Theory Code_Numeral

theory Code_Numeral
imports Divides
```(*  Title:      HOL/Code_Numeral.thy
Author:     Florian Haftmann, TU Muenchen
*)

section ‹Numeric types for code generation onto target language numerals only›

theory Code_Numeral
imports Divides Lifting
begin

subsection ‹Type of target language integers›

typedef integer = "UNIV :: int set"
morphisms int_of_integer integer_of_int ..

setup_lifting type_definition_integer

lemma integer_eq_iff:
"k = l ⟷ int_of_integer k = int_of_integer l"
by transfer rule

lemma integer_eqI:
"int_of_integer k = int_of_integer l ⟹ 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 ⇒ integer ⇒ integer"
is "plus :: int ⇒ int ⇒ int"
.

declare plus_integer.rep_eq [simp]

lift_definition uminus_integer :: "integer ⇒ integer"
is "uminus :: int ⇒ int"
.

declare uminus_integer.rep_eq [simp]

lift_definition minus_integer :: "integer ⇒ integer ⇒ integer"
is "minus :: int ⇒ int ⇒ int"
.

declare minus_integer.rep_eq [simp]

lift_definition times_integer :: "integer ⇒ integer ⇒ integer"
is "times :: int ⇒ int ⇒ int"
.

declare times_integer.rep_eq [simp]

instance proof

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 (=) pcr_integer (of_bool :: bool ⇒ int) (of_bool :: bool ⇒ integer)"
by (unfold of_bool_def [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun (=) pcr_integer (of_nat :: nat ⇒ int) (of_nat :: nat ⇒ integer)"
by (rule transfer_rule_of_nat) transfer_prover+

lemma [transfer_rule]:
"rel_fun (=) pcr_integer (λk :: int. k :: int) (of_int :: int ⇒ integer)"
proof -
have "rel_fun HOL.eq pcr_integer (of_int :: int ⇒ int) (of_int :: int ⇒ 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 ⇒ int) (numeral :: num ⇒ integer)"
by (rule transfer_rule_numeral) transfer_prover+

lemma [transfer_rule]:
"rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ ⇒ _ ⇒ int) (Num.sub :: _ ⇒ _ ⇒ integer)"
by (unfold Num.sub_def [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun pcr_integer (rel_fun (=) pcr_integer) (power :: _ ⇒ _ ⇒ int) (power :: _ ⇒ _ ⇒ integer)"
by (unfold power_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 ⇒ integer"
is "of_nat :: nat ⇒ 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 ⇒ 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"

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 ⇒ 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 ⇒ integer"
is "abs :: int ⇒ int"
.

declare abs_integer.rep_eq [simp]

lift_definition sgn_integer :: "integer ⇒ integer"
is "sgn :: int ⇒ int"
.

declare sgn_integer.rep_eq [simp]

lift_definition less_eq_integer :: "integer ⇒ integer ⇒ bool"
is "less_eq :: int ⇒ int ⇒ bool"
.

lift_definition less_integer :: "integer ⇒ integer ⇒ bool"
is "less :: int ⇒ int ⇒ bool"
.

lift_definition equal_integer :: "integer ⇒ integer ⇒ bool"
is "HOL.equal :: int ⇒ int ⇒ 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 :: _ ⇒ _ ⇒ int) (min :: _ ⇒ _ ⇒ integer)"
by (unfold min_def [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ ⇒ _ ⇒ int) (max :: _ ⇒ _ ⇒ 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 ≤ 0 ⟹ 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 ⇒ integer ⇒ integer"
is "divide :: int ⇒ int ⇒ int"
.

declare divide_integer.rep_eq [simp]

lift_definition modulo_integer :: "integer ⇒ integer ⇒ integer"
is "modulo :: int ⇒ int ⇒ int"
.

declare modulo_integer.rep_eq [simp]

lift_definition euclidean_size_integer :: "integer ⇒ nat"
is "euclidean_size :: int ⇒ nat"
.

declare euclidean_size_integer.rep_eq [simp]

lift_definition division_segment_integer :: "integer ⇒ integer"
is "division_segment :: int ⇒ int"
.

declare division_segment_integer.rep_eq [simp]

instance
by (standard; transfer)
(use mult_le_mono2 [of 1] in ‹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›)

end

lemma [code]:
"euclidean_size = nat_of_integer ∘ abs"

lemma [code]:
"division_segment (k :: integer) = (if k ≥ 0 then 1 else - 1)"

instance integer :: ring_parity
by (standard; transfer) (simp_all add: of_nat_div division_segment_int_def)

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_integer pcr_integer) (push_bit :: _ ⇒ _ ⇒ int) (push_bit :: _ ⇒ _ ⇒ integer)"
by (unfold push_bit_eq_mult [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_integer pcr_integer) (take_bit :: _ ⇒ _ ⇒ int) (take_bit :: _ ⇒ _ ⇒ integer)"
by (unfold take_bit_eq_mod [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_integer pcr_integer) (drop_bit :: _ ⇒ _ ⇒ int) (drop_bit :: _ ⇒ _ ⇒ integer)"
by (unfold drop_bit_eq_div [abs_def]) transfer_prover

instantiation integer :: unique_euclidean_semiring_numeral
begin

definition divmod_integer :: "num ⇒ num ⇒ integer × integer"
where
divmod_integer'_def: "divmod_integer m n = (numeral m div numeral n, numeral m mod numeral n)"

definition divmod_step_integer :: "num ⇒ integer × integer ⇒ integer × integer"
where
"divmod_step_integer l qr = (let (q, r) = qr
in if r ≥ 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 ≥ numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))" for l and qr :: "integer × integer"
by (fact divmod_step_integer_def)
qed (transfer,
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 ‹Code theorems for target language integers›

text ‹Constructors›

definition Pos :: "num ⇒ 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 ⇒ integer"
where
[simp, code_abbrev]: "Neg n = - Pos n"

lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (λn. - numeral n) Neg"
by (simp add: Neg_def [abs_def]) transfer_prover

code_datatype "0::integer" Pos Neg

text ‹A further pair of constructors for generated computations›

context
begin

qualified definition positive :: "num ⇒ integer"
where [simp]: "positive = numeral"

qualified definition negative :: "num ⇒ integer"
where [simp]: "negative = uminus ∘ numeral"

lemma [code_computation_unfold]:
"numeral = positive"
"Pos = positive"
"Neg = negative"

end

text ‹Auxiliary operations›

lift_definition dup :: "integer ⇒ integer"
is "λk::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 ⇒ num ⇒ integer"
is "λm 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 ‹Implementations›

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]:
"¦k¦ = (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 ⇒ integer ⇒ integer × 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"

lemma snd_divmod_integer [simp]:
"snd (divmod_integer k l) = k mod l"

definition divmod_abs :: "integer ⇒ integer ⇒ integer × integer"
where
"divmod_abs k l = (¦k¦ div ¦l¦, ¦k¦ mod ¦l¦)"

lemma fst_divmod_abs [simp]:
"fst (divmod_abs k l) = ¦k¦ div ¦l¦"

lemma snd_divmod_abs [simp]:
"snd (divmod_abs k l) = ¦k¦ mod ¦l¦"

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, ¦j¦)"
"divmod_abs 0 j = (0, 0)"

lemma divmod_integer_code [code]:
"divmod_integer k l =
(if k = 0 then (0, 0) else if l = 0 then (0, k) else
(apsnd ∘ times ∘ 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, ¦l¦ - s))))"
proof -
have aux1: "⋀k l::int. sgn k = sgn l ⟷ k = 0 ∧ l = 0 ∨ 0 < l ∧ 0 < k ∨ l < 0 ∧ k < 0"
have aux2: "⋀q::int. - int_of_integer k = int_of_integer l * q ⟷ 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

definition bit_cut_integer :: "integer ⇒ integer × bool"
where "bit_cut_integer k = (k div 2, odd k)"

lemma bit_cut_integer_code [code]:
"bit_cut_integer k = (if k = 0 then (0, False)
else let (r, s) = Code_Numeral.divmod_abs k 2
in (if k > 0 then r else - r - s, s = 1))"
proof -
have "bit_cut_integer k = (let (r, s) = divmod_integer k 2 in (r, s = 1))"
by (simp add: divmod_integer_def bit_cut_integer_def odd_iff_mod_2_eq_one)
then show ?thesis
qed

lemma equal_integer_code [code]:
"HOL.equal 0 (0::integer) ⟷ True"
"HOL.equal 0 (Pos l) ⟷ False"
"HOL.equal 0 (Neg l) ⟷ False"
"HOL.equal (Pos k) 0 ⟷ False"
"HOL.equal (Pos k) (Pos l) ⟷ HOL.equal k l"
"HOL.equal (Pos k) (Neg l) ⟷ False"
"HOL.equal (Neg k) 0 ⟷ False"
"HOL.equal (Neg k) (Pos l) ⟷ False"
"HOL.equal (Neg k) (Neg l) ⟷ HOL.equal k l"

lemma equal_integer_refl [code nbe]:
"HOL.equal (k::integer) k ⟷ True"
by (fact equal_refl)

lemma less_eq_integer_code [code]:
"0 ≤ (0::integer) ⟷ True"
"0 ≤ Pos l ⟷ True"
"0 ≤ Neg l ⟷ False"
"Pos k ≤ 0 ⟷ False"
"Pos k ≤ Pos l ⟷ k ≤ l"
"Pos k ≤ Neg l ⟷ False"
"Neg k ≤ 0 ⟷ True"
"Neg k ≤ Pos l ⟷ True"
"Neg k ≤ Neg l ⟷ l ≤ k"
by simp_all

lemma less_integer_code [code]:
"0 < (0::integer) ⟷ False"
"0 < Pos l ⟷ True"
"0 < Neg l ⟷ False"
"Pos k < 0 ⟷ False"
"Pos k < Pos l ⟷ k < l"
"Pos k < Neg l ⟷ False"
"Neg k < 0 ⟷ True"
"Neg k < Pos l ⟷ True"
"Neg k < Neg l ⟷ l < k"
by simp_all

lift_definition num_of_integer :: "integer ⇒ num"
is "num_of_nat ∘ nat"
.

lemma num_of_integer_code [code]:
"num_of_integer k = (if k ≤ 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)"
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
qed

lemma nat_of_integer_code [code]:
"nat_of_integer k = (if k ≤ 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 ≥ 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 ‹Serializer setup for target language integers›

code_reserved Eval int Integer abs

code_printing
type_constructor integer ⇀
(SML) "IntInf.int"
and (OCaml) "Big'_int.big'_int"
and (Scala) "BigInt"
and (Eval) "int"
| class_instance integer :: equal ⇀

code_printing
constant "0::integer" ⇀
(SML) "!(0/ :/ IntInf.int)"
and (OCaml) "Big'_int.zero'_big'_int"
and (Scala) "BigInt(0)"

setup ‹
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)
›

code_printing
constant "plus :: integer ⇒ _ ⇒ _" ⇀
(SML) "IntInf.+ ((_), (_))"
and (Scala) infixl 7 "+"
and (Eval) infixl 8 "+"
| constant "uminus :: integer ⇒ _" ⇀
(SML) "IntInf.~"
and (OCaml) "Big'_int.minus'_big'_int"
and (Scala) "!(- _)"
and (Eval) "~/ _"
| constant "minus :: integer ⇒ _" ⇀
(SML) "IntInf.- ((_), (_))"
and (OCaml) "Big'_int.sub'_big'_int"
and (Scala) infixl 7 "-"
and (Eval) infixl 8 "-"
| constant Code_Numeral.dup ⇀
(SML) "IntInf.*/ (2,/ (_))"
and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
and (Scala) "!(2 * _)"
and (Eval) "!(2 * _)"
| constant Code_Numeral.sub ⇀
(SML) "!(raise/ Fail/ \"sub\")"
and (OCaml) "failwith/ \"sub\""
and (Scala) "!sys.error(\"sub\")"
| constant "times :: integer ⇒ _ ⇒ _" ⇀
(SML) "IntInf.* ((_), (_))"
and (OCaml) "Big'_int.mult'_big'_int"
and (Scala) infixl 8 "*"
and (Eval) infixl 9 "*"
| constant Code_Numeral.divmod_abs ⇀
(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 ⇒ _ ⇒ bool" ⇀
(SML) "!((_ : IntInf.int) = _)"
and (OCaml) "Big'_int.eq'_big'_int"
and (Scala) infixl 5 "=="
and (Eval) infixl 6 "="
| constant "less_eq :: integer ⇒ _ ⇒ bool" ⇀
(SML) "IntInf.<= ((_), (_))"
and (OCaml) "Big'_int.le'_big'_int"
and (Scala) infixl 4 "<="
and (Eval) infixl 6 "<="
| constant "less :: integer ⇒ _ ⇒ bool" ⇀
(SML) "IntInf.< ((_), (_))"
and (OCaml) "Big'_int.lt'_big'_int"
and (Scala) infixl 4 "<"
and (Eval) infixl 6 "<"
| constant "abs :: integer ⇒ _" ⇀
(SML) "IntInf.abs"
and (OCaml) "Big'_int.abs'_big'_int"
and (Scala) "_.abs"
and (Eval) "abs"

code_identifier
code_module Code_Numeral ⇀ (SML) Arith and (OCaml) Arith and (Haskell) Arith

subsection ‹Type of target language naturals›

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 ⟷ nat_of_natural m = nat_of_natural n"
by transfer rule

lemma natural_eqI:
"nat_of_natural m = nat_of_natural n ⟹ 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 ⇒ natural ⇒ natural"
is "plus :: nat ⇒ nat ⇒ nat"
.

declare plus_natural.rep_eq [simp]

lift_definition minus_natural :: "natural ⇒ natural ⇒ natural"
is "minus :: nat ⇒ nat ⇒ nat"
.

declare minus_natural.rep_eq [simp]

lift_definition times_natural :: "natural ⇒ natural ⇒ natural"
is "times :: nat ⇒ nat ⇒ nat"
.

declare times_natural.rep_eq [simp]

instance proof

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 (=) pcr_natural (of_bool :: bool ⇒ nat) (of_bool :: bool ⇒ natural)"
by (unfold of_bool_def [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun HOL.eq pcr_natural (λn::nat. n) (of_nat :: nat ⇒ natural)"
proof -
have "rel_fun HOL.eq pcr_natural (of_nat :: nat ⇒ nat) (of_nat :: nat ⇒ 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 ⇒ nat) (numeral :: num ⇒ natural)"
proof -
have "rel_fun HOL.eq pcr_natural (numeral :: num ⇒ nat) (λn. of_nat (numeral n))"
by transfer_prover
then show ?thesis by simp
qed

lemma [transfer_rule]:
"rel_fun pcr_natural (rel_fun (=) pcr_natural) (power :: _ ⇒ _ ⇒ nat) (power :: _ ⇒ _ ⇒ natural)"
by (unfold power_def [abs_def]) transfer_prover

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 ⇒ natural ⇒ bool"
is "less_eq :: nat ⇒ nat ⇒ bool"
.

declare less_eq_natural.rep_eq [termination_simp]

lift_definition less_natural :: "natural ⇒ natural ⇒ bool"
is "less :: nat ⇒ nat ⇒ bool"
.

declare less_natural.rep_eq [termination_simp]

lift_definition equal_natural :: "natural ⇒ natural ⇒ bool"
is "HOL.equal :: nat ⇒ nat ⇒ 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 :: _ ⇒ _ ⇒ nat) (min :: _ ⇒ _ ⇒ natural)"
by (unfold min_def [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ ⇒ _ ⇒ nat) (max :: _ ⇒ _ ⇒ 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 ⇒ natural ⇒ natural"
is "divide :: nat ⇒ nat ⇒ nat"
.

declare divide_natural.rep_eq [simp]

lift_definition modulo_natural :: "natural ⇒ natural ⇒ natural"
is "modulo :: nat ⇒ nat ⇒ nat"
.

declare modulo_natural.rep_eq [simp]

lift_definition euclidean_size_natural :: "natural ⇒ nat"
is "euclidean_size :: nat ⇒ nat"
.

declare euclidean_size_natural.rep_eq [simp]

lift_definition division_segment_natural :: "natural ⇒ natural"
is "division_segment :: nat ⇒ 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"

lemma [code]:
"division_segment (n::natural) = 1"

instance natural :: linordered_semidom
by (standard; transfer) simp_all

instance natural :: semiring_parity
by (standard; transfer) simp_all

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_natural pcr_natural) (push_bit :: _ ⇒ _ ⇒ nat) (push_bit :: _ ⇒ _ ⇒ natural)"
by (unfold push_bit_eq_mult [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_natural pcr_natural) (take_bit :: _ ⇒ _ ⇒ nat) (take_bit :: _ ⇒ _ ⇒ natural)"
by (unfold take_bit_eq_mod [abs_def]) transfer_prover

lemma [transfer_rule]:
"rel_fun (=) (rel_fun pcr_natural pcr_natural) (drop_bit :: _ ⇒ _ ⇒ nat) (drop_bit :: _ ⇒ _ ⇒ natural)"
by (unfold drop_bit_eq_div [abs_def]) transfer_prover

lift_definition natural_of_integer :: "integer ⇒ natural"
is "nat :: int ⇒ nat"
.

lift_definition integer_of_natural :: "natural ⇒ integer"
is "of_nat :: nat ⇒ 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 ‹Inductive representation of target language naturals›

lift_definition Suc :: "natural ⇒ 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 "⋀n. m = of_nat n ⟹ 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 ≠ 0 ⟹ 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 ‹Code refinement for target language naturals›

lift_definition Nat :: "integer ⇒ 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 ∘ integer_of_natural"

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"

lemma [code abstract]:
"integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"

lemma [code]:
"HOL.equal m n ⟷ HOL.equal (integer_of_natural m) (integer_of_natural n)"

lemma [code nbe]: "HOL.equal n (n::natural) ⟷ True"
by (rule equal_class.equal_refl)

lemma [code]: "m ≤ n ⟷ integer_of_natural m ≤ integer_of_natural n"
by transfer simp

lemma [code]: "m < n ⟷ 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 ⇒ _" "minus :: natural ⇒ _"
"times :: natural ⇒ _" "divide :: natural ⇒ _"
"modulo :: natural ⇒ _"
integer_of_natural natural_of_integer

end
```