src/HOL/Code_Numeral.thy
changeset 31205 98370b26c2ce
parent 31203 5c8fb4fd67e0
child 31266 55e70b6d812e
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Code_Numeral.thy	Tue May 19 16:54:55 2009 +0200
@@ -0,0 +1,325 @@
+(* Author: Florian Haftmann, TU Muenchen *)
+
+header {* Type of target language numerals *}
+
+theory Code_Numeral
+imports Nat_Numeral
+begin
+
+text {*
+  Code numerals are isomorphic to HOL @{typ nat} but
+  mapped to target-language builtin numerals.
+*}
+
+subsection {* Datatype of target language numerals *}
+
+typedef (open) code_numeral = "UNIV \<Colon> nat set"
+  morphisms nat_of of_nat by rule
+
+lemma of_nat_nat_of [simp]:
+  "of_nat (nat_of k) = k"
+  by (rule nat_of_inverse)
+
+lemma nat_of_of_nat [simp]:
+  "nat_of (of_nat n) = n"
+  by (rule of_nat_inverse) (rule UNIV_I)
+
+lemma [measure_function]:
+  "is_measure nat_of" by (rule is_measure_trivial)
+
+lemma code_numeral:
+  "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"
+proof
+  fix n :: nat
+  assume "\<And>n\<Colon>code_numeral. PROP P n"
+  then show "PROP P (of_nat n)" .
+next
+  fix n :: code_numeral
+  assume "\<And>n\<Colon>nat. PROP P (of_nat n)"
+  then have "PROP P (of_nat (nat_of n))" .
+  then show "PROP P n" by simp
+qed
+
+lemma code_numeral_case:
+  assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
+  shows P
+  by (rule assms [of "nat_of k"]) simp
+
+lemma code_numeral_induct_raw:
+  assumes "\<And>n. P (of_nat n)"
+  shows "P k"
+proof -
+  from assms have "P (of_nat (nat_of k))" .
+  then show ?thesis by simp
+qed
+
+lemma nat_of_inject [simp]:
+  "nat_of k = nat_of l \<longleftrightarrow> k = l"
+  by (rule nat_of_inject)
+
+lemma of_nat_inject [simp]:
+  "of_nat n = of_nat m \<longleftrightarrow> n = m"
+  by (rule of_nat_inject) (rule UNIV_I)+
+
+instantiation code_numeral :: zero
+begin
+
+definition [simp, code del]:
+  "0 = of_nat 0"
+
+instance ..
+
+end
+
+definition [simp]:
+  "Suc_code_numeral k = of_nat (Suc (nat_of k))"
+
+rep_datatype "0 \<Colon> code_numeral" Suc_code_numeral
+proof -
+  fix P :: "code_numeral \<Rightarrow> bool"
+  fix k :: code_numeral
+  assume "P 0" then have init: "P (of_nat 0)" by simp
+  assume "\<And>k. P k \<Longrightarrow> P (Suc_code_numeral k)"
+    then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc_code_numeral (of_nat n))" .
+    then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Suc n))" by simp
+  from init step have "P (of_nat (nat_of k))"
+    by (induct "nat_of k") simp_all
+  then show "P k" by simp
+qed simp_all
+
+declare code_numeral_case [case_names nat, cases type: code_numeral]
+declare code_numeral.induct [case_names nat, induct type: code_numeral]
+
+lemma code_numeral_decr [termination_simp]:
+  "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Suc 0 < nat_of k"
+  by (cases k) simp
+
+lemma [simp, code]:
+  "code_numeral_size = nat_of"
+proof (rule ext)
+  fix k
+  have "code_numeral_size k = nat_size (nat_of k)"
+    by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
+  also have "nat_size (nat_of k) = nat_of k" by (induct "nat_of k") simp_all
+  finally show "code_numeral_size k = nat_of k" .
+qed
+
+lemma [simp, code]:
+  "size = nat_of"
+proof (rule ext)
+  fix k
+  show "size k = nat_of k"
+  by (induct k) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
+qed
+
+lemmas [code del] = code_numeral.recs code_numeral.cases
+
+lemma [code]:
+  "eq_class.eq k l \<longleftrightarrow> eq_class.eq (nat_of k) (nat_of l)"
+  by (cases k, cases l) (simp add: eq)
+
+lemma [code nbe]:
+  "eq_class.eq (k::code_numeral) k \<longleftrightarrow> True"
+  by (rule HOL.eq_refl)
+
+
+subsection {* Indices as datatype of ints *}
+
+instantiation code_numeral :: number
+begin
+
+definition
+  "number_of = of_nat o nat"
+
+instance ..
+
+end
+
+lemma nat_of_number [simp]:
+  "nat_of (number_of k) = number_of k"
+  by (simp add: number_of_code_numeral_def nat_number_of_def number_of_is_id)
+
+code_datatype "number_of \<Colon> int \<Rightarrow> code_numeral"
+
+
+subsection {* Basic arithmetic *}
+
+instantiation code_numeral :: "{minus, ordered_semidom, semiring_div, linorder}"
+begin
+
+definition [simp, code del]:
+  "(1\<Colon>code_numeral) = of_nat 1"
+
+definition [simp, code del]:
+  "n + m = of_nat (nat_of n + nat_of m)"
+
+definition [simp, code del]:
+  "n - m = of_nat (nat_of n - nat_of m)"
+
+definition [simp, code del]:
+  "n * m = of_nat (nat_of n * nat_of m)"
+
+definition [simp, code del]:
+  "n div m = of_nat (nat_of n div nat_of m)"
+
+definition [simp, code del]:
+  "n mod m = of_nat (nat_of n mod nat_of m)"
+
+definition [simp, code del]:
+  "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
+
+definition [simp, code del]:
+  "n < m \<longleftrightarrow> nat_of n < nat_of m"
+
+instance proof
+qed (auto simp add: code_numeral left_distrib div_mult_self1)
+
+end
+
+lemma zero_code_numeral_code [code inline, code]:
+  "(0\<Colon>code_numeral) = Numeral0"
+  by (simp add: number_of_code_numeral_def Pls_def)
+lemma [code post]: "Numeral0 = (0\<Colon>code_numeral)"
+  using zero_code_numeral_code ..
+
+lemma one_code_numeral_code [code inline, code]:
+  "(1\<Colon>code_numeral) = Numeral1"
+  by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
+lemma [code post]: "Numeral1 = (1\<Colon>code_numeral)"
+  using one_code_numeral_code ..
+
+lemma plus_code_numeral_code [code nbe]:
+  "of_nat n + of_nat m = of_nat (n + m)"
+  by simp
+
+definition subtract_code_numeral :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
+  [simp, code del]: "subtract_code_numeral = op -"
+
+lemma subtract_code_numeral_code [code nbe]:
+  "subtract_code_numeral (of_nat n) (of_nat m) = of_nat (n - m)"
+  by simp
+
+lemma minus_code_numeral_code [code]:
+  "n - m = subtract_code_numeral n m"
+  by simp
+
+lemma times_code_numeral_code [code nbe]:
+  "of_nat n * of_nat m = of_nat (n * m)"
+  by simp
+
+lemma less_eq_code_numeral_code [code nbe]:
+  "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
+  by simp
+
+lemma less_code_numeral_code [code nbe]:
+  "of_nat n < of_nat m \<longleftrightarrow> n < m"
+  by simp
+
+lemma Suc_code_numeral_minus_one: "Suc_code_numeral n - 1 = n" by simp
+
+lemma of_nat_code [code]:
+  "of_nat = Nat.of_nat"
+proof
+  fix n :: nat
+  have "Nat.of_nat n = of_nat n"
+    by (induct n) simp_all
+  then show "of_nat n = Nat.of_nat n"
+    by (rule sym)
+qed
+
+lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
+  by (cases i) auto
+
+definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
+  "nat_of_aux i n = nat_of i + n"
+
+lemma nat_of_aux_code [code]:
+  "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Suc n))"
+  by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
+
+lemma nat_of_code [code]:
+  "nat_of i = nat_of_aux i 0"
+  by (simp add: nat_of_aux_def)
+
+definition div_mod_code_numeral ::  "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
+  [code del]: "div_mod_code_numeral n m = (n div m, n mod m)"
+
+lemma [code]:
+  "div_mod_code_numeral n m = (if m = 0 then (0, n) else (n div m, n mod m))"
+  unfolding div_mod_code_numeral_def by auto
+
+lemma [code]:
+  "n div m = fst (div_mod_code_numeral n m)"
+  unfolding div_mod_code_numeral_def by simp
+
+lemma [code]:
+  "n mod m = snd (div_mod_code_numeral n m)"
+  unfolding div_mod_code_numeral_def by simp
+
+definition int_of :: "code_numeral \<Rightarrow> int" where
+  "int_of = Nat.of_nat o nat_of"
+
+lemma int_of_code [code]:
+  "int_of k = (if k = 0 then 0
+    else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
+  by (auto simp add: int_of_def mod_div_equality')
+
+hide (open) const of_nat nat_of int_of
+
+
+subsection {* Code generator setup *}
+
+text {* Implementation of indices by bounded integers *}
+
+code_type code_numeral
+  (SML "int")
+  (OCaml "int")
+  (Haskell "Int")
+
+code_instance code_numeral :: eq
+  (Haskell -)
+
+setup {*
+  fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
+    false false) ["SML", "OCaml", "Haskell"]
+*}
+
+code_reserved SML Int int
+code_reserved OCaml Pervasives int
+
+code_const "op + \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
+  (SML "Int.+/ ((_),/ (_))")
+  (OCaml "Pervasives.( + )")
+  (Haskell infixl 6 "+")
+
+code_const "subtract_code_numeral \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
+  (SML "Int.max/ (_/ -/ _,/ 0 : int)")
+  (OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ")
+  (Haskell "max/ (_/ -/ _)/ (0 :: Int)")
+
+code_const "op * \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
+  (SML "Int.*/ ((_),/ (_))")
+  (OCaml "Pervasives.( * )")
+  (Haskell infixl 7 "*")
+
+code_const div_mod_code_numeral
+  (SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
+  (OCaml "(fun n -> fun m ->/ if m = 0/ then (0, n) else/ (n '/ m, n mod m))")
+  (Haskell "divMod")
+
+code_const "eq_class.eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
+  (SML "!((_ : Int.int) = _)")
+  (OCaml "!((_ : int) = _)")
+  (Haskell infixl 4 "==")
+
+code_const "op \<le> \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
+  (SML "Int.<=/ ((_),/ (_))")
+  (OCaml "!((_ : int) <= _)")
+  (Haskell infix 4 "<=")
+
+code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
+  (SML "Int.</ ((_),/ (_))")
+  (OCaml "!((_ : int) < _)")
+  (Haskell infix 4 "<")
+
+end