src/HOL/ex/Numeral.thy
 author haftmann Wed, 14 Jul 2010 16:13:14 +0200 changeset 37826 4c0a5e35931a parent 37765 26bdfb7b680b child 38054 acd48ef85bfc permissions -rw-r--r--
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```
(*  Title:      HOL/ex/Numeral.thy
Author:     Florian Haftmann
*)

header {* An experimental alternative numeral representation. *}

theory Numeral
imports Main
begin

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

datatype num = One | Dig0 num | Dig1 num

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

primrec inc :: "num \<Rightarrow> num" where
"inc One = Dig0 One"
| "inc (Dig0 x) = Dig1 x"
| "inc (Dig1 x) = Dig0 (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 (Dig0 x) = nat_of_num x + nat_of_num x"
| "nat_of_num (Dig1 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) = Dig0 (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"}).

It is not entirely clear in which context it is better to use
the one or the other, or whether the construction should be reversed.
*}

subsection {* Numeral operations *}

ML {*
structure Dig_Simps = Named_Thms
(
val name = "numeral"
val description = "Simplification rules for numerals"
)
*}

setup Dig_Simps.setup

instantiation num :: "{plus,times,ord}"
begin

definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where
"m + n = num_of_nat (nat_of_num m + nat_of_num n)"

definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where
"m * n = num_of_nat (nat_of_num m * nat_of_num n)"

definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
"m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"

definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where
"m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"

instance ..

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)

lemma Dig_plus [numeral, simp, code]:
"One + One = Dig0 One"
"One + Dig0 m = Dig1 m"
"One + Dig1 m = Dig0 (m + One)"
"Dig0 n + One = Dig1 n"
"Dig0 n + Dig0 m = Dig0 (n + m)"
"Dig0 n + Dig1 m = Dig1 (n + m)"
"Dig1 n + One = Dig0 (n + One)"
"Dig1 n + Dig0 m = Dig1 (n + m)"
"Dig1 n + Dig1 m = Dig0 (n + m + One)"

lemma Dig_times [numeral, simp, code]:
"One * One = One"
"One * Dig0 n = Dig0 n"
"One * Dig1 n = Dig1 n"
"Dig0 n * One = Dig0 n"
"Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"
"Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"
"Dig1 n * One = Dig1 n"
"Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"
"Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"
left_distrib right_distrib)

lemma Dig_eq:
"One = One \<longleftrightarrow> True"
"One = Dig0 n \<longleftrightarrow> False"
"One = Dig1 n \<longleftrightarrow> False"
"Dig0 m = One \<longleftrightarrow> False"
"Dig1 m = One \<longleftrightarrow> False"
"Dig0 m = Dig0 n \<longleftrightarrow> m = n"
"Dig0 m = Dig1 n \<longleftrightarrow> False"
"Dig1 m = Dig0 n \<longleftrightarrow> False"
"Dig1 m = Dig1 n \<longleftrightarrow> m = n"
by simp_all

lemma less_eq_num_code [numeral, simp, code]:
"One \<le> n \<longleftrightarrow> True"
"Dig0 m \<le> One \<longleftrightarrow> False"
"Dig1 m \<le> One \<longleftrightarrow> False"
"Dig0 m \<le> Dig0 n \<longleftrightarrow> m \<le> n"
"Dig0 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
"Dig1 m \<le> Dig1 n \<longleftrightarrow> m \<le> n"
"Dig1 m \<le> Dig0 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_code [numeral, simp, code]:
"m < One \<longleftrightarrow> False"
"One < One \<longleftrightarrow> False"
"One < Dig0 n \<longleftrightarrow> True"
"One < Dig1 n \<longleftrightarrow> True"
"Dig0 m < Dig0 n \<longleftrightarrow> m < n"
"Dig0 m < Dig1 n \<longleftrightarrow> m \<le> n"
"Dig1 m < Dig1 n \<longleftrightarrow> m < n"
"Dig1 m < Dig0 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_inc: "x + inc y = inc (x + y)"

lemma mult_One: "x * One = x"

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

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

primrec DigM :: "num \<Rightarrow> num" where
"DigM One = One"
| "DigM (Dig0 n) = Dig1 (DigM n)"
| "DigM (Dig1 n) = Dig1 (Dig0 n)"

lemma DigM_plus_one: "DigM n + One = Dig0 n"
by (induct n) simp_all

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

lemma one_plus_DigM: "One + DigM n = Dig0 n"

text {* Squaring and exponentiation *}

primrec square :: "num \<Rightarrow> num" where
"square One = One"
| "square (Dig0 n) = Dig0 (Dig0 (square n))"
| "square (Dig1 n) = Dig1 (Dig0 (square n + n))"

primrec pow :: "num \<Rightarrow> num \<Rightarrow> num"
where
"pow x One = x"
| "pow x (Dig0 y) = square (pow x y)"
| "pow x (Dig1 y) = x * square (pow x y)"

subsection {* Binary numerals *}

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

class semiring_numeral = semiring + monoid_mult
begin

primrec of_num :: "num \<Rightarrow> 'a" where
of_num_One [numeral]: "of_num One = 1"
| "of_num (Dig0 n) = of_num n + of_num n"
| "of_num (Dig1 n) = of_num n + of_num n + 1"

lemma of_num_inc: "of_num (inc x) = of_num x + 1"

lemma of_num_add: "of_num (m + n) = of_num m + of_num n"
apply (induct n rule: num_induct)
done

lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"
apply (induct n rule: num_induct)
done

declare of_num.simps [simp del]

end

text {*
ML stuff and syntax.
*}

ML {*
fun mk_num k =
if k > 1 then
let
val (l, b) = Integer.div_mod k 2;
val bit = (if b = 0 then @{term Dig0} else @{term Dig1});
in bit \$ (mk_num l) end
else if k = 1 then @{term One}
else error ("mk_num " ^ string_of_int k);

fun dest_num @{term One} = 1
| dest_num (@{term Dig0} \$ n) = 2 * dest_num n
| dest_num (@{term Dig1} \$ n) = 2 * dest_num n + 1
| dest_num t = raise TERM ("dest_num", [t]);

(*FIXME these have to gain proper context via morphisms phi*)

fun mk_numeral T k = Const (@{const_name of_num}, @{typ num} --> T)
\$ mk_num k

fun dest_numeral (Const (@{const_name of_num}, Type ("fun", [@{typ num}, T])) \$ t) =
(T, dest_num t)
*}

syntax
"_Numerals" :: "xnum \<Rightarrow> 'a"    ("_")

parse_translation {*
let
fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)
of (0, 1) => Const (@{const_name One}, dummyT)
| (n, 0) => Const (@{const_name Dig0}, dummyT) \$ num_of_int n
| (n, 1) => Const (@{const_name Dig1}, dummyT) \$ num_of_int n
else raise Match;
fun numeral_tr [Free (num, _)] =
let
val _ = leading_zeros = 0 andalso value > 0
orelse error ("Bad numeral: " ^ num);
in Const (@{const_name of_num}, @{typ num} --> dummyT) \$ num_of_int value end
| numeral_tr ts = raise TERM ("numeral_tr", ts);
in [(@{syntax_const "_Numerals"}, numeral_tr)] end
*}

typed_print_translation {*
let
fun dig b n = b + 2 * n;
fun int_of_num' (Const (@{const_syntax Dig0}, _) \$ n) =
dig 0 (int_of_num' n)
| int_of_num' (Const (@{const_syntax Dig1}, _) \$ n) =
dig 1 (int_of_num' n)
| int_of_num' (Const (@{const_syntax One}, _)) = 1;
fun num_tr' show_sorts T [n] =
let
val k = int_of_num' n;
val t' = Syntax.const @{syntax_const "_Numerals"} \$ Syntax.free ("#" ^ string_of_int k);
in case T
of Type (@{type_name fun}, [_, T']) =>
if not (! show_types) andalso can Term.dest_Type T' then t'
else Syntax.const Syntax.constrainC \$ t' \$ Syntax.term_of_typ show_sorts T'
| T' => if T' = dummyT then t' else raise Match
end;
in [(@{const_syntax of_num}, num_tr')] end
*}

subsection {* Class-specific numeral rules *}

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

subsubsection {* Class @{text semiring_numeral} *}

context semiring_numeral
begin

abbreviation "Num1 \<equiv> of_num One"

text {*
Alas, there is still the duplication of @{term 1},
thought the duplicated @{term 0} has disappeared.
We could get rid of it by replacing the constructor
@{term 1} in @{typ num} by two constructors
@{text two} and @{text three}, resulting in a further
blow-up.  But it could be worth the effort.
*}

lemma of_num_plus_one [numeral]:
"of_num n + 1 = of_num (n + One)"

lemma of_num_one_plus [numeral]:
"1 + of_num n = of_num (One + n)"

lemma of_num_plus [numeral]:
"of_num m + of_num n = of_num (m + n)"

lemma of_num_times_one [numeral]:
"of_num n * 1 = of_num n"
by simp

lemma of_num_one_times [numeral]:
"1 * of_num n = of_num n"
by simp

lemma of_num_times [numeral]:
"of_num m * of_num n = of_num (m * n)"
unfolding of_num_mult ..

end

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

context semiring_1
begin

subclass semiring_numeral ..

lemma of_nat_of_num [numeral]: "of_nat (of_num n) = of_num n"
by (induct n)

declare of_nat_1 [numeral]

lemma Dig_plus_zero [numeral]:
"0 + 1 = 1"
"0 + of_num n = of_num n"
"1 + 0 = 1"
"of_num n + 0 = of_num n"
by simp_all

lemma Dig_times_zero [numeral]:
"0 * 1 = 0"
"0 * of_num n = 0"
"1 * 0 = 0"
"of_num n * 0 = 0"
by simp_all

end

lemma nat_of_num_of_num: "nat_of_num = of_num"
proof
fix n
have "of_num n = nat_of_num n"
by (induct n) (simp_all add: of_num.simps)
then show "nat_of_num n = of_num n" by simp
qed

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

context semiring_char_0
begin

lemma of_num_eq_iff [numeral]: "of_num m = of_num n \<longleftrightarrow> m = n"
unfolding of_nat_of_num [symmetric] nat_of_num_of_num [symmetric]
of_nat_eq_iff num_eq_iff ..

lemma of_num_eq_one_iff [numeral]: "of_num n = 1 \<longleftrightarrow> n = One"
using of_num_eq_iff [of n One] by (simp add: of_num_One)

lemma one_eq_of_num_iff [numeral]: "1 = of_num n \<longleftrightarrow> One = n"
using of_num_eq_iff [of One n] by (simp add: of_num_One)

end

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

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

context linordered_semidom
begin

lemma of_num_pos [numeral]: "0 < of_num n"

lemma of_num_less_eq_iff [numeral]: "of_num m \<le> of_num n \<longleftrightarrow> m \<le> n"
proof -
have "of_nat (of_num m) \<le> of_nat (of_num n) \<longleftrightarrow> m \<le> n"
unfolding less_eq_num_def nat_of_num_of_num of_nat_le_iff ..
then show ?thesis by (simp add: of_nat_of_num)
qed

lemma of_num_less_eq_one_iff [numeral]: "of_num n \<le> 1 \<longleftrightarrow> n \<le> One"
using of_num_less_eq_iff [of n One] by (simp add: of_num_One)

lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"
using of_num_less_eq_iff [of One n] by (simp add: of_num_One)

lemma of_num_less_iff [numeral]: "of_num m < of_num n \<longleftrightarrow> m < n"
proof -
have "of_nat (of_num m) < of_nat (of_num n) \<longleftrightarrow> m < n"
unfolding less_num_def nat_of_num_of_num of_nat_less_iff ..
then show ?thesis by (simp add: of_nat_of_num)
qed

lemma of_num_less_one_iff [numeral]: "\<not> of_num n < 1"
using of_num_less_iff [of n One] by (simp add: of_num_One)

lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> One < n"
using of_num_less_iff [of One n] by (simp add: of_num_One)

lemma of_num_nonneg [numeral]: "0 \<le> of_num n"

lemma of_num_less_zero_iff [numeral]: "\<not> of_num n < 0"

lemma of_num_le_zero_iff [numeral]: "\<not> of_num n \<le> 0"

end

context linordered_idom
begin

lemma minus_of_num_less_of_num_iff: "- of_num m < of_num n"
proof -
have "- of_num m < 0" by (simp add: of_num_pos)
also have "0 < of_num n" by (simp add: of_num_pos)
finally show ?thesis .
qed

lemma minus_of_num_less_one_iff: "- of_num n < 1"
using minus_of_num_less_of_num_iff [of n One] by (simp add: of_num_One)

lemma minus_one_less_of_num_iff: "- 1 < of_num n"
using minus_of_num_less_of_num_iff [of One n] by (simp add: of_num_One)

lemma minus_one_less_one_iff: "- 1 < 1"
using minus_of_num_less_of_num_iff [of One One] by (simp add: of_num_One)

lemma minus_of_num_le_of_num_iff: "- of_num m \<le> of_num n"

lemma minus_of_num_le_one_iff: "- of_num n \<le> 1"

lemma minus_one_le_of_num_iff: "- 1 \<le> of_num n"

lemma minus_one_le_one_iff: "- 1 \<le> 1"

lemma of_num_le_minus_of_num_iff: "\<not> of_num m \<le> - of_num n"

lemma one_le_minus_of_num_iff: "\<not> 1 \<le> - of_num n"

lemma of_num_le_minus_one_iff: "\<not> of_num n \<le> - 1"

lemma one_le_minus_one_iff: "\<not> 1 \<le> - 1"

lemma of_num_less_minus_of_num_iff: "\<not> of_num m < - of_num n"

lemma one_less_minus_of_num_iff: "\<not> 1 < - of_num n"

lemma of_num_less_minus_one_iff: "\<not> of_num n < - 1"

lemma one_less_minus_one_iff: "\<not> 1 < - 1"

lemmas le_signed_numeral_special [numeral] =
minus_of_num_le_of_num_iff
minus_of_num_le_one_iff
minus_one_le_of_num_iff
minus_one_le_one_iff
of_num_le_minus_of_num_iff
one_le_minus_of_num_iff
of_num_le_minus_one_iff
one_le_minus_one_iff

lemmas less_signed_numeral_special [numeral] =
minus_of_num_less_of_num_iff
minus_of_num_less_one_iff
minus_one_less_of_num_iff
minus_one_less_one_iff
of_num_less_minus_of_num_iff
one_less_minus_of_num_iff
of_num_less_minus_one_iff
one_less_minus_one_iff

end

subsubsection {*
Structures with subtraction: class @{text semiring_1_minus}
*}

class semiring_minus = semiring + minus + zero +
assumes minus_inverts_plus1: "a + b = c \<Longrightarrow> c - b = a"
assumes minus_minus_zero_inverts_plus1: "a + b = c \<Longrightarrow> b - c = 0 - a"
begin

lemma minus_inverts_plus2: "a + b = c \<Longrightarrow> c - a = b"

lemma minus_minus_zero_inverts_plus2: "a + b = c \<Longrightarrow> a - c = 0 - b"

end

class semiring_1_minus = semiring_1 + semiring_minus
begin

lemma Dig_of_num_pos:
assumes "k + n = m"
shows "of_num m - of_num n = of_num k"
using assms by (simp add: of_num_plus minus_inverts_plus1)

lemma Dig_of_num_zero:
shows "of_num n - of_num n = 0"
by (rule minus_inverts_plus1) simp

lemma Dig_of_num_neg:
assumes "k + m = n"
shows "of_num m - of_num n = 0 - of_num k"
by (rule minus_minus_zero_inverts_plus1) (simp add: of_num_plus assms)

lemmas Dig_plus_eval =
of_num_plus of_num_eq_iff Dig_plus refl [of One, THEN eqTrueI] num.inject

simproc_setup numeral_minus ("of_num m - of_num n") = {*
let
(*TODO proper implicit use of morphism via pattern antiquotations*)
fun cdest_of_num ct = (snd o split_last o snd o Drule.strip_comb) ct;
fun cdest_minus ct = case (rev o snd o Drule.strip_comb) ct of [n, m] => (m, n);
fun attach_num ct = (dest_num (Thm.term_of ct), ct);
fun cdifference t = (pairself (attach_num o cdest_of_num) o cdest_minus) t;
val simplify = MetaSimplifier.rewrite false (map mk_meta_eq @{thms Dig_plus_eval});
fun cert ck cl cj = @{thm eqTrueE} OF [@{thm meta_eq_to_obj_eq} OF [simplify (Drule.list_comb (@{cterm "op = :: num \<Rightarrow> _"},
[Drule.list_comb (@{cterm "op + :: num \<Rightarrow> _"}, [ck, cl]), cj]))]];
in fn phi => fn _ => fn ct => case try cdifference ct
of NONE => (NONE)
| SOME ((k, ck), (l, cl)) => SOME (let val j = k - l in if j = 0
then MetaSimplifier.rewrite false [mk_meta_eq (Morphism.thm phi @{thm Dig_of_num_zero})] ct
else mk_meta_eq (let
val cj = Thm.cterm_of (Thm.theory_of_cterm ct) (mk_num (abs j));
in
(if j > 0 then (Morphism.thm phi @{thm Dig_of_num_pos}) OF [cert cj cl ck]
else (Morphism.thm phi @{thm Dig_of_num_neg}) OF [cert cj ck cl])
end) end)
end
*}

lemma Dig_of_num_minus_zero [numeral]:
"of_num n - 0 = of_num n"

lemma Dig_one_minus_zero [numeral]:
"1 - 0 = 1"

lemma Dig_one_minus_one [numeral]:
"1 - 1 = 0"

lemma Dig_of_num_minus_one [numeral]:
"of_num (Dig0 n) - 1 = of_num (DigM n)"
"of_num (Dig1 n) - 1 = of_num (Dig0 n)"
by (auto intro: minus_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)

lemma Dig_one_minus_of_num [numeral]:
"1 - of_num (Dig0 n) = 0 - of_num (DigM n)"
"1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"
by (auto intro: minus_minus_zero_inverts_plus1 simp add: DigM_plus_one of_num.simps of_num_plus_one)

end

subsubsection {*
Structures with negation: class @{text ring_1}
*}

context ring_1
begin

subclass semiring_1_minus

lemma Dig_zero_minus_of_num [numeral]:
"0 - of_num n = - of_num n"
by simp

lemma Dig_zero_minus_one [numeral]:
"0 - 1 = - 1"
by simp

lemma Dig_uminus_uminus [numeral]:
"- (- of_num n) = of_num n"
by simp

lemma Dig_plus_uminus [numeral]:
"of_num m + - of_num n = of_num m - of_num n"
"- of_num m + of_num n = of_num n - of_num m"
"- of_num m + - of_num n = - (of_num m + of_num n)"
"of_num m - - of_num n = of_num m + of_num n"
"- of_num m - of_num n = - (of_num m + of_num n)"
"- of_num m - - of_num n = of_num n - of_num m"

lemma Dig_times_uminus [numeral]:
"- of_num n * of_num m = - (of_num n * of_num m)"
"of_num n * - of_num m = - (of_num n * of_num m)"
"- of_num n * - of_num m = of_num n * of_num m"
by simp_all

lemma of_int_of_num [numeral]: "of_int (of_num n) = of_num n"
by (induct n)
(simp_all only: of_num.simps semiring_numeral_class.of_num.simps of_int_add, simp_all)

declare of_int_1 [numeral]

end

subsubsection {*
Structures with exponentiation
*}

lemma of_num_square: "of_num (square x) = of_num x * of_num x"
by (induct x)

lemma of_num_pow: "of_num (pow x y) = of_num x ^ of_num y"
by (induct y)

lemma power_of_num [numeral]: "of_num x ^ of_num y = of_num (pow x y)"
unfolding of_num_pow ..

lemma power_zero_of_num [numeral]:
"0 ^ of_num n = (0::'a::semiring_1)"
using of_num_pos [where n=n and ?'a=nat]

lemma power_minus_Dig0 [numeral]:
fixes x :: "'a::ring_1"
shows "(- x) ^ of_num (Dig0 n) = x ^ of_num (Dig0 n)"
by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)

lemma power_minus_Dig1 [numeral]:
fixes x :: "'a::ring_1"
shows "(- x) ^ of_num (Dig1 n) = - (x ^ of_num (Dig1 n))"
by (induct n rule: num_induct) (simp_all add: of_num.simps of_num_inc)

declare power_one [numeral]

subsubsection {*
Greetings to @{typ nat}.
*}

instance nat :: semiring_1_minus proof qed simp_all

lemma Suc_of_num [numeral]: "Suc (of_num n) = of_num (n + One)"
unfolding of_num_plus_one [symmetric] by simp

lemma nat_number:
"1 = Suc 0"
"of_num One = Suc 0"
"of_num (Dig0 n) = Suc (of_num (DigM n))"
"of_num (Dig1 n) = Suc (of_num (Dig0 n))"
by (simp_all add: of_num.simps DigM_plus_one Suc_of_num)

declare diff_0_eq_0 [numeral]

subsection {* Code generator setup for @{typ int} *}

definition Pls :: "num \<Rightarrow> int" where
[simp, code_post]: "Pls n = of_num n"

definition Mns :: "num \<Rightarrow> int" where
[simp, code_post]: "Mns n = - of_num n"

code_datatype "0::int" Pls Mns

lemmas [code_unfold] = Pls_def [symmetric] Mns_def [symmetric]

definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
[simp]: "sub m n = (of_num m - of_num n)"

definition dup :: "int \<Rightarrow> int" where
"dup k = 2 * k"

lemma Dig_sub [code]:
"sub One One = 0"
"sub (Dig0 m) One = of_num (DigM m)"
"sub (Dig1 m) One = of_num (Dig0 m)"
"sub One (Dig0 n) = - of_num (DigM n)"
"sub One (Dig1 n) = - of_num (Dig0 n)"
"sub (Dig0 m) (Dig0 n) = dup (sub m n)"
"sub (Dig1 m) (Dig1 n) = dup (sub m n)"
"sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
"sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
done

lemma dup_code [code]:
"dup 0 = 0"
"dup (Pls n) = Pls (Dig0 n)"
"dup (Mns n) = Mns (Dig0 n)"

lemma [code, code del]:
"(1 :: int) = 1"
"(op + :: int \<Rightarrow> int \<Rightarrow> int) = op +"
"(uminus :: int \<Rightarrow> int) = uminus"
"(op - :: int \<Rightarrow> int \<Rightarrow> int) = op -"
"(op * :: int \<Rightarrow> int \<Rightarrow> int) = op *"
"(eq_class.eq :: int \<Rightarrow> int \<Rightarrow> bool) = eq_class.eq"
"(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"
"(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"
by rule+

lemma one_int_code [code]:
"1 = Pls One"

lemma plus_int_code [code]:
"k + 0 = (k::int)"
"0 + l = (l::int)"
"Pls m + Pls n = Pls (m + n)"
"Pls m - Pls n = sub m n"
"Mns m + Mns n = Mns (m + n)"
"Mns m - Mns n = sub n m"

lemma uminus_int_code [code]:
"uminus 0 = (0::int)"
"uminus (Pls m) = Mns m"
"uminus (Mns m) = Pls m"
by simp_all

lemma minus_int_code [code]:
"k - 0 = (k::int)"
"0 - l = uminus (l::int)"
"Pls m - Pls n = sub m n"
"Pls m - Mns n = Pls (m + n)"
"Mns m - Pls n = Mns (m + n)"
"Mns m - Mns n = sub n m"

lemma times_int_code [code]:
"k * 0 = (0::int)"
"0 * l = (0::int)"
"Pls m * Pls n = Pls (m * n)"
"Pls m * Mns n = Mns (m * n)"
"Mns m * Pls n = Mns (m * n)"
"Mns m * Mns n = Pls (m * n)"

lemma eq_int_code [code]:
"eq_class.eq 0 (0::int) \<longleftrightarrow> True"
"eq_class.eq 0 (Pls l) \<longleftrightarrow> False"
"eq_class.eq 0 (Mns l) \<longleftrightarrow> False"
"eq_class.eq (Pls k) 0 \<longleftrightarrow> False"
"eq_class.eq (Pls k) (Pls l) \<longleftrightarrow> eq_class.eq k l"
"eq_class.eq (Pls k) (Mns l) \<longleftrightarrow> False"
"eq_class.eq (Mns k) 0 \<longleftrightarrow> False"
"eq_class.eq (Mns k) (Pls l) \<longleftrightarrow> False"
"eq_class.eq (Mns k) (Mns l) \<longleftrightarrow> eq_class.eq k l"
using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]

lemma less_eq_int_code [code]:
"0 \<le> (0::int) \<longleftrightarrow> True"
"0 \<le> Pls l \<longleftrightarrow> True"
"0 \<le> Mns l \<longleftrightarrow> False"
"Pls k \<le> 0 \<longleftrightarrow> False"
"Pls k \<le> Pls l \<longleftrightarrow> k \<le> l"
"Pls k \<le> Mns l \<longleftrightarrow> False"
"Mns k \<le> 0 \<longleftrightarrow> True"
"Mns k \<le> Pls l \<longleftrightarrow> True"
"Mns k \<le> Mns l \<longleftrightarrow> l \<le> k"
using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]

lemma less_int_code [code]:
"0 < (0::int) \<longleftrightarrow> False"
"0 < Pls l \<longleftrightarrow> True"
"0 < Mns l \<longleftrightarrow> False"
"Pls k < 0 \<longleftrightarrow> False"
"Pls k < Pls l \<longleftrightarrow> k < l"
"Pls k < Mns l \<longleftrightarrow> False"
"Mns k < 0 \<longleftrightarrow> True"
"Mns k < Pls l \<longleftrightarrow> True"
"Mns k < Mns l \<longleftrightarrow> l < k"
using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]

lemma [code_unfold del]: "(0::int) \<equiv> Numeral0" by simp
lemma [code_unfold del]: "(1::int) \<equiv> Numeral1" by simp
declare zero_is_num_zero [code_unfold del]
declare one_is_num_one [code_unfold del]

hide_const (open) sub dup

subsection {* Numeral equations as default simplification rules *}

declare (in semiring_numeral) of_num_One [simp]
declare (in semiring_numeral) of_num_plus_one [simp]
declare (in semiring_numeral) of_num_one_plus [simp]
declare (in semiring_numeral) of_num_plus [simp]
declare (in semiring_numeral) of_num_times [simp]

declare (in semiring_1) of_nat_of_num [simp]

declare (in semiring_char_0) of_num_eq_iff [simp]
declare (in semiring_char_0) of_num_eq_one_iff [simp]
declare (in semiring_char_0) one_eq_of_num_iff [simp]

declare (in linordered_semidom) of_num_pos [simp]
declare (in linordered_semidom) of_num_less_eq_iff [simp]
declare (in linordered_semidom) of_num_less_eq_one_iff [simp]
declare (in linordered_semidom) one_less_eq_of_num_iff [simp]
declare (in linordered_semidom) of_num_less_iff [simp]
declare (in linordered_semidom) of_num_less_one_iff [simp]
declare (in linordered_semidom) one_less_of_num_iff [simp]
declare (in linordered_semidom) of_num_nonneg [simp]
declare (in linordered_semidom) of_num_less_zero_iff [simp]
declare (in linordered_semidom) of_num_le_zero_iff [simp]

declare (in linordered_idom) le_signed_numeral_special [simp]
declare (in linordered_idom) less_signed_numeral_special [simp]

declare (in semiring_1_minus) Dig_of_num_minus_one [simp]
declare (in semiring_1_minus) Dig_one_minus_of_num [simp]

declare (in ring_1) Dig_plus_uminus [simp]
declare (in ring_1) of_int_of_num [simp]

declare power_of_num [simp]
declare power_zero_of_num [simp]
declare power_minus_Dig0 [simp]
declare power_minus_Dig1 [simp]

declare Suc_of_num [simp]

subsection {* Simplification Procedures *}

subsubsection {* Reorientation of equalities *}

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

simproc_setup reorient_num ("of_num n = x" | "- of_num m = y") = Reorient_Proc.proc

subsubsection {* Constant folding for multiplication in semirings *}

context semiring_numeral
begin

lemma mult_of_num_commute: "x * of_num n = of_num n * x"
by (induct n)
(simp_all only: of_num.simps left_distrib right_distrib mult_1_left mult_1_right)

definition
"commutes_with a b \<longleftrightarrow> a * b = b * a"

lemma commutes_with_commute: "commutes_with a b \<Longrightarrow> a * b = b * a"
unfolding commutes_with_def .

lemma commutes_with_left_commute: "commutes_with a b \<Longrightarrow> a * (b * c) = b * (a * c)"
unfolding commutes_with_def by (simp only: mult_assoc [symmetric])

lemma commutes_with_numeral: "commutes_with x (of_num n)" "commutes_with (of_num n) x"
unfolding commutes_with_def by (simp_all add: mult_of_num_commute)

lemmas mult_ac_numeral =
mult_assoc
commutes_with_commute
commutes_with_left_commute
commutes_with_numeral

end

ML {*
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val assoc_ss = HOL_ss addsimps @{thms mult_ac_numeral}
val eq_reflection = eq_reflection
fun is_numeral (Const(@{const_name of_num}, _) \$ _) = true
| is_numeral _ = false;
end;

structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
*}

simproc_setup semiring_assoc_fold' ("(a::'a::semiring_numeral) * b") =
{* fn phi => fn ss => fn ct =>
Semiring_Times_Assoc.proc ss (Thm.term_of ct) *}

subsection {* Toy examples *}

definition "bar \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat) \<and> #4 * #2 + #7 \<ge> (#8 :: int) - #3"

code_thms bar

export_code bar checking SML OCaml? Haskell?

end
```