(*  Title:      HOL/NatBin.thy

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1999  University of Cambridge

 *)

 header {* Binary arithmetic for the natural numbers *}

 theory NatBin = IntDiv:

 text {*

   Arithmetic for naturals is reduced to that for the non-negative integers.

 *}

 instance nat :: number ..

 defs (overloaded)

   nat_number_of_def:

      "(number_of::bin => nat) v == nat ((number_of :: bin => int) v)"

 subsection{*Function @{term nat}: Coercion from Type @{typ int} to @{typ nat}*}

 declare nat_0 [simp] nat_1 [simp]

 lemma nat_number_of [simp]: "nat (number_of w) = number_of w"

 by (simp add: nat_number_of_def)

 lemma nat_numeral_0_eq_0 [simp]: "Numeral0 = (0::nat)"

 by (simp add: nat_number_of_def)

 lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"

 by (simp add: nat_1 nat_number_of_def)

 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"

 by (simp add: nat_numeral_1_eq_1)

 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"

 apply (unfold nat_number_of_def)

 apply (rule nat_2)

 done

 text{*Distributive laws for type @{text nat}.  The others are in theory

    @{text IntArith}, but these require div and mod to be defined for type

    "int".  They also need some of the lemmas proved above.*}

 lemma nat_div_distrib: "(0::int) <= z ==> nat (z div z') = nat z div nat z'"

 apply (case_tac "0 <= z'")

 apply (auto simp add: div_nonneg_neg_le0 DIVISION_BY_ZERO_DIV)

 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)

 apply (auto elim!: nonneg_eq_int)

 apply (rename_tac m m')

 apply (subgoal_tac "0 <= int m div int m'")

  prefer 2 apply (simp add: nat_numeral_0_eq_0 pos_imp_zdiv_nonneg_iff) 

 apply (rule inj_int [THEN injD], simp)

 apply (rule_tac r = "int (m mod m') " in quorem_div)

  prefer 2 apply force

 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int 

                  zmult_int)

 done

 (*Fails if z'<0: the LHS collapses to (nat z) but the RHS doesn't*)

 lemma nat_mod_distrib:

      "[| (0::int) <= z;  0 <= z' |] ==> nat (z mod z') = nat z mod nat z'"

 apply (case_tac "z' = 0", simp add: DIVISION_BY_ZERO)

 apply (auto elim!: nonneg_eq_int)

 apply (rename_tac m m')

 apply (subgoal_tac "0 <= int m mod int m'")

  prefer 2 apply (simp add: nat_less_iff nat_numeral_0_eq_0 pos_mod_sign) 

 apply (rule inj_int [THEN injD], simp)

 apply (rule_tac q = "int (m div m') " in quorem_mod)

  prefer 2 apply force

 apply (simp add: nat_less_iff [symmetric] quorem_def nat_numeral_0_eq_0 zadd_int zmult_int)

 done

 subsection{*Function @{term int}: Coercion from Type @{typ nat} to @{typ int}*}

 (*"neg" is used in rewrite rules for binary comparisons*)

 lemma int_nat_number_of [simp]:

      "int (number_of v :: nat) =  

          (if neg (number_of v :: int) then 0  

           else (number_of v :: int))"

 by (simp del: nat_number_of

 	 add: neg_nat nat_number_of_def not_neg_nat add_assoc)

 subsubsection{*Successor *}

 lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"

 apply (rule sym)

 apply (simp add: nat_eq_iff int_Suc)

 done

 lemma Suc_nat_number_of_add:

      "Suc (number_of v + n) =  

         (if neg (number_of v :: int) then 1+n else number_of (bin_succ v) + n)" 

 by (simp del: nat_number_of 

          add: nat_number_of_def neg_nat

               Suc_nat_eq_nat_zadd1 number_of_succ) 

 lemma Suc_nat_number_of [simp]:

      "Suc (number_of v) =  

         (if neg (number_of v :: int) then 1 else number_of (bin_succ v))"

 apply (cut_tac n = 0 in Suc_nat_number_of_add)

 apply (simp cong del: if_weak_cong)

 done

 subsubsection{*Addition *}

 (*"neg" is used in rewrite rules for binary comparisons*)

 lemma add_nat_number_of [simp]:

      "(number_of v :: nat) + number_of v' =  

          (if neg (number_of v :: int) then number_of v'  

           else if neg (number_of v' :: int) then number_of v  

           else number_of (bin_add v v'))"

 by (force dest!: neg_nat

           simp del: nat_number_of

           simp add: nat_number_of_def nat_add_distrib [symmetric]) 

 subsubsection{*Subtraction *}

 lemma diff_nat_eq_if:

      "nat z - nat z' =  

         (if neg z' then nat z   

          else let d = z-z' in     

               if neg d then 0 else nat d)"

 apply (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)

 apply (simp add: diff_is_0_eq nat_le_eq_zle)

 done

 lemma diff_nat_number_of [simp]: 

      "(number_of v :: nat) - number_of v' =  

         (if neg (number_of v' :: int) then number_of v  

          else let d = number_of (bin_add v (bin_minus v')) in     

               if neg d then 0 else nat d)"

 by (simp del: nat_number_of add: diff_nat_eq_if nat_number_of_def) 

 subsubsection{*Multiplication *}

 lemma mult_nat_number_of [simp]:

      "(number_of v :: nat) * number_of v' =  

        (if neg (number_of v :: int) then 0 else number_of (bin_mult v v'))"

 by (force dest!: neg_nat

           simp del: nat_number_of

           simp add: nat_number_of_def nat_mult_distrib [symmetric]) 

 subsubsection{*Quotient *}

 lemma div_nat_number_of [simp]:

      "(number_of v :: nat)  div  number_of v' =  

           (if neg (number_of v :: int) then 0  

            else nat (number_of v div number_of v'))"

 by (force dest!: neg_nat

           simp del: nat_number_of

           simp add: nat_number_of_def nat_div_distrib [symmetric]) 

 lemma one_div_nat_number_of [simp]:

      "(Suc 0)  div  number_of v' = (nat (1 div number_of v'))" 

 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 

 subsubsection{*Remainder *}

 lemma mod_nat_number_of [simp]:

      "(number_of v :: nat)  mod  number_of v' =  

         (if neg (number_of v :: int) then 0  

          else if neg (number_of v' :: int) then number_of v  

          else nat (number_of v mod number_of v'))"

 by (force dest!: neg_nat

           simp del: nat_number_of

           simp add: nat_number_of_def nat_mod_distrib [symmetric]) 

 lemma one_mod_nat_number_of [simp]:

      "(Suc 0)  mod  number_of v' =  

         (if neg (number_of v' :: int) then Suc 0

          else nat (1 mod number_of v'))"

 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 

 ML

 {*

 val nat_number_of_def = thm"nat_number_of_def";

 val nat_number_of = thm"nat_number_of";

 val nat_numeral_0_eq_0 = thm"nat_numeral_0_eq_0";

 val nat_numeral_1_eq_1 = thm"nat_numeral_1_eq_1";

 val numeral_1_eq_Suc_0 = thm"numeral_1_eq_Suc_0";

 val numeral_2_eq_2 = thm"numeral_2_eq_2";

 val nat_div_distrib = thm"nat_div_distrib";

 val nat_mod_distrib = thm"nat_mod_distrib";

 val int_nat_number_of = thm"int_nat_number_of";

 val Suc_nat_eq_nat_zadd1 = thm"Suc_nat_eq_nat_zadd1";

 val Suc_nat_number_of_add = thm"Suc_nat_number_of_add";

 val Suc_nat_number_of = thm"Suc_nat_number_of";

 val add_nat_number_of = thm"add_nat_number_of";

 val diff_nat_eq_if = thm"diff_nat_eq_if";

 val diff_nat_number_of = thm"diff_nat_number_of";

 val mult_nat_number_of = thm"mult_nat_number_of";

 val div_nat_number_of = thm"div_nat_number_of";

 val mod_nat_number_of = thm"mod_nat_number_of";

 *}

 subsection{*Comparisons*}

 subsubsection{*Equals (=) *}

 lemma eq_nat_nat_iff:

      "[| (0::int) <= z;  0 <= z' |] ==> (nat z = nat z') = (z=z')"

 by (auto elim!: nonneg_eq_int)

 (*"neg" is used in rewrite rules for binary comparisons*)

 lemma eq_nat_number_of [simp]:

      "((number_of v :: nat) = number_of v') =  

       (if neg (number_of v :: int) then (iszero (number_of v' :: int) | neg (number_of v' :: int))  

        else if neg (number_of v' :: int) then iszero (number_of v :: int)  

        else iszero (number_of (bin_add v (bin_minus v')) :: int))"

 apply (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def

                   eq_nat_nat_iff eq_number_of_eq nat_0 iszero_def

             split add: split_if cong add: imp_cong)

 apply (simp only: nat_eq_iff nat_eq_iff2)

 apply (simp add: not_neg_eq_ge_0 [symmetric])

 done

 subsubsection{*Less-than (<) *}

 (*"neg" is used in rewrite rules for binary comparisons*)

 lemma less_nat_number_of [simp]:

      "((number_of v :: nat) < number_of v') =  

          (if neg (number_of v :: int) then neg (number_of (bin_minus v') :: int)  

           else neg (number_of (bin_add v (bin_minus v')) :: int))"

 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def

                 nat_less_eq_zless less_number_of_eq_neg zless_nat_eq_int_zless

          cong add: imp_cong, simp) 

 (*Maps #n to n for n = 0, 1, 2*)

 lemmas numerals = nat_numeral_0_eq_0 nat_numeral_1_eq_1 numeral_2_eq_2

 subsection{*General Theorems About Powers Involving Binary Numerals*}

 text{*We cannot refer to the number @{term 2} in @{text Ring_and_Field.thy}.

 We cannot prove general results about the numeral @{term "-1"}, so we have to

 use @{term "- 1"} instead.*}

 lemma power2_eq_square: "(a::'a::{comm_semiring_1_cancel,ringpower})\<twosuperior> = a * a"

   by (simp add: numeral_2_eq_2 Power.power_Suc)

 lemma [simp]: "(0::'a::{comm_semiring_1_cancel,ringpower})\<twosuperior> = 0"

   by (simp add: power2_eq_square)

 lemma [simp]: "(1::'a::{comm_semiring_1_cancel,ringpower})\<twosuperior> = 1"

   by (simp add: power2_eq_square)

 text{*Squares of literal numerals will be evaluated.*}

 declare power2_eq_square [of "number_of w", standard, simp]

 lemma zero_le_power2 [simp]: "0 \<le> (a\<twosuperior>::'a::{ordered_idom,ringpower})"

   by (simp add: power2_eq_square zero_le_square)

 lemma zero_less_power2 [simp]:

      "(0 < a\<twosuperior>) = (a \<noteq> (0::'a::{ordered_idom,ringpower}))"

   by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

 lemma zero_eq_power2 [simp]:

      "(a\<twosuperior> = 0) = (a = (0::'a::{ordered_idom,ringpower}))"

   by (force simp add: power2_eq_square mult_eq_0_iff)

 lemma abs_power2 [simp]:

      "abs(a\<twosuperior>) = (a\<twosuperior>::'a::{ordered_idom,ringpower})"

   by (simp add: power2_eq_square abs_mult abs_mult_self)

 lemma power2_abs [simp]:

      "(abs a)\<twosuperior> = (a\<twosuperior>::'a::{ordered_idom,ringpower})"

   by (simp add: power2_eq_square abs_mult_self)

 lemma power2_minus [simp]:

      "(- a)\<twosuperior> = (a\<twosuperior>::'a::{comm_ring_1,ringpower})"

   by (simp add: power2_eq_square)

 lemma power_minus1_even: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,ringpower})"

 apply (induct_tac "n")

 apply (auto simp add: power_Suc power_add)

 done

 lemma power_even_eq: "(a::'a::ringpower) ^ (2*n) = (a^n)^2"

 by (simp add: power_mult power_mult_distrib power2_eq_square)

 lemma power_odd_eq: "(a::int) ^ Suc(2*n) = a * (a^n)^2"

 by (simp add: power_even_eq) 

 lemma power_minus_even [simp]:

      "(-a) ^ (2*n) = (a::'a::{comm_ring_1,ringpower}) ^ (2*n)"

 by (simp add: power_minus1_even power_minus [of a]) 

 lemma zero_le_even_power:

      "0 \<le> (a::'a::{ordered_idom,ringpower}) ^ (2*n)"

 proof (induct "n")

   case 0

     show ?case by (simp add: zero_le_one)

 next

   case (Suc n)

     have "a ^ (2 * Suc n) = (a*a) * a ^ (2*n)" 

       by (simp add: mult_ac power_add power2_eq_square)

     thus ?case

       by (simp add: prems zero_le_square zero_le_mult_iff)

 qed

 lemma odd_power_less_zero:

      "(a::'a::{ordered_idom,ringpower}) < 0 ==> a ^ Suc(2*n) < 0"

 proof (induct "n")

   case 0

     show ?case by (simp add: Power.power_Suc)

 next

   case (Suc n)

     have "a ^ Suc (2 * Suc n) = (a*a) * a ^ Suc(2*n)" 

       by (simp add: mult_ac power_add power2_eq_square Power.power_Suc)

     thus ?case

       by (simp add: prems mult_less_0_iff mult_neg)

 qed

 lemma odd_0_le_power_imp_0_le:

      "0 \<le> a  ^ Suc(2*n) ==> 0 \<le> (a::'a::{ordered_idom,ringpower})"

 apply (insert odd_power_less_zero [of a n]) 

 apply (force simp add: linorder_not_less [symmetric]) 

 done

 subsubsection{*Nat *}

 lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"

 by (simp add: numerals)

 (*Expresses a natural number constant as the Suc of another one.

   NOT suitable for rewriting because n recurs in the condition.*)

 lemmas expand_Suc = Suc_pred' [of "number_of v", standard]

 subsubsection{*Arith *}

 lemma Suc_eq_add_numeral_1: "Suc n = n + 1"

 by (simp add: numerals)

 lemma Suc_eq_add_numeral_1_left: "Suc n = 1 + n"

 by (simp add: numerals)

 (* These two can be useful when m = number_of... *)

 lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"

 apply (case_tac "m")

 apply (simp_all add: numerals)

 done

 lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"

 apply (case_tac "m")

 apply (simp_all add: numerals)

 done

 lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"

 apply (case_tac "m")

 apply (simp_all add: numerals)

 done

 lemma diff_less': "[| 0<n; 0<m |] ==> m - n < (m::nat)"

 by (simp add: diff_less numerals)

 declare diff_less' [of "number_of v", standard, simp]

 subsection{*Comparisons involving (0::nat) *}

 text{*Simplification already does @{term "n<0"}, @{term "n\<le>0"} and @{term "0\<le>n"}.*}

 lemma eq_number_of_0 [simp]:

      "(number_of v = (0::nat)) =  

       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"

 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)

 lemma eq_0_number_of [simp]:

      "((0::nat) = number_of v) =  

       (if neg (number_of v :: int) then True else iszero (number_of v :: int))"

 by (rule trans [OF eq_sym_conv eq_number_of_0])

 lemma less_0_number_of [simp]:

      "((0::nat) < number_of v) = neg (number_of (bin_minus v) :: int)"

 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])

 lemma neg_imp_number_of_eq_0: "neg (number_of v :: int) ==> number_of v = (0::nat)"

 by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric] iszero_0)

 subsection{*Comparisons involving Suc *}

 lemma eq_number_of_Suc [simp]:

      "(number_of v = Suc n) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then False else nat pv = n)"

 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 

                   number_of_pred nat_number_of_def 

             split add: split_if)

 apply (rule_tac x = "number_of v" in spec)

 apply (auto simp add: nat_eq_iff)

 done

 lemma Suc_eq_number_of [simp]:

      "(Suc n = number_of v) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then False else nat pv = n)"

 by (rule trans [OF eq_sym_conv eq_number_of_Suc])

 lemma less_number_of_Suc [simp]:

      "(number_of v < Suc n) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then True else nat pv < n)"

 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 

                   number_of_pred nat_number_of_def  

             split add: split_if)

 apply (rule_tac x = "number_of v" in spec)

 apply (auto simp add: nat_less_iff)

 done

 lemma less_Suc_number_of [simp]:

      "(Suc n < number_of v) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then False else n < nat pv)"

 apply (simp only: simp_thms Let_def neg_eq_less_0 linorder_not_less 

                   number_of_pred nat_number_of_def

             split add: split_if)

 apply (rule_tac x = "number_of v" in spec)

 apply (auto simp add: zless_nat_eq_int_zless)

 done

 lemma le_number_of_Suc [simp]:

      "(number_of v <= Suc n) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then True else nat pv <= n)"

 by (simp add: Let_def less_Suc_number_of linorder_not_less [symmetric])

 lemma le_Suc_number_of [simp]:

      "(Suc n <= number_of v) =  

         (let pv = number_of (bin_pred v) in  

          if neg pv then False else n <= nat pv)"

 by (simp add: Let_def less_number_of_Suc linorder_not_less [symmetric])

 (* Push int(.) inwards: *)

 declare zadd_int [symmetric, simp]

 lemma lemma1: "(m+m = n+n) = (m = (n::int))"

 by auto

 lemma lemma2: "m+m ~= (1::int) + (n + n)"

 apply auto

 apply (drule_tac f = "%x. x mod 2" in arg_cong)

 apply (simp add: zmod_zadd1_eq)

 done

 lemma eq_number_of_BIT_BIT:

      "((number_of (v BIT x) ::int) = number_of (w BIT y)) =  

       (x=y & (((number_of v) ::int) = number_of w))"

 by (simp only: simp_thms number_of_BIT lemma1 lemma2 eq_commute

                OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0

             split add: split_if cong: imp_cong) 

 lemma eq_number_of_BIT_Pls:

      "((number_of (v BIT x) ::int) = number_of bin.Pls) =  

       (x=False & (((number_of v) ::int) = number_of bin.Pls))"

 apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute

             split add: split_if cong: imp_cong)

 apply (rule_tac x = "number_of v" in spec, safe)

 apply (simp_all (no_asm_use))

 apply (drule_tac f = "%x. x mod 2" in arg_cong)

 apply (simp add: zmod_zadd1_eq)

 done

 lemma eq_number_of_BIT_Min:

      "((number_of (v BIT x) ::int) = number_of bin.Min) =  

       (x=True & (((number_of v) ::int) = number_of bin.Min))"

 apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute

             split add: split_if cong: imp_cong)

 apply (rule_tac x = "number_of v" in spec, auto)

 apply (drule_tac f = "%x. x mod 2" in arg_cong, auto)

 done

 lemma eq_number_of_Pls_Min: "(number_of bin.Pls ::int) ~= number_of bin.Min"

 by auto

 subsection{*Literal arithmetic involving powers*}

 lemma nat_power_eq: "(0::int) <= z ==> nat (z^n) = nat z ^ n"

 apply (induct_tac "n")

 apply (simp_all (no_asm_simp) add: nat_mult_distrib)

 done

 lemma power_nat_number_of:

      "(number_of v :: nat) ^ n =  

        (if neg (number_of v :: int) then 0^n else nat ((number_of v :: int) ^ n))"

 by (simp only: simp_thms neg_nat not_neg_eq_ge_0 nat_number_of_def nat_power_eq

          split add: split_if cong: imp_cong)

 declare power_nat_number_of [of _ "number_of w", standard, simp]

 text{*For the integers*}

 lemma zpower_number_of_even:

      "(z::int) ^ number_of (w BIT False) =  

       (let w = z ^ (number_of w) in  w*w)"

 apply (simp del: nat_number_of  add: nat_number_of_def number_of_BIT Let_def)

 apply (simp only: number_of_add) 

 apply (rule_tac x = "number_of w" in spec, clarify)

 apply (case_tac " (0::int) <= x")

 apply (auto simp add: nat_mult_distrib power_even_eq power2_eq_square)

 done

 lemma zpower_number_of_odd:

      "(z::int) ^ number_of (w BIT True) =  

           (if (0::int) <= number_of w                    

            then (let w = z ^ (number_of w) in  z*w*w)    

            else 1)"

 apply (simp del: nat_number_of  add: nat_number_of_def number_of_BIT Let_def)

 apply (simp only: number_of_add nat_numeral_1_eq_1 not_neg_eq_ge_0 neg_eq_less_0) 

 apply (rule_tac x = "number_of w" in spec, clarify)

 apply (auto simp add: nat_add_distrib nat_mult_distrib power_even_eq power2_eq_square neg_nat)

 done

 declare zpower_number_of_even [of "number_of v", standard, simp]

 declare zpower_number_of_odd  [of "number_of v", standard, simp]

 ML

 {*

 val numerals = thms"numerals";

 val numeral_ss = simpset() addsimps numerals;

 val nat_bin_arith_setup =

  [Fast_Arith.map_data 

    (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>

      {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,

       inj_thms = inj_thms,

       lessD = lessD,

       simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,

                                   not_neg_number_of_Pls,

                                   neg_number_of_Min,neg_number_of_BIT]})]

 *}

 setup nat_bin_arith_setup

 (* Enable arith to deal with div/mod k where k is a numeral: *)

 declare split_div[of _ _ "number_of k", standard, arith_split]

 declare split_mod[of _ _ "number_of k", standard, arith_split]

 lemma nat_number_of_Pls: "number_of bin.Pls = (0::nat)"

   by (simp add: number_of_Pls nat_number_of_def)

 lemma nat_number_of_Min: "number_of bin.Min = (0::nat)"

   apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)

   apply (simp add: neg_nat)

   done

 lemma nat_number_of_BIT_True:

   "number_of (w BIT True) =

     (if neg (number_of w :: int) then 0

      else let n = number_of w in Suc (n + n))"

   apply (simp only: nat_number_of_def Let_def split: split_if)

   apply (intro conjI impI)

    apply (simp add: neg_nat neg_number_of_BIT)

   apply (rule int_int_eq [THEN iffD1])

   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)

   apply (simp only: number_of_BIT if_True zadd_assoc)

   done

 lemma nat_number_of_BIT_False:

     "number_of (w BIT False) = (let n::nat = number_of w in n + n)"

   apply (simp only: nat_number_of_def Let_def)

   apply (cases "neg (number_of w :: int)")

    apply (simp add: neg_nat neg_number_of_BIT)

   apply (rule int_int_eq [THEN iffD1])

   apply (simp only: not_neg_nat neg_number_of_BIT int_Suc zadd_int [symmetric] simp_thms)

   apply (simp only: number_of_BIT if_False zadd_0 zadd_assoc)

   done

 lemmas nat_number =

   nat_number_of_Pls nat_number_of_Min

   nat_number_of_BIT_True nat_number_of_BIT_False

 lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"

   by (simp add: Let_def)

 lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring,ringpower})"

 by (simp add: power_mult); 

 lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring,ringpower})"

 by (simp add: power_mult power_Suc); 

 subsection{*Literal arithmetic and @{term of_nat}*}

 lemma of_nat_double:

      "0 \<le> x ==> of_nat (nat (2 * x)) = of_nat (nat x) + of_nat (nat x)"

 by (simp only: mult_2 nat_add_distrib of_nat_add) 

 lemma nat_numeral_m1_eq_0: "-1 = (0::nat)"

 by (simp only:  nat_number_of_def, simp)

 lemma int_double_iff: "(0::int) \<le> 2*x + 1 = (0 \<le> x)"

 by arith

 lemma of_nat_number_of_lemma:

      "of_nat (number_of v :: nat) =  

          (if 0 \<le> (number_of v :: int) 

           then (number_of v :: 'a :: number_ring)

           else 0)"

 apply (induct v, simp, simp add: nat_numeral_m1_eq_0)

 apply (simp only: number_of nat_number_of_def)

 txt{*Generalize in order to eliminate the function @{term number_of} and

 its annoying simprules*}

 apply (erule rev_mp)

 apply (rule_tac x="number_of bin :: int" in spec) 

 apply (rule_tac x="number_of bin :: 'a" in spec) 

 apply (simp add: nat_add_distrib of_nat_double int_double_iff)

 done

 lemma of_nat_number_of_eq [simp]:

      "of_nat (number_of v :: nat) =  

          (if neg (number_of v :: int) then 0  

           else (number_of v :: 'a :: number_ring))"

 by (simp only: of_nat_number_of_lemma neg_def, simp) 

 subsection {*Lemmas for the Combination and Cancellation Simprocs*}

 lemma nat_number_of_add_left:

      "number_of v + (number_of v' + (k::nat)) =  

          (if neg (number_of v :: int) then number_of v' + k  

           else if neg (number_of v' :: int) then number_of v + k  

           else number_of (bin_add v v') + k)"

 by simp

 lemma nat_number_of_mult_left:

      "number_of v * (number_of v' * (k::nat)) =  

          (if neg (number_of v :: int) then 0

           else number_of (bin_mult v v') * k)"

 by simp

 subsubsection{*For @{text combine_numerals}*}

 lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"

 by (simp add: add_mult_distrib)

 subsubsection{*For @{text cancel_numerals}*}

 lemma nat_diff_add_eq1:

      "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"

 by (simp split add: nat_diff_split add: add_mult_distrib)

 lemma nat_diff_add_eq2:

      "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"

 by (simp split add: nat_diff_split add: add_mult_distrib)

 lemma nat_eq_add_iff1:

      "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 lemma nat_eq_add_iff2:

      "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 lemma nat_less_add_iff1:

      "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 lemma nat_less_add_iff2:

      "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 lemma nat_le_add_iff1:

      "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 lemma nat_le_add_iff2:

      "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"

 by (auto split add: nat_diff_split simp add: add_mult_distrib)

 subsubsection{*For @{text cancel_numeral_factors} *}

 lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"

 by auto

 lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"

 by auto

 lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"

 by auto

 lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"

 by auto

 subsubsection{*For @{text cancel_factor} *}

 lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"

 by auto

 lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"

 by auto

 lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"

 by auto

 lemma nat_mult_div_cancel_disj:

      "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"

 by (simp add: nat_mult_div_cancel1)

 ML

 {*

 val eq_nat_nat_iff = thm"eq_nat_nat_iff";

 val eq_nat_number_of = thm"eq_nat_number_of";

 val less_nat_number_of = thm"less_nat_number_of";

 val power2_eq_square = thm "power2_eq_square";

 val zero_le_power2 = thm "zero_le_power2";

 val zero_less_power2 = thm "zero_less_power2";

 val zero_eq_power2 = thm "zero_eq_power2";

 val abs_power2 = thm "abs_power2";

 val power2_abs = thm "power2_abs";

 val power2_minus = thm "power2_minus";

 val power_minus1_even = thm "power_minus1_even";

 val power_minus_even = thm "power_minus_even";

 val zero_le_even_power = thm "zero_le_even_power";

 val odd_power_less_zero = thm "odd_power_less_zero";

 val odd_0_le_power_imp_0_le = thm "odd_0_le_power_imp_0_le";

 val Suc_pred' = thm"Suc_pred'";

 val expand_Suc = thm"expand_Suc";

 val Suc_eq_add_numeral_1 = thm"Suc_eq_add_numeral_1";

 val Suc_eq_add_numeral_1_left = thm"Suc_eq_add_numeral_1_left";

 val add_eq_if = thm"add_eq_if";

 val mult_eq_if = thm"mult_eq_if";

 val power_eq_if = thm"power_eq_if";

 val diff_less' = thm"diff_less'";

 val eq_number_of_0 = thm"eq_number_of_0";

 val eq_0_number_of = thm"eq_0_number_of";

 val less_0_number_of = thm"less_0_number_of";

 val neg_imp_number_of_eq_0 = thm"neg_imp_number_of_eq_0";

 val eq_number_of_Suc = thm"eq_number_of_Suc";

 val Suc_eq_number_of = thm"Suc_eq_number_of";

 val less_number_of_Suc = thm"less_number_of_Suc";

 val less_Suc_number_of = thm"less_Suc_number_of";

 val le_number_of_Suc = thm"le_number_of_Suc";

 val le_Suc_number_of = thm"le_Suc_number_of";

 val eq_number_of_BIT_BIT = thm"eq_number_of_BIT_BIT";

 val eq_number_of_BIT_Pls = thm"eq_number_of_BIT_Pls";

 val eq_number_of_BIT_Min = thm"eq_number_of_BIT_Min";

 val eq_number_of_Pls_Min = thm"eq_number_of_Pls_Min";

 val of_nat_number_of_eq = thm"of_nat_number_of_eq";

 val nat_power_eq = thm"nat_power_eq";

 val power_nat_number_of = thm"power_nat_number_of";

 val zpower_number_of_even = thm"zpower_number_of_even";

 val zpower_number_of_odd = thm"zpower_number_of_odd";

 val nat_number_of_Pls = thm"nat_number_of_Pls";

 val nat_number_of_Min = thm"nat_number_of_Min";

 val nat_number_of_BIT_True = thm"nat_number_of_BIT_True";

 val nat_number_of_BIT_False = thm"nat_number_of_BIT_False";

 val Let_Suc = thm"Let_Suc";

 val nat_number = thms"nat_number";

 val nat_number_of_add_left = thm"nat_number_of_add_left";

 val nat_number_of_mult_left = thm"nat_number_of_mult_left";

 val left_add_mult_distrib = thm"left_add_mult_distrib";

 val nat_diff_add_eq1 = thm"nat_diff_add_eq1";

 val nat_diff_add_eq2 = thm"nat_diff_add_eq2";

 val nat_eq_add_iff1 = thm"nat_eq_add_iff1";

 val nat_eq_add_iff2 = thm"nat_eq_add_iff2";

 val nat_less_add_iff1 = thm"nat_less_add_iff1";

 val nat_less_add_iff2 = thm"nat_less_add_iff2";

 val nat_le_add_iff1 = thm"nat_le_add_iff1";

 val nat_le_add_iff2 = thm"nat_le_add_iff2";

 val nat_mult_le_cancel1 = thm"nat_mult_le_cancel1";

 val nat_mult_less_cancel1 = thm"nat_mult_less_cancel1";

 val nat_mult_eq_cancel1 = thm"nat_mult_eq_cancel1";

 val nat_mult_div_cancel1 = thm"nat_mult_div_cancel1";

 val nat_mult_le_cancel_disj = thm"nat_mult_le_cancel_disj";

 val nat_mult_less_cancel_disj = thm"nat_mult_less_cancel_disj";

 val nat_mult_eq_cancel_disj = thm"nat_mult_eq_cancel_disj";

 val nat_mult_div_cancel_disj = thm"nat_mult_div_cancel_disj";

 val power_minus_even = thm"power_minus_even";

 val zero_le_even_power = thm"zero_le_even_power";

 *}

 subsection {* Configuration of the code generator *}

 ML {*

 infix 7 `*;

 infix 6 `+;

 val op `* = op * : int * int -> int;

 val op `+ = op + : int * int -> int;

 val `~ = ~ : int -> int;

 *}

 types_code

   "int" ("int")

 constdefs

   int_aux :: "int \<Rightarrow> nat \<Rightarrow> int"

   "int_aux i n == (i + int n)"

   nat_aux :: "nat \<Rightarrow> int \<Rightarrow> nat"

   "nat_aux n i == (n + nat i)"

 lemma [code]:

   "int_aux i 0 = i"

   "int_aux i (Suc n) = int_aux (i + 1) n" -- {* tail recursive *}

   "int n = int_aux 0 n"

   by (simp add: int_aux_def)+

 lemma [code]: "nat_aux n i = (if i <= 0 then n else nat_aux (Suc n) (i - 1))"

   by (simp add: nat_aux_def Suc_nat_eq_nat_zadd1) -- {* tail recursive *}

 lemma [code]: "nat i = nat_aux 0 i"

   by (simp add: nat_aux_def)

 consts_code

   "0" :: "int"                  ("0")

   "1" :: "int"                  ("1")

   "uminus" :: "int => int"      ("`~")

   "op +" :: "int => int => int" ("(_ `+/ _)")

   "op *" :: "int => int => int" ("(_ `*/ _)")

   "op <" :: "int => int => bool" ("(_ </ _)")

   "op <=" :: "int => int => bool" ("(_ <=/ _)")

   "neg"                         ("(_ < 0)")

 ML {*

 fun number_of_codegen thy gr s b (Const ("Numeral.number_of",

       Type ("fun", [_, Type ("IntDef.int", [])])) $ bin) =

         (Some (gr, Pretty.str (string_of_int (HOLogic.dest_binum bin)))

         handle TERM _ => None)

   | number_of_codegen thy gr s b (Const ("Numeral.number_of",

       Type ("fun", [_, Type ("nat", [])])) $ bin) =

         Some (Codegen.invoke_codegen thy s b (gr,

           Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT) $

             (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) $ bin)))

   | number_of_codegen _ _ _ _ _ = None;

 *}

 setup {* [Codegen.add_codegen "number_of_codegen" number_of_codegen] *}

 end