(* Title: HOL/Code_Numeral.thy
Author: Florian Haftmann, TU Muenchen
*)
section \<open>Numeric types for code generation onto target language numerals only\<close>
theory Code_Numeral
imports Divides Lifting
begin
subsection \<open>Type of target language integers\<close>
typedef integer = "UNIV :: int set"
morphisms int_of_integer integer_of_int ..
setup_lifting type_definition_integer
lemma integer_eq_iff:
"k = l \<longleftrightarrow> int_of_integer k = int_of_integer l"
by transfer rule
lemma integer_eqI:
"int_of_integer k = int_of_integer l \<Longrightarrow> k = l"
using integer_eq_iff [of k l] by simp
lemma int_of_integer_integer_of_int [simp]:
"int_of_integer (integer_of_int k) = k"
by transfer rule
lemma integer_of_int_int_of_integer [simp]:
"integer_of_int (int_of_integer k) = k"
by transfer rule
instantiation integer :: ring_1
begin
lift_definition zero_integer :: integer
is "0 :: int"
.
declare zero_integer.rep_eq [simp]
lift_definition one_integer :: integer
is "1 :: int"
.
declare one_integer.rep_eq [simp]
lift_definition plus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "plus :: int \<Rightarrow> int \<Rightarrow> int"
.
declare plus_integer.rep_eq [simp]
lift_definition uminus_integer :: "integer \<Rightarrow> integer"
is "uminus :: int \<Rightarrow> int"
.
declare uminus_integer.rep_eq [simp]
lift_definition minus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "minus :: int \<Rightarrow> int \<Rightarrow> int"
.
declare minus_integer.rep_eq [simp]
lift_definition times_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "times :: int \<Rightarrow> int \<Rightarrow> int"
.
declare times_integer.rep_eq [simp]
instance proof
qed (transfer, simp add: algebra_simps)+
end
instance integer :: Rings.dvd ..
lemma [transfer_rule]:
"rel_fun pcr_integer (rel_fun pcr_integer HOL.iff) Rings.dvd Rings.dvd"
unfolding dvd_def by transfer_prover
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
by (rule transfer_rule_of_nat) transfer_prover+
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
proof -
have "rel_fun HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
by (rule transfer_rule_of_int) transfer_prover+
then show ?thesis by (simp add: id_def)
qed
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
by (rule transfer_rule_numeral) transfer_prover+
lemma [transfer_rule]:
"rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
by (unfold Num.sub_def [abs_def]) transfer_prover
lemma int_of_integer_of_nat [simp]:
"int_of_integer (of_nat n) = of_nat n"
by transfer rule
lift_definition integer_of_nat :: "nat \<Rightarrow> integer"
is "of_nat :: nat \<Rightarrow> int"
.
lemma integer_of_nat_eq_of_nat [code]:
"integer_of_nat = of_nat"
by transfer rule
lemma int_of_integer_integer_of_nat [simp]:
"int_of_integer (integer_of_nat n) = of_nat n"
by transfer rule
lift_definition nat_of_integer :: "integer \<Rightarrow> nat"
is Int.nat
.
lemma nat_of_integer_of_nat [simp]:
"nat_of_integer (of_nat n) = n"
by transfer simp
lemma int_of_integer_of_int [simp]:
"int_of_integer (of_int k) = k"
by transfer simp
lemma nat_of_integer_integer_of_nat [simp]:
"nat_of_integer (integer_of_nat n) = n"
by transfer simp
lemma integer_of_int_eq_of_int [simp, code_abbrev]:
"integer_of_int = of_int"
by transfer (simp add: fun_eq_iff)
lemma of_int_integer_of [simp]:
"of_int (int_of_integer k) = (k :: integer)"
by transfer rule
lemma int_of_integer_numeral [simp]:
"int_of_integer (numeral k) = numeral k"
by transfer rule
lemma int_of_integer_sub [simp]:
"int_of_integer (Num.sub k l) = Num.sub k l"
by transfer rule
definition integer_of_num :: "num \<Rightarrow> integer"
where [simp]: "integer_of_num = numeral"
lemma integer_of_num [code]:
"integer_of_num Num.One = 1"
"integer_of_num (Num.Bit0 n) = (let k = integer_of_num n in k + k)"
"integer_of_num (Num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
by (simp_all only: integer_of_num_def numeral.simps Let_def)
lemma integer_of_num_triv:
"integer_of_num Num.One = 1"
"integer_of_num (Num.Bit0 Num.One) = 2"
by simp_all
instantiation integer :: "{linordered_idom, equal}"
begin
lift_definition abs_integer :: "integer \<Rightarrow> integer"
is "abs :: int \<Rightarrow> int"
.
declare abs_integer.rep_eq [simp]
lift_definition sgn_integer :: "integer \<Rightarrow> integer"
is "sgn :: int \<Rightarrow> int"
.
declare sgn_integer.rep_eq [simp]
lift_definition less_eq_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
is "less_eq :: int \<Rightarrow> int \<Rightarrow> bool"
.
lift_definition less_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
is "less :: int \<Rightarrow> int \<Rightarrow> bool"
.
lift_definition equal_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
is "HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool"
.
instance
by standard (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] mult_strict_right_mono linear)+
end
lemma [transfer_rule]:
"rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (min :: _ \<Rightarrow> _ \<Rightarrow> int) (min :: _ \<Rightarrow> _ \<Rightarrow> integer)"
by (unfold min_def [abs_def]) transfer_prover
lemma [transfer_rule]:
"rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ \<Rightarrow> _ \<Rightarrow> int) (max :: _ \<Rightarrow> _ \<Rightarrow> integer)"
by (unfold max_def [abs_def]) transfer_prover
lemma int_of_integer_min [simp]:
"int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
by transfer rule
lemma int_of_integer_max [simp]:
"int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
by transfer rule
lemma nat_of_integer_non_positive [simp]:
"k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
by transfer simp
lemma of_nat_of_integer [simp]:
"of_nat (nat_of_integer k) = max 0 k"
by transfer auto
instantiation integer :: unique_euclidean_ring
begin
lift_definition divide_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "divide :: int \<Rightarrow> int \<Rightarrow> int"
.
declare divide_integer.rep_eq [simp]
lift_definition modulo_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "modulo :: int \<Rightarrow> int \<Rightarrow> int"
.
declare modulo_integer.rep_eq [simp]
lift_definition euclidean_size_integer :: "integer \<Rightarrow> nat"
is "euclidean_size :: int \<Rightarrow> nat"
.
declare euclidean_size_integer.rep_eq [simp]
lift_definition division_segment_integer :: "integer \<Rightarrow> integer"
is "division_segment :: int \<Rightarrow> int"
.
declare division_segment_integer.rep_eq [simp]
instance
by (standard; transfer)
(use mult_le_mono2 [of 1] in \<open>auto simp add: sgn_mult_abs abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib
division_segment_mult division_segment_mod intro: div_eqI\<close>)
end
lemma [code]:
"euclidean_size = nat_of_integer \<circ> abs"
by (simp add: fun_eq_iff nat_of_integer.rep_eq)
lemma [code]:
"division_segment (k :: integer) = (if k \<ge> 0 then 1 else - 1)"
by transfer (simp add: division_segment_int_def)
instance integer :: ring_parity
by (standard; transfer) (simp_all add: of_nat_div division_segment_int_def)
instantiation integer :: unique_euclidean_semiring_numeral
begin
definition divmod_integer :: "num \<Rightarrow> num \<Rightarrow> integer \<times> integer"
where
divmod_integer'_def: "divmod_integer m n = (numeral m div numeral n, numeral m mod numeral n)"
definition divmod_step_integer :: "num \<Rightarrow> integer \<times> integer \<Rightarrow> integer \<times> integer"
where
"divmod_step_integer l qr = (let (q, r) = qr
in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))"
instance proof
show "divmod m n = (numeral m div numeral n :: integer, numeral m mod numeral n)"
for m n by (fact divmod_integer'_def)
show "divmod_step l qr = (let (q, r) = qr
in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))" for l and qr :: "integer \<times> integer"
by (fact divmod_step_integer_def)
qed (transfer,
fact le_add_diff_inverse2
unique_euclidean_semiring_numeral_class.div_less
unique_euclidean_semiring_numeral_class.mod_less
unique_euclidean_semiring_numeral_class.div_positive
unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
unique_euclidean_semiring_numeral_class.pos_mod_bound
unique_euclidean_semiring_numeral_class.pos_mod_sign
unique_euclidean_semiring_numeral_class.mod_mult2_eq
unique_euclidean_semiring_numeral_class.div_mult2_eq
unique_euclidean_semiring_numeral_class.discrete)+
end
declare divmod_algorithm_code [where ?'a = integer,
folded integer_of_num_def, unfolded integer_of_num_triv,
code]
lemma integer_of_nat_0: "integer_of_nat 0 = 0"
by transfer simp
lemma integer_of_nat_1: "integer_of_nat 1 = 1"
by transfer simp
lemma integer_of_nat_numeral:
"integer_of_nat (numeral n) = numeral n"
by transfer simp
subsection \<open>Code theorems for target language integers\<close>
text \<open>Constructors\<close>
definition Pos :: "num \<Rightarrow> integer"
where
[simp, code_post]: "Pos = numeral"
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer numeral Pos"
by simp transfer_prover
lemma Pos_fold [code_unfold]:
"numeral Num.One = Pos Num.One"
"numeral (Num.Bit0 k) = Pos (Num.Bit0 k)"
"numeral (Num.Bit1 k) = Pos (Num.Bit1 k)"
by simp_all
definition Neg :: "num \<Rightarrow> integer"
where
[simp, code_abbrev]: "Neg n = - Pos n"
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
by (simp add: Neg_def [abs_def]) transfer_prover
code_datatype "0::integer" Pos Neg
text \<open>A further pair of constructors for generated computations\<close>
context
begin
qualified definition positive :: "num \<Rightarrow> integer"
where [simp]: "positive = numeral"
qualified definition negative :: "num \<Rightarrow> integer"
where [simp]: "negative = uminus \<circ> numeral"
lemma [code_computation_unfold]:
"numeral = positive"
"Pos = positive"
"Neg = negative"
by (simp_all add: fun_eq_iff)
end
text \<open>Auxiliary operations\<close>
lift_definition dup :: "integer \<Rightarrow> integer"
is "\<lambda>k::int. k + k"
.
lemma dup_code [code]:
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
is "\<lambda>m n. numeral m - numeral n :: int"
.
lemma sub_code [code]:
"sub Num.One Num.One = 0"
"sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
"sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
"sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
"sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
"sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
text \<open>Implementations\<close>
lemma one_integer_code [code, code_unfold]:
"1 = Pos Num.One"
by simp
lemma plus_integer_code [code]:
"k + 0 = (k::integer)"
"0 + l = (l::integer)"
"Pos m + Pos n = Pos (m + n)"
"Pos m + Neg n = sub m n"
"Neg m + Pos n = sub n m"
"Neg m + Neg n = Neg (m + n)"
by (transfer, simp)+
lemma uminus_integer_code [code]:
"uminus 0 = (0::integer)"
"uminus (Pos m) = Neg m"
"uminus (Neg m) = Pos m"
by simp_all
lemma minus_integer_code [code]:
"k - 0 = (k::integer)"
"0 - l = uminus (l::integer)"
"Pos m - Pos n = sub m n"
"Pos m - Neg n = Pos (m + n)"
"Neg m - Pos n = Neg (m + n)"
"Neg m - Neg n = sub n m"
by (transfer, simp)+
lemma abs_integer_code [code]:
"\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
by simp
lemma sgn_integer_code [code]:
"sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
by simp
lemma times_integer_code [code]:
"k * 0 = (0::integer)"
"0 * l = (0::integer)"
"Pos m * Pos n = Pos (m * n)"
"Pos m * Neg n = Neg (m * n)"
"Neg m * Pos n = Neg (m * n)"
"Neg m * Neg n = Pos (m * n)"
by simp_all
definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
where
"divmod_integer k l = (k div l, k mod l)"
lemma fst_divmod_integer [simp]:
"fst (divmod_integer k l) = k div l"
by (simp add: divmod_integer_def)
lemma snd_divmod_integer [simp]:
"snd (divmod_integer k l) = k mod l"
by (simp add: divmod_integer_def)
definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
where
"divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
lemma fst_divmod_abs [simp]:
"fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
by (simp add: divmod_abs_def)
lemma snd_divmod_abs [simp]:
"snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
by (simp add: divmod_abs_def)
lemma divmod_abs_code [code]:
"divmod_abs (Pos k) (Pos l) = divmod k l"
"divmod_abs (Neg k) (Neg l) = divmod k l"
"divmod_abs (Neg k) (Pos l) = divmod k l"
"divmod_abs (Pos k) (Neg l) = divmod k l"
"divmod_abs j 0 = (0, \<bar>j\<bar>)"
"divmod_abs 0 j = (0, 0)"
by (simp_all add: prod_eq_iff)
lemma divmod_integer_code [code]:
"divmod_integer k l =
(if k = 0 then (0, 0) else if l = 0 then (0, k) else
(apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
then divmod_abs k l
else (let (r, s) = divmod_abs k l in
if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
proof -
have aux1: "\<And>k l::int. sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0"
by (auto simp add: sgn_if)
have aux2: "\<And>q::int. - int_of_integer k = int_of_integer l * q \<longleftrightarrow> int_of_integer k = int_of_integer l * - q" by auto
show ?thesis
by (simp add: prod_eq_iff integer_eq_iff case_prod_beta aux1)
(auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
qed
lemma div_integer_code [code]:
"k div l = fst (divmod_integer k l)"
by simp
lemma mod_integer_code [code]:
"k mod l = snd (divmod_integer k l)"
by simp
lemma equal_integer_code [code]:
"HOL.equal 0 (0::integer) \<longleftrightarrow> True"
"HOL.equal 0 (Pos l) \<longleftrightarrow> False"
"HOL.equal 0 (Neg l) \<longleftrightarrow> False"
"HOL.equal (Pos k) 0 \<longleftrightarrow> False"
"HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
"HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
"HOL.equal (Neg k) 0 \<longleftrightarrow> False"
"HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
"HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
by (simp_all add: equal)
lemma equal_integer_refl [code nbe]:
"HOL.equal (k::integer) k \<longleftrightarrow> True"
by (fact equal_refl)
lemma less_eq_integer_code [code]:
"0 \<le> (0::integer) \<longleftrightarrow> True"
"0 \<le> Pos l \<longleftrightarrow> True"
"0 \<le> Neg l \<longleftrightarrow> False"
"Pos k \<le> 0 \<longleftrightarrow> False"
"Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
"Pos k \<le> Neg l \<longleftrightarrow> False"
"Neg k \<le> 0 \<longleftrightarrow> True"
"Neg k \<le> Pos l \<longleftrightarrow> True"
"Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
by simp_all
lemma less_integer_code [code]:
"0 < (0::integer) \<longleftrightarrow> False"
"0 < Pos l \<longleftrightarrow> True"
"0 < Neg l \<longleftrightarrow> False"
"Pos k < 0 \<longleftrightarrow> False"
"Pos k < Pos l \<longleftrightarrow> k < l"
"Pos k < Neg l \<longleftrightarrow> False"
"Neg k < 0 \<longleftrightarrow> True"
"Neg k < Pos l \<longleftrightarrow> True"
"Neg k < Neg l \<longleftrightarrow> l < k"
by simp_all
lift_definition num_of_integer :: "integer \<Rightarrow> num"
is "num_of_nat \<circ> nat"
.
lemma num_of_integer_code [code]:
"num_of_integer k = (if k \<le> 1 then Num.One
else let
(l, j) = divmod_integer k 2;
l' = num_of_integer l;
l'' = l' + l'
in if j = 0 then l'' else l'' + Num.One)"
proof -
{
assume "int_of_integer k mod 2 = 1"
then have "nat (int_of_integer k mod 2) = nat 1" by simp
moreover assume *: "1 < int_of_integer k"
ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
have "num_of_nat (nat (int_of_integer k)) =
num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
by simp
then have "num_of_nat (nat (int_of_integer k)) =
num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
by (simp add: mult_2)
with ** have "num_of_nat (nat (int_of_integer k)) =
num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
by simp
}
note aux = this
show ?thesis
by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
not_le integer_eq_iff less_eq_integer_def
nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
mult_2 [where 'a=nat] aux add_One)
qed
lemma nat_of_integer_code [code]:
"nat_of_integer k = (if k \<le> 0 then 0
else let
(l, j) = divmod_integer k 2;
l' = nat_of_integer l;
l'' = l' + l'
in if j = 0 then l'' else l'' + 1)"
proof -
obtain j where k: "k = integer_of_int j"
proof
show "k = integer_of_int (int_of_integer k)" by simp
qed
have *: "nat j mod 2 = nat_of_integer (of_int j mod 2)" if "j \<ge> 0"
using that by transfer (simp add: nat_mod_distrib)
from k show ?thesis
by (auto simp add: split_def Let_def nat_of_integer_def nat_div_distrib mult_2 [symmetric]
minus_mod_eq_mult_div [symmetric] *)
qed
lemma int_of_integer_code [code]:
"int_of_integer k = (if k < 0 then - (int_of_integer (- k))
else if k = 0 then 0
else let
(l, j) = divmod_integer k 2;
l' = 2 * int_of_integer l
in if j = 0 then l' else l' + 1)"
by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
lemma integer_of_int_code [code]:
"integer_of_int k = (if k < 0 then - (integer_of_int (- k))
else if k = 0 then 0
else let
l = 2 * integer_of_int (k div 2);
j = k mod 2
in if j = 0 then l else l + 1)"
by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
hide_const (open) Pos Neg sub dup divmod_abs
subsection \<open>Serializer setup for target language integers\<close>
code_reserved Eval int Integer abs
code_printing
type_constructor integer \<rightharpoonup>
(SML) "IntInf.int"
and (OCaml) "Big'_int.big'_int"
and (Haskell) "Integer"
and (Scala) "BigInt"
and (Eval) "int"
| class_instance integer :: equal \<rightharpoonup>
(Haskell) -
code_printing
constant "0::integer" \<rightharpoonup>
(SML) "!(0/ :/ IntInf.int)"
and (OCaml) "Big'_int.zero'_big'_int"
and (Haskell) "!(0/ ::/ Integer)"
and (Scala) "BigInt(0)"
setup \<open>
fold (fn target =>
Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
#> Numeral.add_code @{const_name Code_Numeral.Neg} (~) Code_Printer.literal_numeral target)
["SML", "OCaml", "Haskell", "Scala"]
\<close>
code_printing
constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
(SML) "IntInf.+ ((_), (_))"
and (OCaml) "Big'_int.add'_big'_int"
and (Haskell) infixl 6 "+"
and (Scala) infixl 7 "+"
and (Eval) infixl 8 "+"
| constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
(SML) "IntInf.~"
and (OCaml) "Big'_int.minus'_big'_int"
and (Haskell) "negate"
and (Scala) "!(- _)"
and (Eval) "~/ _"
| constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
(SML) "IntInf.- ((_), (_))"
and (OCaml) "Big'_int.sub'_big'_int"
and (Haskell) infixl 6 "-"
and (Scala) infixl 7 "-"
and (Eval) infixl 8 "-"
| constant Code_Numeral.dup \<rightharpoonup>
(SML) "IntInf.*/ (2,/ (_))"
and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
and (Haskell) "!(2 * _)"
and (Scala) "!(2 * _)"
and (Eval) "!(2 * _)"
| constant Code_Numeral.sub \<rightharpoonup>
(SML) "!(raise/ Fail/ \"sub\")"
and (OCaml) "failwith/ \"sub\""
and (Haskell) "error/ \"sub\""
and (Scala) "!sys.error(\"sub\")"
| constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
(SML) "IntInf.* ((_), (_))"
and (OCaml) "Big'_int.mult'_big'_int"
and (Haskell) infixl 7 "*"
and (Scala) infixl 8 "*"
and (Eval) infixl 9 "*"
| constant Code_Numeral.divmod_abs \<rightharpoonup>
(SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
and (OCaml) "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)"
and (Haskell) "divMod/ (abs _)/ (abs _)"
and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
| constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
(SML) "!((_ : IntInf.int) = _)"
and (OCaml) "Big'_int.eq'_big'_int"
and (Haskell) infix 4 "=="
and (Scala) infixl 5 "=="
and (Eval) infixl 6 "="
| constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
(SML) "IntInf.<= ((_), (_))"
and (OCaml) "Big'_int.le'_big'_int"
and (Haskell) infix 4 "<="
and (Scala) infixl 4 "<="
and (Eval) infixl 6 "<="
| constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
(SML) "IntInf.< ((_), (_))"
and (OCaml) "Big'_int.lt'_big'_int"
and (Haskell) infix 4 "<"
and (Scala) infixl 4 "<"
and (Eval) infixl 6 "<"
| constant "abs :: integer \<Rightarrow> _" \<rightharpoonup>
(SML) "IntInf.abs"
and (OCaml) "Big'_int.abs'_big'_int"
and (Haskell) "Prelude.abs"
and (Scala) "_.abs"
and (Eval) "abs"
code_identifier
code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
subsection \<open>Type of target language naturals\<close>
typedef natural = "UNIV :: nat set"
morphisms nat_of_natural natural_of_nat ..
setup_lifting type_definition_natural
lemma natural_eq_iff [termination_simp]:
"m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
by transfer rule
lemma natural_eqI:
"nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
using natural_eq_iff [of m n] by simp
lemma nat_of_natural_of_nat_inverse [simp]:
"nat_of_natural (natural_of_nat n) = n"
by transfer rule
lemma natural_of_nat_of_natural_inverse [simp]:
"natural_of_nat (nat_of_natural n) = n"
by transfer rule
instantiation natural :: "{comm_monoid_diff, semiring_1}"
begin
lift_definition zero_natural :: natural
is "0 :: nat"
.
declare zero_natural.rep_eq [simp]
lift_definition one_natural :: natural
is "1 :: nat"
.
declare one_natural.rep_eq [simp]
lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare plus_natural.rep_eq [simp]
lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare minus_natural.rep_eq [simp]
lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare times_natural.rep_eq [simp]
instance proof
qed (transfer, simp add: algebra_simps)+
end
instance natural :: Rings.dvd ..
lemma [transfer_rule]:
"rel_fun pcr_natural (rel_fun pcr_natural HOL.iff) Rings.dvd Rings.dvd"
unfolding dvd_def by transfer_prover
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
proof -
have "rel_fun HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
by (unfold of_nat_def [abs_def]) transfer_prover
then show ?thesis by (simp add: id_def)
qed
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
proof -
have "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
by transfer_prover
then show ?thesis by simp
qed
lemma nat_of_natural_of_nat [simp]:
"nat_of_natural (of_nat n) = n"
by transfer rule
lemma natural_of_nat_of_nat [simp, code_abbrev]:
"natural_of_nat = of_nat"
by transfer rule
lemma of_nat_of_natural [simp]:
"of_nat (nat_of_natural n) = n"
by transfer rule
lemma nat_of_natural_numeral [simp]:
"nat_of_natural (numeral k) = numeral k"
by transfer rule
instantiation natural :: "{linordered_semiring, equal}"
begin
lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
.
declare less_eq_natural.rep_eq [termination_simp]
lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
.
declare less_natural.rep_eq [termination_simp]
lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
.
instance proof
qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
end
lemma [transfer_rule]:
"rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
by (unfold min_def [abs_def]) transfer_prover
lemma [transfer_rule]:
"rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
by (unfold max_def [abs_def]) transfer_prover
lemma nat_of_natural_min [simp]:
"nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
by transfer rule
lemma nat_of_natural_max [simp]:
"nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
by transfer rule
instantiation natural :: unique_euclidean_semiring
begin
lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare divide_natural.rep_eq [simp]
lift_definition modulo_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "modulo :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare modulo_natural.rep_eq [simp]
lift_definition euclidean_size_natural :: "natural \<Rightarrow> nat"
is "euclidean_size :: nat \<Rightarrow> nat"
.
declare euclidean_size_natural.rep_eq [simp]
lift_definition division_segment_natural :: "natural \<Rightarrow> natural"
is "division_segment :: nat \<Rightarrow> nat"
.
declare division_segment_natural.rep_eq [simp]
instance
by (standard; transfer)
(auto simp add: algebra_simps unit_factor_nat_def gr0_conv_Suc)
end
lemma [code]:
"euclidean_size = nat_of_natural"
by (simp add: fun_eq_iff)
lemma [code]:
"division_segment (n::natural) = 1"
by (simp add: natural_eq_iff)
instance natural :: linordered_semidom
by (standard; transfer) simp_all
instance natural :: semiring_parity
by (standard; transfer) simp_all
lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
is "nat :: int \<Rightarrow> nat"
.
lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
is "of_nat :: nat \<Rightarrow> int"
.
lemma natural_of_integer_of_natural [simp]:
"natural_of_integer (integer_of_natural n) = n"
by transfer simp
lemma integer_of_natural_of_integer [simp]:
"integer_of_natural (natural_of_integer k) = max 0 k"
by transfer auto
lemma int_of_integer_of_natural [simp]:
"int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
by transfer rule
lemma integer_of_natural_of_nat [simp]:
"integer_of_natural (of_nat n) = of_nat n"
by transfer rule
lemma [measure_function]:
"is_measure nat_of_natural"
by (rule is_measure_trivial)
subsection \<open>Inductive representation of target language naturals\<close>
lift_definition Suc :: "natural \<Rightarrow> natural"
is Nat.Suc
.
declare Suc.rep_eq [simp]
old_rep_datatype "0::natural" Suc
by (transfer, fact nat.induct nat.inject nat.distinct)+
lemma natural_cases [case_names nat, cases type: natural]:
fixes m :: natural
assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
shows P
using assms by transfer blast
instantiation natural :: size
begin
definition size_nat where [simp, code]: "size_nat = nat_of_natural"
instance ..
end
lemma natural_decr [termination_simp]:
"n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
by transfer simp
lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
by (rule zero_diff)
lemma Suc_natural_minus_one: "Suc n - 1 = n"
by transfer simp
hide_const (open) Suc
subsection \<open>Code refinement for target language naturals\<close>
lift_definition Nat :: "integer \<Rightarrow> natural"
is nat
.
lemma [code_post]:
"Nat 0 = 0"
"Nat 1 = 1"
"Nat (numeral k) = numeral k"
by (transfer, simp)+
lemma [code abstype]:
"Nat (integer_of_natural n) = n"
by transfer simp
lemma [code]:
"natural_of_nat n = natural_of_integer (integer_of_nat n)"
by transfer simp
lemma [code abstract]:
"integer_of_natural (natural_of_integer k) = max 0 k"
by simp
lemma [code_abbrev]:
"natural_of_integer (Code_Numeral.Pos k) = numeral k"
by transfer simp
lemma [code abstract]:
"integer_of_natural 0 = 0"
by transfer simp
lemma [code abstract]:
"integer_of_natural 1 = 1"
by transfer simp
lemma [code abstract]:
"integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
by transfer simp
lemma [code]:
"nat_of_natural = nat_of_integer \<circ> integer_of_natural"
by transfer (simp add: fun_eq_iff)
lemma [code, code_unfold]:
"case_natural f g n = (if n = 0 then f else g (n - 1))"
by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
declare natural.rec [code del]
lemma [code abstract]:
"integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
by transfer simp
lemma [code abstract]:
"integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
by transfer simp
lemma [code abstract]:
"integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
by transfer simp
lemma [code abstract]:
"integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
by transfer (simp add: zdiv_int)
lemma [code abstract]:
"integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
by transfer (simp add: zmod_int)
lemma [code]:
"HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
by transfer (simp add: equal)
lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
by (rule equal_class.equal_refl)
lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
by transfer simp
lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
by transfer simp
hide_const (open) Nat
lifting_update integer.lifting
lifting_forget integer.lifting
lifting_update natural.lifting
lifting_forget natural.lifting
code_reflect Code_Numeral
datatypes natural
functions "Code_Numeral.Suc" "0 :: natural" "1 :: natural"
"plus :: natural \<Rightarrow> _" "minus :: natural \<Rightarrow> _"
"times :: natural \<Rightarrow> _" "divide :: natural \<Rightarrow> _"
"modulo :: natural \<Rightarrow> _"
integer_of_natural natural_of_integer
end