(* 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 *}
datatype index = index_of_nat nat
lemmas [code func del] = index.recs index.cases
primrec
nat_of_index :: "index \<Rightarrow> nat"
where
"nat_of_index (index_of_nat k) = k"
lemmas [code func del] = nat_of_index.simps
lemma index_id [simp]:
"index_of_nat (nat_of_index n) = n"
by (cases n) simp_all
lemma nat_of_index_inject [simp]:
"nat_of_index n = nat_of_index m \<longleftrightarrow> n = m"
by (cases n) auto
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 [code func]: "size (n\<Colon>index) = 0"
by (cases n) simp_all
subsection {* Indices as datatype of ints *}
instantiation index :: number
begin
definition
"number_of = index_of_nat o nat"
instance ..
end
code_datatype "number_of \<Colon> int \<Rightarrow> index"
subsection {* Basic arithmetic *}
instantiation index :: "{minus, ordered_semidom, Divides.div, linorder}"
begin
definition [simp, code func del]:
"(0\<Colon>index) = index_of_nat 0"
lemma zero_index_code [code inline, code func]:
"(0\<Colon>index) = Numeral0"
by (simp add: number_of_index_def Pls_def)
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 Bit_def)
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
subsection {* ML interface *}
ML {*
structure Index =
struct
fun mk k = @{term index_of_nat} $ HOLogic.mk_number @{typ index} k;
end;
*}
subsection {* Code serialization *}
text {* Pecularity for operations with potentially negative result *}
definition
minus_index' :: "index \<Rightarrow> index \<Rightarrow> index"
where
[code func del]: "minus_index' = op -"
lemma minus_index_code [code func]:
"n - m = (let q = minus_index' n m
in if q < 0 then 0 else q)"
by (simp add: minus_index'_def Let_def)
text {* Implementation of indices by bounded integers *}
code_type index
(SML "int")
(OCaml "int")
(Haskell "Integer")
code_instance index :: eq
(Haskell -)
setup {*
fold (fn target => CodeTarget.add_pretty_numeral target true
@{const_name number_index_inst.number_of_index}
@{const_name Numeral.B0} @{const_name Numeral.B1}
@{const_name Numeral.Pls} @{const_name Numeral.Min}
@{const_name Numeral.Bit}
) ["SML", "OCaml", "Haskell"]
*}
code_reserved SML int
code_reserved OCaml int
code_const "op + \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.+ ((_), (_))")
(OCaml "Pervasives.+")
(Haskell infixl 6 "+")
code_const "minus_index' \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.- ((_), (_))")
(OCaml "Pervasives.-")
(Haskell infixl 6 "-")
code_const "op * \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "Int.* ((_), (_))")
(OCaml "Pervasives.*")
(Haskell infixl 7 "*")
code_const "op = \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
(OCaml "!((_ : Pervasives.int) = _)")
(Haskell infixl 4 "==")
code_const "op \<le> \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.<= ((_), (_))")
(OCaml "!((_ : Pervasives.int) <= _)")
(Haskell infix 4 "<=")
code_const "op < \<Colon> index \<Rightarrow> index \<Rightarrow> bool"
(SML "Int.< ((_), (_))")
(OCaml "!((_ : Pervasives.int) < _)")
(Haskell infix 4 "<")
code_const "op div \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "IntInf.div ((_), (_))")
(OCaml "Big'_int.div'_big'_int")
(Haskell "div")
code_const "op mod \<Colon> index \<Rightarrow> index \<Rightarrow> index"
(SML "IntInf.mod ((_), (_))")
(OCaml "Big'_int.mod'_big'_int")
(Haskell "mod")
code_reserved SML Int
code_reserved OCaml Pervasives
end