src/HOL/HOLCF/Fixrec.thy
 author huffman Mon Dec 06 08:59:58 2010 -0800 (2010-12-06) changeset 41029 f7d8cfa6e7fc parent 40834 a1249aeff5b6 child 41429 cf5f025bc3c7 permissions -rw-r--r--
pcpodef no longer generates _defined lemmas, use _bottom_iff lemmas instead
```     1 (*  Title:      HOLCF/Fixrec.thy
```
```     2     Author:     Amber Telfer and Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header "Package for defining recursive functions in HOLCF"
```
```     6
```
```     7 theory Fixrec
```
```     8 imports Plain_HOLCF
```
```     9 uses
```
```    10   ("Tools/holcf_library.ML")
```
```    11   ("Tools/fixrec.ML")
```
```    12 begin
```
```    13
```
```    14 subsection {* Pattern-match monad *}
```
```    15
```
```    16 default_sort cpo
```
```    17
```
```    18 pcpodef (open) 'a match = "UNIV::(one ++ 'a u) set"
```
```    19 by simp_all
```
```    20
```
```    21 definition
```
```    22   fail :: "'a match" where
```
```    23   "fail = Abs_match (sinl\<cdot>ONE)"
```
```    24
```
```    25 definition
```
```    26   succeed :: "'a \<rightarrow> 'a match" where
```
```    27   "succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))"
```
```    28
```
```    29 lemma matchE [case_names bottom fail succeed, cases type: match]:
```
```    30   "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```    31 unfolding fail_def succeed_def
```
```    32 apply (cases p, rename_tac r)
```
```    33 apply (rule_tac p=r in ssumE, simp add: Abs_match_strict)
```
```    34 apply (rule_tac p=x in oneE, simp, simp)
```
```    35 apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match)
```
```    36 done
```
```    37
```
```    38 lemma succeed_defined [simp]: "succeed\<cdot>x \<noteq> \<bottom>"
```
```    39 by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff)
```
```    40
```
```    41 lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
```
```    42 by (simp add: fail_def Abs_match_bottom_iff)
```
```    43
```
```    44 lemma succeed_eq [simp]: "(succeed\<cdot>x = succeed\<cdot>y) = (x = y)"
```
```    45 by (simp add: succeed_def cont_Abs_match Abs_match_inject)
```
```    46
```
```    47 lemma succeed_neq_fail [simp]:
```
```    48   "succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x"
```
```    49 by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
```
```    50
```
```    51 subsubsection {* Run operator *}
```
```    52
```
```    53 definition
```
```    54   run :: "'a match \<rightarrow> 'a::pcpo" where
```
```    55   "run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))"
```
```    56
```
```    57 text {* rewrite rules for run *}
```
```    58
```
```    59 lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
```
```    60 unfolding run_def
```
```    61 by (simp add: cont_Rep_match Rep_match_strict)
```
```    62
```
```    63 lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
```
```    64 unfolding run_def fail_def
```
```    65 by (simp add: cont_Rep_match Abs_match_inverse)
```
```    66
```
```    67 lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x"
```
```    68 unfolding run_def succeed_def
```
```    69 by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
```
```    70
```
```    71 subsubsection {* Monad plus operator *}
```
```    72
```
```    73 definition
```
```    74   mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where
```
```    75   "mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))"
```
```    76
```
```    77 abbreviation
```
```    78   mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match"  (infixr "+++" 65)  where
```
```    79   "m1 +++ m2 == mplus\<cdot>m1\<cdot>m2"
```
```    80
```
```    81 text {* rewrite rules for mplus *}
```
```    82
```
```    83 lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
```
```    84 unfolding mplus_def
```
```    85 by (simp add: cont_Rep_match Rep_match_strict)
```
```    86
```
```    87 lemma mplus_fail [simp]: "fail +++ m = m"
```
```    88 unfolding mplus_def fail_def
```
```    89 by (simp add: cont_Rep_match Abs_match_inverse)
```
```    90
```
```    91 lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x"
```
```    92 unfolding mplus_def succeed_def
```
```    93 by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
```
```    94
```
```    95 lemma mplus_fail2 [simp]: "m +++ fail = m"
```
```    96 by (cases m, simp_all)
```
```    97
```
```    98 lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
```
```    99 by (cases x, simp_all)
```
```   100
```
```   101 subsection {* Match functions for built-in types *}
```
```   102
```
```   103 default_sort pcpo
```
```   104
```
```   105 definition
```
```   106   match_bottom :: "'a \<rightarrow> 'c match \<rightarrow> 'c match"
```
```   107 where
```
```   108   "match_bottom = (\<Lambda> x k. seq\<cdot>x\<cdot>fail)"
```
```   109
```
```   110 definition
```
```   111   match_Pair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
```
```   112 where
```
```   113   "match_Pair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)"
```
```   114
```
```   115 definition
```
```   116   match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
```
```   117 where
```
```   118   "match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)"
```
```   119
```
```   120 definition
```
```   121   match_sinl :: "'a \<oplus> 'b \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
```
```   122 where
```
```   123   "match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)"
```
```   124
```
```   125 definition
```
```   126   match_sinr :: "'a \<oplus> 'b \<rightarrow> ('b \<rightarrow> 'c match) \<rightarrow> 'c match"
```
```   127 where
```
```   128   "match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)"
```
```   129
```
```   130 definition
```
```   131   match_up :: "'a::cpo u \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
```
```   132 where
```
```   133   "match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)"
```
```   134
```
```   135 definition
```
```   136   match_ONE :: "one \<rightarrow> 'c match \<rightarrow> 'c match"
```
```   137 where
```
```   138   "match_ONE = (\<Lambda> ONE k. k)"
```
```   139
```
```   140 definition
```
```   141   match_TT :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
```
```   142 where
```
```   143   "match_TT = (\<Lambda> x k. If x then k else fail)"
```
```   144
```
```   145 definition
```
```   146   match_FF :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
```
```   147 where
```
```   148   "match_FF = (\<Lambda> x k. If x then fail else k)"
```
```   149
```
```   150 lemma match_bottom_simps [simp]:
```
```   151   "match_bottom\<cdot>x\<cdot>k = (if x = \<bottom> then \<bottom> else fail)"
```
```   152 by (simp add: match_bottom_def)
```
```   153
```
```   154 lemma match_Pair_simps [simp]:
```
```   155   "match_Pair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y"
```
```   156 by (simp_all add: match_Pair_def)
```
```   157
```
```   158 lemma match_spair_simps [simp]:
```
```   159   "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x, y:)\<cdot>k = k\<cdot>x\<cdot>y"
```
```   160   "match_spair\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   161 by (simp_all add: match_spair_def)
```
```   162
```
```   163 lemma match_sinl_simps [simp]:
```
```   164   "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x)\<cdot>k = k\<cdot>x"
```
```   165   "y \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>y)\<cdot>k = fail"
```
```   166   "match_sinl\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   167 by (simp_all add: match_sinl_def)
```
```   168
```
```   169 lemma match_sinr_simps [simp]:
```
```   170   "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x)\<cdot>k = fail"
```
```   171   "y \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>y)\<cdot>k = k\<cdot>y"
```
```   172   "match_sinr\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   173 by (simp_all add: match_sinr_def)
```
```   174
```
```   175 lemma match_up_simps [simp]:
```
```   176   "match_up\<cdot>(up\<cdot>x)\<cdot>k = k\<cdot>x"
```
```   177   "match_up\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   178 by (simp_all add: match_up_def)
```
```   179
```
```   180 lemma match_ONE_simps [simp]:
```
```   181   "match_ONE\<cdot>ONE\<cdot>k = k"
```
```   182   "match_ONE\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   183 by (simp_all add: match_ONE_def)
```
```   184
```
```   185 lemma match_TT_simps [simp]:
```
```   186   "match_TT\<cdot>TT\<cdot>k = k"
```
```   187   "match_TT\<cdot>FF\<cdot>k = fail"
```
```   188   "match_TT\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   189 by (simp_all add: match_TT_def)
```
```   190
```
```   191 lemma match_FF_simps [simp]:
```
```   192   "match_FF\<cdot>FF\<cdot>k = k"
```
```   193   "match_FF\<cdot>TT\<cdot>k = fail"
```
```   194   "match_FF\<cdot>\<bottom>\<cdot>k = \<bottom>"
```
```   195 by (simp_all add: match_FF_def)
```
```   196
```
```   197 subsection {* Mutual recursion *}
```
```   198
```
```   199 text {*
```
```   200   The following rules are used to prove unfolding theorems from
```
```   201   fixed-point definitions of mutually recursive functions.
```
```   202 *}
```
```   203
```
```   204 lemma Pair_equalI: "\<lbrakk>x \<equiv> fst p; y \<equiv> snd p\<rbrakk> \<Longrightarrow> (x, y) \<equiv> p"
```
```   205 by simp
```
```   206
```
```   207 lemma Pair_eqD1: "(x, y) = (x', y') \<Longrightarrow> x = x'"
```
```   208 by simp
```
```   209
```
```   210 lemma Pair_eqD2: "(x, y) = (x', y') \<Longrightarrow> y = y'"
```
```   211 by simp
```
```   212
```
```   213 lemma def_cont_fix_eq:
```
```   214   "\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F\<rbrakk> \<Longrightarrow> f = F f"
```
```   215 by (simp, subst fix_eq, simp)
```
```   216
```
```   217 lemma def_cont_fix_ind:
```
```   218   "\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F x)\<rbrakk> \<Longrightarrow> P f"
```
```   219 by (simp add: fix_ind)
```
```   220
```
```   221 text {* lemma for proving rewrite rules *}
```
```   222
```
```   223 lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
```
```   224 by simp
```
```   225
```
```   226
```
```   227 subsection {* Initializing the fixrec package *}
```
```   228
```
```   229 use "Tools/holcf_library.ML"
```
```   230 use "Tools/fixrec.ML"
```
```   231
```
```   232 setup {* Fixrec.setup *}
```
```   233
```
```   234 setup {*
```
```   235   Fixrec.add_matchers
```
```   236     [ (@{const_name up}, @{const_name match_up}),
```
```   237       (@{const_name sinl}, @{const_name match_sinl}),
```
```   238       (@{const_name sinr}, @{const_name match_sinr}),
```
```   239       (@{const_name spair}, @{const_name match_spair}),
```
```   240       (@{const_name Pair}, @{const_name match_Pair}),
```
```   241       (@{const_name ONE}, @{const_name match_ONE}),
```
```   242       (@{const_name TT}, @{const_name match_TT}),
```
```   243       (@{const_name FF}, @{const_name match_FF}),
```
```   244       (@{const_name UU}, @{const_name match_bottom}) ]
```
```   245 *}
```
```   246
```
```   247 hide_const (open) succeed fail run
```
```   248
```
```   249 end
```