1(* Title: HOL/HOLCF/Fixrec.thy 2 Author: Amber Telfer and Brian Huffman 3*) 4 5section "Package for defining recursive functions in HOLCF" 6 7theory Fixrec 8imports Cprod Sprod Ssum Up One Tr Fix 9keywords "fixrec" :: thy_defn 10begin 11 12subsection \<open>Pattern-match monad\<close> 13 14default_sort cpo 15 16pcpodef 'a match = "UNIV::(one ++ 'a u) set" 17by simp_all 18 19definition 20 fail :: "'a match" where 21 "fail = Abs_match (sinl\<cdot>ONE)" 22 23definition 24 succeed :: "'a \<rightarrow> 'a match" where 25 "succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))" 26 27lemma matchE [case_names bottom fail succeed, cases type: match]: 28 "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" 29unfolding fail_def succeed_def 30apply (cases p, rename_tac r) 31apply (rule_tac p=r in ssumE, simp add: Abs_match_strict) 32apply (rule_tac p=x in oneE, simp, simp) 33apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match) 34done 35 36lemma succeed_defined [simp]: "succeed\<cdot>x \<noteq> \<bottom>" 37by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff) 38 39lemma fail_defined [simp]: "fail \<noteq> \<bottom>" 40by (simp add: fail_def Abs_match_bottom_iff) 41 42lemma succeed_eq [simp]: "(succeed\<cdot>x = succeed\<cdot>y) = (x = y)" 43by (simp add: succeed_def cont_Abs_match Abs_match_inject) 44 45lemma succeed_neq_fail [simp]: 46 "succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x" 47by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject) 48 49subsubsection \<open>Run operator\<close> 50 51definition 52 run :: "'a match \<rightarrow> 'a::pcpo" where 53 "run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))" 54 55text \<open>rewrite rules for run\<close> 56 57lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>" 58unfolding run_def 59by (simp add: cont_Rep_match Rep_match_strict) 60 61lemma run_fail [simp]: "run\<cdot>fail = \<bottom>" 62unfolding run_def fail_def 63by (simp add: cont_Rep_match Abs_match_inverse) 64 65lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x" 66unfolding run_def succeed_def 67by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse) 68 69subsubsection \<open>Monad plus operator\<close> 70 71definition 72 mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where 73 "mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))" 74 75abbreviation 76 mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match" (infixr "+++" 65) where 77 "m1 +++ m2 == mplus\<cdot>m1\<cdot>m2" 78 79text \<open>rewrite rules for mplus\<close> 80 81lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>" 82unfolding mplus_def 83by (simp add: cont_Rep_match Rep_match_strict) 84 85lemma mplus_fail [simp]: "fail +++ m = m" 86unfolding mplus_def fail_def 87by (simp add: cont_Rep_match Abs_match_inverse) 88 89lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x" 90unfolding mplus_def succeed_def 91by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse) 92 93lemma mplus_fail2 [simp]: "m +++ fail = m" 94by (cases m, simp_all) 95 96lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)" 97by (cases x, simp_all) 98 99subsection \<open>Match functions for built-in types\<close> 100 101default_sort pcpo 102 103definition 104 match_bottom :: "'a \<rightarrow> 'c match \<rightarrow> 'c match" 105where 106 "match_bottom = (\<Lambda> x k. seq\<cdot>x\<cdot>fail)" 107 108definition 109 match_Pair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match" 110where 111 "match_Pair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)" 112 113definition 114 match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match" 115where 116 "match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)" 117 118definition 119 match_sinl :: "'a \<oplus> 'b \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match" 120where 121 "match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)" 122 123definition 124 match_sinr :: "'a \<oplus> 'b \<rightarrow> ('b \<rightarrow> 'c match) \<rightarrow> 'c match" 125where 126 "match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)" 127 128definition 129 match_up :: "'a::cpo u \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match" 130where 131 "match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)" 132 133definition 134 match_ONE :: "one \<rightarrow> 'c match \<rightarrow> 'c match" 135where 136 "match_ONE = (\<Lambda> ONE k. k)" 137 138definition 139 match_TT :: "tr \<rightarrow> 'c match \<rightarrow> 'c match" 140where 141 "match_TT = (\<Lambda> x k. If x then k else fail)" 142 143definition 144 match_FF :: "tr \<rightarrow> 'c match \<rightarrow> 'c match" 145where 146 "match_FF = (\<Lambda> x k. If x then fail else k)" 147 148lemma match_bottom_simps [simp]: 149 "match_bottom\<cdot>x\<cdot>k = (if x = \<bottom> then \<bottom> else fail)" 150by (simp add: match_bottom_def) 151 152lemma match_Pair_simps [simp]: 153 "match_Pair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y" 154by (simp_all add: match_Pair_def) 155 156lemma match_spair_simps [simp]: 157 "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x, y:)\<cdot>k = k\<cdot>x\<cdot>y" 158 "match_spair\<cdot>\<bottom>\<cdot>k = \<bottom>" 159by (simp_all add: match_spair_def) 160 161lemma match_sinl_simps [simp]: 162 "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x)\<cdot>k = k\<cdot>x" 163 "y \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>y)\<cdot>k = fail" 164 "match_sinl\<cdot>\<bottom>\<cdot>k = \<bottom>" 165by (simp_all add: match_sinl_def) 166 167lemma match_sinr_simps [simp]: 168 "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x)\<cdot>k = fail" 169 "y \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>y)\<cdot>k = k\<cdot>y" 170 "match_sinr\<cdot>\<bottom>\<cdot>k = \<bottom>" 171by (simp_all add: match_sinr_def) 172 173lemma match_up_simps [simp]: 174 "match_up\<cdot>(up\<cdot>x)\<cdot>k = k\<cdot>x" 175 "match_up\<cdot>\<bottom>\<cdot>k = \<bottom>" 176by (simp_all add: match_up_def) 177 178lemma match_ONE_simps [simp]: 179 "match_ONE\<cdot>ONE\<cdot>k = k" 180 "match_ONE\<cdot>\<bottom>\<cdot>k = \<bottom>" 181by (simp_all add: match_ONE_def) 182 183lemma match_TT_simps [simp]: 184 "match_TT\<cdot>TT\<cdot>k = k" 185 "match_TT\<cdot>FF\<cdot>k = fail" 186 "match_TT\<cdot>\<bottom>\<cdot>k = \<bottom>" 187by (simp_all add: match_TT_def) 188 189lemma match_FF_simps [simp]: 190 "match_FF\<cdot>FF\<cdot>k = k" 191 "match_FF\<cdot>TT\<cdot>k = fail" 192 "match_FF\<cdot>\<bottom>\<cdot>k = \<bottom>" 193by (simp_all add: match_FF_def) 194 195subsection \<open>Mutual recursion\<close> 196 197text \<open> 198 The following rules are used to prove unfolding theorems from 199 fixed-point definitions of mutually recursive functions. 200\<close> 201 202lemma Pair_equalI: "\<lbrakk>x \<equiv> fst p; y \<equiv> snd p\<rbrakk> \<Longrightarrow> (x, y) \<equiv> p" 203by simp 204 205lemma Pair_eqD1: "(x, y) = (x', y') \<Longrightarrow> x = x'" 206by simp 207 208lemma Pair_eqD2: "(x, y) = (x', y') \<Longrightarrow> y = y'" 209by simp 210 211lemma def_cont_fix_eq: 212 "\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F\<rbrakk> \<Longrightarrow> f = F f" 213by (simp, subst fix_eq, simp) 214 215lemma def_cont_fix_ind: 216 "\<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" 217by (simp add: fix_ind) 218 219text \<open>lemma for proving rewrite rules\<close> 220 221lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q" 222by simp 223 224 225subsection \<open>Initializing the fixrec package\<close> 226 227ML_file \<open>Tools/holcf_library.ML\<close> 228ML_file \<open>Tools/fixrec.ML\<close> 229 230method_setup fixrec_simp = \<open> 231 Scan.succeed (SIMPLE_METHOD' o Fixrec.fixrec_simp_tac) 232\<close> "pattern prover for fixrec constants" 233 234setup \<open> 235 Fixrec.add_matchers 236 [ (\<^const_name>\<open>up\<close>, \<^const_name>\<open>match_up\<close>), 237 (\<^const_name>\<open>sinl\<close>, \<^const_name>\<open>match_sinl\<close>), 238 (\<^const_name>\<open>sinr\<close>, \<^const_name>\<open>match_sinr\<close>), 239 (\<^const_name>\<open>spair\<close>, \<^const_name>\<open>match_spair\<close>), 240 (\<^const_name>\<open>Pair\<close>, \<^const_name>\<open>match_Pair\<close>), 241 (\<^const_name>\<open>ONE\<close>, \<^const_name>\<open>match_ONE\<close>), 242 (\<^const_name>\<open>TT\<close>, \<^const_name>\<open>match_TT\<close>), 243 (\<^const_name>\<open>FF\<close>, \<^const_name>\<open>match_FF\<close>), 244 (\<^const_name>\<open>bottom\<close>, \<^const_name>\<open>match_bottom\<close>) ] 245\<close> 246 247hide_const (open) succeed fail run 248 249end 250