(* Title: HOL/Code_Numeral.thy
Author: Florian Haftmann, TU Muenchen
*)
header {* Numeric types for code generation onto target language numerals only *}
theory Code_Numeral
imports Nat_Transfer Divides Lifting
begin
subsection {* Type of target language integers *}
typedef integer = "UNIV \<Colon> int set"
morphisms int_of_integer integer_of_int ..
setup_lifting (no_code) 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
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
by (unfold of_nat_def [abs_def]) 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 (unfold of_int_of_nat [abs_def]) 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)"
proof -
have "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (\<lambda>n. of_int (numeral n))"
by transfer_prover
then show ?thesis by simp
qed
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
instantiation integer :: "{ring_div, equal, linordered_idom}"
begin
lift_definition div_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "Divides.div :: int \<Rightarrow> int \<Rightarrow> int"
.
declare div_integer.rep_eq [simp]
lift_definition mod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
is "Divides.mod :: int \<Rightarrow> int \<Rightarrow> int"
.
declare mod_integer.rep_eq [simp]
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 proof
qed (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
instance integer :: semiring_numeral_div
by intro_classes (transfer,
fact semiring_numeral_div_class.diff_invert_add1
semiring_numeral_div_class.le_add_diff_inverse2
semiring_numeral_div_class.mult_div_cancel
semiring_numeral_div_class.div_less
semiring_numeral_div_class.mod_less
semiring_numeral_div_class.div_positive
semiring_numeral_div_class.mod_less_eq_dividend
semiring_numeral_div_class.pos_mod_bound
semiring_numeral_div_class.pos_mod_sign
semiring_numeral_div_class.mod_mult2_eq
semiring_numeral_div_class.div_mult2_eq
semiring_numeral_div_class.discrete)+
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 \<Rightarrow> integer"
where
[simp, code_abbrev]: "Pos = numeral"
lemma [transfer_rule]:
"rel_fun HOL.eq pcr_integer numeral Pos"
by simp transfer_prover
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 {* Auxiliary operations *}
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 {* 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]:
"\<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 [simp]:
"fst (divmod_integer k l) = k div l"
by (simp add: divmod_integer_def)
lemma snd_divmod [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 integer_of_num :: "num \<Rightarrow> integer"
is "numeral :: num \<Rightarrow> int"
.
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 (transfer, simp only: numeral.simps Let_def)+
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 = integer_of_int j"
proof
show "k = integer_of_int (int_of_integer k)" by simp
qed
moreover have "2 * (j div 2) = j - j mod 2"
by (simp add: zmult_div_cancel mult.commute)
ultimately show ?thesis
by (auto simp add: split_def Let_def mod_integer_def nat_of_integer_def not_le
nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
(auto simp add: mult_2 [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 zmult_div_cancel)
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, j) = divmod_int k 2;
l' = 2 * integer_of_int l
in if j = 0 then l' else l' + 1)"
by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
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 \<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"
and (OCaml) "Big'_int.zero'_big'_int"
and (Haskell) "0"
and (Scala) "BigInt(0)"
setup {*
fold (Numeral.add_code @{const_name Code_Numeral.Pos}
false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
*}
setup {*
fold (Numeral.add_code @{const_name Code_Numeral.Neg}
true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
*}
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 "<"
code_identifier
code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
subsection {* Type of target language naturals *}
typedef natural = "UNIV \<Colon> nat set"
morphisms nat_of_natural natural_of_nat ..
setup_lifting (no_code) 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
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 :: "{semiring_div, equal, linordered_semiring}"
begin
lift_definition div_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "Divides.div :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare div_natural.rep_eq [simp]
lift_definition mod_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
is "Divides.mod :: nat \<Rightarrow> nat \<Rightarrow> nat"
.
declare mod_natural.rep_eq [simp]
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
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 {* Inductive representation of target language naturals *}
lift_definition Suc :: "natural \<Rightarrow> natural"
is Nat.Suc
.
declare Suc.rep_eq [simp]
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
lemma [simp, code]:
"size_natural = nat_of_natural"
proof (rule ext)
fix n
show "size_natural n = nat_of_natural n"
by (induct n) simp_all
qed
lemma [simp, code]:
"size = nat_of_natural"
proof (rule ext)
fix n
show "size n = nat_of_natural n"
by (induct n) simp_all
qed
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 simp
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 \<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 abstract]:
"integer_of_natural (natural_of_nat n) = of_nat n"
by 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 add: of_nat_mult)
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 (simp add: equal)
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 integer_of_natural natural_of_integer
end