(*  Title:      HOL/ex/Numeral.thy

    ID:         $Id$

    Author:     Florian Haftmann

An experimental alternative numeral representation.

*)

theory Numeral

imports Int Inductive

begin

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

text {*

  We construct @{text num} as a copy of strictly positive

  natural numbers.

*}

typedef (open) num = "\<lambda>n\<Colon>nat. n > 0"

  morphisms nat_of_num num_of_nat_abs

  by (auto simp add: mem_def)

text {*

  A totalized abstraction function.  It is not entirely clear

  whether this is really useful.

*}

definition num_of_nat :: "nat \<Rightarrow> num" where

  "num_of_nat n = (if n = 0 then num_of_nat_abs 1 else num_of_nat_abs n)"

lemma num_cases [case_names nat, cases type: num]:

  assumes "(\<And>n\<Colon>nat. m = num_of_nat n \<Longrightarrow> 0 < n \<Longrightarrow> P)"

  shows P

apply (rule num_of_nat_abs_cases)

apply (unfold mem_def)

using assms unfolding num_of_nat_def

apply auto

done

lemma num_of_nat_zero: "num_of_nat 0 = num_of_nat 1"

  by (simp add: num_of_nat_def)

lemma num_of_nat_inverse: "nat_of_num (num_of_nat n) = (if n = 0 then 1 else n)"

  apply (simp add: num_of_nat_def)

  apply (subst num_of_nat_abs_inverse)

  apply (auto simp add: mem_def num_of_nat_abs_inverse)

  done

lemma num_of_nat_inject:

  "num_of_nat m = num_of_nat n \<longleftrightarrow> m = n \<or> (m = 0 \<or> m = 1) \<and> (n = 0 \<or> n = 1)"

by (auto simp add: num_of_nat_def num_of_nat_abs_inject [unfolded mem_def])

lemma split_num_all:

  "(\<And>m. PROP P m) \<equiv> (\<And>n. PROP P (num_of_nat n))"

proof

  fix n

  assume "\<And>m\<Colon>num. PROP P m"

  then show "PROP P (num_of_nat n)" .

next

  fix m

  have nat_of_num: "\<And>m. nat_of_num m \<noteq> 0"

    using nat_of_num by (auto simp add: mem_def)

  have nat_of_num_inverse: "\<And>m. num_of_nat (nat_of_num m) = m"

    by (auto simp add: num_of_nat_def nat_of_num_inverse nat_of_num)

  assume "\<And>n. PROP P (num_of_nat n)"

  then have "PROP P (num_of_nat (nat_of_num m))" .

  then show "PROP P m" unfolding nat_of_num_inverse .

qed

subsection {* Digit representation for @{typ num} *}

instantiation num :: "{semiring, monoid_mult}"

begin

definition one_num :: num where

  [code func del]: "1 = num_of_nat 1"

definition plus_num :: "num \<Rightarrow> num \<Rightarrow> num" where

  [code func del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"

definition times_num :: "num \<Rightarrow> num \<Rightarrow> num" where

  [code func del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"

definition Dig0 :: "num \<Rightarrow> num" where

  [code func del]: "Dig0 n = n + n"

definition Dig1 :: "num \<Rightarrow> num" where

  [code func del]: "Dig1 n = n + n + 1"

instance proof

qed (simp_all add: one_num_def plus_num_def times_num_def

  split_num_all num_of_nat_inverse num_of_nat_zero add_ac mult_ac nat_distrib)

end

text {*

  The following proofs seem horribly complicated.

  Any room for simplification!?

*}

lemma nat_dig_cases [case_names 0 1 dig0 dig1]:

  fixes n :: nat

  assumes "n = 0 \<Longrightarrow> P"

  and "n = 1 \<Longrightarrow> P"

  and "\<And>m. m > 0 \<Longrightarrow> n = m + m \<Longrightarrow> P"

  and "\<And>m. m > 0 \<Longrightarrow> n = Suc (m + m) \<Longrightarrow> P"

  shows P

using assms proof (induct n)

  case 0 then show ?case by simp

next

  case (Suc n)

  show P proof (rule Suc.hyps)

    assume "n = 0"

    then have "Suc n = 1" by simp

    then show P by (rule Suc.prems(2))

  next

    assume "n = 1"

    have "1 > (0\<Colon>nat)" by simp

    moreover from `n = 1` have "Suc n = 1 + 1" by simp

    ultimately show P by (rule Suc.prems(3))

  next

    fix m

    assume "0 < m" and "n = m + m"

    note `0 < m`

    moreover from `n = m + m` have "Suc n = Suc (m + m)" by simp

    ultimately show P by (rule Suc.prems(4))

  next

    fix m

    assume "0 < m" and "n = Suc (m + m)"

    have "0 < Suc m" by simp

    moreover from `n = Suc (m + m)` have "Suc n = Suc m + Suc m" by simp

    ultimately show P by (rule Suc.prems(3))

  qed

qed

lemma num_induct_raw:

  fixes n :: nat

  assumes not0: "n > 0"

  assumes "P 1"

  and "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (n + n)"

  and "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc (n + n))"

  shows "P n"

using not0 proof (induct n rule: less_induct)

  case (less n)

  show "P n" proof (cases n rule: nat_dig_cases)

    case 0 then show ?thesis using less by simp

  next

    case 1 then show ?thesis using assms by simp

  next

    case (dig0 m)

    then show ?thesis apply simp

      apply (rule assms(3)) apply assumption

      apply (rule less)

      apply simp_all

    done

  next

    case (dig1 m)

    then show ?thesis apply simp

      apply (rule assms(4)) apply assumption

      apply (rule less)

      apply simp_all

    done

  qed

qed

lemma num_of_nat_Suc: "num_of_nat (Suc n) = (if n = 0 then 1 else num_of_nat n + 1)"

  by (cases n) (auto simp add: one_num_def plus_num_def num_of_nat_inverse)

lemma num_induct [case_names 1 Suc, induct type: num]:

  fixes P :: "num \<Rightarrow> bool"

  assumes 1: "P 1"

    and Suc: "\<And>n. P n \<Longrightarrow> P (n + 1)"

  shows "P n"

proof (cases n)

  case (nat m) then show ?thesis by (induct m arbitrary: n)

    (auto simp: num_of_nat_Suc intro: 1 Suc split: split_if_asm)

qed

rep_datatype "1::num" Dig0 Dig1 proof -

  fix P m

  assume 1: "P 1"

    and Dig0: "\<And>m. P m \<Longrightarrow> P (Dig0 m)"

    and Dig1: "\<And>m. P m \<Longrightarrow> P (Dig1 m)"

  obtain n where "0 < n" and m: "m = num_of_nat n"

    by (cases m) auto

  from `0 < n` have "P (num_of_nat n)" proof (induct n rule: num_induct_raw)

    case 1 from `0 < n` show ?case .

  next

    case 2 with 1 show ?case by (simp add: one_num_def)

  next

    case (3 n) then have "P (num_of_nat n)" by auto

    then have "P (Dig0 (num_of_nat n))" by (rule Dig0)

    with 3 show ?case by (simp add: Dig0_def plus_num_def num_of_nat_inverse)

  next

    case (4 n) then have "P (num_of_nat n)" by auto

    then have "P (Dig1 (num_of_nat n))" by (rule Dig1)

    with 4 show ?case by (simp add: Dig1_def one_num_def plus_num_def num_of_nat_inverse)

  qed

  with m show "P m" by simp

next

  fix m n

  show "Dig0 m = Dig0 n \<longleftrightarrow> m = n"

    apply (cases m) apply (cases n)

    by (auto simp add: Dig0_def plus_num_def num_of_nat_inverse num_of_nat_inject)

next

  fix m n

  show "Dig1 m = Dig1 n \<longleftrightarrow> m = n"

    apply (cases m) apply (cases n)

    by (auto simp add: Dig1_def plus_num_def num_of_nat_inverse num_of_nat_inject)

next

  fix n

  show "1 \<noteq> Dig0 n"

    apply (cases n)

    by (auto simp add: Dig0_def one_num_def plus_num_def num_of_nat_inverse num_of_nat_inject)

next

  fix n

  show "1 \<noteq> Dig1 n"

    apply (cases n)

    by (auto simp add: Dig1_def one_num_def plus_num_def num_of_nat_inverse num_of_nat_inject)

next

  fix m n

  have "\<And>n m. n + n \<noteq> Suc (m + m)"

  proof -

    fix n m

    show "n + n \<noteq> Suc (m + m)"

    proof (induct m arbitrary: n)

      case 0 then show ?case by (cases n) simp_all

    next

      case (Suc m) then show ?case by (cases n) simp_all

    qed

  qed

  then show "Dig0 n \<noteq> Dig1 m"

    apply (cases n) apply (cases m)

    by (auto simp add: Dig0_def Dig1_def one_num_def plus_num_def num_of_nat_inverse num_of_nat_inject)

qed

text {*

  From now on, there are two possible models for @{typ num}:

  as positive naturals (rules @{text "num_induct"}, @{text "num_cases"})

  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 {* Binary numerals *}

text {*

  We embed binary representations into a generic algebraic

  structure using @{text of_num}

*}

ML {*

structure DigSimps =

  NamedThmsFun(val name = "numeral"; val description = "Simplification rules for numerals")

*}

setup DigSimps.setup

class semiring_numeral = semiring + monoid_mult

begin

primrec of_num :: "num \<Rightarrow> 'a" where

  of_num_one [numeral]: "of_num 1 = 1"

  | "of_num (Dig0 n) = of_num n + of_num n"

  | "of_num (Dig1 n) = of_num n + of_num n + 1"

declare of_num.simps [simp del]

end

text {*

  ML stuff and syntax.

*}

ML {*

fun mk_num 1 = @{term "1::num"}

  | mk_num k =

      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;

fun dest_num @{term "1::num"} = 1

  | dest_num (@{term Dig0} $ n) = 2 * dest_num n

  | dest_num (@{term Dig1} $ n) = 2 * dest_num n + 1;

(*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 HOL.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 [("_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 HOL.one}, _)) = 1;

  fun num_tr' show_sorts T [n] =

    let

      val k = int_of_num' n;

      val t' = Syntax.const "_Numerals" $ Syntax.free ("#" ^ string_of_int k);

    in case T

     of Type ("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 {* Numeral operations *}

text {*

  First, addition and multiplication on digits.

*}

lemma Dig_plus [numeral, simp, code]:

  "1 + 1 = Dig0 1"

  "1 + Dig0 m = Dig1 m"

  "1 + Dig1 m = Dig0 (m + 1)"

  "Dig0 n + 1 = Dig1 n"

  "Dig0 n + Dig0 m = Dig0 (n + m)"

  "Dig0 n + Dig1 m = Dig1 (n + m)"

  "Dig1 n + 1 = Dig0 (n + 1)"

  "Dig1 n + Dig0 m = Dig1 (n + m)"

  "Dig1 n + Dig1 m = Dig0 (n + m + 1)"

  by (simp_all add: add_ac Dig0_def Dig1_def)

lemma Dig_times [numeral, simp, code]:

  "1 * 1 = (1::num)"

  "1 * Dig0 n = Dig0 n"

  "1 * Dig1 n = Dig1 n"

  "Dig0 n * 1 = Dig0 n"

  "Dig0 n * Dig0 m = Dig0 (n * Dig0 m)"

  "Dig0 n * Dig1 m = Dig0 (n * Dig1 m)"

  "Dig1 n * 1 = Dig1 n"

  "Dig1 n * Dig0 m = Dig0 (n * Dig0 m + m)"

  "Dig1 n * Dig1 m = Dig1 (n * Dig1 m + m)"

  by (simp_all add: left_distrib right_distrib add_ac Dig0_def Dig1_def)

text {*

  @{const of_num} is a morphism.

*}

context semiring_numeral

begin

abbreviation "Num1 \<equiv> of_num 1"

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 + 1)"

  by (rule sym, induct n) (simp_all add: Dig_plus of_num.simps add_ac)

lemma of_num_one_plus [numeral]:

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

  unfolding of_num_plus_one [symmetric] add_commute ..

lemma of_num_plus [numeral]:

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

  by (induct n rule: num_induct)

    (simp_all add: Dig_plus of_num_one semigroup_add_class.add_assoc [symmetric, of m]

    add_ac of_num_plus_one [symmetric])

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)"

  by (induct n rule: num_induct)

    (simp_all add: of_num_plus [symmetric]

    semiring_class.right_distrib right_distrib of_num_one)

end

text {*

  Structures with a @{term 0}.

*}

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" apply (induct n)

    apply (simp_all add: of_num.simps)

    using nat_of_num

    apply (simp_all add: one_num_def plus_num_def Dig0_def Dig1_def num_of_nat_inverse mem_def)

    done

  then show "nat_of_num n = of_num n" by simp

qed

text {*

  Equality.

*}

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 nat_of_num_inject ..

lemma of_num_eq_one_iff [numeral]:

  "of_num n = 1 \<longleftrightarrow> n = 1"

proof -

  have "of_num n = of_num 1 \<longleftrightarrow> n = 1" unfolding of_num_eq_iff ..

  then show ?thesis by (simp add: of_num_one)

qed

lemma one_eq_of_num_iff [numeral]:

  "1 = of_num n \<longleftrightarrow> n = 1"

  unfolding of_num_eq_one_iff [symmetric] by auto

end

text {*

  Comparisons.  Could be perhaps more general than here.

*}

lemma (in ordered_semidom) of_num_pos: "0 < of_num n"

proof -

  have "(0::nat) < of_num n"

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

  then have "of_nat 0 \<noteq> of_nat (of_num n)" 

    by (cases n) (simp_all only: semiring_numeral_class.of_num.simps of_nat_eq_iff)

  then have "0 \<noteq> of_num n"

    by (simp add: of_nat_of_num)

  moreover have "0 \<le> of_nat (of_num n)" by simp

  ultimately show ?thesis by (simp add: of_nat_of_num)

qed

instantiation num :: linorder

begin

definition less_eq_num :: "num \<Rightarrow> num \<Rightarrow> bool" where

  [code func del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"

definition less_num :: "num \<Rightarrow> num \<Rightarrow> bool" where

  [code func del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"

instance proof

qed (auto simp add: less_eq_num_def less_num_def

  split_num_all num_of_nat_inverse num_of_nat_inject split: split_if_asm)

end

lemma less_eq_num_code [numeral, simp, code]:

  "(1::num) \<le> n \<longleftrightarrow> True"

  "Dig0 m \<le> 1 \<longleftrightarrow> False"

  "Dig1 m \<le> 1 \<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 of_num_pos [of n, where ?'a = nat] of_num_pos [of m, where ?'a = nat]

  by (auto simp add: less_eq_num_def less_num_def nat_of_num_of_num of_num.simps)

lemma less_num_code [numeral, simp, code]:

  "m < (1::num) \<longleftrightarrow> False"

  "(1::num) < 1 \<longleftrightarrow> False"

  "1 < Dig0 n \<longleftrightarrow> True"

  "1 < 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 of_num_pos [of n, where ?'a = nat] of_num_pos [of m, where ?'a = nat]

  by (auto simp add: less_eq_num_def less_num_def nat_of_num_of_num of_num.simps)

context ordered_semidom

begin

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 = 1"

proof -

  have "of_num n \<le> of_num 1 \<longleftrightarrow> n = 1"

    by (cases n) (simp_all add: of_num_less_eq_iff)

  then show ?thesis by (simp add: of_num_one)

qed

lemma one_less_eq_of_num_iff [numeral]: "1 \<le> of_num n"

proof -

  have "of_num 1 \<le> of_num n"

    by (cases n) (simp_all add: of_num_less_eq_iff)

  then show ?thesis by (simp add: of_num_one)

qed

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"

proof -

  have "\<not> of_num n < of_num 1"

    by (cases n) (simp_all add: of_num_less_iff)

  then show ?thesis by (simp add: of_num_one)

qed

lemma one_less_of_num_iff [numeral]: "1 < of_num n \<longleftrightarrow> n \<noteq> 1"

proof -

  have "of_num 1 < of_num n \<longleftrightarrow> n \<noteq> 1"

    by (cases n) (simp_all add: of_num_less_iff)

  then show ?thesis by (simp add: of_num_one)

qed

end

text {*

  Structures with subtraction @{term "op -"}.

*}

text {* A decrement function *}

primrec dec :: "num \<Rightarrow> num" where

  "dec 1 = 1"

  | "dec (Dig0 n) = (case n of 1 \<Rightarrow> 1 | _ \<Rightarrow> Dig1 (dec n))"

  | "dec (Dig1 n) = Dig0 n"

declare dec.simps [simp del, code del]

lemma Dig_dec [numeral, simp, code]:

  "dec 1 = 1"

  "dec (Dig0 1) = 1"

  "dec (Dig0 (Dig0 n)) = Dig1 (dec (Dig0 n))"

  "dec (Dig0 (Dig1 n)) = Dig1 (Dig0 n)"

  "dec (Dig1 n) = Dig0 n"

  by (simp_all add: dec.simps)

lemma Dig_dec_plus_one:

  "dec n + 1 = (if n = 1 then Dig0 1 else n)"

  by (induct n)

    (auto simp add: Dig_plus dec.simps,

     auto simp add: Dig_plus split: num.splits)

lemma Dig_one_plus_dec:

  "1 + dec n = (if n = 1 then Dig0 1 else n)"

  unfolding add_commute [of 1] Dig_dec_plus_one ..

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 "1::num", 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 (dec (Dig0 n))"

  "of_num (Dig1 n) - 1 = of_num (Dig0 n)"

  by (auto intro: minus_inverts_plus1 simp add: Dig_dec_plus_one of_num.simps of_num_plus_one)

lemma Dig_one_minus_of_num [numeral]:

  "1 - of_num (Dig0 n) = 0 - of_num (dec (Dig0 n))"

  "1 - of_num (Dig1 n) = 0 - of_num (Dig0 n)"

  by (auto intro: minus_minus_zero_inverts_plus1 simp add: Dig_dec_plus_one of_num.simps of_num_plus_one)

end

context ring_1

begin

subclass semiring_1_minus

  by unfold_locales (simp_all add: ring_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 add: minus_mult_left [symmetric] minus_mult_right [symmetric])

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

text {*

  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 + 1)"

  unfolding of_num_plus_one [symmetric] by simp

lemma nat_number:

  "1 = Suc 0"

  "of_num 1 = Suc 0"

  "of_num (Dig0 n) = Suc (of_num (dec (Dig0 n)))"

  "of_num (Dig1 n) = Suc (of_num (Dig0 n))"

  by (simp_all add: of_num.simps Dig_dec_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 inline] = Pls_def [symmetric] Mns_def [symmetric]

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

  [simp, code func del]: "sub m n = (of_num m - of_num n)"

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

  [code func del]: "dup k = 2 * k"

lemma Dig_sub [code]:

  "sub 1 1 = 0"

  "sub (Dig0 m) 1 = of_num (dec (Dig0 m))"

  "sub (Dig1 m) 1 = of_num (Dig0 m)"

  "sub 1 (Dig0 n) = - of_num (dec (Dig0 n))"

  "sub 1 (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"

  apply (simp_all add: dup_def ring_simps)

  apply (simp_all add: of_num_plus Dig_one_plus_dec)[4]

  apply (simp_all add: of_num.simps)

  done

lemma dup_code [code]:

  "dup 0 = 0"

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

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

  by (simp_all add: dup_def of_num.simps)

lemma [code func, code func 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 *"

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

  "(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 1"

  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 - Pls n = sub m n"

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

  "Mns m - Mns n = sub n m"

  by (simp_all add: of_num_plus [symmetric])

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 add: of_num_plus [symmetric])

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 add: of_num_times [symmetric])

lemma eq_int_code [code]:

  "0 = (0::int) \<longleftrightarrow> True"

  "0 = Pls l \<longleftrightarrow> False"

  "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> False"

  "Mns k = Pls l \<longleftrightarrow> False"

  "Mns k = Mns l \<longleftrightarrow> k = l"

  using of_num_pos [of l, where ?'a = int] of_num_pos [of k, where ?'a = int]

  by (simp_all add: of_num_eq_iff)

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]

  by (simp_all add: of_num_less_eq_iff)

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]

  by (simp_all add: of_num_less_iff)

lemma [code inline del]: "(0::int) \<equiv> Numeral0" by simp

lemma [code inline del]: "(1::int) \<equiv> Numeral1" by simp

declare zero_is_num_zero [code inline del]

declare one_is_num_one [code inline del]

hide (open) const sub dup

subsection {* Numeral equations as default simplification rules *}

text {* TODO.  Be more precise here with respect to subsumed facts. *}

declare (in semiring_numeral) numeral [simp]

declare (in semiring_1) numeral [simp]

declare (in semiring_char_0) numeral [simp]

declare (in ring_1) numeral [simp]

thm numeral

text {* Toy examples *}

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

code_thms bar

export_code bar in Haskell file -

export_code bar in OCaml module_name Foo file -

ML {* @{code bar} *}

end