(* ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
header {* Type of indices *}
theory Code_Index
imports ATP_Linkup
begin
text {*
Indices are isomorphic to HOL @{typ nat} but
mapped to target-language builtin integers.
*}
subsection {* Datatype of indices *}
typedef index = "UNIV \<Colon> nat set"
morphisms nat_of_index index_of_nat by rule
lemma index_of_nat_nat_of_index [simp]:
"index_of_nat (nat_of_index k) = k"
by (rule nat_of_index_inverse)
lemma nat_of_index_index_of_nat [simp]:
"nat_of_index (index_of_nat n) = n"
by (rule index_of_nat_inverse)
(unfold index_def, rule UNIV_I)
lemma index:
"(\<And>n\<Colon>index. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (index_of_nat n))"
proof
fix n :: nat
assume "\<And>n\<Colon>index. PROP P n"
then show "PROP P (index_of_nat n)" .
next
fix n :: index
assume "\<And>n\<Colon>nat. PROP P (index_of_nat n)"
then have "PROP P (index_of_nat (nat_of_index n))" .
then show "PROP P n" by simp
qed
lemma index_case:
assumes "\<And>n. k = index_of_nat n \<Longrightarrow> P"
shows P
by (rule assms [of "nat_of_index k"]) simp
lemma index_induct_raw:
assumes "\<And>n. P (index_of_nat n)"
shows "P k"
proof -
from assms have "P (index_of_nat (nat_of_index k))" .
then show ?thesis by simp
qed
lemma nat_of_index_inject [simp]:
"nat_of_index k = nat_of_index l \<longleftrightarrow> k = l"
by (rule nat_of_index_inject)
lemma index_of_nat_inject [simp]:
"index_of_nat n = index_of_nat m \<longleftrightarrow> n = m"
by (auto intro!: index_of_nat_inject simp add: index_def)
instantiation index :: zero
begin
definition [simp, code func del]:
"0 = index_of_nat 0"
instance ..
end
definition [simp]:
"Suc_index k = index_of_nat (Suc (nat_of_index k))"
rep_datatype "0 \<Colon> index" Suc_index
proof -
fix P :: "index \<Rightarrow> bool"
fix k :: index
assume "P 0" then have init: "P (index_of_nat 0)" by simp
assume "\<And>k. P k \<Longrightarrow> P (Suc_index k)"
then have "\<And>n. P (index_of_nat n) \<Longrightarrow> P (Suc_index (index_of_nat n))" .
then have step: "\<And>n. P (index_of_nat n) \<Longrightarrow> P (index_of_nat (Suc n))" by simp
from init step have "P (index_of_nat (nat_of_index k))"
by (induct "nat_of_index k") simp_all
then show "P k" by simp
qed simp_all
lemmas [code func del] = index.recs index.cases
declare index_case [case_names nat, cases type: index]
declare index.induct [case_names nat, induct type: index]
lemma [code func]:
"index_size = nat_of_index"
proof (rule ext)
fix k
have "index_size k = nat_size (nat_of_index k)"
by (induct k rule: index.induct) (simp_all del: zero_index_def Suc_index_def, simp_all)
also have "nat_size (nat_of_index k) = nat_of_index k" by (induct "nat_of_index k") simp_all
finally show "index_size k = nat_of_index k" .
qed
lemma [code func]:
"size = nat_of_index"
proof (rule ext)
fix k
show "size k = nat_of_index k"
by (induct k) (simp_all del: zero_index_def Suc_index_def, simp_all)
qed
lemma [code func]:
"k = l \<longleftrightarrow> nat_of_index k = nat_of_index l"
by (cases k, cases l) simp
subsection {* Indices as datatype of ints *}
instantiation index :: number
begin
definition
"number_of = index_of_nat o nat"
instance ..
end
lemma nat_of_index_number [simp]:
"nat_of_index (number_of k) = number_of k"
by (simp add: number_of_index_def nat_number_of_def number_of_is_id)
code_datatype "number_of \<Colon> int \<Rightarrow> index"
subsection {* Basic arithmetic *}
instantiation index :: "{minus, ordered_semidom, Divides.div, linorder}"
begin
lemma zero_index_code [code inline, code func]:
"(0\<Colon>index) = Numeral0"
by (simp add: number_of_index_def Pls_def)
lemma [code post]: "Numeral0 = (0\<Colon>index)"
using zero_index_code ..
definition [simp, code func del]:
"(1\<Colon>index) = index_of_nat 1"
lemma one_index_code [code inline, code func]:
"(1\<Colon>index) = Numeral1"
by (simp add: number_of_index_def Pls_def Bit1_def)
lemma [code post]: "Numeral1 = (1\<Colon>index)"
using one_index_code ..
definition [simp, code func del]:
"n + m = index_of_nat (nat_of_index n + nat_of_index m)"
lemma plus_index_code [code func]:
"index_of_nat n + index_of_nat m = index_of_nat (n + m)"
by simp
definition [simp, code func del]:
"n - m = index_of_nat (nat_of_index n - nat_of_index m)"
definition [simp, code func del]:
"n * m = index_of_nat (nat_of_index n * nat_of_index m)"
lemma times_index_code [code func]:
"index_of_nat n * index_of_nat m = index_of_nat (n * m)"
by simp
definition [simp, code func del]:
"n div m = index_of_nat (nat_of_index n div nat_of_index m)"
definition [simp, code func del]:
"n mod m = index_of_nat (nat_of_index n mod nat_of_index m)"
lemma div_index_code [code func]:
"index_of_nat n div index_of_nat m = index_of_nat (n div m)"
by simp
lemma mod_index_code [code func]:
"index_of_nat n mod index_of_nat m = index_of_nat (n mod m)"
by simp
definition [simp, code func del]:
"n \<le> m \<longleftrightarrow> nat_of_index n \<le> nat_of_index m"
definition [simp, code func del]:
"n < m \<longleftrightarrow> nat_of_index n < nat_of_index m"
lemma less_eq_index_code [code func]:
"index_of_nat n \<le> index_of_nat m \<longleftrightarrow> n \<le> m"
by simp
lemma less_index_code [code func]:
"index_of_nat n < index_of_nat m \<longleftrightarrow> n < m"
by simp
instance by default (auto simp add: left_distrib index)
end
lemma Suc_index_minus_one: "Suc_index n - 1 = n" by simp
lemma index_of_nat_code [code]:
"index_of_nat = of_nat"
proof
fix n :: nat
have "of_nat n = index_of_nat n"
by (induct n) simp_all
then show "index_of_nat n = of_nat n"
by (rule sym)
qed
lemma index_not_eq_zero: "i \<noteq> index_of_nat 0 \<longleftrightarrow> i \<ge> 1"
by (cases i) auto
definition
nat_of_index_aux :: "index \<Rightarrow> nat \<Rightarrow> nat"
where
"nat_of_index_aux i n = nat_of_index i + n"
lemma nat_of_index_aux_code [code]:
"nat_of_index_aux i n = (if i = 0 then n else nat_of_index_aux (i - 1) (Suc n))"
by (auto simp add: nat_of_index_aux_def index_not_eq_zero)
lemma nat_of_index_code [code]:
"nat_of_index i = nat_of_index_aux i 0"
by (simp add: nat_of_index_aux_def)
subsection {* ML interface *}
ML {*
structure Index =
struct
fun mk k = HOLogic.mk_number @{typ index} k;
end;
*}
subsection {* Specialized @{term "op - \<Colon> index \<Rightarrow> index \<Rightarrow> index"},
@{term "op div \<Colon> index \<Rightarrow> index \<Rightarrow> index"} and @{term "op mod \<Colon> index \<Rightarrow> index \<Rightarrow> index"}
operations *}
definition
minus_index_aux :: "index \<Rightarrow> index \<Rightarrow> index"
where
[code func del]: "minus_index_aux = op -"
lemma [code func]: "op - = minus_index_aux"
using minus_index_aux_def ..
definition
div_mod_index :: "index \<Rightarrow> index \<Rightarrow> index \<times> index"
where
[code func del]: "div_mod_index n m = (n div m, n mod m)"
lemma [code func]:
"div_mod_index n m = (if m = 0 then (0, n) else (n div m, n mod m))"
unfolding div_mod_index_def by auto
lemma [code func]:
"n div m = fst (div_mod_index n m)"
unfolding div_mod_index_def by simp
lemma [code func]:
"n mod m = snd (div_mod_index n m)"
unfolding div_mod_index_def by simp
subsection {* Code serialization *}
text {* Implementation of indices by bounded integers *}
code_type index
(SML "int")
(OCaml "int")
(Haskell "Int")
code_instance index :: eq
(Haskell -)
setup {*
fold (Numeral.add_code @{const_name number_index_inst.number_of_index}
false false) ["SML", "OCaml", "Haskell"]
*}
code_reserved SML Int int
code_reserved OCaml Pervasives int
code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.+/ ((_),/ (_))")
(OCaml "Pervasives.( + )")
(Haskell infixl 6 "+")
code_const "minus_index_aux \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.max/ (_/ -/ _,/ 0 : int)")
(OCaml "Pervasives.max/ (_/ -/ _)/ (0 : int) ")
(Haskell "max/ (_/ -/ _)/ (0 :: Int)")
code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.*/ ((_),/ (_))")
(OCaml "Pervasives.( * )")
(Haskell infixl 7 "*")
code_const div_mod_index
(SML "(fn n => fn m =>/ (n div m, n mod m))")
(OCaml "(fun n -> fun m ->/ (n '/ m, n mod m))")
(Haskell "divMod")
code_const "op = \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
(OCaml "!((_ : int) = _)")
(Haskell infixl 4 "==")
code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.<=/ ((_),/ (_))")
(OCaml "!((_ : int) <= _)")
(Haskell infix 4 "<=")
code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.</ ((_),/ (_))")
(OCaml "!((_ : int) < _)")
(Haskell infix 4 "<")
end