Theory Code_Index

Up to index of Isabelle/HOL/ex

theory Code_Index
imports PreList
begin

(*  ID:         $Id: Code_Index.thy,v 1.2 2007/11/08 12:21:12 haftmann Exp $
    Author:     Florian Haftmann, TU Muenchen
*)

header {* Type of indices *}

theory Code_Index
imports PreList
begin

text {*
  Indices are isomorphic to HOL @{typ int} but
  mapped to target-language builtin integers
*}

subsection {* Datatype of indices *}

datatype index = index_of_int int

lemmas [code func del] = index.recs index.cases

fun
  int_of_index :: "index => int"
where
  "int_of_index (index_of_int k) = k"
lemmas [code func del] = int_of_index.simps

lemma index_id [simp]:
  "index_of_int (int_of_index k) = k"
  by (cases k) simp_all

lemma index:
  "(!!k::index. PROP P k) ≡ (!!k::int. PROP P (index_of_int k))"
proof
  fix k :: int
  assume "!!k::index. PROP P k"
  then show "PROP P (index_of_int k)" .
next
  fix k :: index
  assume "!!k::int. PROP P (index_of_int k)"
  then have "PROP P (index_of_int (int_of_index k))" .
  then show "PROP P k" by simp
qed

lemma [code func]: "size (k::index) = 0"
  by (cases k) simp_all


subsection {* Built-in integers as datatype on numerals *}

instance index :: number
  "number_of ≡ index_of_int" ..

code_datatype "number_of :: int => index"

lemma number_of_index_id [simp]:
  "number_of (int_of_index k) = k"
  unfolding number_of_index_def by simp

lemma number_of_index_shift:
  "number_of k = index_of_int (number_of k)"
  by (simp add: number_of_is_id number_of_index_def)

lemma int_of_index_number_of [simp]:
  "int_of_index (number_of k) = number_of k"
  unfolding number_of_index_def number_of_is_id by simp


subsection {* Basic arithmetic *}

instance index :: zero
  [simp]: "0 ≡ index_of_int 0" ..
lemmas [code func del] = zero_index_def

instance index :: one
  [simp]: "1 ≡ index_of_int 1" ..
lemmas [code func del] = one_index_def

instance index :: plus
  [simp]: "k + l ≡ index_of_int (int_of_index k + int_of_index l)" ..
lemmas [code func del] = plus_index_def
lemma plus_index_code [code func]:
  "index_of_int k + index_of_int l = index_of_int (k + l)"
  unfolding plus_index_def by simp

instance index :: minus
  [simp]: "- k ≡ index_of_int (- int_of_index k)"
  [simp]: "k - l ≡ index_of_int (int_of_index k - int_of_index l)" ..
lemmas [code func del] = uminus_index_def minus_index_def
lemma uminus_index_code [code func]:
  "- index_of_int k ≡ index_of_int (- k)"
  unfolding uminus_index_def by simp
lemma minus_index_code [code func]:
  "index_of_int k - index_of_int l = index_of_int (k - l)"
  unfolding minus_index_def by simp

instance index :: times
  [simp]: "k * l ≡ index_of_int (int_of_index k * int_of_index l)" ..
lemmas [code func del] = times_index_def
lemma times_index_code [code func]:
  "index_of_int k * index_of_int l = index_of_int (k * l)"
  unfolding times_index_def by simp

instance index :: ord
  [simp]: "k ≤ l ≡ int_of_index k ≤ int_of_index l"
  [simp]: "k < l ≡ int_of_index k < int_of_index l" ..
lemmas [code func del] = less_eq_index_def less_index_def
lemma less_eq_index_code [code func]:
  "index_of_int k ≤ index_of_int l <-> k ≤ l"
  unfolding less_eq_index_def by simp
lemma less_index_code [code func]:
  "index_of_int k < index_of_int l <-> k < l"
  unfolding less_index_def by simp

instance index :: "Divides.div"
  [simp]: "k div l ≡ index_of_int (int_of_index k div int_of_index l)"
  [simp]: "k mod l ≡ index_of_int (int_of_index k mod int_of_index l)" ..

instance index :: ring_1
  by default (auto simp add: left_distrib right_distrib)

lemma of_nat_index: "of_nat n = index_of_int (of_nat n)"
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n)
  then have "int_of_index (index_of_int (int n))
    = int_of_index (of_nat n)" by simp
  then have "int n = int_of_index (of_nat n)" by simp
  then show ?case by simp
qed

instance index :: number_ring
  by default
    (simp_all add: left_distrib number_of_index_def of_int_of_nat of_nat_index)

lemma zero_index_code [code inline, code func]:
  "(0::index) = Numeral0"
  by simp

lemma one_index_code [code inline, code func]:
  "(1::index) = Numeral1"
  by simp

instance index :: abs
  "¦k¦ ≡ if k < 0 then -k else k" ..

lemma index_of_int [code func]:
  "index_of_int k = (if k = 0 then 0
    else if k = -1 then -1
    else let (l, m) = divAlg (k, 2) in 2 * index_of_int l +
      (if m = 0 then 0 else 1))"
  by (simp add: number_of_index_shift Let_def split_def divAlg_mod_div) arith

lemma int_of_index [code func]:
  "int_of_index k = (if k = 0 then 0
    else if k = -1 then -1
    else let l = k div 2; m = k mod 2 in 2 * int_of_index l +
      (if m = 0 then 0 else 1))"
  by (auto simp add: number_of_index_shift Let_def split_def) arith


subsection {* Conversion to and from @{typ nat} *}

definition
  nat_of_index :: "index => nat"
where
  [code func del]: "nat_of_index = nat o int_of_index"

definition
  nat_of_index_aux :: "index => nat => nat" where
  [code func del]: "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 nat_of_index_def)

lemma nat_of_index_code [code]:
  "nat_of_index i = nat_of_index_aux i 0"
  by (simp add: nat_of_index_aux_def)

definition
  index_of_nat :: "nat => index"
where
  [code func del]: "index_of_nat = index_of_int o of_nat"

lemma index_of_nat [code func]:
  "index_of_nat 0 = 0"
  "index_of_nat (Suc n) = index_of_nat n + 1"
  unfolding index_of_nat_def by simp_all

lemma index_nat_id [simp]:
  "nat_of_index (index_of_nat n) = n"
  "index_of_nat (nat_of_index i) = (if i ≤ 0 then 0 else i)"
  unfolding index_of_nat_def nat_of_index_def by simp_all


subsection {* ML interface *}

ML {*
structure Index =
struct

fun mk k = @{term index_of_int} $ HOLogic.mk_number @{typ index} k;

end;
*}


subsection {* Code serialization *}

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 + :: index => index => index"
  (SML "Int.+ ((_), (_))")
  (OCaml "Pervasives.+")
  (Haskell infixl 6 "+")

code_const "uminus :: index => index"
  (SML "Int.~")
  (OCaml "Pervasives.~-")
  (Haskell "negate")

code_const "op - :: index => index => index"
  (SML "Int.- ((_), (_))")
  (OCaml "Pervasives.-")
  (Haskell infixl 6 "-")

code_const "op * :: index => index => index"
  (SML "Int.* ((_), (_))")
  (OCaml "Pervasives.*")
  (Haskell infixl 7 "*")

code_const "op = :: index => index => bool"
  (SML "!((_ : Int.int) = _)")
  (OCaml "!((_ : Pervasives.int) = _)")
  (Haskell infixl 4 "==")

code_const "op ≤ :: index => index => bool"
  (SML "Int.<= ((_), (_))")
  (OCaml "!((_ : Pervasives.int) <= _)")
  (Haskell infix 4 "<=")

code_const "op < :: index => index => bool"
  (SML "Int.< ((_), (_))")
  (OCaml "!((_ : Pervasives.int) < _)")
  (Haskell infix 4 "<")

code_reserved SML Int
code_reserved OCaml Pervasives

end

Datatype of indices

lemma

  index_rec f1.0 (index_of_int int) = f1.0 int
  index_case f1.0 (index_of_int int) = f1.0 int

lemma

  int_of_index (index_of_int k) = k

lemma index_id:

  index_of_int (int_of_index k) = k

lemma index:

  (!!k. PROP P k) == (!!k. PROP P (index_of_int k))

lemma

  size k = 0

Built-in integers as datatype on numerals

lemma number_of_index_id:

  number_of (int_of_index k) = k

lemma number_of_index_shift:

  number_of k = index_of_int (number_of k)

lemma int_of_index_number_of:

  int_of_index (number_of k) = number_of k

Basic arithmetic

lemma

  0 == index_of_int 0

lemma

  1 == index_of_int 1

lemma

  k + l == index_of_int (int_of_index k + int_of_index l)

lemma plus_index_code:

  index_of_int k + index_of_int l = index_of_int (k + l)

lemma

  - k == index_of_int (- int_of_index k)
  k - l == index_of_int (int_of_index k - int_of_index l)

lemma uminus_index_code:

  - index_of_int k == index_of_int (- k)

lemma minus_index_code:

  index_of_int k - index_of_int l = index_of_int (k - l)

lemma

  k * l == index_of_int (int_of_index k * int_of_index l)

lemma times_index_code:

  index_of_int k * index_of_int l = index_of_int (k * l)

lemma

  k  l == int_of_index k  int_of_index l
  k < l == int_of_index k < int_of_index l

lemma less_eq_index_code:

  (index_of_int k  index_of_int l) = (k  l)

lemma less_index_code:

  (index_of_int k < index_of_int l) = (k < l)

lemma of_nat_index:

  of_nat n = index_of_int (int n)

lemma zero_index_code:

  0 = Numeral0

lemma one_index_code:

  1 = Numeral1

lemma index_of_int:

  index_of_int k =
  (if k = 0 then 0
   else if k = -1 then -1
        else let (l, m) = divAlg (k, 2)
             in 2 * index_of_int l + (if m = 0 then 0 else 1))

lemma int_of_index:

  int_of_index k =
  (if k = 0 then 0
   else if k = -1 then -1
        else let l = k div 2; m = k mod 2
             in 2 * int_of_index l + (if m = 0 then 0 else 1))

Conversion to and from @{typ nat}

lemma nat_of_index_aux_code:

  nat_of_index_aux i n = (if i  0 then n else nat_of_index_aux (i - 1) (Suc n))

lemma nat_of_index_code:

  nat_of_index i = nat_of_index_aux i 0

lemma index_of_nat:

  index_of_nat 0 = 0
  index_of_nat (Suc n) = index_of_nat n + 1

lemma index_nat_id:

  nat_of_index (index_of_nat n) = n
  index_of_nat (nat_of_index i) = (if i  0 then 0 else i)

ML interface

Code serialization