src/HOLCF/Fixrec.thy
changeset 40774 0437dbc127b3
parent 40773 6c12f5e24e34
child 40775 ed7a4eadb2f6
     1.1 --- a/src/HOLCF/Fixrec.thy	Sat Nov 27 14:34:54 2010 -0800
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,252 +0,0 @@
     1.4 -(*  Title:      HOLCF/Fixrec.thy
     1.5 -    Author:     Amber Telfer and Brian Huffman
     1.6 -*)
     1.7 -
     1.8 -header "Package for defining recursive functions in HOLCF"
     1.9 -
    1.10 -theory Fixrec
    1.11 -imports Plain_HOLCF
    1.12 -uses
    1.13 -  ("Tools/holcf_library.ML")
    1.14 -  ("Tools/fixrec.ML")
    1.15 -begin
    1.16 -
    1.17 -subsection {* Pattern-match monad *}
    1.18 -
    1.19 -default_sort cpo
    1.20 -
    1.21 -pcpodef (open) 'a match = "UNIV::(one ++ 'a u) set"
    1.22 -by simp_all
    1.23 -
    1.24 -definition
    1.25 -  fail :: "'a match" where
    1.26 -  "fail = Abs_match (sinl\<cdot>ONE)"
    1.27 -
    1.28 -definition
    1.29 -  succeed :: "'a \<rightarrow> 'a match" where
    1.30 -  "succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))"
    1.31 -
    1.32 -lemma matchE [case_names bottom fail succeed, cases type: match]:
    1.33 -  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
    1.34 -unfolding fail_def succeed_def
    1.35 -apply (cases p, rename_tac r)
    1.36 -apply (rule_tac p=r in ssumE, simp add: Abs_match_strict)
    1.37 -apply (rule_tac p=x in oneE, simp, simp)
    1.38 -apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match)
    1.39 -done
    1.40 -
    1.41 -lemma succeed_defined [simp]: "succeed\<cdot>x \<noteq> \<bottom>"
    1.42 -by (simp add: succeed_def cont_Abs_match Abs_match_defined)
    1.43 -
    1.44 -lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
    1.45 -by (simp add: fail_def Abs_match_defined)
    1.46 -
    1.47 -lemma succeed_eq [simp]: "(succeed\<cdot>x = succeed\<cdot>y) = (x = y)"
    1.48 -by (simp add: succeed_def cont_Abs_match Abs_match_inject)
    1.49 -
    1.50 -lemma succeed_neq_fail [simp]:
    1.51 -  "succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x"
    1.52 -by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
    1.53 -
    1.54 -subsubsection {* Run operator *}
    1.55 -
    1.56 -definition
    1.57 -  run :: "'a match \<rightarrow> 'a::pcpo" where
    1.58 -  "run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))"
    1.59 -
    1.60 -text {* rewrite rules for run *}
    1.61 -
    1.62 -lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
    1.63 -unfolding run_def
    1.64 -by (simp add: cont_Rep_match Rep_match_strict)
    1.65 -
    1.66 -lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
    1.67 -unfolding run_def fail_def
    1.68 -by (simp add: cont_Rep_match Abs_match_inverse)
    1.69 -
    1.70 -lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x"
    1.71 -unfolding run_def succeed_def
    1.72 -by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
    1.73 -
    1.74 -subsubsection {* Monad plus operator *}
    1.75 -
    1.76 -definition
    1.77 -  mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where
    1.78 -  "mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))"
    1.79 -
    1.80 -abbreviation
    1.81 -  mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match"  (infixr "+++" 65)  where
    1.82 -  "m1 +++ m2 == mplus\<cdot>m1\<cdot>m2"
    1.83 -
    1.84 -text {* rewrite rules for mplus *}
    1.85 -
    1.86 -lemmas cont2cont_Rep_match = cont_Rep_match [THEN cont_compose]
    1.87 -
    1.88 -lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
    1.89 -unfolding mplus_def
    1.90 -by (simp add: cont2cont_Rep_match Rep_match_strict)
    1.91 -
    1.92 -lemma mplus_fail [simp]: "fail +++ m = m"
    1.93 -unfolding mplus_def fail_def
    1.94 -by (simp add: cont2cont_Rep_match Abs_match_inverse)
    1.95 -
    1.96 -lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x"
    1.97 -unfolding mplus_def succeed_def
    1.98 -by (simp add: cont2cont_Rep_match cont_Abs_match Abs_match_inverse)
    1.99 -
   1.100 -lemma mplus_fail2 [simp]: "m +++ fail = m"
   1.101 -by (cases m, simp_all)
   1.102 -
   1.103 -lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
   1.104 -by (cases x, simp_all)
   1.105 -
   1.106 -subsection {* Match functions for built-in types *}
   1.107 -
   1.108 -default_sort pcpo
   1.109 -
   1.110 -definition
   1.111 -  match_bottom :: "'a \<rightarrow> 'c match \<rightarrow> 'c match"
   1.112 -where
   1.113 -  "match_bottom = (\<Lambda> x k. seq\<cdot>x\<cdot>fail)"
   1.114 -
   1.115 -definition
   1.116 -  match_Pair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
   1.117 -where
   1.118 -  "match_Pair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)"
   1.119 -
   1.120 -definition
   1.121 -  match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
   1.122 -where
   1.123 -  "match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)"
   1.124 -
   1.125 -definition
   1.126 -  match_sinl :: "'a \<oplus> 'b \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
   1.127 -where
   1.128 -  "match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)"
   1.129 -
   1.130 -definition
   1.131 -  match_sinr :: "'a \<oplus> 'b \<rightarrow> ('b \<rightarrow> 'c match) \<rightarrow> 'c match"
   1.132 -where
   1.133 -  "match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)"
   1.134 -
   1.135 -definition
   1.136 -  match_up :: "'a::cpo u \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
   1.137 -where
   1.138 -  "match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)"
   1.139 -
   1.140 -definition
   1.141 -  match_ONE :: "one \<rightarrow> 'c match \<rightarrow> 'c match"
   1.142 -where
   1.143 -  "match_ONE = (\<Lambda> ONE k. k)"
   1.144 -
   1.145 -definition
   1.146 -  match_TT :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
   1.147 -where
   1.148 -  "match_TT = (\<Lambda> x k. If x then k else fail)"
   1.149 - 
   1.150 -definition
   1.151 -  match_FF :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
   1.152 -where
   1.153 -  "match_FF = (\<Lambda> x k. If x then fail else k)"
   1.154 -
   1.155 -lemma match_bottom_simps [simp]:
   1.156 -  "match_bottom\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.157 -  "x \<noteq> \<bottom> \<Longrightarrow> match_bottom\<cdot>x\<cdot>k = fail"
   1.158 -by (simp_all add: match_bottom_def)
   1.159 -
   1.160 -lemma match_Pair_simps [simp]:
   1.161 -  "match_Pair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y"
   1.162 -by (simp_all add: match_Pair_def)
   1.163 -
   1.164 -lemma match_spair_simps [simp]:
   1.165 -  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x, y:)\<cdot>k = k\<cdot>x\<cdot>y"
   1.166 -  "match_spair\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.167 -by (simp_all add: match_spair_def)
   1.168 -
   1.169 -lemma match_sinl_simps [simp]:
   1.170 -  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x)\<cdot>k = k\<cdot>x"
   1.171 -  "y \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>y)\<cdot>k = fail"
   1.172 -  "match_sinl\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.173 -by (simp_all add: match_sinl_def)
   1.174 -
   1.175 -lemma match_sinr_simps [simp]:
   1.176 -  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x)\<cdot>k = fail"
   1.177 -  "y \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>y)\<cdot>k = k\<cdot>y"
   1.178 -  "match_sinr\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.179 -by (simp_all add: match_sinr_def)
   1.180 -
   1.181 -lemma match_up_simps [simp]:
   1.182 -  "match_up\<cdot>(up\<cdot>x)\<cdot>k = k\<cdot>x"
   1.183 -  "match_up\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.184 -by (simp_all add: match_up_def)
   1.185 -
   1.186 -lemma match_ONE_simps [simp]:
   1.187 -  "match_ONE\<cdot>ONE\<cdot>k = k"
   1.188 -  "match_ONE\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.189 -by (simp_all add: match_ONE_def)
   1.190 -
   1.191 -lemma match_TT_simps [simp]:
   1.192 -  "match_TT\<cdot>TT\<cdot>k = k"
   1.193 -  "match_TT\<cdot>FF\<cdot>k = fail"
   1.194 -  "match_TT\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.195 -by (simp_all add: match_TT_def)
   1.196 -
   1.197 -lemma match_FF_simps [simp]:
   1.198 -  "match_FF\<cdot>FF\<cdot>k = k"
   1.199 -  "match_FF\<cdot>TT\<cdot>k = fail"
   1.200 -  "match_FF\<cdot>\<bottom>\<cdot>k = \<bottom>"
   1.201 -by (simp_all add: match_FF_def)
   1.202 -
   1.203 -subsection {* Mutual recursion *}
   1.204 -
   1.205 -text {*
   1.206 -  The following rules are used to prove unfolding theorems from
   1.207 -  fixed-point definitions of mutually recursive functions.
   1.208 -*}
   1.209 -
   1.210 -lemma Pair_equalI: "\<lbrakk>x \<equiv> fst p; y \<equiv> snd p\<rbrakk> \<Longrightarrow> (x, y) \<equiv> p"
   1.211 -by simp
   1.212 -
   1.213 -lemma Pair_eqD1: "(x, y) = (x', y') \<Longrightarrow> x = x'"
   1.214 -by simp
   1.215 -
   1.216 -lemma Pair_eqD2: "(x, y) = (x', y') \<Longrightarrow> y = y'"
   1.217 -by simp
   1.218 -
   1.219 -lemma def_cont_fix_eq:
   1.220 -  "\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F\<rbrakk> \<Longrightarrow> f = F f"
   1.221 -by (simp, subst fix_eq, simp)
   1.222 -
   1.223 -lemma def_cont_fix_ind:
   1.224 -  "\<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"
   1.225 -by (simp add: fix_ind)
   1.226 -
   1.227 -text {* lemma for proving rewrite rules *}
   1.228 -
   1.229 -lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
   1.230 -by simp
   1.231 -
   1.232 -
   1.233 -subsection {* Initializing the fixrec package *}
   1.234 -
   1.235 -use "Tools/holcf_library.ML"
   1.236 -use "Tools/fixrec.ML"
   1.237 -
   1.238 -setup {* Fixrec.setup *}
   1.239 -
   1.240 -setup {*
   1.241 -  Fixrec.add_matchers
   1.242 -    [ (@{const_name up}, @{const_name match_up}),
   1.243 -      (@{const_name sinl}, @{const_name match_sinl}),
   1.244 -      (@{const_name sinr}, @{const_name match_sinr}),
   1.245 -      (@{const_name spair}, @{const_name match_spair}),
   1.246 -      (@{const_name Pair}, @{const_name match_Pair}),
   1.247 -      (@{const_name ONE}, @{const_name match_ONE}),
   1.248 -      (@{const_name TT}, @{const_name match_TT}),
   1.249 -      (@{const_name FF}, @{const_name match_FF}),
   1.250 -      (@{const_name UU}, @{const_name match_bottom}) ]
   1.251 -*}
   1.252 -
   1.253 -hide_const (open) succeed fail run
   1.254 -
   1.255 -end