1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Code_Numeral.thy Tue May 19 16:54:55 2009 +0200
1.3 @@ -0,0 +1,325 @@
1.4 +(* Author: Florian Haftmann, TU Muenchen *)
1.5 +
1.6 +header {* Type of target language numerals *}
1.7 +
1.8 +theory Code_Numeral
1.9 +imports Nat_Numeral
1.10 +begin
1.11 +
1.12 +text {*
1.13 + Code numerals are isomorphic to HOL @{typ nat} but
1.14 + mapped to target-language builtin numerals.
1.15 +*}
1.16 +
1.17 +subsection {* Datatype of target language numerals *}
1.18 +
1.19 +typedef (open) code_numeral = "UNIV \<Colon> nat set"
1.20 + morphisms nat_of of_nat by rule
1.21 +
1.22 +lemma of_nat_nat_of [simp]:
1.23 + "of_nat (nat_of k) = k"
1.24 + by (rule nat_of_inverse)
1.25 +
1.26 +lemma nat_of_of_nat [simp]:
1.27 + "nat_of (of_nat n) = n"
1.28 + by (rule of_nat_inverse) (rule UNIV_I)
1.29 +
1.30 +lemma [measure_function]:
1.31 + "is_measure nat_of" by (rule is_measure_trivial)
1.32 +
1.33 +lemma code_numeral:
1.34 + "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"
1.35 +proof
1.36 + fix n :: nat
1.37 + assume "\<And>n\<Colon>code_numeral. PROP P n"
1.38 + then show "PROP P (of_nat n)" .
1.39 +next
1.40 + fix n :: code_numeral
1.41 + assume "\<And>n\<Colon>nat. PROP P (of_nat n)"
1.42 + then have "PROP P (of_nat (nat_of n))" .
1.43 + then show "PROP P n" by simp
1.44 +qed
1.45 +
1.46 +lemma code_numeral_case:
1.47 + assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
1.48 + shows P
1.49 + by (rule assms [of "nat_of k"]) simp
1.50 +
1.51 +lemma code_numeral_induct_raw:
1.52 + assumes "\<And>n. P (of_nat n)"
1.53 + shows "P k"
1.54 +proof -
1.55 + from assms have "P (of_nat (nat_of k))" .
1.56 + then show ?thesis by simp
1.57 +qed
1.58 +
1.59 +lemma nat_of_inject [simp]:
1.60 + "nat_of k = nat_of l \<longleftrightarrow> k = l"
1.61 + by (rule nat_of_inject)
1.62 +
1.63 +lemma of_nat_inject [simp]:
1.64 + "of_nat n = of_nat m \<longleftrightarrow> n = m"
1.65 + by (rule of_nat_inject) (rule UNIV_I)+
1.66 +
1.67 +instantiation code_numeral :: zero
1.68 +begin
1.69 +
1.70 +definition [simp, code del]:
1.71 + "0 = of_nat 0"
1.72 +
1.73 +instance ..
1.74 +
1.75 +end
1.76 +
1.77 +definition [simp]:
1.78 + "Suc_code_numeral k = of_nat (Suc (nat_of k))"
1.79 +
1.80 +rep_datatype "0 \<Colon> code_numeral" Suc_code_numeral
1.81 +proof -
1.82 + fix P :: "code_numeral \<Rightarrow> bool"
1.83 + fix k :: code_numeral
1.84 + assume "P 0" then have init: "P (of_nat 0)" by simp
1.85 + assume "\<And>k. P k \<Longrightarrow> P (Suc_code_numeral k)"
1.86 + then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc_code_numeral (of_nat n))" .
1.87 + then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Suc n))" by simp
1.88 + from init step have "P (of_nat (nat_of k))"
1.89 + by (induct "nat_of k") simp_all
1.90 + then show "P k" by simp
1.91 +qed simp_all
1.92 +
1.93 +declare code_numeral_case [case_names nat, cases type: code_numeral]
1.94 +declare code_numeral.induct [case_names nat, induct type: code_numeral]
1.95 +
1.96 +lemma code_numeral_decr [termination_simp]:
1.97 + "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Suc 0 < nat_of k"
1.98 + by (cases k) simp
1.99 +
1.100 +lemma [simp, code]:
1.101 + "code_numeral_size = nat_of"
1.102 +proof (rule ext)
1.103 + fix k
1.104 + have "code_numeral_size k = nat_size (nat_of k)"
1.105 + by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
1.106 + also have "nat_size (nat_of k) = nat_of k" by (induct "nat_of k") simp_all
1.107 + finally show "code_numeral_size k = nat_of k" .
1.108 +qed
1.109 +
1.110 +lemma [simp, code]:
1.111 + "size = nat_of"
1.112 +proof (rule ext)
1.113 + fix k
1.114 + show "size k = nat_of k"
1.115 + by (induct k) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
1.116 +qed
1.117 +
1.118 +lemmas [code del] = code_numeral.recs code_numeral.cases
1.119 +
1.120 +lemma [code]:
1.121 + "eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of k) (nat_of l)"
1.122 + by (cases k, cases l) (simp add: eq)
1.123 +
1.124 +lemma [code nbe]:
1.125 + "eq_class.eq (k::code_numeral) k \<longleftrightarrow> True"
1.126 + by (rule HOL.eq_refl)
1.127 +
1.128 +
1.129 +subsection {* Indices as datatype of ints *}
1.130 +
1.131 +instantiation code_numeral :: number
1.132 +begin
1.133 +
1.134 +definition
1.135 + "number_of = of_nat o nat"
1.136 +
1.137 +instance ..
1.138 +
1.139 +end
1.140 +
1.141 +lemma nat_of_number [simp]:
1.142 + "nat_of (number_of k) = number_of k"
1.143 + by (simp add: number_of_code_numeral_def nat_number_of_def number_of_is_id)
1.144 +
1.145 +code_datatype "number_of \<Colon> int \<Rightarrow> code_numeral"
1.146 +
1.147 +
1.148 +subsection {* Basic arithmetic *}
1.149 +
1.150 +instantiation code_numeral :: "{minus, ordered_semidom, semiring_div, linorder}"
1.151 +begin
1.152 +
1.153 +definition [simp, code del]:
1.154 + "(1\<Colon>code_numeral) = of_nat 1"
1.155 +
1.156 +definition [simp, code del]:
1.157 + "n + m = of_nat (nat_of n + nat_of m)"
1.158 +
1.159 +definition [simp, code del]:
1.160 + "n - m = of_nat (nat_of n - nat_of m)"
1.161 +
1.162 +definition [simp, code del]:
1.163 + "n * m = of_nat (nat_of n * nat_of m)"
1.164 +
1.165 +definition [simp, code del]:
1.166 + "n div m = of_nat (nat_of n div nat_of m)"
1.167 +
1.168 +definition [simp, code del]:
1.169 + "n mod m = of_nat (nat_of n mod nat_of m)"
1.170 +
1.171 +definition [simp, code del]:
1.172 + "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
1.173 +
1.174 +definition [simp, code del]:
1.175 + "n < m \<longleftrightarrow> nat_of n < nat_of m"
1.176 +
1.177 +instance proof
1.178 +qed (auto simp add: code_numeral left_distrib div_mult_self1)
1.179 +
1.180 +end
1.181 +
1.182 +lemma zero_code_numeral_code [code inline, code]:
1.183 + "(0\<Colon>code_numeral) = Numeral0"
1.184 + by (simp add: number_of_code_numeral_def Pls_def)
1.185 +lemma [code post]: "Numeral0 = (0\<Colon>code_numeral)"
1.186 + using zero_code_numeral_code ..
1.187 +
1.188 +lemma one_code_numeral_code [code inline, code]:
1.189 + "(1\<Colon>code_numeral) = Numeral1"
1.190 + by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
1.191 +lemma [code post]: "Numeral1 = (1\<Colon>code_numeral)"
1.192 + using one_code_numeral_code ..
1.193 +
1.194 +lemma plus_code_numeral_code [code nbe]:
1.195 + "of_nat n + of_nat m = of_nat (n + m)"
1.196 + by simp
1.197 +
1.198 +definition subtract_code_numeral :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
1.199 + [simp, code del]: "subtract_code_numeral = op -"
1.200 +
1.201 +lemma subtract_code_numeral_code [code nbe]:
1.202 + "subtract_code_numeral (of_nat n) (of_nat m) = of_nat (n - m)"
1.203 + by simp
1.204 +
1.205 +lemma minus_code_numeral_code [code]:
1.206 + "n - m = subtract_code_numeral n m"
1.207 + by simp
1.208 +
1.209 +lemma times_code_numeral_code [code nbe]:
1.210 + "of_nat n * of_nat m = of_nat (n * m)"
1.211 + by simp
1.212 +
1.213 +lemma less_eq_code_numeral_code [code nbe]:
1.214 + "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
1.215 + by simp
1.216 +
1.217 +lemma less_code_numeral_code [code nbe]:
1.218 + "of_nat n < of_nat m \<longleftrightarrow> n < m"
1.219 + by simp
1.220 +
1.221 +lemma Suc_code_numeral_minus_one: "Suc_code_numeral n - 1 = n" by simp
1.222 +
1.223 +lemma of_nat_code [code]:
1.224 + "of_nat = Nat.of_nat"
1.225 +proof
1.226 + fix n :: nat
1.227 + have "Nat.of_nat n = of_nat n"
1.228 + by (induct n) simp_all
1.229 + then show "of_nat n = Nat.of_nat n"
1.230 + by (rule sym)
1.231 +qed
1.232 +
1.233 +lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
1.234 + by (cases i) auto
1.235 +
1.236 +definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
1.237 + "nat_of_aux i n = nat_of i + n"
1.238 +
1.239 +lemma nat_of_aux_code [code]:
1.240 + "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Suc n))"
1.241 + by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
1.242 +
1.243 +lemma nat_of_code [code]:
1.244 + "nat_of i = nat_of_aux i 0"
1.245 + by (simp add: nat_of_aux_def)
1.246 +
1.247 +definition div_mod_code_numeral :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
1.248 + [code del]: "div_mod_code_numeral n m = (n div m, n mod m)"
1.249 +
1.250 +lemma [code]:
1.251 + "div_mod_code_numeral n m = (if m = 0 then (0, n) else (n div m, n mod m))"
1.252 + unfolding div_mod_code_numeral_def by auto
1.253 +
1.254 +lemma [code]:
1.255 + "n div m = fst (div_mod_code_numeral n m)"
1.256 + unfolding div_mod_code_numeral_def by simp
1.257 +
1.258 +lemma [code]:
1.259 + "n mod m = snd (div_mod_code_numeral n m)"
1.260 + unfolding div_mod_code_numeral_def by simp
1.261 +
1.262 +definition int_of :: "code_numeral \<Rightarrow> int" where
1.263 + "int_of = Nat.of_nat o nat_of"
1.264 +
1.265 +lemma int_of_code [code]:
1.266 + "int_of k = (if k = 0 then 0
1.267 + else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
1.268 + by (auto simp add: int_of_def mod_div_equality')
1.269 +
1.270 +hide (open) const of_nat nat_of int_of
1.271 +
1.272 +
1.273 +subsection {* Code generator setup *}
1.274 +
1.275 +text {* Implementation of indices by bounded integers *}
1.276 +
1.277 +code_type code_numeral
1.278 + (SML "int")
1.279 + (OCaml "int")
1.280 + (Haskell "Int")
1.281 +
1.282 +code_instance code_numeral :: eq
1.283 + (Haskell -)
1.284 +
1.285 +setup {*
1.286 + fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
1.287 + false false) ["SML", "OCaml", "Haskell"]
1.288 +*}
1.289 +
1.290 +code_reserved SML Int int
1.291 +code_reserved OCaml Pervasives int
1.292 +
1.293 +code_const "op + \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
1.294 + (SML "Int.+/ ((_),/ (_))")
1.295 + (OCaml "Pervasives.( + )")
1.296 + (Haskell infixl 6 "+")
1.297 +
1.298 +code_const "subtract_code_numeral \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
1.299 + (SML "Int.max/ (_/ -/ _,/ 0 : int)")
1.300 + (OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ")
1.301 + (Haskell "max/ (_/ -/ _)/ (0 :: Int)")
1.302 +
1.303 +code_const "op * \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
1.304 + (SML "Int.*/ ((_),/ (_))")
1.305 + (OCaml "Pervasives.( * )")
1.306 + (Haskell infixl 7 "*")
1.307 +
1.308 +code_const div_mod_code_numeral
1.309 + (SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
1.310 + (OCaml "(fun n -> fun m ->/ if m = 0/ then (0, n) else/ (n '/ m, n mod m))")
1.311 + (Haskell "divMod")
1.312 +
1.313 +code_const "eq_class.eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
1.314 + (SML "!((_ : Int.int) = _)")
1.315 + (OCaml "!((_ : int) = _)")
1.316 + (Haskell infixl 4 "==")
1.317 +
1.318 +code_const "op \<le> \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
1.319 + (SML "Int.<=/ ((_),/ (_))")
1.320 + (OCaml "!((_ : int) <= _)")
1.321 + (Haskell infix 4 "<=")
1.322 +
1.323 +code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
1.324 + (SML "Int.</ ((_),/ (_))")
1.325 + (OCaml "!((_ : int) < _)")
1.326 + (Haskell infix 4 "<")
1.327 +
1.328 +end