(*  Title:      HOL/Num.thy

    Author:     Florian Haftmann

    Author:     Brian Huffman

*)

header {* Binary Numerals *}

theory Num

imports Datatype Power

begin

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

datatype num = One | Bit0 num | Bit1 num

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

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

  "inc One = Bit0 One" |

  "inc (Bit0 x) = Bit1 x" |

  "inc (Bit1 x) = Bit0 (inc x)"

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

primrec nat_of_num :: "num \<Rightarrow> nat" where

  "nat_of_num One = Suc 0" |

  "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |

  "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"

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

  "num_of_nat 0 = One" |

  "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"

lemma nat_of_num_pos: "0 < nat_of_num x"

  by (induct x) simp_all

lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"

  by (induct x) simp_all

lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"

  by (induct x) simp_all

lemma num_of_nat_double:

  "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"

  by (induct n) simp_all

text {*

  Type @{typ num} is isomorphic to the strictly positive

  natural numbers.

*}

lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"

  by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)

lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"

  by (induct n) (simp_all add: nat_of_num_inc)

lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"

  apply safe

  apply (drule arg_cong [where f=num_of_nat])

  apply (simp add: nat_of_num_inverse)

  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"

    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"}).

*}

subsection {* Numeral operations *}

instantiation num :: "{plus,times,linorder}"

begin

definition [code del]:

  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"

definition [code del]:

  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"

definition [code del]:

  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"

definition [code del]:

  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"

instance

  by (default, auto simp add: less_num_def less_eq_num_def num_eq_iff)

end

lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"

  unfolding plus_num_def

  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 add_num_simps [simp, code]:

  "One + One = Bit0 One"

  "One + Bit0 n = Bit1 n"

  "One + Bit1 n = Bit0 (n + One)"

  "Bit0 m + One = Bit1 m"

  "Bit0 m + Bit0 n = Bit0 (m + n)"

  "Bit0 m + Bit1 n = Bit1 (m + n)"

  "Bit1 m + One = Bit0 (m + One)"

  "Bit1 m + Bit0 n = Bit1 (m + n)"

  "Bit1 m + Bit1 n = Bit0 (m + n + One)"

  by (simp_all add: num_eq_iff nat_of_num_add)

lemma mult_num_simps [simp, code]:

  "m * One = m"

  "One * n = n"

  "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"

  "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"

  "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"

  "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"

  by (simp_all add: num_eq_iff nat_of_num_add

    nat_of_num_mult left_distrib right_distrib)

lemma eq_num_simps:

  "One = One \<longleftrightarrow> True"

  "One = Bit0 n \<longleftrightarrow> False"

  "One = Bit1 n \<longleftrightarrow> False"

  "Bit0 m = One \<longleftrightarrow> False"

  "Bit1 m = One \<longleftrightarrow> False"

  "Bit0 m = Bit0 n \<longleftrightarrow> m = n"

  "Bit0 m = Bit1 n \<longleftrightarrow> False"

  "Bit1 m = Bit0 n \<longleftrightarrow> False"

  "Bit1 m = Bit1 n \<longleftrightarrow> m = n"

  by simp_all

lemma le_num_simps [simp, code]:

  "One \<le> n \<longleftrightarrow> True"

  "Bit0 m \<le> One \<longleftrightarrow> False"

  "Bit1 m \<le> One \<longleftrightarrow> False"

  "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"

  "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"

  "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"

  "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"

  using nat_of_num_pos [of n] nat_of_num_pos [of m]

  by (auto simp add: less_eq_num_def less_num_def)

lemma less_num_simps [simp, code]:

  "m < One \<longleftrightarrow> False"

  "One < Bit0 n \<longleftrightarrow> True"

  "One < Bit1 n \<longleftrightarrow> True"

  "Bit0 m < Bit0 n \<longleftrightarrow> m < n"

  "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"

  "Bit1 m < Bit1 n \<longleftrightarrow> m < n"

  "Bit1 m < Bit0 n \<longleftrightarrow> m < n"

  using nat_of_num_pos [of n] nat_of_num_pos [of m]

  by (auto simp add: less_eq_num_def less_num_def)

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

lemma add_One: "x + One = inc x"

  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)

lemma add_One_commute: "One + n = n + One"

  by (induct n) simp_all

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

lemma num_of_nat_One:

  "n \<le> 1 \<Longrightarrow> num_of_nat n = Num.One"

  by (cases n) simp_all

lemma num_of_nat_plus_distrib:

  "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"

  by (induct n) (auto simp add: add_One add_One_commute add_inc)

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

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

  "BitM One = One" |

  "BitM (Bit0 n) = Bit1 (BitM n)" |

  "BitM (Bit1 n) = Bit1 (Bit0 n)"

lemma BitM_plus_one: "BitM n + One = Bit0 n"

  by (induct n) simp_all

lemma one_plus_BitM: "One + BitM n = Bit0 n"

  unfolding add_One_commute BitM_plus_one ..

text {* Squaring and exponentiation *}

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

  "sqr One = One" |

  "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |

  "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"

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

  "pow x One = x" |

  "pow x (Bit0 y) = sqr (pow x y)" |

  "pow x (Bit1 y) = x * sqr (pow x y)"

lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"

  by (induct x, simp_all add: algebra_simps nat_of_num_add)

lemma sqr_conv_mult: "sqr x = x * x"

  by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)

subsection {* Numary numerals *}

text {*

  We embed numary representations into a generic algebraic

  structure using @{text numeral}.

*}

class numeral = one + semigroup_add

begin

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

  numeral_One: "numeral One = 1" |

  numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |

  numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"

lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"

  apply (induct x)

  apply simp

  apply (simp add: add_assoc [symmetric], simp add: add_assoc)

  apply (simp add: add_assoc [symmetric], simp add: add_assoc)

  done

lemma numeral_inc: "numeral (inc x) = numeral x + 1"

proof (induct x)

  case (Bit1 x)

  have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"

    by (simp only: one_plus_numeral_commute)

  with Bit1 show ?case

    by (simp add: add_assoc)

qed simp_all

declare numeral.simps [simp del]

abbreviation "Numeral1 \<equiv> numeral One"

declare numeral_One [code_post]

end

text {* Negative numerals. *}

class neg_numeral = numeral + group_add

begin

definition neg_numeral :: "num \<Rightarrow> 'a" where

  "neg_numeral k = - numeral k"

end

text {* Numeral syntax. *}

syntax

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

parse_translation {*

let

  fun num_of_int n = if n > 0 then case IntInf.quotRem (n, 2)

     of (0, 1) => Syntax.const @{const_name One}

      | (n, 0) => Syntax.const @{const_name Bit0} $ num_of_int n

      | (n, 1) => Syntax.const @{const_name Bit1} $ num_of_int n

    else raise Match;

  val pos = Syntax.const @{const_name numeral}

  val neg = Syntax.const @{const_name neg_numeral}

  val one = Syntax.const @{const_name Groups.one}

  val zero = Syntax.const @{const_name Groups.zero}

  fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =

        c $ numeral_tr [t] $ u

    | numeral_tr [Const (num, _)] =

        let

          val {value, ...} = Lexicon.read_xnum num;

        in

          if value = 0 then zero else

          if value > 0

          then pos $ num_of_int value

          else neg $ num_of_int (~value)

        end

    | numeral_tr ts = raise TERM ("numeral_tr", ts);

in [("_Numeral", numeral_tr)] end

*}

typed_print_translation (advanced) {*

let

  fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n

    | dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1

    | dest_num (Const (@{const_syntax One}, _)) = 1;

  fun num_tr' sign ctxt T [n] =

    let

      val k = dest_num n;

      val t' = Syntax.const @{syntax_const "_Numeral"} $

        Syntax.free (sign ^ string_of_int k);

    in

      case T of

        Type (@{type_name fun}, [_, T']) =>

          if not (Config.get ctxt show_types) andalso can Term.dest_Type T' then t'

          else Syntax.const @{syntax_const "_constrain"} $ t' $ Syntax_Phases.term_of_typ ctxt T'

      | T' => if T' = dummyT then t' else raise Match

    end;

in [(@{const_syntax numeral}, num_tr' ""),

    (@{const_syntax neg_numeral}, num_tr' "-")] end

*}

subsection {* Class-specific numeral rules *}

text {*

  @{const numeral} is a morphism.

*}

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

context numeral

begin

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

  by (induct n rule: num_induct)

     (simp_all only: numeral_One add_One add_inc numeral_inc add_assoc)

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

  by (rule numeral_add [symmetric])

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

  using numeral_add [of n One] by (simp add: numeral_One)

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

  using numeral_add [of One n] by (simp add: numeral_One)

lemma one_add_one: "1 + 1 = 2"

  using numeral_add [of One One] by (simp add: numeral_One)

lemmas add_numeral_special =

  numeral_plus_one one_plus_numeral one_add_one

end

subsubsection {*

  Structures with negation: class @{text neg_numeral}

*}

context neg_numeral

begin

text {* Numerals form an abelian subgroup. *}

inductive is_num :: "'a \<Rightarrow> bool" where

  "is_num 1" |

  "is_num x \<Longrightarrow> is_num (- x)" |

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

lemma is_num_numeral: "is_num (numeral k)"

  by (induct k, simp_all add: numeral.simps is_num.intros)

lemma is_num_add_commute:

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

  apply (induct x rule: is_num.induct)

  apply (induct y rule: is_num.induct)

  apply simp

  apply (rule_tac a=x in add_left_imp_eq)

  apply (rule_tac a=x in add_right_imp_eq)

  apply (simp add: add_assoc minus_add_cancel)

  apply (simp add: add_assoc [symmetric], simp add: add_assoc)

  apply (rule_tac a=x in add_left_imp_eq)

  apply (rule_tac a=x in add_right_imp_eq)

  apply (simp add: add_assoc minus_add_cancel add_minus_cancel)

  apply (simp add: add_assoc, simp add: add_assoc [symmetric])

  done

lemma is_num_add_left_commute:

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

  by (simp only: add_assoc [symmetric] is_num_add_commute)

lemmas is_num_normalize =

  add_assoc is_num_add_commute is_num_add_left_commute

  is_num.intros is_num_numeral

  diff_minus minus_add add_minus_cancel minus_add_cancel

definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"

definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"

definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"

definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where

  "sub k l = numeral k - numeral l"

lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"

  by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)

lemma dbl_simps [simp]:

  "dbl (neg_numeral k) = neg_numeral (Bit0 k)"

  "dbl 0 = 0"

  "dbl 1 = 2"

  "dbl (numeral k) = numeral (Bit0 k)"

  unfolding dbl_def neg_numeral_def numeral.simps

  by (simp_all add: minus_add)

lemma dbl_inc_simps [simp]:

  "dbl_inc (neg_numeral k) = neg_numeral (BitM k)"

  "dbl_inc 0 = 1"

  "dbl_inc 1 = 3"

  "dbl_inc (numeral k) = numeral (Bit1 k)"

  unfolding dbl_inc_def neg_numeral_def numeral.simps numeral_BitM

  by (simp_all add: is_num_normalize)

lemma dbl_dec_simps [simp]:

  "dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"

  "dbl_dec 0 = -1"

  "dbl_dec 1 = 1"

  "dbl_dec (numeral k) = numeral (BitM k)"

  unfolding dbl_dec_def neg_numeral_def numeral.simps numeral_BitM

  by (simp_all add: is_num_normalize)

lemma sub_num_simps [simp]:

  "sub One One = 0"

  "sub One (Bit0 l) = neg_numeral (BitM l)"

  "sub One (Bit1 l) = neg_numeral (Bit0 l)"

  "sub (Bit0 k) One = numeral (BitM k)"

  "sub (Bit1 k) One = numeral (Bit0 k)"

  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"

  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"

  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"

  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"

  unfolding dbl_def dbl_dec_def dbl_inc_def sub_def

  unfolding neg_numeral_def numeral.simps numeral_BitM

  by (simp_all add: is_num_normalize)

lemma add_neg_numeral_simps:

  "numeral m + neg_numeral n = sub m n"

  "neg_numeral m + numeral n = sub n m"

  "neg_numeral m + neg_numeral n = neg_numeral (m + n)"

  unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps

  by (simp_all add: is_num_normalize)

lemma add_neg_numeral_special:

  "1 + neg_numeral m = sub One m"

  "neg_numeral m + 1 = sub One m"

  unfolding sub_def diff_minus neg_numeral_def numeral_add numeral.simps

  by (simp_all add: is_num_normalize)

lemma diff_numeral_simps:

  "numeral m - numeral n = sub m n"

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

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

  "neg_numeral m - neg_numeral n = sub n m"

  unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps

  by (simp_all add: is_num_normalize)

lemma diff_numeral_special:

  "1 - numeral n = sub One n"

  "1 - neg_numeral n = numeral (One + n)"

  "numeral m - 1 = sub m One"

  "neg_numeral m - 1 = neg_numeral (m + One)"

  unfolding neg_numeral_def sub_def diff_minus numeral_add numeral.simps

  by (simp_all add: is_num_normalize)

lemma minus_one: "- 1 = -1"

  unfolding neg_numeral_def numeral.simps ..

lemma minus_numeral: "- numeral n = neg_numeral n"

  unfolding neg_numeral_def ..

lemma minus_neg_numeral: "- neg_numeral n = numeral n"

  unfolding neg_numeral_def by simp

lemmas minus_numeral_simps [simp] =

  minus_one minus_numeral minus_neg_numeral

end

subsubsection {*

  Structures with multiplication: class @{text semiring_numeral}

*}

class semiring_numeral = semiring + monoid_mult

begin

subclass numeral ..

lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"

  apply (induct n rule: num_induct)

  apply (simp add: numeral_One)

  apply (simp add: mult_inc numeral_inc numeral_add numeral_inc right_distrib)

  done

lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"

  by (rule numeral_mult [symmetric])

end

subsubsection {*

  Structures with a zero: class @{text semiring_1}

*}

context semiring_1

begin

subclass semiring_numeral ..

lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"

  by (induct n,

    simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)

end

lemma nat_of_num_numeral: "nat_of_num = numeral"

proof

  fix n

  have "numeral n = nat_of_num n"

    by (induct n) (simp_all add: numeral.simps)

  then show "nat_of_num n = numeral n" by simp

qed

subsubsection {*

  Equality: class @{text semiring_char_0}

*}

context semiring_char_0

begin

lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"

  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]

    of_nat_eq_iff num_eq_iff ..

lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"

  by (rule numeral_eq_iff [of n One, unfolded numeral_One])

lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"

  by (rule numeral_eq_iff [of One n, unfolded numeral_One])

lemma numeral_neq_zero: "numeral n \<noteq> 0"

  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]

  by (simp add: nat_of_num_pos)

lemma zero_neq_numeral: "0 \<noteq> numeral n"

  unfolding eq_commute [of 0] by (rule numeral_neq_zero)

lemmas eq_numeral_simps [simp] =

  numeral_eq_iff

  numeral_eq_one_iff

  one_eq_numeral_iff

  numeral_neq_zero

  zero_neq_numeral

end

subsubsection {*

  Comparisons: class @{text linordered_semidom}

*}

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

context linordered_semidom

begin

lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"

proof -

  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"

    unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..

  then show ?thesis by simp

qed

lemma one_le_numeral: "1 \<le> numeral n"

using numeral_le_iff [of One n] by (simp add: numeral_One)

lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"

using numeral_le_iff [of n One] by (simp add: numeral_One)

lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"

proof -

  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"

    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..

  then show ?thesis by simp

qed

lemma not_numeral_less_one: "\<not> numeral n < 1"

  using numeral_less_iff [of n One] by (simp add: numeral_One)

lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"

  using numeral_less_iff [of One n] by (simp add: numeral_One)

lemma zero_le_numeral: "0 \<le> numeral n"

  by (induct n) (simp_all add: numeral.simps)

lemma zero_less_numeral: "0 < numeral n"

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

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

  by (simp add: not_le zero_less_numeral)

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

  by (simp add: not_less zero_le_numeral)

lemmas le_numeral_extra =

  zero_le_one not_one_le_zero

  order_refl [of 0] order_refl [of 1]

lemmas less_numeral_extra =

  zero_less_one not_one_less_zero

  less_irrefl [of 0] less_irrefl [of 1]

lemmas le_numeral_simps [simp] =

  numeral_le_iff

  one_le_numeral

  numeral_le_one_iff

  zero_le_numeral

  not_numeral_le_zero

lemmas less_numeral_simps [simp] =

  numeral_less_iff

  one_less_numeral_iff

  not_numeral_less_one

  zero_less_numeral

  not_numeral_less_zero

end

subsubsection {*

  Multiplication and negation: class @{text ring_1}

*}

context ring_1

begin

subclass neg_numeral ..

lemma mult_neg_numeral_simps:

  "neg_numeral m * neg_numeral n = numeral (m * n)"

  "neg_numeral m * numeral n = neg_numeral (m * n)"

  "numeral m * neg_numeral n = neg_numeral (m * n)"

  unfolding neg_numeral_def mult_minus_left mult_minus_right

  by (simp_all only: minus_minus numeral_mult)

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

  unfolding neg_numeral_def numeral.simps mult_minus_left by simp

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

  unfolding neg_numeral_def numeral.simps mult_minus_right by simp

end

subsubsection {*

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

*}

context ring_1

begin

definition iszero :: "'a \<Rightarrow> bool"

  where "iszero z \<longleftrightarrow> z = 0"

lemma iszero_0 [simp]: "iszero 0"

  by (simp add: iszero_def)

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

  by (simp add: iszero_def)

lemma not_iszero_Numeral1: "\<not> iszero Numeral1"

  by (simp add: numeral_One)

lemma iszero_neg_numeral [simp]:

  "iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"

  unfolding iszero_def neg_numeral_def

  by (rule neg_equal_0_iff_equal)

lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"

  unfolding iszero_def by (rule eq_iff_diff_eq_0)

text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared

@{text "[simp]"} by default, because for rings of characteristic zero,

better simp rules are possible. For a type like integers mod @{text

"n"}, type-instantiated versions of these rules should be added to the

simplifier, along with a type-specific rule for deciding propositions

of the form @{text "iszero (numeral w)"}.

bh: Maybe it would not be so bad to just declare these as simp

rules anyway? I should test whether these rules take precedence over

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

*}

lemma eq_numeral_iff_iszero:

  "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"

  "numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"

  "neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"

  "neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"

  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"

  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"

  "neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"

  "1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"

  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"

  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"

  "neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"

  "0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"

  unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special

  by simp_all

end

subsubsection {*

  Equality and negation: class @{text ring_char_0}

*}

class ring_char_0 = ring_1 + semiring_char_0

begin

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

  by (simp add: iszero_def)

lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"

  by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)

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

  unfolding neg_numeral_def eq_neg_iff_add_eq_0

  by (simp add: numeral_plus_numeral)

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

  by (rule numeral_neq_neg_numeral [symmetric])

lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"

  unfolding neg_numeral_def neg_0_equal_iff_equal by simp

lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"

  unfolding neg_numeral_def neg_equal_0_iff_equal by simp

lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"

  using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)

lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"

  using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)

lemmas eq_neg_numeral_simps [simp] =

  neg_numeral_eq_iff

  numeral_neq_neg_numeral neg_numeral_neq_numeral

  one_neq_neg_numeral neg_numeral_neq_one

  zero_neq_neg_numeral neg_numeral_neq_zero

end

subsubsection {*

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

*}

context linordered_idom

begin

subclass ring_char_0 ..

lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"

  by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)

lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"

  by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)

lemma neg_numeral_less_zero: "neg_numeral n < 0"

  by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)

lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"

  by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)

lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"

  by (simp only: not_less neg_numeral_le_zero)

lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"

  by (simp only: not_le neg_numeral_less_zero)

lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"

  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)

lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"

  by (simp only: less_imp_le neg_numeral_less_numeral)

lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"

  by (simp only: not_less neg_numeral_le_numeral)

lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"

  by (simp only: not_le neg_numeral_less_numeral)

lemma neg_numeral_less_one: "neg_numeral m < 1"

  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])

lemma neg_numeral_le_one: "neg_numeral m \<le> 1"

  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])

lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"

  by (simp only: not_less neg_numeral_le_one)

lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"

  by (simp only: not_le neg_numeral_less_one)

lemma sub_non_negative:

  "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"

  by (simp only: sub_def le_diff_eq) simp

lemma sub_positive:

  "sub n m > 0 \<longleftrightarrow> n > m"

  by (simp only: sub_def less_diff_eq) simp

lemma sub_non_positive:

  "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"

  by (simp only: sub_def diff_le_eq) simp

lemma sub_negative:

  "sub n m < 0 \<longleftrightarrow> n < m"

  by (simp only: sub_def diff_less_eq) simp

lemmas le_neg_numeral_simps [simp] =

  neg_numeral_le_iff

  neg_numeral_le_numeral not_numeral_le_neg_numeral

  neg_numeral_le_zero not_zero_le_neg_numeral

  neg_numeral_le_one not_one_le_neg_numeral

lemmas less_neg_numeral_simps [simp] =

  neg_numeral_less_iff

  neg_numeral_less_numeral not_numeral_less_neg_numeral

  neg_numeral_less_zero not_zero_less_neg_numeral

  neg_numeral_less_one not_one_less_neg_numeral

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

  by simp

lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"

  by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)

end

subsubsection {*

  Natural numbers

*}

lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"

  unfolding numeral_plus_one [symmetric] by simp

lemma nat_number:

  "1 = Suc 0"

  "numeral One = Suc 0"

  "numeral (Bit0 n) = Suc (numeral (BitM n))"

  "numeral (Bit1 n) = Suc (numeral (Bit0 n))"

  by (simp_all add: numeral.simps BitM_plus_one)

subsubsection {*

  Structures with exponentiation

*}

context semiring_numeral

begin

lemma numeral_sqr: "numeral (sqr n) = numeral n * numeral n"

  by (simp add: sqr_conv_mult numeral_mult)

lemma numeral_pow: "numeral (pow m n) = numeral m ^ numeral n"

  by (induct n, simp_all add: numeral_class.numeral.simps

    power_add numeral_sqr numeral_mult)

lemma power_numeral [simp]: "numeral m ^ numeral n = numeral (pow m n)"

  by (rule numeral_pow [symmetric])

end

context semiring_1

begin

lemma power_zero_numeral [simp]: "(0::'a) ^ numeral n = 0"

  by (induct n, simp_all add: numeral_class.numeral.simps power_add)

end

context ring_1

begin

lemma power_minus_Bit0: "(- x) ^ numeral (Bit0 n) = x ^ numeral (Bit0 n)"

  by (induct n, simp_all add: numeral_class.numeral.simps power_add)

lemma power_minus_Bit1: "(- x) ^ numeral (Bit1 n) = - (x ^ numeral (Bit1 n))"

  by (simp only: nat_number(4) power_Suc power_minus_Bit0 mult_minus_left)

lemma power_neg_numeral_Bit0 [simp]:

  "neg_numeral m ^ numeral (Bit0 n) = numeral (pow m (Bit0 n))"

  by (simp only: neg_numeral_def power_minus_Bit0 power_numeral)

lemma power_neg_numeral_Bit1 [simp]:

  "neg_numeral m ^ numeral (Bit1 n) = neg_numeral (pow m (Bit1 n))"

  by (simp only: neg_numeral_def power_minus_Bit1 power_numeral pow.simps)

end

subsection {* Numeral equations as default simplification rules *}

declare (in numeral) numeral_One [simp]

declare (in numeral) numeral_plus_numeral [simp]

declare (in numeral) add_numeral_special [simp]

declare (in neg_numeral) add_neg_numeral_simps [simp]

declare (in neg_numeral) add_neg_numeral_special [simp]

declare (in neg_numeral) diff_numeral_simps [simp]

declare (in neg_numeral) diff_numeral_special [simp]

declare (in semiring_numeral) numeral_times_numeral [simp]

declare (in ring_1) mult_neg_numeral_simps [simp]

subsection {* Setting up simprocs *}

lemma numeral_reorient:

  "(numeral w = x) = (x = numeral w)"

  by auto

lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"

  by simp

lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"

  by simp

lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"

  by simp

lemma inverse_numeral_1:

  "inverse Numeral1 = (Numeral1::'a::division_ring)"

  by simp

text{*Theorem lists for the cancellation simprocs. The use of a numary

numeral for 1 reduces the number of special cases.*}

lemmas mult_1s =

  mult_numeral_1 mult_numeral_1_right 

  mult_minus1 mult_minus1_right

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

lemmas arith_special = (* already declared simp above *)

  add_numeral_special add_neg_numeral_special

  diff_numeral_special minus_one

(* rules already in simpset *)

lemmas arith_extra_simps =

  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right

  minus_numeral minus_neg_numeral minus_zero minus_one

  diff_numeral_simps diff_0 diff_0_right

  numeral_times_numeral mult_neg_numeral_simps

  mult_zero_left mult_zero_right

  abs_numeral abs_neg_numeral

text {*

  For making a minimal simpset, one must include these default simprules.

  Also include @{text simp_thms}.

*}

lemmas arith_simps =

  add_num_simps mult_num_simps sub_num_simps

  BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps

  abs_zero abs_one arith_extra_simps

text {* Simplification of relational operations *}

lemmas eq_numeral_extra =

  zero_neq_one one_neq_zero

lemmas rel_simps =

  le_num_simps less_num_simps eq_num_simps

  le_numeral_simps le_neg_numeral_simps le_numeral_extra

  less_numeral_simps less_neg_numeral_simps less_numeral_extra

  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra

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

lemma add_numeral_left [simp]:

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

  by (simp_all add: add_assoc [symmetric])