# Theory Num

theory Num
imports Transfer
```(*  Title:      HOL/Num.thy
Author:     Florian Haftmann
Author:     Brian Huffman
*)

section ‹Binary Numerals›

theory Num
imports BNF_Least_Fixpoint Transfer
begin

subsection ‹The ‹num› type›

datatype num = One | Bit0 num | Bit1 num

text ‹Increment function for type @{typ num}›

primrec inc :: "num ⇒ 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 ⇒ 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 ⇒ 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 ≠ 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 ⟹ 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 ⟹ nat_of_num (num_of_nat n) = n"
by (induct n) (simp_all add: nat_of_num_inc)

lemma num_eq_iff: "x = y ⟷ 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 ⇒ bool"
assumes One: "P One"
and inc: "⋀x. P x ⟹ 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
from One show ?case 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 ‹num_induct›) and as digit representation (rules
‹num.induct›, ‹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 ≤ n ⟷ nat_of_num m ≤ nat_of_num n"

definition [code del]: "m < n ⟷ 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

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))"

lemma eq_num_simps:
"One = One ⟷ True"
"One = Bit0 n ⟷ False"
"One = Bit1 n ⟷ False"
"Bit0 m = One ⟷ False"
"Bit1 m = One ⟷ False"
"Bit0 m = Bit0 n ⟷ m = n"
"Bit0 m = Bit1 n ⟷ False"
"Bit1 m = Bit0 n ⟷ False"
"Bit1 m = Bit1 n ⟷ m = n"
by simp_all

lemma le_num_simps [simp, code]:
"One ≤ n ⟷ True"
"Bit0 m ≤ One ⟷ False"
"Bit1 m ≤ One ⟷ False"
"Bit0 m ≤ Bit0 n ⟷ m ≤ n"
"Bit0 m ≤ Bit1 n ⟷ m ≤ n"
"Bit1 m ≤ Bit1 n ⟷ m ≤ n"
"Bit1 m ≤ Bit0 n ⟷ 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 ⟷ False"
"One < Bit0 n ⟷ True"
"One < Bit1 n ⟷ True"
"Bit0 m < Bit0 n ⟷ m < n"
"Bit0 m < Bit1 n ⟷ m ≤ n"
"Bit1 m < Bit1 n ⟷ m < n"
"Bit1 m < Bit0 n ⟷ 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 ≤ num.One ⟷ x = num.One"

text ‹Rules using ‹One› and ‹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 ≤ 1 ⟹ num_of_nat n = One"
by (cases n) simp_all

lemma num_of_nat_plus_distrib:
"0 < m ⟹ 0 < n ⟹ num_of_nat (m + n) = num_of_nat m + num_of_nat n"

text ‹A double-and-decrement function.›

primrec BitM :: "num ⇒ 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 ⇒ num"
where
"sqr One = One"
| "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))"
| "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"

primrec pow :: "num ⇒ num ⇒ 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 ‹numeral›.
›

class numeral = one + semigroup_add
begin

primrec numeral :: "num ⇒ '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)"

lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
proof (induct x)
case One
then show ?case by simp
next
case Bit0
next
case Bit1
qed

lemma numeral_inc: "numeral (inc x) = numeral x + 1"
proof (induct x)
case One
then show ?case by simp
next
case Bit0
then show ?case by simp
next
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

declare numeral.simps [simp del]

abbreviation "Numeral1 ≡ numeral One"

declare numeral_One [code_post]

end

text ‹Numeral syntax.›

syntax
"_Numeral" :: "num_const ⇒ 'a"    ("_")

ML_file "Tools/numeral.ML"

parse_translation ‹
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
›

typed_print_translation ‹
let
fun num_tr' ctxt T [n] =
let
val k = Numeral.dest_num_syntax 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
›

subsection ‹Class-specific numeral rules›

text ‹@{const numeral} is a morphism.›

subsubsection ‹Structures with addition: class ‹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 ‹neg_numeral››

class neg_numeral = numeral + group_add
begin

lemma uminus_numeral_One: "- Numeral1 = - 1"

text ‹Numerals form an abelian subgroup.›

inductive is_num :: "'a ⇒ bool"
where
"is_num 1"
| "is_num x ⟹ is_num (- x)"
| "is_num x ⟹ is_num y ⟹ 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: "is_num x ⟹ is_num y ⟹ x + y = y + x"
apply (induct x rule: is_num.induct)
apply (induct y rule: is_num.induct)
apply simp
done

lemma is_num_add_left_commute: "is_num x ⟹ is_num y ⟹ x + (y + z) = y + (x + z)"

lemmas is_num_normalize =
is_num.intros is_num_numeral

definition dbl :: "'a ⇒ 'a"
where "dbl x = x + x"

definition dbl_inc :: "'a ⇒ 'a"
where "dbl_inc x = x + x + 1"

definition dbl_dec :: "'a ⇒ 'a"
where "dbl_dec x = x + x - 1"

definition sub :: "num ⇒ num ⇒ '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)"

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

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 m + - numeral n = sub m n"
"- numeral m + numeral n = sub n m"
"- numeral m + - numeral n = - (numeral m + numeral n)"

"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"

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"

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"

end

subsubsection ‹Structures with multiplication: class ‹semiring_numeral››

class semiring_numeral = semiring + monoid_mult
begin

subclass numeral ..

lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
by (induct n rule: num_induct)

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

lemma mult_2: "2 * z = z + z"

lemma mult_2_right: "z * 2 = z + z"

end

subsubsection ‹Structures with a zero: class ‹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 numeral_unfold_funpow:
"numeral k = (op + 1 ^^ numeral k) 0"
unfolding of_nat_def [symmetric] by simp

end

lemma transfer_rule_numeral:
fixes R :: "'a::semiring_1 ⇒ 'b::semiring_1 ⇒ bool"
assumes [transfer_rule]: "R 0 0" "R 1 1"
"rel_fun R (rel_fun R R) plus plus"
shows "rel_fun HOL.eq R numeral numeral"
apply (subst (2) numeral_unfold_funpow [abs_def])
apply (subst (1) numeral_unfold_funpow [abs_def])
apply transfer_prover
done

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))"

subsubsection ‹Equality: class ‹semiring_char_0››

context semiring_char_0
begin

lemma numeral_eq_iff: "numeral m = numeral n ⟷ m = n"
by (simp only: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
of_nat_eq_iff num_eq_iff)

lemma numeral_eq_one_iff: "numeral n = 1 ⟷ n = One"
by (rule numeral_eq_iff [of n One, unfolded numeral_One])

lemma one_eq_numeral_iff: "1 = numeral n ⟷ One = n"
by (rule numeral_eq_iff [of One n, unfolded numeral_One])

lemma numeral_neq_zero: "numeral n ≠ 0"
by (simp add: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric] nat_of_num_pos)

lemma zero_neq_numeral: "0 ≠ 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 ‹linordered_semidom››

text ‹Could be perhaps more general than here.›

context linordered_semidom
begin

lemma numeral_le_iff: "numeral m ≤ numeral n ⟷ m ≤ n"
proof -
have "of_nat (numeral m) ≤ of_nat (numeral n) ⟷ m ≤ n"
by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
then show ?thesis by simp
qed

lemma one_le_numeral: "1 ≤ numeral n"
using numeral_le_iff [of One n] by (simp add: numeral_One)

lemma numeral_le_one_iff: "numeral n ≤ 1 ⟷ n ≤ One"
using numeral_le_iff [of n One] by (simp add: numeral_One)

lemma numeral_less_iff: "numeral m < numeral n ⟷ m < n"
proof -
have "of_nat (numeral m) < of_nat (numeral n) ⟷ 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: "¬ numeral n < 1"
using numeral_less_iff [of n One] by (simp add: numeral_One)

lemma one_less_numeral_iff: "1 < numeral n ⟷ One < n"
using numeral_less_iff [of One n] by (simp add: numeral_One)

lemma zero_le_numeral: "0 ≤ numeral n"
by (induct n) (simp_all add: numeral.simps)

lemma zero_less_numeral: "0 < numeral n"

lemma not_numeral_le_zero: "¬ numeral n ≤ 0"

lemma not_numeral_less_zero: "¬ 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

lemma min_0_1 [simp]:
fixes min' :: "'a ⇒ 'a ⇒ 'a"
defines "min' ≡ 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 ⇒ 'a ⇒ 'a"
defines "max' ≡ 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 ‹Multiplication and negation: class ‹ring_1››

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)"
by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)

lemma mult_minus1 [simp]: "- 1 * z = - z"

lemma mult_minus1_right [simp]: "z * - 1 = - z"

end

subsubsection ‹Equality using ‹iszero› for rings with non-zero characteristic›

context ring_1
begin

definition iszero :: "'a ⇒ bool"
where "iszero z ⟷ z = 0"

lemma iszero_0 [simp]: "iszero 0"

lemma not_iszero_1 [simp]: "¬ iszero 1"

lemma not_iszero_Numeral1: "¬ iszero Numeral1"

lemma not_iszero_neg_1 [simp]: "¬ iszero (- 1)"

lemma not_iszero_neg_Numeral1: "¬ iszero (- Numeral1)"

lemma iszero_neg_numeral [simp]: "iszero (- numeral w) ⟷ iszero (numeral w)"
unfolding iszero_def by (rule neg_equal_0_iff_equal)

lemma eq_iff_iszero_diff: "x = y ⟷ iszero (x - y)"
unfolding iszero_def by (rule eq_iff_diff_eq_0)

text ‹
The ‹eq_numeral_iff_iszero› lemmas are not declared ‹[simp]› by default,
because for rings of characteristic zero, better simp rules are possible.
For a type like integers mod ‹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 ‹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 ‹ring_char_0›
rules in the simplifier.
›

lemma eq_numeral_iff_iszero:
"numeral x = numeral y ⟷ iszero (sub x y)"
"numeral x = - numeral y ⟷ iszero (numeral (x + y))"
"- numeral x = numeral y ⟷ iszero (numeral (x + y))"
"- numeral x = - numeral y ⟷ iszero (sub y x)"
"numeral x = 1 ⟷ iszero (sub x One)"
"1 = numeral y ⟷ iszero (sub One y)"
"- numeral x = 1 ⟷ iszero (numeral (x + One))"
"1 = - numeral y ⟷ iszero (numeral (One + y))"
"numeral x = 0 ⟷ iszero (numeral x)"
"0 = numeral y ⟷ iszero (numeral y)"
"- numeral x = 0 ⟷ iszero (numeral x)"
"0 = - numeral y ⟷ iszero (numeral y)"
unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
by simp_all

end

subsubsection ‹Equality and negation: class ‹ring_char_0››

context ring_char_0
begin

lemma not_iszero_numeral [simp]: "¬ iszero (numeral w)"

lemma neg_numeral_eq_iff: "- numeral m = - numeral n ⟷ m = n"
by simp

lemma numeral_neq_neg_numeral: "numeral m ≠ - numeral n"

lemma neg_numeral_neq_numeral: "- numeral m ≠ numeral n"
by (rule numeral_neq_neg_numeral [symmetric])

lemma zero_neq_neg_numeral: "0 ≠ - numeral n"
by simp

lemma neg_numeral_neq_zero: "- numeral n ≠ 0"
by simp

lemma one_neq_neg_numeral: "1 ≠ - numeral n"
using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)

lemma neg_numeral_neq_one: "- numeral n ≠ 1"
using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)

lemma neg_one_neq_numeral: "- 1 ≠ numeral n"
using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)

lemma numeral_neq_neg_one: "numeral n ≠ - 1"
using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)

lemma neg_one_eq_numeral_iff: "- 1 = - numeral n ⟷ 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 ⟷ n = One"
using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)

lemma neg_one_neq_zero: "- 1 ≠ 0"
by simp

lemma zero_neq_neg_one: "0 ≠ - 1"
by simp

lemma neg_one_neq_one: "- 1 ≠ 1"
using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)

lemma one_neq_neg_one: "1 ≠ - 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 ‹Structures with negation and order: class ‹linordered_idom››

context linordered_idom
begin

subclass ring_char_0 ..

lemma neg_numeral_le_iff: "- numeral m ≤ - numeral n ⟷ n ≤ m"
by (simp only: neg_le_iff_le numeral_le_iff)

lemma neg_numeral_less_iff: "- numeral m < - numeral n ⟷ 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 ≤ 0"
by (simp only: neg_le_0_iff_le zero_le_numeral)

lemma not_zero_less_neg_numeral: "¬ 0 < - numeral n"
by (simp only: not_less neg_numeral_le_zero)

lemma not_zero_le_neg_numeral: "¬ 0 ≤ - 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 ≤ numeral n"
by (simp only: less_imp_le neg_numeral_less_numeral)

lemma not_numeral_less_neg_numeral: "¬ numeral m < - numeral n"
by (simp only: not_less neg_numeral_le_numeral)

lemma not_numeral_le_neg_numeral: "¬ numeral m ≤ - 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 ≤ 1"
by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])

lemma not_one_less_neg_numeral: "¬ 1 < - numeral m"
by (simp only: not_less neg_numeral_le_one)

lemma not_one_le_neg_numeral: "¬ 1 ≤ - numeral m"
by (simp only: not_le neg_numeral_less_one)

lemma not_numeral_less_neg_one: "¬ numeral m < - 1"
using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)

lemma not_numeral_le_neg_one: "¬ numeral m ≤ - 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 ≤ 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 ⟷ m ≠ One"
by (cases m) simp_all

lemma neg_numeral_le_neg_one: "- numeral m ≤ - 1"
by simp

lemma not_neg_one_less_neg_numeral: "¬ - 1 < - numeral m"
by simp

lemma not_neg_one_le_neg_numeral_iff: "¬ - 1 ≤ - numeral m ⟷ m ≠ One"
by (cases m) simp_all

lemma sub_non_negative: "sub n m ≥ 0 ⟷ n ≥ m"
by (simp only: sub_def le_diff_eq) simp

lemma sub_positive: "sub n m > 0 ⟷ n > m"
by (simp only: sub_def less_diff_eq) simp

lemma sub_non_positive: "sub n m ≤ 0 ⟷ n ≤ m"
by (simp only: sub_def diff_le_eq) simp

lemma sub_negative: "sub n m < 0 ⟷ 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 ≤ 0"
"- 1 ≤ 1"
"¬ 0 ≤ - 1"
"¬ 1 ≤ - 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"
"¬ 0 < - 1"
"¬ 1 < - 1"

lemma abs_numeral [simp]: "¦numeral n¦ = numeral n"
by simp

lemma abs_neg_numeral [simp]: "¦- numeral n¦ = numeral n"
by (simp only: abs_minus_cancel abs_numeral)

lemma abs_neg_one [simp]: "¦- 1¦ = 1"
by simp

end

subsubsection ‹Natural numbers›

lemma Suc_1 [simp]: "Suc 1 = 2"

lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
unfolding Suc_eq_plus1 by (rule numeral_plus_one)

definition pred_numeral :: "num ⇒ nat"
where [code del]: "pred_numeral k = numeral k - 1"

lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"

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 One = 0"
"pred_numeral (Bit0 k) = numeral (BitM k)"
"pred_numeral (Bit1 k) = numeral (Bit0 k)"
by (simp_all only: pred_numeral_def eval_nat_numeral 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)

lemma Suc_nat_number_of_add: "Suc (numeral v + n) = numeral (v + One) + n"
by simp

lemma numerals: "Numeral1 = (1::nat)" "2 = Suc (Suc 0)"
by (rule numeral_One) (rule numeral_2_eq_2)

lemmas numeral_nat = eval_nat_numeral BitM.simps One_nat_def

text ‹Comparisons involving @{term Suc}.›

lemma eq_numeral_Suc [simp]: "numeral k = Suc n ⟷ pred_numeral k = n"

lemma Suc_eq_numeral [simp]: "Suc n = numeral k ⟷ n = pred_numeral k"

lemma less_numeral_Suc [simp]: "numeral k < Suc n ⟷ pred_numeral k < n"

lemma less_Suc_numeral [simp]: "Suc n < numeral k ⟷ n < pred_numeral k"

lemma le_numeral_Suc [simp]: "numeral k ≤ Suc n ⟷ pred_numeral k ≤ n"

lemma le_Suc_numeral [simp]: "Suc n ≤ numeral k ⟷ n ≤ 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 case_nat} and @{term rec_nat}.›

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

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

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

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

text ‹Case analysis on @{term "n < 2"}.›
lemma less_2_cases: "n < 2 ⟹ n = 0 ∨ n = Suc 0"

text ‹Removal of Small Numerals: 0, 1 and (in additive positions) 2.›
text ‹bh: Are these rules really a good idea?›

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 ‹Can be used to eliminate long strings of Sucs, but not by default.›
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
by simp

subsection ‹Particular lemmas concerning @{term 2}›

context linordered_field
begin

subclass field_char_0 ..

lemma half_gt_zero_iff: "0 < a / 2 ⟷ 0 < a"

lemma half_gt_zero [simp]: "0 < a ⟹ 0 < a / 2"

end

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"
for a :: "'a::semiring_numeral"
by simp

lemma mult_numeral_1_right: "a * Numeral1 = a"
for a :: "'a::semiring_numeral"
by simp

lemma divide_numeral_1: "a / Numeral1 = a"
for 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.
›

lemma mult_1s:
"Numeral1 * a = a"
"a * Numeral1 = a"
"- Numeral1 * b = - b"
"b * - Numeral1 = - b"
for a :: "'a::semiring_numeral" and b :: "'b::ring_1"
by simp_all

setup ‹
(fn Const (@{const_name numeral}, _) \$ _ => true
| Const (@{const_name uminus}, _) \$ (Const (@{const_name numeral}, _) \$ _) => true
| _ => false)
›

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

subsubsection ‹Simplification of arithmetic operations on integer constants›

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

lemmas arith_extra_simps = (* rules already in simpset *)
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 ‹
For making a minimal simpset, one must include these default simprules.
Also include ‹simp_thms›.
›

lemmas arith_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

lemmas of_nat_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_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)"
― ‹Unfold all ‹let›s involving constants›
unfolding Let_def ..

lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
― ‹Unfold all ‹let›s involving constants›
unfolding Let_def ..

declaration ‹
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 (
@{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
arith_special numeral_One of_nat_simps uminus_numeral_One}
@{thms Suc_numeral Let_numeral Let_neg_numeral Let_0 Let_1
le_Suc_numeral le_numeral_Suc less_Suc_numeral less_numeral_Suc
Suc_eq_numeral eq_numeral_Suc mult_Suc mult_Suc_right of_nat_numeral}
#> Lin_Arith.set_number_of number_of)
end
›

subsubsection ‹Simplification of arithmetic when nested to the right›

lemma add_numeral_left [simp]: "numeral v + (numeral w + z) = (numeral(v + w) + z)"

"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)"

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)"

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

subsection ‹Code module namespace›

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

subsection ‹Printing of evaluated natural numbers as numerals›

lemma [code_post]:
"Suc 0 = 1"
"Suc 1 = 2"
"Suc (numeral n) = numeral (Num.inc n)"

lemmas [code_post] = Num.inc.simps

end
```