260
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   261
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   262
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   263
lemma vimage_id [simp]: "id -` A = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   264
by (simp add: id_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   265
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   266
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
   267
by (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   268
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   269
lemma image_vimage_subset: "f ` (f -` A) <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   270
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   271
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   272
lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   273
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   274
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   275
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   276
by (simp add: surj_range)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   277
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   278
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   279
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   280
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   281
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   282
apply (unfold surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   283
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   284
done
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_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   287
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   288
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   289
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   290
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   291
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   292
done
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 image_Int_subset: "f`(A Int B) <= f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   295
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   296
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   297
lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   298
by blast
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   299
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   300
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   301
   "[| 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
   302
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   303
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   304
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   305
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   306
   "[| 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
   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 image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   311
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   312
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   313
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   314
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   316
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   317
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   318
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   319
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   320
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   321
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   322
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   324
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   325
lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   326
by blast
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   327
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   328
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   329
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   332
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   333
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   334
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   335
(*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
   336
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   337
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
   338
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   339
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   340
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   341
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   342
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   343
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   344
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   345
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   346
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   347
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   348
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   349
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   350
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   351
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
   352
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   353
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   354
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   355
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   356
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   357
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   358
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   359
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   360
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   361
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   362
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   363
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   364
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   365
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   366
(* f(x := f x) = f *)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   367
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   368
declare fun_upd_triv [iff]
13585
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)"
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   371
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   372
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   373
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   374
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   375
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   376
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   377
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   378
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   379
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   380
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   381
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   382
by (simp add: expand_fun_eq)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   383
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   384
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
   385
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   386
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   387
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
   388
by(fastsimp simp:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   389
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   390
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   391
     "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
   392
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   393
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   394
subsection{* @{text override_on} *}
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   395
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   396
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   397
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   398
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   399
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
   400
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   401
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   402
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
   403
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   404
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   405
subsection{* swap *}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   406
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   407
constdefs
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   408
  swap :: "['a, 'a, 'a => 'b] => ('a => 'b)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   409
   "swap a b f == f(a := f b, b:= f a)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   410
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   411
lemma swap_self: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   412
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   413
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   414
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   415
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   416
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   417
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   418
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   419
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   420
lemma inj_on_imp_inj_on_swap:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   421
     "[|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
   422
by (simp add: inj_on_def swap_def, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   423
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   424
lemma inj_on_swap_iff [simp]:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   425
  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
   426
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   427
  assume "inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   428
  with A have "inj_on (swap a b (swap a b f)) A" 
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   429
    by (iprover intro: inj_on_imp_inj_on_swap) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   430
  thus "inj_on f A" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   431
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   432
  assume "inj_on f A"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   433
  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
   434
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   435
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   436
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   437
apply (simp add: surj_def swap_def, clarify)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   438
apply (rule_tac P = "y = f b" in case_split_thm, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   439
apply (rule_tac P = "y = f a" in case_split_thm, auto)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   440
  --{*We don't yet have @{text case_tac}*}
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   441
done
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   442
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   443
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   444
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   445
  assume "surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   446
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   447
  thus "surj f" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   448
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   449
  assume "surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   450
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   451
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   452
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   453
lemma bij_swap_iff: "bij (swap a b f) = bij f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   454
by (simp add: bij_def)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   455
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   456
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   457
subsection {* Order on functions *}
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   458
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   459
instance "fun" :: (type, order) order
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   460
  le_fun_def: "f \<le> g \<equiv> \<forall>x. f x \<le> g x"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   461
  less_fun_def: "f < g \<equiv> f \<le> g \<and> f \<noteq> g"
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   462
  by default
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   463
    (auto simp add: le_fun_def less_fun_def intro: order_trans, rule ext, auto intro: order_antisym)
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   464
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   465
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
   466
  unfolding le_fun_def by simp
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   467
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   468
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
   469
  unfolding le_fun_def by simp
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
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
   472
  unfolding le_fun_def by simp
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   473
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   474
instance "fun" :: (type, ord) ord ..
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
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   477
subsection {* ML legacy bindings *} 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   478
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   479
text{*The ML section includes some compatibility bindings and a simproc
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   480
for function updates, in addition to the usual ML-bindings of theorems.*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   481
ML
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   482
{*
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   483
val id_def = thm "id_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   484
val inj_on_def = thm "inj_on_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   485
val surj_def = thm "surj_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   486
val bij_def = thm "bij_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   487
val fun_upd_def = thm "fun_upd_def";
11451
8abfb4f7bd02 partial restructuring to reduce dependence on Axiom of Choice
paulson
parents: 11123
diff changeset
   488
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   489
val o_def = thm "comp_def";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   490
val injI = thm "inj_onI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   491
val inj_inverseI = thm "inj_on_inverseI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   492
val set_cs = claset() delrules [equalityI];
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   493
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   494
val print_translation = [("Pi", dependent_tr' ("@Pi", "op funcset"))];
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   495
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   496
(* 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
   497
local
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15510
diff changeset
   498
  fun gen_fun_upd NONE T _ _ = NONE
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15510
diff changeset
   499
    | gen_fun_upd (SOME f) T x y = SOME (Const ("Fun.fun_upd",T) $ f $ x $ y)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   500
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   501
  fun find_double (t as Const ("Fun.fun_upd",T) $ f $ x $ y) =
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   502
    let
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   503
      fun find (Const ("Fun.fun_upd",T) $ g $ v $ w) =
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15510
diff changeset
   504
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15510
diff changeset
   505
        | find t = NONE
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   506
    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
   507
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16733
diff changeset
   508
  val current_ss = simpset ()
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16733
diff changeset
   509
  fun fun_upd_prover ss =
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16733
diff changeset
   510
    rtac eq_reflection 1 THEN rtac ext 1 THEN
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17589
diff changeset
   511
    simp_tac (Simplifier.inherit_context ss current_ss) 1
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   512
in
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   513
  val fun_upd2_simproc =
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   514
    Simplifier.simproc (Theory.sign_of (the_context ()))
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   515
      "fun_upd2" ["f(v := w, x := y)"]
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19656
diff changeset
   516
      (fn _ => fn ss => fn t =>
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15510
diff changeset
   517
        case find_double t of (T, NONE) => NONE
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16733
diff changeset
   518
        | (T, SOME rhs) =>
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19656
diff changeset
   519
            SOME (Goal.prove (Simplifier.the_context ss) [] []
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19656
diff changeset
   520
              (Term.equals T $ t $ rhs) (K (fun_upd_prover ss))))
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   521
end;
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   522
Addsimprocs[fun_upd2_simproc];
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   523
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   524
val expand_fun_eq = thm "expand_fun_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   525
val apply_inverse = thm "apply_inverse";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   526
val id_apply = thm "id_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   527
val o_apply = thm "o_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   528
val o_assoc = thm "o_assoc";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   529
val id_o = thm "id_o";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   530
val o_id = thm "o_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   531
val image_compose = thm "image_compose";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   532
val image_eq_UN = thm "image_eq_UN";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   533
val UN_o = thm "UN_o";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   534
val datatype_injI = thm "datatype_injI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   535
val injD = thm "injD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   536
val inj_eq = thm "inj_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   537
val inj_onI = thm "inj_onI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   538
val inj_on_inverseI = thm "inj_on_inverseI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   539
val inj_onD = thm "inj_onD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   540
val inj_on_iff = thm "inj_on_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   541
val comp_inj_on = thm "comp_inj_on";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   542
val inj_on_contraD = thm "inj_on_contraD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   543
val inj_singleton = thm "inj_singleton";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   544
val subset_inj_on = thm "subset_inj_on";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   545
val surjI = thm "surjI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   546
val surj_range = thm "surj_range";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   547
val surjD = thm "surjD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   548
val surjE = thm "surjE";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   549
val comp_surj = thm "comp_surj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   550
val bijI = thm "bijI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   551
val bij_is_inj = thm "bij_is_inj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   552
val bij_is_surj = thm "bij_is_surj";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   553
val image_ident = thm "image_ident";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   554
val image_id = thm "image_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   555
val vimage_ident = thm "vimage_ident";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   556
val vimage_id = thm "vimage_id";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   557
val vimage_image_eq = thm "vimage_image_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   558
val image_vimage_subset = thm "image_vimage_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   559
val image_vimage_eq = thm "image_vimage_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   560
val surj_image_vimage_eq = thm "surj_image_vimage_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   561
val inj_vimage_image_eq = thm "inj_vimage_image_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   562
val vimage_subsetD = thm "vimage_subsetD";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   563
val vimage_subsetI = thm "vimage_subsetI";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   564
val vimage_subset_eq = thm "vimage_subset_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   565
val image_Int_subset = thm "image_Int_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   566
val image_diff_subset = thm "image_diff_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   567
val inj_on_image_Int = thm "inj_on_image_Int";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   568
val inj_on_image_set_diff = thm "inj_on_image_set_diff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   569
val image_Int = thm "image_Int";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   570
val image_set_diff = thm "image_set_diff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   571
val inj_image_mem_iff = thm "inj_image_mem_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   572
val inj_image_subset_iff = thm "inj_image_subset_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   573
val inj_image_eq_iff = thm "inj_image_eq_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   574
val image_UN = thm "image_UN";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   575
val image_INT = thm "image_INT";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   576
val bij_image_INT = thm "bij_image_INT";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   577
val surj_Compl_image_subset = thm "surj_Compl_image_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   578
val inj_image_Compl_subset = thm "inj_image_Compl_subset";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   579
val bij_image_Compl_eq = thm "bij_image_Compl_eq";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   580
val fun_upd_idem_iff = thm "fun_upd_idem_iff";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   581
val fun_upd_idem = thm "fun_upd_idem";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   582
val fun_upd_apply = thm "fun_upd_apply";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   583
val fun_upd_same = thm "fun_upd_same";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   584
val fun_upd_other = thm "fun_upd_other";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   585
val fun_upd_upd = thm "fun_upd_upd";
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   586
val fun_upd_twist = thm "fun_upd_twist";
13637
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   587
val range_ex1_eq = thm "range_ex1_eq";
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   588
*}
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   589
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   590
end