src/HOL/Num.thy

generalized tends over powr; added DERIV rule for powr

(*  Title:      HOL/Num.thy
    Author:     Florian Haftmann
    Author:     Brian Huffman
*)

section {* Binary Numerals *}

theory Num
imports BNF_Least_Fixpoint
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 distrib_right distrib_left)

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

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 {* Binary numerals *}

text {*
  We embed binary 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 numeral_code [code]:
  "numeral One = 1"
  "numeral (Bit0 n) = (let m = numeral n in m + m)"
  "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
  by (simp_all add: Let_def)
  
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 {* Numeral syntax. *}

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

ML_file "Tools/numeral.ML"

parse_translation {*
  let
    fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
          c $ numeral_tr [t] $ u
      | numeral_tr [Const (num, _)] =
          (Numeral.mk_number_syntax o #value o Lexicon.read_num) num
      | numeral_tr ts = raise TERM ("numeral_tr", ts);
  in [(@{syntax_const "_Numeral"}, K numeral_tr)] end
*}

typed_print_translation {*
  let
    fun 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' ctxt T [n] =
      let
        val k = dest_num n;
        val t' =
          Syntax.const @{syntax_const "_Numeral"} $
            Syntax.free (string_of_int k);
      in
        (case T of
          Type (@{type_name fun}, [_, T']) =>
            if Printer.type_emphasis ctxt T' then
              Syntax.const @{syntax_const "_constrain"} $ t' $
                Syntax_Phases.term_of_typ ctxt T'
            else t'
        | _ => if T = dummyT then t' else raise Match)
      end;
  in
   [(@{const_syntax numeral}, num_tr')]
  end
*}


subsection {* Class-specific numeral rules *}

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

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

class neg_numeral = numeral + group_add
begin

lemma uminus_numeral_One:
  "- Numeral1 = - 1"
  by (simp add: numeral_One)

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)
  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)
  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
  minus_add

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 (- numeral k) = - dbl (numeral k)"
  "dbl 0 = 0"
  "dbl 1 = 2"
  "dbl (- 1) = - 2"
  "dbl (numeral k) = numeral (Bit0 k)"
  by (simp_all add: dbl_def numeral.simps minus_add)

lemma dbl_inc_simps [simp]:
  "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
  "dbl_inc 0 = 1"
  "dbl_inc 1 = 3"
  "dbl_inc (- 1) = - 1"
  "dbl_inc (numeral k) = numeral (Bit1 k)"
  by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)

lemma dbl_dec_simps [simp]:
  "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
  "dbl_dec 0 = - 1"
  "dbl_dec 1 = 1"
  "dbl_dec (- 1) = - 3"
  "dbl_dec (numeral k) = numeral (BitM k)"
  by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)

lemma sub_num_simps [simp]:
  "sub One One = 0"
  "sub One (Bit0 l) = - numeral (BitM l)"
  "sub One (Bit1 l) = - numeral (Bit0 l)"
  "sub (Bit0 k) One = numeral (BitM k)"
  "sub (Bit1 k) One = numeral (Bit0 k)"
  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
  by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
    numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)

lemma add_neg_numeral_simps:
  "numeral m + - numeral n = sub m n"
  "- numeral m + numeral n = sub n m"
  "- numeral m + - numeral n = - (numeral m + numeral n)"
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
    del: add_uminus_conv_diff add: diff_conv_add_uminus)

lemma add_neg_numeral_special:
  "1 + - numeral m = sub One m"
  "- numeral m + 1 = sub One m"
  "numeral m + - 1 = sub m One"
  "- 1 + numeral n = sub n One"
  "- 1 + - numeral n = - numeral (inc n)"
  "- numeral m + - 1 = - numeral (inc m)"
  "1 + - 1 = 0"
  "- 1 + 1 = 0"
  "- 1 + - 1 = - 2"
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
    del: add_uminus_conv_diff add: diff_conv_add_uminus)

lemma diff_numeral_simps:
  "numeral m - numeral n = sub m n"
  "numeral m - - numeral n = numeral (m + n)"
  "- numeral m - numeral n = - numeral (m + n)"
  "- numeral m - - numeral n = sub n m"
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
    del: add_uminus_conv_diff add: diff_conv_add_uminus)

lemma diff_numeral_special:
  "1 - numeral n = sub One n"
  "numeral m - 1 = sub m One"
  "1 - - numeral n = numeral (One + n)"
  "- numeral m - 1 = - numeral (m + One)"
  "- 1 - numeral n = - numeral (inc n)"
  "numeral m - - 1 = numeral (inc m)"
  "- 1 - - numeral n = sub n One"
  "- numeral m - - 1 = sub One m"
  "1 - 1 = 0"
  "- 1 - 1 = - 2"
  "1 - - 1 = 2"
  "- 1 - - 1 = 0"
  by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
    del: add_uminus_conv_diff add: diff_conv_add_uminus)

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 distrib_left)
  done

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

lemma mult_2: "2 * z = z + z"
  unfolding one_add_one [symmetric] distrib_right by simp

lemma mult_2_right: "z * 2 = z + z"
  unfolding one_add_one [symmetric] distrib_left by simp

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 [code_abbrev]:
  "nat_of_num = numeral"
proof
  fix n
  have "numeral n = nat_of_num n"
    by (induct n) (simp_all add: numeral.simps)
  then show "nat_of_num n = numeral n" by simp
qed

lemma nat_of_num_code [code]:
  "nat_of_num One = 1"
  "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
  "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
  by (simp_all add: Let_def)

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:
  "- numeral m * - numeral n = numeral (m * n)"
  "- numeral m * numeral n = - numeral (m * n)"
  "numeral m * - numeral n = - numeral (m * n)"
  unfolding mult_minus_left mult_minus_right
  by (simp_all only: minus_minus numeral_mult)

lemma mult_minus1 [simp]: "- 1 * z = - z"
  unfolding numeral.simps mult_minus_left by simp

lemma mult_minus1_right [simp]: "z * - 1 = - z"
  unfolding 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 not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
  by (simp add: iszero_def)

lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
  by (simp add: numeral_One)

lemma iszero_neg_numeral [simp]:
  "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
  unfolding iszero_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 = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
  "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
  "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
  "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
  "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
  "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
  "0 = - 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: "- numeral m = - numeral n \<longleftrightarrow> m = n"
  by simp

lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
  unfolding eq_neg_iff_add_eq_0
  by (simp add: numeral_plus_numeral)

lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
  by (rule numeral_neq_neg_numeral [symmetric])

lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
  unfolding neg_0_equal_iff_equal by simp

lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
  unfolding neg_equal_0_iff_equal by simp

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

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

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

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

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

lemma numeral_eq_neg_one_iff:
  "- numeral n = - 1 \<longleftrightarrow> n = One"
  using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)

lemma neg_one_neq_zero:
  "- 1 \<noteq> 0"
  by simp

lemma zero_neq_neg_one:
  "0 \<noteq> - 1"
  by simp

lemma neg_one_neq_one:
  "- 1 \<noteq> 1"
  using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)

lemma one_neq_neg_one:
  "1 \<noteq> - 1"
  using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)

lemmas eq_neg_numeral_simps [simp] =
  neg_numeral_eq_iff
  numeral_neq_neg_numeral neg_numeral_neq_numeral
  one_neq_neg_numeral neg_numeral_neq_one
  zero_neq_neg_numeral neg_numeral_neq_zero
  neg_one_neq_numeral numeral_neq_neg_one
  neg_one_eq_numeral_iff numeral_eq_neg_one_iff
  neg_one_neq_zero zero_neq_neg_one
  neg_one_neq_one one_neq_neg_one

end

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

context linordered_idom
begin

subclass ring_char_0 ..

lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
  by (simp only: neg_le_iff_le numeral_le_iff)

lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
  by (simp only: neg_less_iff_less numeral_less_iff)

lemma neg_numeral_less_zero: "- numeral n < 0"
  by (simp only: neg_less_0_iff_less zero_less_numeral)

lemma neg_numeral_le_zero: "- numeral n \<le> 0"
  by (simp only: neg_le_0_iff_le zero_le_numeral)

lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
  by (simp only: not_less neg_numeral_le_zero)

lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
  by (simp only: not_le neg_numeral_less_zero)

lemma neg_numeral_less_numeral: "- numeral m < numeral n"
  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)

lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
  by (simp only: less_imp_le neg_numeral_less_numeral)

lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
  by (simp only: not_less neg_numeral_le_numeral)

lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
  by (simp only: not_le neg_numeral_less_numeral)
  
lemma neg_numeral_less_one: "- numeral m < 1"
  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])

lemma neg_numeral_le_one: "- numeral m \<le> 1"
  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])

lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
  by (simp only: not_less neg_numeral_le_one)

lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
  by (simp only: not_le neg_numeral_less_one)

lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
  using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)

lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
  using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)

lemma neg_one_less_numeral: "- 1 < numeral m"
  using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)

lemma neg_one_le_numeral: "- 1 \<le> numeral m"
  using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)

lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
  by (cases m) simp_all

lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
  by simp

lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
  by simp

lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
  by (cases m) simp_all

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
  neg_one_le_numeral not_numeral_le_neg_one
  neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff

lemma le_minus_one_simps [simp]:
  "- 1 \<le> 0"
  "- 1 \<le> 1"
  "\<not> 0 \<le> - 1"
  "\<not> 1 \<le> - 1"
  by simp_all

lemmas less_neg_numeral_simps [simp] =
  neg_numeral_less_iff
  neg_numeral_less_numeral not_numeral_less_neg_numeral
  neg_numeral_less_zero not_zero_less_neg_numeral
  neg_numeral_less_one not_one_less_neg_numeral
  neg_one_less_numeral not_numeral_less_neg_one
  neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral

lemma less_minus_one_simps [simp]:
  "- 1 < 0"
  "- 1 < 1"
  "\<not> 0 < - 1"
  "\<not> 1 < - 1"
  by (simp_all add: less_le)

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

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

lemma abs_neg_one [simp]:
  "abs (- 1) = 1"
  by simp

end

subsubsection {*
  Natural numbers
*}

lemma Suc_1 [simp]: "Suc 1 = 2"
  unfolding Suc_eq_plus1 by (rule one_add_one)

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

definition pred_numeral :: "num \<Rightarrow> nat"
  where [code del]: "pred_numeral k = numeral k - 1"

lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
  unfolding pred_numeral_def by simp

lemma eval_nat_numeral:
  "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)

lemma pred_numeral_simps [simp]:
  "pred_numeral One = 0"
  "pred_numeral (Bit0 k) = numeral (BitM k)"
  "pred_numeral (Bit1 k) = numeral (Bit0 k)"
  unfolding pred_numeral_def eval_nat_numeral
  by (simp_all only: diff_Suc_Suc diff_0)

lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
  by (simp add: eval_nat_numeral)

lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
  by (simp add: eval_nat_numeral)

lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
  by (simp only: numeral_One One_nat_def)

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

(*Maps #n to n for n = 1, 2*)
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2

text {* Comparisons involving @{term Suc}. *}

lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
  by (simp add: numeral_eq_Suc)

lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
  by (simp add: numeral_eq_Suc)

lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
  by (simp add: numeral_eq_Suc)

lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
  by (simp add: numeral_eq_Suc)

lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
  by (simp add: numeral_eq_Suc)

lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
  by (simp add: numeral_eq_Suc)

lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
  by (simp add: numeral_eq_Suc)

lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
  by (simp add: numeral_eq_Suc)

lemma max_Suc_numeral [simp]:
  "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
  by (simp add: numeral_eq_Suc)

lemma max_numeral_Suc [simp]:
  "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
  by (simp add: numeral_eq_Suc)

lemma min_Suc_numeral [simp]:
  "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
  by (simp add: numeral_eq_Suc)

lemma min_numeral_Suc [simp]:
  "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
  by (simp add: numeral_eq_Suc)

text {* For @{term case_nat} and @{term rec_nat}. *}

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

lemma case_nat_add_eq_if [simp]:
  "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
  by (simp add: numeral_eq_Suc)

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

lemma rec_nat_add_eq_if [simp]:
  "rec_nat a f (numeral v + n) =
    (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
  by (simp add: numeral_eq_Suc Let_def)

text {* Case analysis on @{term "n < 2"} *}

lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
  by (auto simp add: numeral_2_eq_2)

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

lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
  by simp

lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
  by simp

text {* Can be used to eliminate long strings of Sucs, but not by default. *}

lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
  by simp

lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)


subsection {* Particular lemmas concerning @{term 2} *}

context linordered_field
begin

lemma half_gt_zero_iff:
  "0 < a / 2 \<longleftrightarrow> 0 < a" (is "?P \<longleftrightarrow> ?Q")
  by (auto simp add: field_simps)

lemma half_gt_zero [simp]:
  "0 < a \<Longrightarrow> 0 < a / 2"
  by (simp add: half_gt_zero_iff)

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

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

setup {*
  Reorient_Proc.add
    (fn Const (@{const_name numeral}, _) $ _ => true
    | Const (@{const_name uminus}, _) $ (Const (@{const_name numeral}, _) $ _) => true
    | _ => false)
*}

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


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

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

(* rules already in simpset *)
lemmas arith_extra_simps =
  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
  minus_zero
  diff_numeral_simps diff_0 diff_0_right
  numeral_times_numeral mult_neg_numeral_simps
  mult_zero_left mult_zero_right
  abs_numeral abs_neg_numeral

text {*
  For making a minimal simpset, one must include these default simprules.
  Also include @{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

lemmas more_arith_simps =
  neg_le_iff_le
  minus_zero left_minus right_minus
  mult_1_left mult_1_right
  mult_minus_left mult_minus_right
  minus_add_distrib minus_minus mult.assoc

lemmas of_nat_simps =
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult

text {* Simplification of relational operations *}

lemmas eq_numeral_extra =
  zero_neq_one one_neq_zero

lemmas rel_simps =
  le_num_simps less_num_simps eq_num_simps
  le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
  less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra

lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
  -- {* Unfold all @{text let}s involving constants *}
  unfolding Let_def ..

lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
  -- {* Unfold all @{text let}s involving constants *}
  unfolding Let_def ..

declaration {*
let 
  fun number_of ctxt T n =
    if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort numeral}))
    then raise CTERM ("number_of", [])
    else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
in
  K (
    Lin_Arith.add_simps (@{thms arith_simps} @ @{thms more_arith_simps}
      @ @{thms rel_simps}
      @ @{thms pred_numeral_simps}
      @ @{thms arith_special numeral_One}
      @ @{thms of_nat_simps})
    #> Lin_Arith.add_simps [@{thm Suc_numeral},
      @{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},
      @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
      @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
      @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},
      @{thm mult_Suc}, @{thm mult_Suc_right},
      @{thm of_nat_numeral}]
    #> Lin_Arith.set_number_of number_of)
end
*}


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

lemma add_numeral_left [simp]:
  "numeral v + (numeral w + z) = (numeral(v + w) + z)"
  by (simp_all add: add.assoc [symmetric])

lemma add_neg_numeral_left [simp]:
  "numeral v + (- numeral w + y) = (sub v w + y)"
  "- numeral v + (numeral w + y) = (sub w v + y)"
  "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
  by (simp_all add: add.assoc [symmetric])

lemma mult_numeral_left [simp]:
  "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
  "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
  "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
  "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
  by (simp_all add: mult.assoc [symmetric])

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


subsection {* code module namespace *}

code_identifier
  code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

end