theory Target_Numeral
imports Main Code_Nat
begin
subsection {* Type of target language numerals *}
typedef (open) int = "UNIV \<Colon> int set"
morphisms int_of of_int ..
hide_type (open) int
hide_const (open) of_int
lemma int_eq_iff:
"k = l \<longleftrightarrow> int_of k = int_of l"
using int_of_inject [of k l] ..
lemma int_eqI:
"int_of k = int_of l \<Longrightarrow> k = l"
using int_eq_iff [of k l] by simp
lemma int_of_int [simp]:
"int_of (Target_Numeral.of_int k) = k"
using of_int_inverse [of k] by simp
lemma of_int_of [simp]:
"Target_Numeral.of_int (int_of k) = k"
using int_of_inverse [of k] by simp
hide_fact (open) int_eq_iff int_eqI
instantiation Target_Numeral.int :: ring_1
begin
definition
"0 = Target_Numeral.of_int 0"
lemma int_of_zero [simp]:
"int_of 0 = 0"
by (simp add: zero_int_def)
definition
"1 = Target_Numeral.of_int 1"
lemma int_of_one [simp]:
"int_of 1 = 1"
by (simp add: one_int_def)
definition
"k + l = Target_Numeral.of_int (int_of k + int_of l)"
lemma int_of_plus [simp]:
"int_of (k + l) = int_of k + int_of l"
by (simp add: plus_int_def)
definition
"- k = Target_Numeral.of_int (- int_of k)"
lemma int_of_uminus [simp]:
"int_of (- k) = - int_of k"
by (simp add: uminus_int_def)
definition
"k - l = Target_Numeral.of_int (int_of k - int_of l)"
lemma int_of_minus [simp]:
"int_of (k - l) = int_of k - int_of l"
by (simp add: minus_int_def)
definition
"k * l = Target_Numeral.of_int (int_of k * int_of l)"
lemma int_of_times [simp]:
"int_of (k * l) = int_of k * int_of l"
by (simp add: times_int_def)
instance proof
qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps)
end
lemma int_of_of_nat [simp]:
"int_of (of_nat n) = of_nat n"
by (induct n) simp_all
definition nat_of :: "Target_Numeral.int \<Rightarrow> nat" where
"nat_of k = Int.nat (int_of k)"
lemma nat_of_of_nat [simp]:
"nat_of (of_nat n) = n"
by (simp add: nat_of_def)
lemma int_of_of_int [simp]:
"int_of (of_int k) = k"
by (induct k) (simp_all, simp only: neg_numeral_def numeral_One int_of_uminus int_of_one)
lemma of_int_of_int [simp, code_abbrev]:
"Target_Numeral.of_int = of_int"
by rule (simp add: Target_Numeral.int_eq_iff)
lemma int_of_numeral [simp]:
"int_of (numeral k) = numeral k"
using int_of_of_int [of "numeral k"] by simp
lemma int_of_neg_numeral [simp]:
"int_of (neg_numeral k) = neg_numeral k"
by (simp only: neg_numeral_def int_of_uminus) simp
lemma int_of_sub [simp]:
"int_of (Num.sub k l) = Num.sub k l"
by (simp only: Num.sub_def int_of_minus int_of_numeral)
instantiation Target_Numeral.int :: "{ring_div, equal, linordered_idom}"
begin
definition
"k div l = of_int (int_of k div int_of l)"
lemma int_of_div [simp]:
"int_of (k div l) = int_of k div int_of l"
by (simp add: div_int_def)
definition
"k mod l = of_int (int_of k mod int_of l)"
lemma int_of_mod [simp]:
"int_of (k mod l) = int_of k mod int_of l"
by (simp add: mod_int_def)
definition
"\<bar>k\<bar> = of_int \<bar>int_of k\<bar>"
lemma int_of_abs [simp]:
"int_of \<bar>k\<bar> = \<bar>int_of k\<bar>"
by (simp add: abs_int_def)
definition
"sgn k = of_int (sgn (int_of k))"
lemma int_of_sgn [simp]:
"int_of (sgn k) = sgn (int_of k)"
by (simp add: sgn_int_def)
definition
"k \<le> l \<longleftrightarrow> int_of k \<le> int_of l"
definition
"k < l \<longleftrightarrow> int_of k < int_of l"
definition
"HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)"
instance proof
qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps
less_eq_int_def less_int_def equal_int_def equal)
end
lemma int_of_min [simp]:
"int_of (min k l) = min (int_of k) (int_of l)"
by (simp add: min_def less_eq_int_def)
lemma int_of_max [simp]:
"int_of (max k l) = max (int_of k) (int_of l)"
by (simp add: max_def less_eq_int_def)
subsection {* Code theorems for target language numerals *}
text {* Constructors *}
definition Pos :: "num \<Rightarrow> Target_Numeral.int" where
[simp, code_abbrev]: "Pos = numeral"
definition Neg :: "num \<Rightarrow> Target_Numeral.int" where
[simp, code_abbrev]: "Neg = neg_numeral"
code_datatype "0::Target_Numeral.int" Pos Neg
text {* Auxiliary operations *}
definition dup :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int" where
[simp]: "dup k = k + k"
lemma dup_code [code]:
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
unfolding Pos_def Neg_def neg_numeral_def
by (simp_all add: numeral_Bit0)
definition sub :: "num \<Rightarrow> num \<Rightarrow> Target_Numeral.int" where
[simp]: "sub m n = numeral m - numeral n"
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"
unfolding sub_def dup_def numeral.simps Pos_def Neg_def
neg_numeral_def numeral_BitM
by (simp_all only: algebra_simps add.comm_neutral)
text {* Implementations *}
lemma one_int_code [code, code_unfold]:
"1 = Pos Num.One"
by simp
lemma plus_int_code [code]:
"k + 0 = (k::Target_Numeral.int)"
"0 + l = (l::Target_Numeral.int)"
"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 simp_all
lemma uminus_int_code [code]:
"uminus 0 = (0::Target_Numeral.int)"
"uminus (Pos m) = Neg m"
"uminus (Neg m) = Pos m"
by simp_all
lemma minus_int_code [code]:
"k - 0 = (k::Target_Numeral.int)"
"0 - l = uminus (l::Target_Numeral.int)"
"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 simp_all
lemma times_int_code [code]:
"k * 0 = (0::Target_Numeral.int)"
"0 * l = (0::Target_Numeral.int)"
"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 :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" where
"divmod k l = (k div l, k mod l)"
lemma fst_divmod [simp]:
"fst (divmod k l) = k div l"
by (simp add: divmod_def)
lemma snd_divmod [simp]:
"snd (divmod k l) = k mod l"
by (simp add: divmod_def)
definition divmod_abs :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" 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_terminate_code [code]:
"divmod_abs (Neg k) (Neg l) = divmod_abs (Pos k) (Pos l)"
"divmod_abs (Neg k) (Pos l) = divmod_abs (Pos k) (Pos l)"
"divmod_abs (Pos k) (Neg l) = divmod_abs (Pos k) (Pos l)"
"divmod_abs j 0 = (0, \<bar>j\<bar>)"
"divmod_abs 0 j = (0, 0)"
by (simp_all add: prod_eq_iff)
lemma divmod_abs_rec_code [code]:
"divmod_abs (Pos k) (Pos l) =
(let j = sub k l in
if j < 0 then (0, Pos k)
else let (q, r) = divmod_abs j (Pos l) in (q + 1, r))"
by (auto simp add: prod_eq_iff Target_Numeral.int_eq_iff Let_def prod_case_beta
sub_non_negative sub_negative div_pos_pos_trivial mod_pos_pos_trivial div_pos_geq mod_pos_geq)
lemma divmod_code [code]: "divmod 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 k = int_of l * q \<longleftrightarrow> int_of k = int_of l * - q" by auto
show ?thesis
by (simp add: prod_eq_iff Target_Numeral.int_eq_iff prod_case_beta aux1)
(auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
qed
lemma div_int_code [code]:
"k div l = fst (divmod k l)"
by simp
lemma div_mod_code [code]:
"k mod l = snd (divmod k l)"
by simp
lemma equal_int_code [code]:
"HOL.equal 0 (0::Target_Numeral.int) \<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 Target_Numeral.int_eq_iff)
lemma equal_int_refl [code nbe]:
"HOL.equal (k::Target_Numeral.int) k \<longleftrightarrow> True"
by (fact equal_refl)
lemma less_eq_int_code [code]:
"0 \<le> (0::Target_Numeral.int) \<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 add: less_eq_int_def)
lemma less_int_code [code]:
"0 < (0::Target_Numeral.int) \<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 add: less_int_def)
lemma nat_of_code [code]:
"nat_of (Neg k) = 0"
"nat_of 0 = 0"
"nat_of (Pos k) = nat_of_num k"
by (simp_all add: nat_of_def nat_of_num_numeral)
lemma int_of_code [code]:
"int_of (Neg k) = neg_numeral k"
"int_of 0 = 0"
"int_of (Pos k) = numeral k"
by simp_all
lemma of_int_code [code]:
"Target_Numeral.of_int (Int.Neg k) = neg_numeral k"
"Target_Numeral.of_int 0 = 0"
"Target_Numeral.of_int (Int.Pos k) = numeral k"
by simp_all
definition num_of_int :: "Target_Numeral.int \<Rightarrow> num" where
"num_of_int = num_of_nat \<circ> nat_of"
lemma num_of_int_code [code]:
"num_of_int k = (if k \<le> 1 then Num.One
else let
(l, j) = divmod k 2;
l' = num_of_int l + num_of_int l
in if j = 0 then l' else l' + Num.One)"
proof -
{
assume "int_of k mod 2 = 1"
then have "nat (int_of k mod 2) = nat 1" by simp
moreover assume *: "1 < int_of k"
ultimately have **: "nat (int_of k) mod 2 = 1" by (simp add: nat_mod_distrib)
have "num_of_nat (nat (int_of k)) =
num_of_nat (2 * (nat (int_of k) div 2) + nat (int_of k) mod 2)"
by simp
then have "num_of_nat (nat (int_of k)) =
num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + nat (int_of k) mod 2)"
by (simp add: mult_2)
with ** have "num_of_nat (nat (int_of k)) =
num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + 1)"
by simp
}
note aux = this
show ?thesis
by (auto simp add: num_of_int_def nat_of_def Let_def prod_case_beta
not_le Target_Numeral.int_eq_iff less_eq_int_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
hide_const (open) int_of nat_of Pos Neg sub dup divmod_abs num_of_int
subsection {* Serializer setup for target language numerals *}
code_type Target_Numeral.int
(SML "IntInf.int")
(OCaml "Big'_int.big'_int")
(Haskell "Integer")
(Scala "BigInt")
(Eval "int")
code_instance Target_Numeral.int :: equal
(Haskell -)
code_const "0::Target_Numeral.int"
(SML "0")
(OCaml "Big'_int.zero'_big'_int")
(Haskell "0")
(Scala "BigInt(0)")
setup {*
fold (Numeral.add_code @{const_name Target_Numeral.Pos}
false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
*}
setup {*
fold (Numeral.add_code @{const_name Target_Numeral.Neg}
true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
*}
code_const "plus :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
(SML "IntInf.+ ((_), (_))")
(OCaml "Big'_int.add'_big'_int")
(Haskell infixl 6 "+")
(Scala infixl 7 "+")
(Eval infixl 8 "+")
code_const "uminus :: Target_Numeral.int \<Rightarrow> _"
(SML "IntInf.~")
(OCaml "Big'_int.minus'_big'_int")
(Haskell "negate")
(Scala "!(- _)")
(Eval "~/ _")
code_const "minus :: Target_Numeral.int \<Rightarrow> _"
(SML "IntInf.- ((_), (_))")
(OCaml "Big'_int.sub'_big'_int")
(Haskell infixl 6 "-")
(Scala infixl 7 "-")
(Eval infixl 8 "-")
code_const Target_Numeral.dup
(SML "IntInf.*/ (2,/ (_))")
(OCaml "Big'_int.mult'_big'_int/ 2")
(Haskell "!(2 * _)")
(Scala "!(2 * _)")
(Eval "!(2 * _)")
code_const Target_Numeral.sub
(SML "!(raise/ Fail/ \"sub\")")
(OCaml "failwith/ \"sub\"")
(Haskell "error/ \"sub\"")
(Scala "!error(\"sub\")")
code_const "times :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
(SML "IntInf.* ((_), (_))")
(OCaml "Big'_int.mult'_big'_int")
(Haskell infixl 7 "*")
(Scala infixl 8 "*")
(Eval infixl 9 "*")
code_const Target_Numeral.divmod_abs
(SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
(OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
(Haskell "divMod/ (abs _)/ (abs _)")
(Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
(Eval "Integer.div'_mod/ (abs _)/ (abs _)")
code_const "HOL.equal :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
(SML "!((_ : IntInf.int) = _)")
(OCaml "Big'_int.eq'_big'_int")
(Haskell infix 4 "==")
(Scala infixl 5 "==")
(Eval infixl 6 "=")
code_const "less_eq :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
(SML "IntInf.<= ((_), (_))")
(OCaml "Big'_int.le'_big'_int")
(Haskell infix 4 "<=")
(Scala infixl 4 "<=")
(Eval infixl 6 "<=")
code_const "less :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
(SML "IntInf.< ((_), (_))")
(OCaml "Big'_int.lt'_big'_int")
(Haskell infix 4 "<")
(Scala infixl 4 "<")
(Eval infixl 6 "<")
ML {*
structure Target_Numeral =
struct
val T = @{typ "Target_Numeral.int"};
end;
*}
code_reserved Eval Target_Numeral
code_const "Code_Evaluation.term_of \<Colon> Target_Numeral.int \<Rightarrow> term"
(Eval "HOLogic.mk'_number/ Target'_Numeral.T")
code_modulename SML
Target_Numeral Arith
code_modulename OCaml
Target_Numeral Arith
code_modulename Haskell
Target_Numeral Arith
subsection {* Implementation for @{typ int} *}
code_datatype Target_Numeral.int_of
lemma [code, code del]:
"Target_Numeral.of_int = Target_Numeral.of_int" ..
lemma [code]:
"Target_Numeral.of_int (Target_Numeral.int_of k) = k"
by (simp add: Target_Numeral.int_eq_iff)
declare Int.Pos_def [code]
lemma [code_abbrev]:
"Target_Numeral.int_of (Target_Numeral.Pos k) = Int.Pos k"
by simp
declare Int.Neg_def [code]
lemma [code_abbrev]:
"Target_Numeral.int_of (Target_Numeral.Neg k) = Int.Neg k"
by simp
lemma [code]:
"0 = Target_Numeral.int_of 0"
by simp
lemma [code]:
"1 = Target_Numeral.int_of 1"
by simp
lemma [code]:
"k + l = Target_Numeral.int_of (of_int k + of_int l)"
by simp
lemma [code]:
"- k = Target_Numeral.int_of (- of_int k)"
by simp
lemma [code]:
"k - l = Target_Numeral.int_of (of_int k - of_int l)"
by simp
lemma [code]:
"Int.dup k = Target_Numeral.int_of (Target_Numeral.dup (of_int k))"
by simp
lemma [code, code del]:
"Int.sub = Int.sub" ..
lemma [code]:
"k * l = Target_Numeral.int_of (of_int k * of_int l)"
by simp
lemma [code]:
"pdivmod k l = map_pair Target_Numeral.int_of Target_Numeral.int_of
(Target_Numeral.divmod_abs (of_int k) (of_int l))"
by (simp add: prod_eq_iff pdivmod_def)
lemma [code]:
"k div l = Target_Numeral.int_of (of_int k div of_int l)"
by simp
lemma [code]:
"k mod l = Target_Numeral.int_of (of_int k mod of_int l)"
by simp
lemma [code]:
"HOL.equal k l = HOL.equal (of_int k :: Target_Numeral.int) (of_int l)"
by (simp add: equal Target_Numeral.int_eq_iff)
lemma [code]:
"k \<le> l \<longleftrightarrow> (of_int k :: Target_Numeral.int) \<le> of_int l"
by (simp add: less_eq_int_def)
lemma [code]:
"k < l \<longleftrightarrow> (of_int k :: Target_Numeral.int) < of_int l"
by (simp add: less_int_def)
lemma (in ring_1) of_int_code:
"of_int k = (if k = 0 then 0
else if k < 0 then - of_int (- k)
else let
(l, j) = divmod_int k 2;
l' = 2 * of_int l
in if j = 0 then l' else l' + 1)"
proof -
from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
show ?thesis
by (simp add: Let_def divmod_int_mod_div mod_2_not_eq_zero_eq_one_int
of_int_add [symmetric]) (simp add: * mult_commute)
qed
declare of_int_code [code]
subsection {* Implementation for @{typ nat} *}
definition of_nat :: "nat \<Rightarrow> Target_Numeral.int" where
[code_abbrev]: "of_nat = Nat.of_nat"
hide_const (open) of_nat
lemma int_of_nat [simp]:
"Target_Numeral.int_of (Target_Numeral.of_nat n) = of_nat n"
by (simp add: of_nat_def)
lemma [code abstype]:
"Target_Numeral.nat_of (Target_Numeral.of_nat n) = n"
by (simp add: nat_of_def)
lemma [code_abbrev]:
"nat (Int.Pos k) = nat_of_num k"
by (simp add: nat_of_num_numeral)
lemma [code abstract]:
"Target_Numeral.of_nat 0 = 0"
by (simp add: Target_Numeral.int_eq_iff)
lemma [code abstract]:
"Target_Numeral.of_nat 1 = 1"
by (simp add: Target_Numeral.int_eq_iff)
lemma [code abstract]:
"Target_Numeral.of_nat (m + n) = of_nat m + of_nat n"
by (simp add: Target_Numeral.int_eq_iff)
lemma [code abstract]:
"Target_Numeral.of_nat (Code_Nat.dup n) = Target_Numeral.dup (of_nat n)"
by (simp add: Target_Numeral.int_eq_iff Code_Nat.dup_def)
lemma [code, code del]:
"Code_Nat.sub = Code_Nat.sub" ..
lemma [code abstract]:
"Target_Numeral.of_nat (m - n) = max 0 (of_nat m - of_nat n)"
by (simp add: Target_Numeral.int_eq_iff)
lemma [code abstract]:
"Target_Numeral.of_nat (m * n) = of_nat m * of_nat n"
by (simp add: Target_Numeral.int_eq_iff of_nat_mult)
lemma [code abstract]:
"Target_Numeral.of_nat (m div n) = of_nat m div of_nat n"
by (simp add: Target_Numeral.int_eq_iff zdiv_int)
lemma [code abstract]:
"Target_Numeral.of_nat (m mod n) = of_nat m mod of_nat n"
by (simp add: Target_Numeral.int_eq_iff zmod_int)
lemma [code]:
"Divides.divmod_nat m n = (m div n, m mod n)"
by (simp add: prod_eq_iff)
lemma [code]:
"HOL.equal m n = HOL.equal (of_nat m :: Target_Numeral.int) (of_nat n)"
by (simp add: equal Target_Numeral.int_eq_iff)
lemma [code]:
"m \<le> n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) \<le> of_nat n"
by (simp add: less_eq_int_def)
lemma [code]:
"m < n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) < of_nat n"
by (simp add: less_int_def)
lemma num_of_nat_code [code]:
"num_of_nat = Target_Numeral.num_of_int \<circ> Target_Numeral.of_nat"
by (simp add: fun_eq_iff num_of_int_def of_nat_def)
lemma (in semiring_1) of_nat_code:
"of_nat n = (if n = 0 then 0
else let
(m, q) = divmod_nat n 2;
m' = 2 * of_nat m
in if q = 0 then m' else m' + 1)"
proof -
from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
show ?thesis
by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat
of_nat_add [symmetric])
(simp add: * mult_commute of_nat_mult add_commute)
qed
declare of_nat_code [code]
text {* Conversions between @{typ nat} and @{typ int} *}
definition int :: "nat \<Rightarrow> int" where
[code_abbrev]: "int = of_nat"
hide_const (open) int
lemma [code]:
"Target_Numeral.int n = Target_Numeral.int_of (of_nat n)"
by (simp add: int_def)
lemma [code abstract]:
"Target_Numeral.of_nat (nat k) = max 0 (Target_Numeral.of_int k)"
by (simp add: of_nat_def of_int_of_nat max_def)
end