(*  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