(*  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"

 proof

   assume "nat_of_num x = nat_of_num y"

   then have "num_of_nat (nat_of_num x) = num_of_nat (nat_of_num y)" by simp

   then show "x = y" by (simp add: nat_of_num_inverse)

 qed simp

 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 {*

   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

   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 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)"

   by (simp_all add: num_eq_iff nat_of_num_add)

 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)"

   by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult

                     left_distrib right_distrib)

 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"

   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)

 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_One: "x * One = x"

   by (simp add: num_eq_iff nat_of_num_mult)

 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 {* 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"

   by (simp add: add_One_commute DigM_plus_one)

 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 n) = of_num n + 1"

   by (induct n) (simp_all add: add_ac)

 lemma of_num_add: "of_num (m + n) = of_num m + of_num n"

   by (induct n rule: num_induct) (simp_all add: add_One add_inc of_num_inc add_ac)

 lemma of_num_mult: "of_num (m * n) = of_num m * of_num n"

   by (induct n rule: num_induct) (simp_all add: mult_One mult_inc of_num_add of_num_inc right_distrib)

 declare of_num.simps [simp del]

 end

 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]);

 fun mk_numeral phi T k = Morphism.term phi (Const (@{const_name of_num}, @{typ num} --> T))

   $ mk_num k

 fun dest_numeral phi (u $ t) =

   if Term.aconv_untyped (u, Morphism.term phi (Const (@{const_name of_num}, dummyT)))

   then (range_type (fastype_of u), dest_num t)

   else raise TERM ("dest_numeral", [u, 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, value, ...} = Syntax.read_xnum num;

           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 (advanced) {*

 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' ctxt 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 (Config.get ctxt 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 *}

 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}, although 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)"

   by (simp only: of_num_add of_num_One)

 lemma of_num_one_plus [numeral]:

   "1 + of_num n = of_num (One + n)"

   by (simp only: of_num_add of_num_One)

 lemma of_num_plus [numeral]:

   "of_num m + of_num n = of_num (m + n)"

   by (simp only: of_num_add)

 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)

     (simp_all add: semiring_numeral_class.of_num.simps of_num.simps add_ac)

 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 {*

   Perhaps the underlying structure could even 

   be more general than @{text linordered_semidom}.

 *}

 context linordered_semidom

 begin

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

   by (induct n) (simp_all add: of_num.simps add_pos_pos)

 lemma of_num_not_zero [numeral]: "of_num n \<noteq> 0"

   using of_num_pos [of n] by simp

 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"

   by (induct n) (simp_all add: of_num.simps add_nonneg_nonneg)

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

   by (simp add: not_less of_num_nonneg)

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

   by (simp add: not_le of_num_pos)

 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_not_equal_of_num: "- of_num m \<noteq> of_num n"

   using minus_of_num_less_of_num_iff [of m n] by simp

 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"

   by (simp add: less_imp_le minus_of_num_less_of_num_iff)

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

   by (simp add: less_imp_le minus_of_num_less_one_iff)

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

   by (simp add: less_imp_le minus_one_less_of_num_iff)

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

   by (simp add: less_imp_le minus_one_less_one_iff)

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

   by (simp add: not_le minus_of_num_less_of_num_iff)

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

   by (simp add: not_le minus_of_num_less_one_iff)

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

   by (simp add: not_le minus_one_less_of_num_iff)

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

   by (simp add: not_le minus_one_less_one_iff)

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

   by (simp add: not_less minus_of_num_le_of_num_iff)

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

   by (simp add: not_less minus_of_num_le_one_iff)

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

   by (simp add: not_less minus_one_le_of_num_iff)

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

   by (simp add: not_less minus_one_le_one_iff)

 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_not_equal_of_num

   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"

   by (simp add: add_ac minus_inverts_plus1 [of b a])

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

   by (simp add: add_ac minus_minus_zero_inverts_plus1 [of b a])

 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"

   by (simp add: minus_inverts_plus1)

 lemma Dig_one_minus_zero [numeral]:

   "1 - 0 = 1"

   by (simp add: minus_inverts_plus1)

 lemma Dig_one_minus_one [numeral]:

   "1 - 1 = 0"

   by (simp add: minus_inverts_plus1)

 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 proof

 qed (simp_all add: algebra_simps)

 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"

   by (simp_all add: diff_minus add_commute)

 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)

    (simp_all add: of_num.simps of_num_add algebra_simps)

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

 by (induct y)

    (simp_all add: of_num.simps of_num_square of_num_mult power_add)

 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]

   by (simp add: power_0_left)

 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 {* Proof tools setup *}

 subsubsection {* 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_not_zero [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]

 subsubsection {* Reorientation of equalities *}

 setup {*

   Reorient_Proc.add

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

 text {* Reversing standard setup *}

 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]

 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 *"

   "(HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool) = HOL.equal"

   "(op \<le> :: int \<Rightarrow> int \<Rightarrow> bool) = op \<le>"

   "(op < :: int \<Rightarrow> int \<Rightarrow> bool) = op <"

   by rule+

 text {* Constructors *}

 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]

 text {* Auxiliary operations *}

 definition dup :: "int \<Rightarrow> int" where

   [simp]: "dup k = k + k"

 lemma Dig_dup [code]:

   "dup 0 = 0"

   "dup (Pls n) = Pls (Dig0 n)"

   "dup (Mns n) = Mns (Dig0 n)"

   by (simp_all add: of_num.simps)

 definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where

   [simp]: "sub m n = (of_num m - of_num n)"

 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"

   by (simp_all add: algebra_simps num_eq_iff nat_of_num_add)

 text {* Implementations *}

 lemma one_int_code [code]:

   "1 = Pls One"

   by (simp add: of_num_One)

 lemma plus_int_code [code]:

   "k + 0 = (k::int)"

   "0 + l = (l::int)"

   "Pls m + Pls n = Pls (m + n)"

   "Pls m + Mns n = sub m n"

   "Mns m + Pls n = sub n m"

   "Mns m + Mns n = Mns (m + n)"

   by simp_all

 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"

   by simp_all

 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)"

   by simp_all

 lemma eq_int_code [code]:

   "HOL.equal 0 (0::int) \<longleftrightarrow> True"

   "HOL.equal 0 (Pls l) \<longleftrightarrow> False"

   "HOL.equal 0 (Mns l) \<longleftrightarrow> False"

   "HOL.equal (Pls k) 0 \<longleftrightarrow> False"

   "HOL.equal (Pls k) (Pls l) \<longleftrightarrow> HOL.equal k l"

   "HOL.equal (Pls k) (Mns l) \<longleftrightarrow> False"

   "HOL.equal (Mns k) 0 \<longleftrightarrow> False"

   "HOL.equal (Mns k) (Pls l) \<longleftrightarrow> False"

   "HOL.equal (Mns k) (Mns l) \<longleftrightarrow> HOL.equal k l"

   by (auto simp add: equal dest: sym)

 lemma [code nbe]:

   "HOL.equal (k::int) k \<longleftrightarrow> True"

   by (fact equal_refl)

 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"

   by simp_all

 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"

   by simp_all

 hide_const (open) sub dup

 text {* Pretty literals *}

 ML {*

 local open Code_Thingol in

 fun add_code print target =

   let

     fun dest_num one' dig0' dig1' thm =

       let

         fun dest_dig (IConst (c, _)) = if c = dig0' then 0

               else if c = dig1' then 1

               else Code_Printer.eqn_error thm "Illegal numeral expression: illegal dig"

           | dest_dig _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal digit";

         fun dest_num (IConst (c, _)) = if c = one' then 1

               else Code_Printer.eqn_error thm "Illegal numeral expression: illegal leading digit"

           | dest_num (t1 `$ t2) = 2 * dest_num t2 + dest_dig t1

           | dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";

       in dest_num end;

     fun pretty sgn literals [one', dig0', dig1'] _ thm _ _ [(t, _)] =

       (Code_Printer.str o print literals o sgn o dest_num one' dig0' dig1' thm) t

     fun add_syntax (c, sgn) = Code_Target.add_const_syntax target c

       (SOME (Code_Printer.complex_const_syntax

         (1, ([@{const_name One}, @{const_name Dig0}, @{const_name Dig1}],

           pretty sgn))));

   in

     add_syntax (@{const_name Pls}, I)

     #> add_syntax (@{const_name Mns}, (fn k => ~ k))

   end;

 end

 *}

 hide_const (open) One Dig0 Dig1

 subsection {* Toy examples *}

 definition "foo \<longleftrightarrow> #4 * #2 + #7 = (#8 :: nat)"

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

 code_thms foo bar

 export_code foo bar checking SML OCaml? Haskell? Scala?

 text {* This is an ad-hoc @{text Code_Integer} setup. *}

 setup {*

   fold (add_code Code_Printer.literal_numeral)

     [Code_ML.target_SML, Code_ML.target_OCaml, Code_Haskell.target, Code_Scala.target]

 *}

 code_type int

   (SML "IntInf.int")

   (OCaml "Big'_int.big'_int")

   (Haskell "Integer")

   (Scala "BigInt")

   (Eval "int")

 code_const "0::int"

   (SML "0/ :/ IntInf.int")

   (OCaml "Big'_int.zero")

   (Haskell "0")

   (Scala "BigInt(0)")

   (Eval "0/ :/ int")

 code_const Int.pred

   (SML "IntInf.- ((_), 1)")

   (OCaml "Big'_int.pred'_big'_int")

   (Haskell "!(_/ -/ 1)")

   (Scala "!(_ -/ 1)")

   (Eval "!(_/ -/ 1)")

 code_const Int.succ

   (SML "IntInf.+ ((_), 1)")

   (OCaml "Big'_int.succ'_big'_int")

   (Haskell "!(_/ +/ 1)")

   (Scala "!(_ +/ 1)")

   (Eval "!(_/ +/ 1)")

 code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"

   (SML "IntInf.+ ((_), (_))")

   (OCaml "Big'_int.add'_big'_int")

   (Haskell infixl 6 "+")

   (Scala infixl 7 "+")

   (Eval infixl 8 "+")

 code_const "uminus \<Colon> int \<Rightarrow> int"

   (SML "IntInf.~")

   (OCaml "Big'_int.minus'_big'_int")

   (Haskell "negate")

   (Scala "!(- _)")

   (Eval "~/ _")

 code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"

   (SML "IntInf.- ((_), (_))")

   (OCaml "Big'_int.sub'_big'_int")

   (Haskell infixl 6 "-")

   (Scala infixl 7 "-")

   (Eval infixl 8 "-")

 code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"

   (SML "IntInf.* ((_), (_))")

   (OCaml "Big'_int.mult'_big'_int")

   (Haskell infixl 7 "*")

   (Scala infixl 8 "*")

   (Eval infixl 9 "*")

 code_const pdivmod

   (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")

   (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")

   (Haskell "divMod/ (abs _)/ (abs _)")

   (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")

   (Eval "Integer.div'_mod/ (abs _)/ (abs _)")

 code_const "HOL.equal \<Colon> int \<Rightarrow> int \<Rightarrow> bool"

   (SML "!((_ : IntInf.int) = _)")

   (OCaml "Big'_int.eq'_big'_int")

   (Haskell infix 4 "==")

   (Scala infixl 5 "==")

   (Eval infixl 6 "=")

 code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"

   (SML "IntInf.<= ((_), (_))")

   (OCaml "Big'_int.le'_big'_int")

   (Haskell infix 4 "<=")

   (Scala infixl 4 "<=")

   (Eval infixl 6 "<=")

 code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"

   (SML "IntInf.< ((_), (_))")

   (OCaml "Big'_int.lt'_big'_int")

   (Haskell infix 4 "<")

   (Scala infixl 4 "<")

   (Eval infixl 6 "<")

 code_const Code_Numeral.int_of

   (SML "IntInf.fromInt")

   (OCaml "_")

   (Haskell "toInteger")

   (Scala "!_.as'_BigInt")

   (Eval "_")

 export_code foo bar checking SML OCaml? Haskell? Scala?

 end