src/HOLCF/Fixrec.thy
author huffman
Fri, 08 Jul 2005 02:41:35 +0200
changeset 16754 1b979f8b7e8e
parent 16551 7abf8a713613
child 16758 c32334d98fcd
permissions -rw-r--r--
renamed upE1 to upE
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     1
(*  Title:      HOLCF/Fixrec.thy
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     2
    ID:         $Id$
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     3
    Author:     Amber Telfer and Brian Huffman
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     4
*)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     5
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     6
header "Package for defining recursive functions in HOLCF"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     7
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
     8
theory Fixrec
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
     9
imports Sprod Ssum Up One Tr Fix
16417
9bc16273c2d4 migrated theory headers to new format
haftmann
parents: 16401
diff changeset
    10
uses ("fixrec_package.ML")
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    11
begin
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    12
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    13
subsection {* Maybe monad type *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    14
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    15
types 'a maybe = "one ++ 'a u"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    16
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    17
constdefs
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    18
  fail :: "'a maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    19
  "fail \<equiv> sinl\<cdot>ONE"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    20
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    21
  return :: "'a \<rightarrow> 'a maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    22
  "return \<equiv> sinr oo up"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    23
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    24
lemma maybeE:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    25
  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = return\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    26
apply (unfold fail_def return_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    27
apply (rule_tac p=p in ssumE, simp)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    28
apply (rule_tac p=x in oneE, simp, simp)
16754
1b979f8b7e8e renamed upE1 to upE
huffman
parents: 16551
diff changeset
    29
apply (rule_tac p=y in upE, simp, simp)
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    30
done
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    31
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    32
subsection {* Monadic bind operator *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    33
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    34
constdefs
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    35
  bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    36
  "bind \<equiv> \<Lambda> m f. sscase\<cdot>sinl\<cdot>(fup\<cdot>f)\<cdot>m"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    37
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    38
syntax
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    39
  "_bind" :: "'a maybe \<Rightarrow> ('a \<rightarrow> 'b maybe) \<Rightarrow> 'b maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    40
    ("(_ >>= _)" [50, 51] 50)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    41
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    42
translations "m >>= k" == "bind\<cdot>m\<cdot>k"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    43
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    44
nonterminals
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    45
  maybebind maybebinds
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    46
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    47
syntax 
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    48
  "_MBIND"  :: "pttrn \<Rightarrow> 'a maybe \<Rightarrow> maybebind"         ("(2_ <-/ _)" 10)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    49
  ""        :: "maybebind \<Rightarrow> maybebinds"                ("_")
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    50
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    51
  "_MBINDS" :: "[maybebind, maybebinds] \<Rightarrow> maybebinds"  ("_;/ _")
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    52
  "_MDO"    :: "[maybebinds, 'a maybe] \<Rightarrow> 'a maybe"     ("(do _;/ (_))" 10)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    53
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    54
translations
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    55
  "_MDO (_MBINDS b bs) e" == "_MDO b (_MDO bs e)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    56
  "do (x,y) <- m; e" == "m >>= (LAM <x,y>. e)" 
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    57
  "do x <- m; e"            == "m >>= (LAM x. e)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    58
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    59
text {* monad laws *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    60
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    61
lemma bind_strict [simp]: "UU >>= f = UU"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    62
by (simp add: bind_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    63
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    64
lemma bind_fail [simp]: "fail >>= f = fail"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    65
by (simp add: bind_def fail_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    66
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    67
lemma left_unit [simp]: "(return\<cdot>a) >>= k = k\<cdot>a"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    68
by (simp add: bind_def return_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    69
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    70
lemma right_unit [simp]: "m >>= return = m"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    71
by (rule_tac p=m in maybeE, simp_all)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    72
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    73
lemma bind_assoc [simp]:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    74
 "(do a <- m; b <- k\<cdot>a; h\<cdot>b) = (do b <- (do a <- m; k\<cdot>a); h\<cdot>b)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    75
by (rule_tac p=m in maybeE, simp_all)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    76
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    77
subsection {* Run operator *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    78
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    79
constdefs
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    80
  run:: "'a maybe \<rightarrow> 'a"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    81
  "run \<equiv> sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    82
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    83
text {* rewrite rules for run *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    84
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    85
lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    86
by (simp add: run_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    87
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    88
lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    89
by (simp add: run_def fail_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    90
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    91
lemma run_return [simp]: "run\<cdot>(return\<cdot>x) = x"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    92
by (simp add: run_def return_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    93
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    94
subsection {* Monad plus operator *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    95
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    96
constdefs
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    97
  mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    98
  "mplus \<equiv> \<Lambda> m1 m2. sscase\<cdot>(\<Lambda> x. m2)\<cdot>(fup\<cdot>return)\<cdot>m1"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
    99
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   100
syntax "+++" :: "'a maybe \<Rightarrow> 'a maybe \<Rightarrow> 'a maybe" (infixr 65)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   101
translations "x +++ y" == "mplus\<cdot>x\<cdot>y"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   102
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   103
text {* rewrite rules for mplus *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   104
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   105
lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   106
by (simp add: mplus_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   107
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   108
lemma mplus_fail [simp]: "fail +++ m = m"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   109
by (simp add: mplus_def fail_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   110
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   111
lemma mplus_return [simp]: "return\<cdot>x +++ m = return\<cdot>x"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   112
by (simp add: mplus_def return_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   113
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   114
lemma mplus_fail2 [simp]: "m +++ fail = m"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   115
by (rule_tac p=m in maybeE, simp_all)
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   116
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   117
lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   118
by (rule_tac p=x in maybeE, simp_all)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   119
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   120
subsection {* Match functions for built-in types *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   121
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   122
constdefs
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   123
  match_cpair :: "'a \<times> 'b \<rightarrow> ('a \<times> 'b) maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   124
  "match_cpair \<equiv> csplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   125
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   126
  match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<times> 'b) maybe"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   127
  "match_spair \<equiv> ssplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   128
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   129
  match_sinl :: "'a \<oplus> 'b \<rightarrow> 'a maybe"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   130
  "match_sinl \<equiv> sscase\<cdot>return\<cdot>(\<Lambda> y. fail)"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   131
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   132
  match_sinr :: "'a \<oplus> 'b \<rightarrow> 'b maybe"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   133
  "match_sinr \<equiv> sscase\<cdot>(\<Lambda> x. fail)\<cdot>return"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   134
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   135
  match_up :: "'a u \<rightarrow> 'a maybe"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   136
  "match_up \<equiv> fup\<cdot>return"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   137
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   138
  match_ONE :: "one \<rightarrow> unit maybe"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   139
  "match_ONE \<equiv> flift1 (\<lambda>u. return\<cdot>())"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   140
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   141
  match_TT :: "tr \<rightarrow> unit maybe"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   142
  "match_TT \<equiv> flift1 (\<lambda>b. if b then return\<cdot>() else fail)"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   143
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   144
  match_FF :: "tr \<rightarrow> unit maybe"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   145
  "match_FF \<equiv> flift1 (\<lambda>b. if b then fail else return\<cdot>())"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   146
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   147
lemma match_cpair_simps [simp]:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   148
  "match_cpair\<cdot><x,y> = return\<cdot><x,y>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   149
by (simp add: match_cpair_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   150
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   151
lemma match_spair_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   152
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x,y:) = return\<cdot><x,y>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   153
  "match_spair\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   154
by (simp_all add: match_spair_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   155
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   156
lemma match_sinl_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   157
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x) = return\<cdot>x"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   158
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>x) = fail"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   159
  "match_sinl\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   160
by (simp_all add: match_sinl_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   161
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   162
lemma match_sinr_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   163
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>x) = return\<cdot>x"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   164
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x) = fail"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   165
  "match_sinr\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   166
by (simp_all add: match_sinr_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   167
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   168
lemma match_up_simps [simp]:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   169
  "match_up\<cdot>(up\<cdot>x) = return\<cdot>x"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   170
  "match_up\<cdot>\<bottom> = \<bottom>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   171
by (simp_all add: match_up_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   172
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   173
lemma match_ONE_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   174
  "match_ONE\<cdot>ONE = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   175
  "match_ONE\<cdot>\<bottom> = \<bottom>"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   176
by (simp_all add: ONE_def match_ONE_def)
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   177
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   178
lemma match_TT_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   179
  "match_TT\<cdot>TT = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   180
  "match_TT\<cdot>FF = fail"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   181
  "match_TT\<cdot>\<bottom> = \<bottom>"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   182
by (simp_all add: TT_def FF_def match_TT_def)
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   183
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   184
lemma match_FF_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   185
  "match_FF\<cdot>FF = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   186
  "match_FF\<cdot>TT = fail"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   187
  "match_FF\<cdot>\<bottom> = \<bottom>"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   188
by (simp_all add: TT_def FF_def match_FF_def)
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   189
16401
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   190
subsection {* Mutual recursion *}
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   191
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   192
text {*
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   193
  The following rules are used to prove unfolding theorems from
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   194
  fixed-point definitions of mutually recursive functions.
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   195
*}
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   196
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   197
lemma cpair_equalI: "\<lbrakk>x \<equiv> cfst\<cdot>p; y \<equiv> csnd\<cdot>p\<rbrakk> \<Longrightarrow> <x,y> \<equiv> p"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   198
by (simp add: surjective_pairing_Cprod2)
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   199
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   200
lemma cpair_eqD1: "<x,y> = <x',y'> \<Longrightarrow> x = x'"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   201
by simp
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   202
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   203
lemma cpair_eqD2: "<x,y> = <x',y'> \<Longrightarrow> y = y'"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   204
by simp
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   205
16463
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   206
text {* lemma for proving rewrite rules *}
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   207
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   208
lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   209
by simp
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   210
16401
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   211
ML {*
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   212
val cpair_equalI = thm "cpair_equalI";
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   213
val cpair_eqD1 = thm "cpair_eqD1";
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   214
val cpair_eqD2 = thm "cpair_eqD2";
16463
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   215
val ssubst_lhs = thm "ssubst_lhs";
16401
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   216
*}
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   217
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   218
subsection {* Intitializing the fixrec package *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   219
16229
77cae9c8e73e use fixrec_package.ML
huffman
parents: 16221
diff changeset
   220
use "fixrec_package.ML"
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   222
end