(* Title: HOL/Num.thy
Author: Florian Haftmann
Author: Brian Huffman
*)
section \<open>Binary Numerals\<close>
theory Num
imports BNF_Least_Fixpoint
begin
subsection \<open>The \<open>num\<close> type\<close>
datatype num = One | Bit0 num | Bit1 num
text \<open>Increment function for type @{typ num}\<close>
primrec inc :: "num \<Rightarrow> num" where
"inc One = Bit0 One" |
"inc (Bit0 x) = Bit1 x" |
"inc (Bit1 x) = Bit0 (inc x)"
text \<open>Converting between type @{typ num} and type @{typ nat}\<close>
primrec nat_of_num :: "num \<Rightarrow> nat" where
"nat_of_num One = Suc 0" |
"nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |
"nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
primrec num_of_nat :: "nat \<Rightarrow> num" where
"num_of_nat 0 = One" |
"num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
lemma nat_of_num_pos: "0 < nat_of_num x"
by (induct x) simp_all
lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
by (induct x) simp_all
lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
by (induct x) simp_all
lemma num_of_nat_double:
"0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
by (induct n) simp_all
text \<open>
Type @{typ num} is isomorphic to the strictly positive
natural numbers.
\<close>
lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
by (induct n) (simp_all add: nat_of_num_inc)
lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
apply safe
apply (drule arg_cong [where f=num_of_nat])
apply (simp add: nat_of_num_inverse)
done
lemma num_induct [case_names One inc]:
fixes P :: "num \<Rightarrow> bool"
assumes One: "P One"
and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
shows "P x"
proof -
obtain n where n: "Suc n = nat_of_num x"
by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
have "P (num_of_nat (Suc n))"
proof (induct n)
case 0 show ?case using One by simp
next
case (Suc n)
then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
then show "P (num_of_nat (Suc (Suc n)))" by simp
qed
with n show "P x"
by (simp add: nat_of_num_inverse)
qed
text \<open>
From now on, there are two possible models for @{typ num}:
as positive naturals (rule \<open>num_induct\<close>)
and as digit representation (rules \<open>num.induct\<close>, \<open>num.cases\<close>).
\<close>
subsection \<open>Numeral operations\<close>
instantiation num :: "{plus,times,linorder}"
begin
definition [code del]:
"m + n = num_of_nat (nat_of_num m + nat_of_num n)"
definition [code del]:
"m * n = num_of_nat (nat_of_num m * nat_of_num n)"
definition [code del]:
"m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
definition [code del]:
"m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
instance
by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
end
lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
unfolding plus_num_def
by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
unfolding times_num_def
by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
lemma add_num_simps [simp, code]:
"One + One = Bit0 One"
"One + Bit0 n = Bit1 n"
"One + Bit1 n = Bit0 (n + One)"
"Bit0 m + One = Bit1 m"
"Bit0 m + Bit0 n = Bit0 (m + n)"
"Bit0 m + Bit1 n = Bit1 (m + n)"
"Bit1 m + One = Bit0 (m + One)"
"Bit1 m + Bit0 n = Bit1 (m + n)"
"Bit1 m + Bit1 n = Bit0 (m + n + One)"
by (simp_all add: num_eq_iff nat_of_num_add)
lemma mult_num_simps [simp, code]:
"m * One = m"
"One * n = n"
"Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
"Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
"Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
"Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
by (simp_all add: num_eq_iff nat_of_num_add
nat_of_num_mult distrib_right distrib_left)
lemma eq_num_simps:
"One = One \<longleftrightarrow> True"
"One = Bit0 n \<longleftrightarrow> False"
"One = Bit1 n \<longleftrightarrow> False"
"Bit0 m = One \<longleftrightarrow> False"
"Bit1 m = One \<longleftrightarrow> False"
"Bit0 m = Bit0 n \<longleftrightarrow> m = n"
"Bit0 m = Bit1 n \<longleftrightarrow> False"
"Bit1 m = Bit0 n \<longleftrightarrow> False"
"Bit1 m = Bit1 n \<longleftrightarrow> m = n"
by simp_all
lemma le_num_simps [simp, code]:
"One \<le> n \<longleftrightarrow> True"
"Bit0 m \<le> One \<longleftrightarrow> False"
"Bit1 m \<le> One \<longleftrightarrow> False"
"Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
"Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
"Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
"Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
using nat_of_num_pos [of n] nat_of_num_pos [of m]
by (auto simp add: less_eq_num_def less_num_def)
lemma less_num_simps [simp, code]:
"m < One \<longleftrightarrow> False"
"One < Bit0 n \<longleftrightarrow> True"
"One < Bit1 n \<longleftrightarrow> True"
"Bit0 m < Bit0 n \<longleftrightarrow> m < n"
"Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
"Bit1 m < Bit1 n \<longleftrightarrow> m < n"
"Bit1 m < Bit0 n \<longleftrightarrow> m < n"
using nat_of_num_pos [of n] nat_of_num_pos [of m]
by (auto simp add: less_eq_num_def less_num_def)
lemma le_num_One_iff: "x \<le> num.One \<longleftrightarrow> x = num.One"
by (simp add: antisym_conv)
text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors\<close>
lemma add_One: "x + One = inc x"
by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
lemma add_One_commute: "One + n = n + One"
by (induct n) simp_all
lemma add_inc: "x + inc y = inc (x + y)"
by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
lemma mult_inc: "x * inc y = x * y + x"
by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
text \<open>The @{const num_of_nat} conversion\<close>
lemma num_of_nat_One:
"n \<le> 1 \<Longrightarrow> num_of_nat n = One"
by (cases n) simp_all
lemma num_of_nat_plus_distrib:
"0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
by (induct n) (auto simp add: add_One add_One_commute add_inc)
text \<open>A double-and-decrement function\<close>
primrec BitM :: "num \<Rightarrow> num" where
"BitM One = One" |
"BitM (Bit0 n) = Bit1 (BitM n)" |
"BitM (Bit1 n) = Bit1 (Bit0 n)"
lemma BitM_plus_one: "BitM n + One = Bit0 n"
by (induct n) simp_all
lemma one_plus_BitM: "One + BitM n = Bit0 n"
unfolding add_One_commute BitM_plus_one ..
text \<open>Squaring and exponentiation\<close>
primrec sqr :: "num \<Rightarrow> num" where
"sqr One = One" |
"sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |
"sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
"pow x One = x" |
"pow x (Bit0 y) = sqr (pow x y)" |
"pow x (Bit1 y) = sqr (pow x y) * x"
lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
by (induct x, simp_all add: algebra_simps nat_of_num_add)
lemma sqr_conv_mult: "sqr x = x * x"
by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
subsection \<open>Binary numerals\<close>
text \<open>
We embed binary representations into a generic algebraic
structure using \<open>numeral\<close>.
\<close>
class numeral = one + semigroup_add
begin
primrec numeral :: "num \<Rightarrow> 'a" where
numeral_One: "numeral One = 1" |
numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |
numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
lemma numeral_code [code]:
"numeral One = 1"
"numeral (Bit0 n) = (let m = numeral n in m + m)"
"numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
by (simp_all add: Let_def)
lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
apply (induct x)
apply simp
apply (simp add: add.assoc [symmetric], simp add: add.assoc)
apply (simp add: add.assoc [symmetric], simp add: add.assoc)
done
lemma numeral_inc: "numeral (inc x) = numeral x + 1"
proof (induct x)
case (Bit1 x)
have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
by (simp only: one_plus_numeral_commute)
with Bit1 show ?case
by (simp add: add.assoc)
qed simp_all
declare numeral.simps [simp del]
abbreviation "Numeral1 \<equiv> numeral One"
declare numeral_One [code_post]
end
text \<open>Numeral syntax.\<close>
syntax
"_Numeral" :: "num_const \<Rightarrow> 'a" ("_")
ML_file "Tools/numeral.ML"
parse_translation \<open>
let
fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
c $ numeral_tr [t] $ u
| numeral_tr [Const (num, _)] =
(Numeral.mk_number_syntax o #value o Lexicon.read_num) num
| numeral_tr ts = raise TERM ("numeral_tr", ts);
in [(@{syntax_const "_Numeral"}, K numeral_tr)] end
\<close>
typed_print_translation \<open>
let
fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n
| dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1
| dest_num (Const (@{const_syntax One}, _)) = 1;
fun num_tr' ctxt T [n] =
let
val k = dest_num n;
val t' =
Syntax.const @{syntax_const "_Numeral"} $
Syntax.free (string_of_int k);
in
(case T of
Type (@{type_name fun}, [_, T']) =>
if Printer.type_emphasis ctxt T' then
Syntax.const @{syntax_const "_constrain"} $ t' $
Syntax_Phases.term_of_typ ctxt T'
else t'
| _ => if T = dummyT then t' else raise Match)
end;
in
[(@{const_syntax numeral}, num_tr')]
end
\<close>
subsection \<open>Class-specific numeral rules\<close>
text \<open>
@{const numeral} is a morphism.
\<close>
subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
context numeral
begin
lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
by (induct n rule: num_induct)
(simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
by (rule numeral_add [symmetric])
lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
using numeral_add [of n One] by (simp add: numeral_One)
lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
using numeral_add [of One n] by (simp add: numeral_One)
lemma one_add_one: "1 + 1 = 2"
using numeral_add [of One One] by (simp add: numeral_One)
lemmas add_numeral_special =
numeral_plus_one one_plus_numeral one_add_one
end
subsubsection \<open>
Structures with negation: class \<open>neg_numeral\<close>
\<close>
class neg_numeral = numeral + group_add
begin
lemma uminus_numeral_One:
"- Numeral1 = - 1"
by (simp add: numeral_One)
text \<open>Numerals form an abelian subgroup.\<close>
inductive is_num :: "'a \<Rightarrow> bool" where
"is_num 1" |
"is_num x \<Longrightarrow> is_num (- x)" |
"\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
lemma is_num_numeral: "is_num (numeral k)"
by (induct k, simp_all add: numeral.simps is_num.intros)
lemma is_num_add_commute:
"\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
apply (induct x rule: is_num.induct)
apply (induct y rule: is_num.induct)
apply simp
apply (rule_tac a=x in add_left_imp_eq)
apply (rule_tac a=x in add_right_imp_eq)
apply (simp add: add.assoc)
apply (simp add: add.assoc [symmetric], simp add: add.assoc)
apply (rule_tac a=x in add_left_imp_eq)
apply (rule_tac a=x in add_right_imp_eq)
apply (simp add: add.assoc)
apply (simp add: add.assoc, simp add: add.assoc [symmetric])
done
lemma is_num_add_left_commute:
"\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
by (simp only: add.assoc [symmetric] is_num_add_commute)
lemmas is_num_normalize =
add.assoc is_num_add_commute is_num_add_left_commute
is_num.intros is_num_numeral
minus_add
definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
"sub k l = numeral k - numeral l"
lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
lemma dbl_simps [simp]:
"dbl (- numeral k) = - dbl (numeral k)"
"dbl 0 = 0"
"dbl 1 = 2"
"dbl (- 1) = - 2"
"dbl (numeral k) = numeral (Bit0 k)"
by (simp_all add: dbl_def numeral.simps minus_add)
lemma dbl_inc_simps [simp]:
"dbl_inc (- numeral k) = - dbl_dec (numeral k)"
"dbl_inc 0 = 1"
"dbl_inc 1 = 3"
"dbl_inc (- 1) = - 1"
"dbl_inc (numeral k) = numeral (Bit1 k)"
by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
lemma dbl_dec_simps [simp]:
"dbl_dec (- numeral k) = - dbl_inc (numeral k)"
"dbl_dec 0 = - 1"
"dbl_dec 1 = 1"
"dbl_dec (- 1) = - 3"
"dbl_dec (numeral k) = numeral (BitM k)"
by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
lemma sub_num_simps [simp]:
"sub One One = 0"
"sub One (Bit0 l) = - numeral (BitM l)"
"sub One (Bit1 l) = - numeral (Bit0 l)"
"sub (Bit0 k) One = numeral (BitM k)"
"sub (Bit1 k) One = numeral (Bit0 k)"
"sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
"sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
"sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
"sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_simps:
"numeral m + - numeral n = sub m n"
"- numeral m + numeral n = sub n m"
"- numeral m + - numeral n = - (numeral m + numeral n)"
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma add_neg_numeral_special:
"1 + - numeral m = sub One m"
"- numeral m + 1 = sub One m"
"numeral m + - 1 = sub m One"
"- 1 + numeral n = sub n One"
"- 1 + - numeral n = - numeral (inc n)"
"- numeral m + - 1 = - numeral (inc m)"
"1 + - 1 = 0"
"- 1 + 1 = 0"
"- 1 + - 1 = - 2"
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_simps:
"numeral m - numeral n = sub m n"
"numeral m - - numeral n = numeral (m + n)"
"- numeral m - numeral n = - numeral (m + n)"
"- numeral m - - numeral n = sub n m"
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
del: add_uminus_conv_diff add: diff_conv_add_uminus)
lemma diff_numeral_special:
"1 - numeral n = sub One n"
"numeral m - 1 = sub m One"
"1 - - numeral n = numeral (One + n)"
"- numeral m - 1 = - numeral (m + One)"
"- 1 - numeral n = - numeral (inc n)"
"numeral m - - 1 = numeral (inc m)"
"- 1 - - numeral n = sub n One"
"- numeral m - - 1 = sub One m"
"1 - 1 = 0"
"- 1 - 1 = - 2"
"1 - - 1 = 2"
"- 1 - - 1 = 0"
by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
del: add_uminus_conv_diff add: diff_conv_add_uminus)
end
subsubsection \<open>
Structures with multiplication: class \<open>semiring_numeral\<close>
\<close>
class semiring_numeral = semiring + monoid_mult
begin
subclass numeral ..
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
apply (induct n rule: num_induct)
apply (simp add: numeral_One)
apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
done
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
by (rule numeral_mult [symmetric])
lemma mult_2: "2 * z = z + z"
unfolding one_add_one [symmetric] distrib_right by simp
lemma mult_2_right: "z * 2 = z + z"
unfolding one_add_one [symmetric] distrib_left by simp
end
subsubsection \<open>
Structures with a zero: class \<open>semiring_1\<close>
\<close>
context semiring_1
begin
subclass semiring_numeral ..
lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
by (induct n,
simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
end
lemma nat_of_num_numeral [code_abbrev]:
"nat_of_num = numeral"
proof
fix n
have "numeral n = nat_of_num n"
by (induct n) (simp_all add: numeral.simps)
then show "nat_of_num n = numeral n" by simp
qed
lemma nat_of_num_code [code]:
"nat_of_num One = 1"
"nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
"nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
by (simp_all add: Let_def)
subsubsection \<open>
Equality: class \<open>semiring_char_0\<close>
\<close>
context semiring_char_0
begin
lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
of_nat_eq_iff num_eq_iff ..
lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
by (rule numeral_eq_iff [of n One, unfolded numeral_One])
lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
by (rule numeral_eq_iff [of One n, unfolded numeral_One])
lemma numeral_neq_zero: "numeral n \<noteq> 0"
unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
by (simp add: nat_of_num_pos)
lemma zero_neq_numeral: "0 \<noteq> numeral n"
unfolding eq_commute [of 0] by (rule numeral_neq_zero)
lemmas eq_numeral_simps [simp] =
numeral_eq_iff
numeral_eq_one_iff
one_eq_numeral_iff
numeral_neq_zero
zero_neq_numeral
end
subsubsection \<open>
Comparisons: class \<open>linordered_semidom\<close>
\<close>
text \<open>Could be perhaps more general than here.\<close>
context linordered_semidom
begin
lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
proof -
have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
then show ?thesis by simp
qed
lemma one_le_numeral: "1 \<le> numeral n"
using numeral_le_iff [of One n] by (simp add: numeral_One)
lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
using numeral_le_iff [of n One] by (simp add: numeral_One)
lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
proof -
have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
then show ?thesis by simp
qed
lemma not_numeral_less_one: "\<not> numeral n < 1"
using numeral_less_iff [of n One] by (simp add: numeral_One)
lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
using numeral_less_iff [of One n] by (simp add: numeral_One)
lemma zero_le_numeral: "0 \<le> numeral n"
by (induct n) (simp_all add: numeral.simps)
lemma zero_less_numeral: "0 < numeral n"
by (induct n) (simp_all add: numeral.simps add_pos_pos)
lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
by (simp add: not_le zero_less_numeral)
lemma not_numeral_less_zero: "\<not> numeral n < 0"
by (simp add: not_less zero_le_numeral)
lemmas le_numeral_extra =
zero_le_one not_one_le_zero
order_refl [of 0] order_refl [of 1]
lemmas less_numeral_extra =
zero_less_one not_one_less_zero
less_irrefl [of 0] less_irrefl [of 1]
lemmas le_numeral_simps [simp] =
numeral_le_iff
one_le_numeral
numeral_le_one_iff
zero_le_numeral
not_numeral_le_zero
lemmas less_numeral_simps [simp] =
numeral_less_iff
one_less_numeral_iff
not_numeral_less_one
zero_less_numeral
not_numeral_less_zero
lemma min_0_1 [simp]:
fixes min' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" defines "min' \<equiv> min" shows
"min' 0 1 = 0"
"min' 1 0 = 0"
"min' 0 (numeral x) = 0"
"min' (numeral x) 0 = 0"
"min' 1 (numeral x) = 1"
"min' (numeral x) 1 = 1"
by(simp_all add: min'_def min_def le_num_One_iff)
lemma max_0_1 [simp]:
fixes max' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" defines "max' \<equiv> max" shows
"max' 0 1 = 1"
"max' 1 0 = 1"
"max' 0 (numeral x) = numeral x"
"max' (numeral x) 0 = numeral x"
"max' 1 (numeral x) = numeral x"
"max' (numeral x) 1 = numeral x"
by(simp_all add: max'_def max_def le_num_One_iff)
end
subsubsection \<open>
Multiplication and negation: class \<open>ring_1\<close>
\<close>
context ring_1
begin
subclass neg_numeral ..
lemma mult_neg_numeral_simps:
"- numeral m * - numeral n = numeral (m * n)"
"- numeral m * numeral n = - numeral (m * n)"
"numeral m * - numeral n = - numeral (m * n)"
unfolding mult_minus_left mult_minus_right
by (simp_all only: minus_minus numeral_mult)
lemma mult_minus1 [simp]: "- 1 * z = - z"
unfolding numeral.simps mult_minus_left by simp
lemma mult_minus1_right [simp]: "z * - 1 = - z"
unfolding numeral.simps mult_minus_right by simp
end
subsubsection \<open>
Equality using \<open>iszero\<close> for rings with non-zero characteristic
\<close>
context ring_1
begin
definition iszero :: "'a \<Rightarrow> bool"
where "iszero z \<longleftrightarrow> z = 0"
lemma iszero_0 [simp]: "iszero 0"
by (simp add: iszero_def)
lemma not_iszero_1 [simp]: "\<not> iszero 1"
by (simp add: iszero_def)
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
by (simp add: numeral_One)
lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
by (simp add: iszero_def)
lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
by (simp add: numeral_One)
lemma iszero_neg_numeral [simp]:
"iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
unfolding iszero_def
by (rule neg_equal_0_iff_equal)
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
unfolding iszero_def by (rule eq_iff_diff_eq_0)
text \<open>The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared
\<open>[simp]\<close> by default, because for rings of characteristic zero,
better simp rules are possible. For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules should be added to the
simplifier, along with a type-specific rule for deciding propositions
of the form \<open>iszero (numeral w)\<close>.
bh: Maybe it would not be so bad to just declare these as simp
rules anyway? I should test whether these rules take precedence over
the \<open>ring_char_0\<close> rules in the simplifier.
\<close>
lemma eq_numeral_iff_iszero:
"numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
"numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
"- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
"- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
"numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
"1 = numeral y \<longleftrightarrow> iszero (sub One y)"
"- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
"1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
"numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
"0 = numeral y \<longleftrightarrow> iszero (numeral y)"
"- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
"0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
by simp_all
end
subsubsection \<open>
Equality and negation: class \<open>ring_char_0\<close>
\<close>
class ring_char_0 = ring_1 + semiring_char_0
begin
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
by (simp add: iszero_def)
lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
by simp
lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
unfolding eq_neg_iff_add_eq_0
by (simp add: numeral_plus_numeral)
lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
by (rule numeral_neq_neg_numeral [symmetric])
lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
unfolding neg_0_equal_iff_equal by simp
lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
unfolding neg_equal_0_iff_equal by simp
lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
lemma neg_one_neq_numeral:
"- 1 \<noteq> numeral n"
using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
lemma numeral_neq_neg_one:
"numeral n \<noteq> - 1"
using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
lemma neg_one_eq_numeral_iff:
"- 1 = - numeral n \<longleftrightarrow> n = One"
using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
lemma numeral_eq_neg_one_iff:
"- numeral n = - 1 \<longleftrightarrow> n = One"
using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
lemma neg_one_neq_zero:
"- 1 \<noteq> 0"
by simp
lemma zero_neq_neg_one:
"0 \<noteq> - 1"
by simp
lemma neg_one_neq_one:
"- 1 \<noteq> 1"
using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
lemma one_neq_neg_one:
"1 \<noteq> - 1"
using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
lemmas eq_neg_numeral_simps [simp] =
neg_numeral_eq_iff
numeral_neq_neg_numeral neg_numeral_neq_numeral
one_neq_neg_numeral neg_numeral_neq_one
zero_neq_neg_numeral neg_numeral_neq_zero
neg_one_neq_numeral numeral_neq_neg_one
neg_one_eq_numeral_iff numeral_eq_neg_one_iff
neg_one_neq_zero zero_neq_neg_one
neg_one_neq_one one_neq_neg_one
end
subsubsection \<open>
Structures with negation and order: class \<open>linordered_idom\<close>
\<close>
context linordered_idom
begin
subclass ring_char_0 ..
lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
by (simp only: neg_le_iff_le numeral_le_iff)
lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
by (simp only: neg_less_iff_less numeral_less_iff)
lemma neg_numeral_less_zero: "- numeral n < 0"
by (simp only: neg_less_0_iff_less zero_less_numeral)
lemma neg_numeral_le_zero: "- numeral n \<le> 0"
by (simp only: neg_le_0_iff_le zero_le_numeral)
lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
by (simp only: not_less neg_numeral_le_zero)
lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
by (simp only: not_le neg_numeral_less_zero)
lemma neg_numeral_less_numeral: "- numeral m < numeral n"
using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
by (simp only: less_imp_le neg_numeral_less_numeral)
lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
by (simp only: not_less neg_numeral_le_numeral)
lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
by (simp only: not_le neg_numeral_less_numeral)
lemma neg_numeral_less_one: "- numeral m < 1"
by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
lemma neg_numeral_le_one: "- numeral m \<le> 1"
by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
by (simp only: not_less neg_numeral_le_one)
lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
by (simp only: not_le neg_numeral_less_one)
lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
lemma neg_one_less_numeral: "- 1 < numeral m"
using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
lemma neg_one_le_numeral: "- 1 \<le> numeral m"
using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
by (cases m) simp_all
lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
by simp
lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
by simp
lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
by (cases m) simp_all
lemma sub_non_negative:
"sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
by (simp only: sub_def le_diff_eq) simp
lemma sub_positive:
"sub n m > 0 \<longleftrightarrow> n > m"
by (simp only: sub_def less_diff_eq) simp
lemma sub_non_positive:
"sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
by (simp only: sub_def diff_le_eq) simp
lemma sub_negative:
"sub n m < 0 \<longleftrightarrow> n < m"
by (simp only: sub_def diff_less_eq) simp
lemmas le_neg_numeral_simps [simp] =
neg_numeral_le_iff
neg_numeral_le_numeral not_numeral_le_neg_numeral
neg_numeral_le_zero not_zero_le_neg_numeral
neg_numeral_le_one not_one_le_neg_numeral
neg_one_le_numeral not_numeral_le_neg_one
neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
lemma le_minus_one_simps [simp]:
"- 1 \<le> 0"
"- 1 \<le> 1"
"\<not> 0 \<le> - 1"
"\<not> 1 \<le> - 1"
by simp_all
lemmas less_neg_numeral_simps [simp] =
neg_numeral_less_iff
neg_numeral_less_numeral not_numeral_less_neg_numeral
neg_numeral_less_zero not_zero_less_neg_numeral
neg_numeral_less_one not_one_less_neg_numeral
neg_one_less_numeral not_numeral_less_neg_one
neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
lemma less_minus_one_simps [simp]:
"- 1 < 0"
"- 1 < 1"
"\<not> 0 < - 1"
"\<not> 1 < - 1"
by (simp_all add: less_le)
lemma abs_numeral [simp]: "\<bar>numeral n\<bar> = numeral n"
by simp
lemma abs_neg_numeral [simp]: "\<bar>- numeral n\<bar> = numeral n"
by (simp only: abs_minus_cancel abs_numeral)
lemma abs_neg_one [simp]: "\<bar>- 1\<bar> = 1"
by simp
end
subsubsection \<open>
Natural numbers
\<close>
lemma Suc_1 [simp]: "Suc 1 = 2"
unfolding Suc_eq_plus1 by (rule one_add_one)
lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
unfolding Suc_eq_plus1 by (rule numeral_plus_one)
definition pred_numeral :: "num \<Rightarrow> nat"
where [code del]: "pred_numeral k = numeral k - 1"
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
unfolding pred_numeral_def by simp
lemma eval_nat_numeral:
"numeral One = Suc 0"
"numeral (Bit0 n) = Suc (numeral (BitM n))"
"numeral (Bit1 n) = Suc (numeral (Bit0 n))"
by (simp_all add: numeral.simps BitM_plus_one)
lemma pred_numeral_simps [simp]:
"pred_numeral One = 0"
"pred_numeral (Bit0 k) = numeral (BitM k)"
"pred_numeral (Bit1 k) = numeral (Bit0 k)"
unfolding pred_numeral_def eval_nat_numeral
by (simp_all only: diff_Suc_Suc diff_0)
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
by (simp add: eval_nat_numeral)
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
by (simp add: eval_nat_numeral)
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
by (simp only: numeral_One One_nat_def)
lemma Suc_nat_number_of_add:
"Suc (numeral v + n) = numeral (v + One) + n"
by simp
(*Maps #n to n for n = 1, 2*)
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
text \<open>Comparisons involving @{term Suc}.\<close>
lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
by (simp add: numeral_eq_Suc)
lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
by (simp add: numeral_eq_Suc)
lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
by (simp add: numeral_eq_Suc)
lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
by (simp add: numeral_eq_Suc)
lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
by (simp add: numeral_eq_Suc)
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
by (simp add: numeral_eq_Suc)
lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
by (simp add: numeral_eq_Suc)
lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
by (simp add: numeral_eq_Suc)
lemma max_Suc_numeral [simp]:
"max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
by (simp add: numeral_eq_Suc)
lemma max_numeral_Suc [simp]:
"max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
by (simp add: numeral_eq_Suc)
lemma min_Suc_numeral [simp]:
"min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
by (simp add: numeral_eq_Suc)
lemma min_numeral_Suc [simp]:
"min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
by (simp add: numeral_eq_Suc)
text \<open>For @{term case_nat} and @{term rec_nat}.\<close>
lemma case_nat_numeral [simp]:
"case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
by (simp add: numeral_eq_Suc)
lemma case_nat_add_eq_if [simp]:
"case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
by (simp add: numeral_eq_Suc)
lemma rec_nat_numeral [simp]:
"rec_nat a f (numeral v) =
(let pv = pred_numeral v in f pv (rec_nat a f pv))"
by (simp add: numeral_eq_Suc Let_def)
lemma rec_nat_add_eq_if [simp]:
"rec_nat a f (numeral v + n) =
(let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
by (simp add: numeral_eq_Suc Let_def)
text \<open>Case analysis on @{term "n < 2"}\<close>
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
by (auto simp add: numeral_2_eq_2)
text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2\<close>
text \<open>bh: Are these rules really a good idea?\<close>
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
by simp
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
by simp
text \<open>Can be used to eliminate long strings of Sucs, but not by default.\<close>
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
by simp
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
subsection \<open>Particular lemmas concerning @{term 2}\<close>
context linordered_field
begin
lemma half_gt_zero_iff:
"0 < a / 2 \<longleftrightarrow> 0 < a" (is "?P \<longleftrightarrow> ?Q")
by (auto simp add: field_simps)
lemma half_gt_zero [simp]:
"0 < a \<Longrightarrow> 0 < a / 2"
by (simp add: half_gt_zero_iff)
end
subsection \<open>Numeral equations as default simplification rules\<close>
declare (in numeral) numeral_One [simp]
declare (in numeral) numeral_plus_numeral [simp]
declare (in numeral) add_numeral_special [simp]
declare (in neg_numeral) add_neg_numeral_simps [simp]
declare (in neg_numeral) add_neg_numeral_special [simp]
declare (in neg_numeral) diff_numeral_simps [simp]
declare (in neg_numeral) diff_numeral_special [simp]
declare (in semiring_numeral) numeral_times_numeral [simp]
declare (in ring_1) mult_neg_numeral_simps [simp]
subsection \<open>Setting up simprocs\<close>
lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
by simp
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
by simp
lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
by simp
lemma inverse_numeral_1:
"inverse Numeral1 = (Numeral1::'a::division_ring)"
by simp
text\<open>Theorem lists for the cancellation simprocs. The use of a binary
numeral for 1 reduces the number of special cases.\<close>
lemma mult_1s:
fixes a :: "'a::semiring_numeral"
and b :: "'b::ring_1"
shows "Numeral1 * a = a"
"a * Numeral1 = a"
"- Numeral1 * b = - b"
"b * - Numeral1 = - b"
by simp_all
setup \<open>
Reorient_Proc.add
(fn Const (@{const_name numeral}, _) $ _ => true
| Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
| _ => false)
\<close>
simproc_setup reorient_numeral
("numeral w = x" | "- numeral w = y") = Reorient_Proc.proc
subsubsection \<open>Simplification of arithmetic operations on integer constants.\<close>
lemmas arith_special = (* already declared simp above *)
add_numeral_special add_neg_numeral_special
diff_numeral_special
(* rules already in simpset *)
lemmas arith_extra_simps =
numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
minus_zero
diff_numeral_simps diff_0 diff_0_right
numeral_times_numeral mult_neg_numeral_simps
mult_zero_left mult_zero_right
abs_numeral abs_neg_numeral
text \<open>
For making a minimal simpset, one must include these default simprules.
Also include \<open>simp_thms\<close>.
\<close>
lemmas arith_simps =
add_num_simps mult_num_simps sub_num_simps
BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
abs_zero abs_one arith_extra_simps
lemmas more_arith_simps =
neg_le_iff_le
minus_zero left_minus right_minus
mult_1_left mult_1_right
mult_minus_left mult_minus_right
minus_add_distrib minus_minus mult.assoc
lemmas of_nat_simps =
of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
text \<open>Simplification of relational operations\<close>
lemmas eq_numeral_extra =
zero_neq_one one_neq_zero
lemmas rel_simps =
le_num_simps less_num_simps eq_num_simps
le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
unfolding Let_def ..
lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
\<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
unfolding Let_def ..
declaration \<open>
let
fun number_of ctxt T n =
if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort numeral}))
then raise CTERM ("number_of", [])
else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
in
K (
Lin_Arith.add_simps (@{thms arith_simps} @ @{thms more_arith_simps}
@ @{thms rel_simps}
@ @{thms pred_numeral_simps}
@ @{thms arith_special numeral_One}
@ @{thms of_nat_simps})
#> Lin_Arith.add_simps [@{thm Suc_numeral},
@{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
@{thm le_Suc_numeral}, @{thm le_numeral_Suc},
@{thm less_Suc_numeral}, @{thm less_numeral_Suc},
@{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
@{thm mult_Suc}, @{thm mult_Suc_right},
@{thm of_nat_numeral}]
#> Lin_Arith.set_number_of number_of)
end
\<close>
subsubsection \<open>Simplification of arithmetic when nested to the right.\<close>
lemma add_numeral_left [simp]:
"numeral v + (numeral w + z) = (numeral(v + w) + z)"
by (simp_all add: add.assoc [symmetric])
lemma add_neg_numeral_left [simp]:
"numeral v + (- numeral w + y) = (sub v w + y)"
"- numeral v + (numeral w + y) = (sub w v + y)"
"- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
by (simp_all add: add.assoc [symmetric])
lemma mult_numeral_left [simp]:
"numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
"- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
"numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
"- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
by (simp_all add: mult.assoc [symmetric])
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
subsection \<open>code module namespace\<close>
code_identifier
code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
end