src/HOL/Num.thy
 author huffman Fri, 30 Mar 2012 16:43:07 +0200 changeset 47227 bc18fa07053b parent 47226 ea712316fc87 child 47228 4f4d85c3516f permissions -rw-r--r--
tuned proof
```
(*  Title:      HOL/Num.thy
Author:     Florian Haftmann
Author:     Brian Huffman
*)

theory Num
imports Datatype
begin

subsection {* The @{text num} type *}

datatype num = One | Bit0 num | Bit1 num

text {* Increment function for type @{typ num} *}

primrec inc :: "num \<Rightarrow> num" where
"inc One = Bit0 One" |
"inc (Bit0 x) = Bit1 x" |
"inc (Bit1 x) = Bit0 (inc x)"

text {* Converting between type @{typ num} and type @{typ nat} *}

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 {*
Type @{typ num} is isomorphic to the strictly positive
natural numbers.
*}

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])
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"
qed

text {*
From now on, there are two possible models for @{typ num}:
as positive naturals (rule @{text "num_induct"})
and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
*}

subsection {* Numeral operations *}

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 (default, 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

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)

"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)"

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))"
nat_of_num_mult left_distrib right_distrib)

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)

text {* Rules using @{text One} and @{text inc} as constructors *}

lemma add_One: "x + One = inc x"

lemma add_One_commute: "One + n = n + One"
by (induct n) simp_all

lemma add_inc: "x + inc y = inc (x + y)"

lemma mult_inc: "x * inc y = x * y + x"

text {* The @{const num_of_nat} conversion *}

lemma num_of_nat_One:
"n \<le> 1 \<Longrightarrow> num_of_nat n = Num.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"

text {* A double-and-decrement function *}

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"

text {* Squaring and exponentiation *}

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"

lemma sqr_conv_mult: "sqr x = x * x"
by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)

subsection {* Binary numerals *}

text {*
We embed binary representations into a generic algebraic
structure using @{text numeral}.
*}

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 one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
apply (induct x)
apply simp
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
qed simp_all

declare numeral.simps [simp del]

abbreviation "Numeral1 \<equiv> numeral One"

declare numeral_One [code_post]

end

text {* Negative numerals. *}

class neg_numeral = numeral + group_add
begin

definition neg_numeral :: "num \<Rightarrow> 'a" where
"neg_numeral k = - numeral k"

end

text {* Numeral syntax. *}

syntax
"_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")

parse_translation {*
let
fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
of (0, 1) => Syntax.const @{const_name One}
| (n, 0) => Syntax.const @{const_name Bit0} \$ num_of_int n
| (n, 1) => Syntax.const @{const_name Bit1} \$ num_of_int n
else raise Match;
val pos = Syntax.const @{const_name numeral}
val neg = Syntax.const @{const_name neg_numeral}
val one = Syntax.const @{const_name Groups.one}
val zero = Syntax.const @{const_name Groups.zero}
fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) \$ t \$ u] =
c \$ numeral_tr [t] \$ u
| numeral_tr [Const (num, _)] =
let
val {value, ...} = Lexicon.read_xnum num;
in
if value = 0 then zero else
if value > 0
then pos \$ num_of_int value
else neg \$ num_of_int (~value)
end
| numeral_tr ts = raise TERM ("numeral_tr", ts);
in [("_Numeral", numeral_tr)] end
*}

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' sign ctxt T [n] =
let
val k = dest_num n;
val t' = Syntax.const @{syntax_const "_Numeral"} \$
Syntax.free (sign ^ string_of_int k);
in
case T of
Type (@{type_name fun}, [_, T']) =>
if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'
else Syntax.const @{syntax_const "_constrain"} \$ t' \$ Syntax_Phases.term_of_typ ctxt T'
| T' => if T' = dummyT then t' else raise Match
end;
in [(@{const_syntax numeral}, num_tr' ""),
(@{const_syntax neg_numeral}, num_tr' "-")] end
*}

subsection {* Class-specific numeral rules *}

text {*
@{const numeral} is a morphism.
*}

subsubsection {* Structures with addition: class @{text numeral} *}

context numeral
begin

lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
by (induct n rule: num_induct)

lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"

lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"

lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"

lemma one_add_one: "1 + 1 = 2"

end

subsubsection {*
Structures with negation: class @{text neg_numeral}
*}

context neg_numeral
begin

text {* Numerals form an abelian subgroup. *}

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)

"\<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
done

"\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"

lemmas is_num_normalize =
is_num.intros is_num_numeral

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 (neg_numeral k) = neg_numeral (Bit0 k)"
"dbl 0 = 0"
"dbl 1 = 2"
"dbl (numeral k) = numeral (Bit0 k)"
unfolding dbl_def neg_numeral_def numeral.simps

lemma dbl_inc_simps [simp]:
"dbl_inc (neg_numeral k) = neg_numeral (BitM k)"
"dbl_inc 0 = 1"
"dbl_inc 1 = 3"
"dbl_inc (numeral k) = numeral (Bit1 k)"
unfolding dbl_inc_def neg_numeral_def numeral.simps numeral_BitM

lemma dbl_dec_simps [simp]:
"dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"
"dbl_dec 0 = -1"
"dbl_dec 1 = 1"
"dbl_dec (numeral k) = numeral (BitM k)"
unfolding dbl_dec_def neg_numeral_def numeral.simps numeral_BitM

lemma sub_num_simps [simp]:
"sub One One = 0"
"sub One (Bit0 l) = neg_numeral (BitM l)"
"sub One (Bit1 l) = neg_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)"
unfolding dbl_def dbl_dec_def dbl_inc_def sub_def
unfolding neg_numeral_def numeral.simps numeral_BitM

"numeral m + neg_numeral n = sub m n"
"neg_numeral m + numeral n = sub n m"
"neg_numeral m + neg_numeral n = neg_numeral (m + n)"
unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps

"1 + neg_numeral m = sub One m"
"neg_numeral m + 1 = sub One m"
unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps

lemma diff_numeral_simps:
"numeral m - numeral n = sub m n"
"numeral m - neg_numeral n = numeral (m + n)"
"neg_numeral m - numeral n = neg_numeral (m + n)"
"neg_numeral m - neg_numeral n = sub n m"
unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps

lemma diff_numeral_special:
"1 - numeral n = sub One n"
"1 - neg_numeral n = numeral (One + n)"
"numeral m - 1 = sub m One"
"neg_numeral m - 1 = neg_numeral (m + One)"
unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps

lemma minus_one: "- 1 = -1"
unfolding neg_numeral_def numeral.simps ..

lemma minus_numeral: "- numeral n = neg_numeral n"
unfolding neg_numeral_def ..

lemma minus_neg_numeral: "- neg_numeral n = numeral n"
unfolding neg_numeral_def by simp

lemmas minus_numeral_simps [simp] =
minus_one minus_numeral minus_neg_numeral

end

subsubsection {*
Structures with multiplication: class @{text semiring_numeral}
*}

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)
done

lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
by (rule numeral_mult [symmetric])

end

subsubsection {*
Structures with a zero: class @{text semiring_1}
*}

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)

lemma mult_2: "2 * z = z + z"
unfolding one_add_one [symmetric] left_distrib by simp

lemma mult_2_right: "z * 2 = z + z"
unfolding one_add_one [symmetric] right_distrib by simp

end

lemma nat_of_num_numeral: "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

subsubsection {*
Equality: class @{text semiring_char_0}
*}

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]

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 {*
Comparisons: class @{text linordered_semidom}
*}

text {*  Could be perhaps more general than here. *}

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"

lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"

lemma not_numeral_less_zero: "\<not> numeral n < 0"

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

end

subsubsection {*
Multiplication and negation: class @{text ring_1}
*}

context ring_1
begin

subclass neg_numeral ..

lemma mult_neg_numeral_simps:
"neg_numeral m * neg_numeral n = numeral (m * n)"
"neg_numeral m * numeral n = neg_numeral (m * n)"
"numeral m * neg_numeral n = neg_numeral (m * n)"
unfolding neg_numeral_def mult_minus_left mult_minus_right
by (simp_all only: minus_minus numeral_mult)

lemma mult_minus1 [simp]: "-1 * z = - z"
unfolding neg_numeral_def numeral.simps mult_minus_left by simp

lemma mult_minus1_right [simp]: "z * -1 = - z"
unfolding neg_numeral_def numeral.simps mult_minus_right by simp

end

subsubsection {*
Equality using @{text iszero} for rings with non-zero characteristic
*}

context ring_1
begin

definition iszero :: "'a \<Rightarrow> bool"
where "iszero z \<longleftrightarrow> z = 0"

lemma iszero_0 [simp]: "iszero 0"

lemma not_iszero_1 [simp]: "\<not> iszero 1"

lemma not_iszero_Numeral1: "\<not> iszero Numeral1"

lemma iszero_neg_numeral [simp]:
"iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"
unfolding iszero_def neg_numeral_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 {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
@{text "[simp]"} by default, because for rings of characteristic zero,
better simp rules are possible. For a type like integers mod @{text
"n"}, type-instantiated versions of these rules should be added to the
simplifier, along with a type-specific rule for deciding propositions
of the form @{text "iszero (numeral w)"}.

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 @{text "ring_char_0"} rules in the simplifier.
*}

lemma eq_numeral_iff_iszero:
"numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
"numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"
"neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
"neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"
"numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
"1 = numeral y \<longleftrightarrow> iszero (sub One y)"
"neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
"1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"
"numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
"0 = numeral y \<longleftrightarrow> iszero (numeral y)"
"neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
"0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"
unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
by simp_all

end

subsubsection {*
Equality and negation: class @{text ring_char_0}
*}

class ring_char_0 = ring_1 + semiring_char_0
begin

lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"

lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"
by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)

lemma numeral_neq_neg_numeral: "numeral m \<noteq> neg_numeral n"

lemma neg_numeral_neq_numeral: "neg_numeral m \<noteq> numeral n"
by (rule numeral_neq_neg_numeral [symmetric])

lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"
unfolding neg_numeral_def neg_0_equal_iff_equal by simp

lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"
unfolding neg_numeral_def neg_equal_0_iff_equal by simp

lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"
using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)

lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"
using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)

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

end

subsubsection {*
Structures with negation and order: class @{text linordered_idom}
*}

context linordered_idom
begin

subclass ring_char_0 ..

lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"
by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)

lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"
by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)

lemma neg_numeral_less_zero: "neg_numeral n < 0"
by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)

lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"
by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)

lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"
by (simp only: not_less neg_numeral_le_zero)

lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"
by (simp only: not_le neg_numeral_less_zero)

lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"
using neg_numeral_less_zero zero_less_numeral by (rule less_trans)

lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"
by (simp only: less_imp_le neg_numeral_less_numeral)

lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"
by (simp only: not_less neg_numeral_le_numeral)

lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"
by (simp only: not_le neg_numeral_less_numeral)

lemma neg_numeral_less_one: "neg_numeral m < 1"
by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])

lemma neg_numeral_le_one: "neg_numeral m \<le> 1"
by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])

lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"
by (simp only: not_less neg_numeral_le_one)

lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"
by (simp only: not_le neg_numeral_less_one)

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

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

lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
by simp

lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"
by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)

end

subsubsection {*
Natural numbers
*}

lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
unfolding numeral_plus_one [symmetric] by simp

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))"

lemma pred_numeral_simps [simp]:
"pred_numeral Num.One = 0"
"pred_numeral (Num.Bit0 k) = numeral (Num.BitM k)"
"pred_numeral (Num.Bit1 k) = numeral (Num.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)"

lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"

lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
by (simp only: numeral_One One_nat_def)

"Suc (numeral v + n) = numeral (v + Num.One) + n"
by simp

(*Maps #n to n for n = 1, 2*)
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2

text {* Comparisons involving @{term Suc}. *}

lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"

lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"

lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"

lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"

lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"

lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"

lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"

lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"

lemma max_Suc_numeral [simp]:
"max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"

lemma max_numeral_Suc [simp]:
"max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"

lemma min_Suc_numeral [simp]:
"min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"

lemma min_numeral_Suc [simp]:
"min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"

text {* For @{term nat_case} and @{term nat_rec}. *}

lemma nat_case_numeral [simp]:
"nat_case a f (numeral v) = (let pv = pred_numeral v in f pv)"

"nat_case a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"

lemma nat_rec_numeral [simp]:
"nat_rec a f (numeral v) =
(let pv = pred_numeral v in f pv (nat_rec a f pv))"

"nat_rec a f (numeral v + n) =
(let pv = pred_numeral v in f (pv + n) (nat_rec a f (pv + n)))"

subsection {* Numeral equations as default simplification rules *}

declare (in numeral) numeral_One [simp]
declare (in numeral) numeral_plus_numeral [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 {* Setting up simprocs *}

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{*Theorem lists for the cancellation simprocs. The use of a binary
numeral for 1 reduces the number of special cases.*}

lemmas mult_1s =
mult_numeral_1 mult_numeral_1_right
mult_minus1 mult_minus1_right

setup {*
(fn Const (@{const_name numeral}, _) \$ _ => true
| Const (@{const_name neg_numeral}, _) \$ _ => true
| _ => false)
*}

simproc_setup reorient_numeral
("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc

subsubsection {* Simplification of arithmetic operations on integer constants. *}

lemmas arith_special = (* already declared simp above *)
diff_numeral_special minus_one

(* rules already in simpset *)
lemmas arith_extra_simps =
minus_numeral minus_neg_numeral minus_zero minus_one
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 {*
For making a minimal simpset, one must include these default simprules.
Also include @{text simp_thms}.
*}

lemmas arith_simps =
BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
abs_zero abs_one arith_extra_simps

text {* Simplification of relational operations *}

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_numeral_extra
less_numeral_simps less_neg_numeral_simps less_numeral_extra
eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra

subsubsection {* Simplification of arithmetic when nested to the right. *}

"numeral v + (numeral w + z) = (numeral(v + w) + z)"

"numeral v + (neg_numeral w + y) = (sub v w + y)"
"neg_numeral v + (numeral w + y) = (sub w v + y)"
"neg_numeral v + (neg_numeral w + y) = (neg_numeral(v + w) + y)"

lemma mult_numeral_left [simp]:
"numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
"neg_numeral v * (numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
"numeral v * (neg_numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
"neg_numeral v * (neg_numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"

hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec

subsection {* code module namespace *}

code_modulename SML
Num Arith

code_modulename OCaml
Num Arith