src/HOL/Fun.thy
author wenzelm
Sat, 13 Mar 2010 14:44:47 +0100
changeset 35743 c506c029a082
parent 35584 768f8d92b767
child 36176 3fe7e97ccca8
permissions -rw-r--r--
adapted to localized typedef: handle single global interpretation only;
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
7f5a4cd08209 expanded tabs; renamed subtype to typedef;
clasohm
parents: 923
diff changeset
     2
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     3
    Copyright   1994  University of Cambridge
18154
0c05abaf6244 add header
huffman
parents: 17956
diff changeset
     4
*)
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     5
18154
0c05abaf6244 add header
huffman
parents: 17956
diff changeset
     6
header {* Notions about functions *}
923
ff1574a81019 new version of HOL with curried function application
clasohm
parents:
diff changeset
     7
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
     8
theory Fun
32139
e271a64f03ff moved complete_lattice &c. into separate theory
haftmann
parents: 31949
diff changeset
     9
imports Complete_Lattice
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15111
diff changeset
    10
begin
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    11
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    12
text{*As a simplification rule, it replaces all function equalities by
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    13
  first-order equalities.*}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    14
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    15
apply (rule iffI)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    16
apply (simp (no_asm_simp))
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    17
apply (rule ext)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    18
apply (simp (no_asm_simp))
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    19
done
5305
513925de8962 cleanup for Fun.thy:
oheimb
parents: 4830
diff changeset
    20
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    21
lemma apply_inverse:
26357
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
    22
  "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    23
  by auto
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    24
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
    25
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    26
subsection {* The Identity Function @{text id} *}
6171
cd237a10cbf8 inj is now a translation of inj_on
paulson
parents: 5852
diff changeset
    27
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    28
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    29
  id :: "'a \<Rightarrow> 'a"
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    30
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    31
  "id = (\<lambda>x. x)"
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
    32
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    33
lemma id_apply [simp]: "id x = x"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    34
  by (simp add: id_def)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    35
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    36
lemma image_ident [simp]: "(%x. x) ` Y = Y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    37
by blast
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    38
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    39
lemma image_id [simp]: "id ` Y = Y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    40
by (simp add: id_def)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    41
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    42
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    43
by blast
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    44
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    45
lemma vimage_id [simp]: "id -` A = A"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    46
by (simp add: id_def)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    47
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    48
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    49
subsection {* The Composition Operator @{text "f \<circ> g"} *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
    50
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    51
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    52
  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    53
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
    54
  "f o g = (\<lambda>x. f (g x))"
11123
15ffc08f905e removed whitespace
oheimb
parents: 10826
diff changeset
    55
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20044
diff changeset
    56
notation (xsymbols)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    57
  comp  (infixl "\<circ>" 55)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    58
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20044
diff changeset
    59
notation (HTML output)
19656
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    60
  comp  (infixl "\<circ>" 55)
09be06943252 tuned concrete syntax -- abbreviation/const_syntax;
wenzelm
parents: 19536
diff changeset
    61
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    62
text{*compatibility*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    63
lemmas o_def = comp_def
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
    64
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    65
lemma o_apply [simp]: "(f o g) x = f (g x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    66
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    67
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    68
lemma o_assoc: "f o (g o h) = f o g o h"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    69
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    70
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    71
lemma id_o [simp]: "id o g = g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    72
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    73
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    74
lemma o_id [simp]: "f o id = f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    75
by (simp add: comp_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    76
34150
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    77
lemma o_eq_dest:
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    78
  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    79
  by (simp only: o_def) (fact fun_cong)
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    80
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    81
lemma o_eq_elim:
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    82
  "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    83
  by (erule meta_mp) (fact o_eq_dest) 
22acb8b38639 moved lemmas o_eq_dest, o_eq_elim here
haftmann
parents: 34101
diff changeset
    84
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    85
lemma image_compose: "(f o g) ` r = f`(g`r)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    86
by (simp add: comp_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    87
33044
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32998
diff changeset
    88
lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32998
diff changeset
    89
  by auto
fd0a9c794ec1 Some new lemmas concerning sets
paulson
parents: 32998
diff changeset
    90
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    91
lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    92
by (unfold comp_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    93
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
    94
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26357
diff changeset
    95
subsection {* The Forward Composition Operator @{text fcomp} *}
26357
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
    96
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
    97
definition
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
    98
  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
    99
where
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   100
  "f o> g = (\<lambda>x. g (f x))"
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   101
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   102
lemma fcomp_apply:  "(f o> g) x = g (f x)"
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   103
  by (simp add: fcomp_def)
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   104
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   105
lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   106
  by (simp add: fcomp_def)
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   107
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   108
lemma id_fcomp [simp]: "id o> g = g"
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   109
  by (simp add: fcomp_def)
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   110
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   111
lemma fcomp_id [simp]: "f o> id = f"
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   112
  by (simp add: fcomp_def)
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   113
31202
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 31080
diff changeset
   114
code_const fcomp
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 31080
diff changeset
   115
  (Eval infixl 1 "#>")
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 31080
diff changeset
   116
26588
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26357
diff changeset
   117
no_notation fcomp (infixl "o>" 60)
d83271bfaba5 removed syntax from monad combinators; renamed mbind to scomp
haftmann
parents: 26357
diff changeset
   118
26357
19b153ebda0b added forward composition
haftmann
parents: 26342
diff changeset
   119
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   120
subsection {* Injectivity and Surjectivity *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   121
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   122
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   123
  inj_on :: "['a => 'b, 'a set] => bool" where
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   124
  -- "injective"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   125
  "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   126
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   127
text{*A common special case: functions injective over the entire domain type.*}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   128
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   129
abbreviation
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   130
  "inj f == inj_on f UNIV"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   131
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   132
definition
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   133
  bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 27330
diff changeset
   134
  [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   135
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   136
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   137
  surj :: "('a => 'b) => bool" where
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   138
  -- "surjective"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   139
  "surj f == ! y. ? x. y=f(x)"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   140
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   141
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   142
  bij :: "('a => 'b) => bool" where
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   143
  -- "bijective"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   144
  "bij f == inj f & surj f"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   145
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   146
lemma injI:
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   147
  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   148
  shows "inj f"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   149
  using assms unfolding inj_on_def by auto
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   150
31775
2b04504fcb69 uniformly capitialized names for subdirectories
haftmann
parents: 31604
diff changeset
   151
text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   152
lemma datatype_injI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   153
    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   154
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   155
13637
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   156
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
   157
  by (unfold inj_on_def, blast)
02aa63636ab8 - Added range_ex1_eq
berghofe
parents: 13585
diff changeset
   158
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   159
lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   160
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   161
32988
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   162
lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   163
by (force simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   164
32988
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   165
lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   166
by (simp add: inj_on_eq_iff)
d1d4d7a08a66 Inv -> inv_onto, inv abbr. inv_onto UNIV.
nipkow
parents: 32961
diff changeset
   167
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   168
lemma inj_on_id[simp]: "inj_on id A"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   169
  by (simp add: inj_on_def) 
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   170
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   171
lemma inj_on_id2[simp]: "inj_on (%x. x) A"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   172
by (simp add: inj_on_def) 
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   173
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   174
lemma surj_id[simp]: "surj id"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   175
by (simp add: surj_def) 
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   176
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   177
lemma bij_id[simp]: "bij id"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   178
by (simp add: bij_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   179
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   180
lemma inj_onI:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   181
    "(!! 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
   182
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   183
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   184
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
   185
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   186
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   187
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
   188
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   189
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   190
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
   191
by (blast dest!: inj_onD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   192
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   193
lemma comp_inj_on:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   194
     "[| 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
   195
by (simp add: comp_def inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   196
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   197
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   198
apply(simp add:inj_on_def image_def)
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   199
apply blast
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   200
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   201
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   202
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
   203
  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
   204
apply(unfold inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   205
apply blast
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   206
done
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15303
diff changeset
   207
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   208
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
   209
by (unfold inj_on_def, blast)
12258
5da24e7e9aba got rid of theory Inverse_Image;
wenzelm
parents: 12114
diff changeset
   210
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   211
lemma inj_singleton: "inj (%s. {s})"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   212
by (simp add: inj_on_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   213
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   214
lemma inj_on_empty[iff]: "inj_on f {}"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   215
by(simp add: inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   216
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   217
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
   218
by (unfold inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   219
15111
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   220
lemma inj_on_Un:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   221
 "inj_on f (A Un B) =
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   222
  (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
   223
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   224
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   225
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   226
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   227
lemma inj_on_insert[iff]:
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   228
  "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
   229
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   230
apply (blast intro:sym)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   231
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   232
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   233
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   234
apply(unfold inj_on_def)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   235
apply (blast)
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   236
done
c108189645f8 added some inj_on thms
nipkow
parents: 14565
diff changeset
   237
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   238
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   239
apply (simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   240
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   241
done
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 surj_range: "surj f ==> range f = UNIV"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   244
by (auto simp add: surj_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 surjD: "surj f ==> EX x. y = f x"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   247
by (simp add: surj_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 surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   250
by (simp add: surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   251
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   252
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   253
apply (simp add: comp_def surj_def, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   254
apply (drule_tac x = y in spec, clarify)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   255
apply (drule_tac x = x in spec, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   256
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   257
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   258
lemma bijI: "[| inj f; surj f |] ==> bij f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   259
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   260
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   261
lemma bij_is_inj: "bij f ==> inj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   262
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   263
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   264
lemma bij_is_surj: "bij f ==> surj f"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   265
by (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   266
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   267
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   268
by (simp add: bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   269
32337
7887cb2848bb new lemma bij_comp
nipkow
parents: 32139
diff changeset
   270
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
7887cb2848bb new lemma bij_comp
nipkow
parents: 32139
diff changeset
   271
by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
7887cb2848bb new lemma bij_comp
nipkow
parents: 32139
diff changeset
   272
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   273
lemma bij_betw_trans:
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   274
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   275
by(auto simp add:bij_betw_def comp_inj_on)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   276
26105
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   277
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   278
proof -
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   279
  have i: "inj_on f A" and s: "f ` A = B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   280
    using assms by(auto simp:bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   281
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   282
  { fix a b assume P: "?P b a"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   283
    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   284
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   285
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   286
  } note g = this
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   287
  have "inj_on ?g B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   288
  proof(rule inj_onI)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   289
    fix x y assume "x:B" "y:B" "?g x = ?g y"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   290
    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   291
    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   292
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   293
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   294
  moreover have "?g ` B = A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   295
  proof(auto simp:image_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   296
    fix b assume "b:B"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   297
    with s obtain a where P: "?P b a" unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   298
    thus "?g b \<in> A" using g[OF P] by auto
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   299
  next
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   300
    fix a assume "a:A"
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   301
    then obtain b where P: "?P b a" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   302
    then have "b:B" using s unfolding image_def by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   303
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   304
  qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   305
  ultimately show ?thesis by(auto simp:bij_betw_def)
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   306
qed
ae06618225ec moved bij_betw from Library/FuncSet to Fun, redistributed some lemmas, and
nipkow
parents: 25886
diff changeset
   307
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   308
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   309
by (simp add: surj_range)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   310
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   311
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   312
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   313
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   314
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   315
apply (unfold surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   316
apply (blast intro: sym)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   317
done
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 vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   320
by (unfold 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 vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   323
apply (unfold bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   324
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   325
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   326
31438
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   327
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   328
by(blast dest: inj_onD)
a1c4c1500abe A few finite lemmas
nipkow
parents: 31202
diff changeset
   329
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   330
lemma inj_on_image_Int:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   331
   "[| 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
   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
lemma inj_on_image_set_diff:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   336
   "[| 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
   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
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
   341
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   342
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   343
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   344
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   345
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   346
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   347
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   348
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   349
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   350
by (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   351
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   352
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   353
by (blast dest: injD)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   354
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   355
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   356
lemma image_INT:
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   357
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   358
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   359
apply (simp add: inj_on_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   360
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   361
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   362
(*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
   363
  it doesn't matter whether A is empty*)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   364
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
   365
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   366
apply (simp add: inj_on_def surj_def, blast)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   367
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   368
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   369
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   370
by (auto simp add: surj_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   371
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   372
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   373
by (auto simp add: inj_on_def)
5852
4d7320490be4 the function space operator
paulson
parents: 5608
diff changeset
   374
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   375
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   376
apply (simp add: bij_def)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   377
apply (rule equalityI)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   378
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
   379
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   380
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   381
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
35580
0f74806cab22 Rewrite rules for images of minus of intervals
hoelzl
parents: 35416
diff changeset
   382
  by (auto intro!: inj_onI)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   383
35584
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   384
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   385
  by (auto intro!: inj_onI dest: strict_mono_eq)
768f8d92b767 generalized inj_uminus; added strict_mono_imp_inj_on
hoelzl
parents: 35580
diff changeset
   386
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   387
subsection{*Function Updating*}
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   388
35416
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   389
definition
d8d7d1b785af replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents: 35115
diff changeset
   390
  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   391
  "fun_upd f a b == % x. if x=a then b else f x"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   392
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   393
nonterminals
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   394
  updbinds updbind
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   395
syntax
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   396
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   397
  ""         :: "updbind => updbinds"             ("_")
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   398
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   399
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   400
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   401
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   402
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   403
  "f(x:=y)" == "CONST fun_upd f x y"
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   404
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   405
(* Hint: to define the sum of two functions (or maps), use sum_case.
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   406
         A nice infix syntax could be defined (in Datatype.thy or below) by
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   407
notation
446c5063e4fd modernized translations;
wenzelm
parents: 34209
diff changeset
   408
  sum_case  (infixr "'(+')"80)
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   409
*)
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   410
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   411
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   412
apply (simp add: fun_upd_def, safe)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   413
apply (erule subst)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   414
apply (rule_tac [2] ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   415
done
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   416
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   417
(* f x = y ==> f(x:=y) = f *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   418
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   419
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   420
(* f(x := f x) = f *)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   421
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16973
diff changeset
   422
declare fun_upd_triv [iff]
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   423
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   424
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
   425
by (simp add: fun_upd_def)
13585
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   426
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   427
(* fun_upd_apply supersedes these two,   but they are useful
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   428
   if fun_upd_apply is intentionally removed from the simpset *)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   429
lemma fun_upd_same: "(f(x:=y)) x = y"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   430
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   431
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   432
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   433
by simp
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   434
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   435
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   436
by (simp add: expand_fun_eq)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   437
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   438
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
   439
by (rule ext, auto)
db4005b40cc6 Converted Fun to Isar style.
paulson
parents: 12460
diff changeset
   440
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   441
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   442
by (fastsimp simp:inj_on_def image_def)
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15140
diff changeset
   443
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   444
lemma fun_upd_image:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   445
     "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
   446
by auto
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   447
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   448
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   449
by (auto intro: ext)
31080
21ffc770ebc0 lemmas by Andreas Lochbihler
nipkow
parents: 30301
diff changeset
   450
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   451
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   452
subsection {* @{text override_on} *}
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   453
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   454
definition
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   455
  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   456
where
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   457
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   458
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   459
lemma override_on_emptyset[simp]: "override_on f g {} = f"
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   460
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   461
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   462
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
   463
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   464
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   465
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
   466
by(simp add:override_on_def)
13910
f9a9ef16466f Added thms
nipkow
parents: 13637
diff changeset
   467
26147
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   468
ae2bf929e33c moved some set lemmas to Set.thy
haftmann
parents: 26105
diff changeset
   469
subsection {* @{text swap} *}
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   470
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   471
definition
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   472
  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
   473
where
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   474
  "swap a b f = f (a := f b, b:= f a)"
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   475
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   476
lemma swap_self [simp]: "swap a a f = f"
15691
900cf45ff0a6 _(_|_) is now override_on
nipkow
parents: 15531
diff changeset
   477
by (simp add: swap_def)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   478
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   479
lemma swap_commute: "swap a b f = swap b a f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   480
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   481
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   482
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   483
by (rule ext, simp add: fun_upd_def swap_def)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   484
34145
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   485
lemma swap_triple:
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   486
  assumes "a \<noteq> c" and "b \<noteq> c"
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   487
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   488
  using assms by (simp add: expand_fun_eq swap_def)
402b7c74799d add lemma swap_triple
huffman
parents: 34101
diff changeset
   489
34101
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   490
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   491
by (rule ext, simp add: fun_upd_def swap_def)
d689f0b33047 declare swap_self [simp], add lemma comp_swap
huffman
parents: 33318
diff changeset
   492
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   493
lemma inj_on_imp_inj_on_swap:
22744
5cbe966d67a2 Isar definitions are now added explicitly to code theorem table
haftmann
parents: 22577
diff changeset
   494
  "[|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
   495
by (simp add: inj_on_def swap_def, blast)
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   496
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   497
lemma inj_on_swap_iff [simp]:
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   498
  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
   499
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   500
  assume "inj_on (swap a b f) A"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   501
  with A have "inj_on (swap a b (swap a b f)) A" 
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17084
diff changeset
   502
    by (iprover intro: inj_on_imp_inj_on_swap) 
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   503
  thus "inj_on f A" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   504
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   505
  assume "inj_on f A"
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   506
  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
   507
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   508
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   509
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   510
apply (simp add: surj_def swap_def, clarify)
27125
0733f575b51e tuned proofs -- case_tac *is* available here;
wenzelm
parents: 27106
diff changeset
   511
apply (case_tac "y = f b", blast)
0733f575b51e tuned proofs -- case_tac *is* available here;
wenzelm
parents: 27106
diff changeset
   512
apply (case_tac "y = f a", auto)
15510
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   513
done
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   514
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   515
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   516
proof 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   517
  assume "surj (swap a b f)"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   518
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   519
  thus "surj f" by simp 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   520
next
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   521
  assume "surj f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   522
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   523
qed
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   524
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   525
lemma bij_swap_iff: "bij (swap a b f) = bij f"
9de204d7b699 new foldSet proofs
paulson
parents: 15439
diff changeset
   526
by (simp add: bij_def)
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   527
27188
e47b069cab35 hide -> hide (open)
nipkow
parents: 27165
diff changeset
   528
hide (open) const swap
21547
9c9fdf4c2949 moved order arities for fun and bool to Fun/Orderings
haftmann
parents: 21327
diff changeset
   529
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   530
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   531
subsection {* Inversion of injective functions *}
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   532
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   533
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   534
"the_inv_into A f == %x. THE y. y : A & f y = x"
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   535
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   536
lemma the_inv_into_f_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   537
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   538
apply (simp add: the_inv_into_def inj_on_def)
34209
c7f621786035 killed a few warnings
krauss
parents: 34153
diff changeset
   539
apply blast
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   540
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   541
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   542
lemma f_the_inv_into_f:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   543
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   544
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   545
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   546
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   547
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   548
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   549
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   550
lemma the_inv_into_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   551
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   552
apply (simp add: the_inv_into_def)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   553
apply (rule the1I2)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   554
 apply(blast dest: inj_onD)
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   555
apply blast
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   556
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   557
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   558
lemma the_inv_into_onto[simp]:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   559
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   560
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   561
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   562
lemma the_inv_into_f_eq:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   563
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   564
  apply (erule subst)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   565
  apply (erule the_inv_into_f_f, assumption)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   566
  done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   567
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   568
lemma the_inv_into_comp:
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   569
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   570
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   571
apply (rule the_inv_into_f_eq)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   572
  apply (fast intro: comp_inj_on)
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   573
 apply (simp add: f_the_inv_into_f the_inv_into_into)
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   574
apply (simp add: the_inv_into_into)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   575
done
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   576
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   577
lemma inj_on_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   578
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   579
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   580
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   581
lemma bij_betw_the_inv_into:
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   582
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   583
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
32961
61431a41ddd5 added the_inv_onto
nipkow
parents: 32740
diff changeset
   584
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   585
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   586
  "the_inv f \<equiv> the_inv_into UNIV f"
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   587
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   588
lemma the_inv_f_f:
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   589
  assumes "inj f"
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   590
  shows "the_inv f (f x) = x" using assms UNIV_I
33057
764547b68538 inv_onto -> inv_into
nipkow
parents: 33044
diff changeset
   591
  by (rule the_inv_into_f_f)
32998
31b19fa0de0b Renamed inv to the_inv and turned it into an abbreviation (based on the_inv_onto).
berghofe
parents: 32988
diff changeset
   592
31949
3f933687fae9 moved Inductive.myinv to Fun.inv; tuned
haftmann
parents: 31775
diff changeset
   593
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   594
subsection {* Proof tool setup *} 
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   595
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   596
text {* simplifies terms of the form
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   597
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   598
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   599
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   600
let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   601
  fun gen_fun_upd NONE T _ _ = NONE
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   602
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   603
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   604
  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
   605
    let
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   606
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   607
            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
   608
        | find t = NONE
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   609
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   610
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   611
  fun proc ss ct =
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   612
    let
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   613
      val ctxt = Simplifier.the_context ss
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   614
      val t = Thm.term_of ct
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   615
    in
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   616
      case find_double t of
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   617
        (T, NONE) => NONE
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   618
      | (T, SOME rhs) =>
27330
1af2598b5f7d Logic.all/mk_equals/mk_implies;
wenzelm
parents: 27188
diff changeset
   619
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
24017
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   620
            (fn _ =>
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   621
              rtac eq_reflection 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   622
              rtac ext 1 THEN
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   623
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   624
    end
363287741ebe simproc_setup fun_upd2;
wenzelm
parents: 23878
diff changeset
   625
in proc end
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   626
*}
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   627
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22744
diff changeset
   628
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   629
subsection {* Code generator setup *}
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   630
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   631
types_code
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   632
  "fun"  ("(_ ->/ _)")
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   633
attach (term_of) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   634
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   635
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   636
attach (test) {*
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   637
fun gen_fun_type aF aT bG bT i =
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   638
  let
32740
9dd0a2f83429 explicit indication of Unsynchronized.ref;
wenzelm
parents: 32554
diff changeset
   639
    val tab = Unsynchronized.ref [];
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   640
    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   641
      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   642
  in
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   643
    (fn x =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   644
       case AList.lookup op = (!tab) x of
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   645
         NONE =>
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   646
           let val p as (y, _) = bG i
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   647
           in (tab := (x, p) :: !tab; y) end
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   648
       | SOME (y, _) => y,
28711
60e51a045755 Replaced arbitrary by undefined.
berghofe
parents: 28562
diff changeset
   649
     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
25886
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   650
  end;
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   651
*}
7753e0d81b7a Added test data generator for function type (from Pure/codegen.ML).
berghofe
parents: 24286
diff changeset
   652
21870
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   653
code_const "op \<circ>"
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   654
  (SML infixl 5 "o")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   655
  (Haskell infixr 9 ".")
c701cdacf69b infix syntax for generated code for composition
haftmann
parents: 21547
diff changeset
   656
21906
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   657
code_const "id"
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   658
  (Haskell "id")
db805c70a519 explizit serialization for Haskell id
haftmann
parents: 21870
diff changeset
   659
2912
3fac3e8d5d3e moved inj and surj from Set to Fun and Inv -> inv.
nipkow
parents: 1475
diff changeset
   660
end