(*  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

imports IntDiv

begin

text {*

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

*}

instantiation nat :: number

begin

definition

  nat_number_of_def [code inline]: "number_of v = nat (number_of (v\<Colon>int))"

instance ..

end

abbreviation (xsymbols)

  square :: "'a::power => 'a"  ("(_\<twosuperior>)" [1000] 999) where

  "x\<twosuperior> == x^2"

notation (latex output)

  square  ("(_\<twosuperior>)" [1000] 999)

notation (HTML output)

  square  ("(_\<twosuperior>)" [1000] 999)

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 of_nat_eq_iff [where 'a=int, THEN iffD1], 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

                 of_nat_add [symmetric] of_nat_mult [symmetric]

            del: of_nat_add of_nat_mult)

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 int_int_eq [THEN iffD1], 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

                 of_nat_add [symmetric] of_nat_mult [symmetric]

            del: of_nat_add of_nat_mult)

done

text{*Suggested by Matthias Daum*}

lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

apply (subgoal_tac "nat x div nat k < nat x")

 apply (simp (asm_lr) add: nat_div_distrib [symmetric])

apply (rule Divides.div_less_dividend, simp_all) 

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) =  

         (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 (Int.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 (Int.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 (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)

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 (v + uminus 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 (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]) 

subsubsection{* Divisibility *}

lemmas dvd_eq_mod_eq_0_number_of =

  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]

declare dvd_eq_mod_eq_0_number_of [simp]

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 (v + uminus 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 (uminus v') :: int)  

          else neg (number_of (v + uminus 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 add: Pls_def)

(*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{*Powers with Numeric Exponents*}

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::recpower)\<twosuperior> = a * a"

  by (simp add: numeral_2_eq_2 Power.power_Suc)

lemma zero_power2 [simp]: "(0::'a::{semiring_1,recpower})\<twosuperior> = 0"

  by (simp add: power2_eq_square)

lemma one_power2 [simp]: "(1::'a::{semiring_1,recpower})\<twosuperior> = 1"

  by (simp add: power2_eq_square)

lemma power3_eq_cube: "(x::'a::recpower) ^ 3 = x * x * x"

  apply (subgoal_tac "3 = Suc (Suc (Suc 0))")

  apply (erule ssubst)

  apply (simp add: power_Suc mult_ac)

  apply (unfold nat_number_of_def)

  apply (subst nat_eq_iff)

  apply simp

done

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

lemmas power2_eq_square_number_of =

    power2_eq_square [of "number_of w", standard]

declare power2_eq_square_number_of [simp]

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

  by (simp add: power2_eq_square)

lemma zero_less_power2[simp]:

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

  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

lemma power2_less_0[simp]:

  fixes a :: "'a::{ordered_idom,recpower}"

  shows "~ (a\<twosuperior> < 0)"

by (force simp add: power2_eq_square mult_less_0_iff) 

lemma zero_eq_power2[simp]:

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

  by (force simp add: power2_eq_square mult_eq_0_iff)

lemma abs_power2[simp]:

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

  by (simp add: power2_eq_square abs_mult abs_mult_self)

lemma power2_abs[simp]:

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

  by (simp add: power2_eq_square abs_mult_self)

lemma power2_minus[simp]:

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

  by (simp add: power2_eq_square)

lemma power2_le_imp_le:

  fixes x y :: "'a::{ordered_semidom,recpower}"

  shows "\<lbrakk>x\<twosuperior> \<le> y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x \<le> y"

unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)

lemma power2_less_imp_less:

  fixes x y :: "'a::{ordered_semidom,recpower}"

  shows "\<lbrakk>x\<twosuperior> < y\<twosuperior>; 0 \<le> y\<rbrakk> \<Longrightarrow> x < y"

by (rule power_less_imp_less_base)

lemma power2_eq_imp_eq:

  fixes x y :: "'a::{ordered_semidom,recpower}"

  shows "\<lbrakk>x\<twosuperior> = y\<twosuperior>; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> x = y"

unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base, simp)

lemma power_minus1_even[simp]: "(- 1) ^ (2*n) = (1::'a::{comm_ring_1,recpower})"

apply (induct "n")

apply (auto simp add: power_Suc power_add)

done

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

by (subst mult_commute) (simp add: power_mult)

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,recpower}) ^ (2*n)"

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

lemma zero_le_even_power'[simp]:

     "0 \<le> (a::'a::{ordered_idom,recpower}) ^ (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_mult_iff)

qed

lemma odd_power_less_zero:

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

proof (induct "n")

  case 0

  then 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_neg)

qed

lemma odd_0_le_power_imp_0_le:

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

apply (insert odd_power_less_zero [of a n]) 

apply (force simp add: linorder_not_less [symmetric]) 

done

text{*Simprules for comparisons where common factors can be cancelled.*}

lemmas zero_compare_simps =

    add_strict_increasing add_strict_increasing2 add_increasing

    zero_le_mult_iff zero_le_divide_iff 

    zero_less_mult_iff zero_less_divide_iff 

    mult_le_0_iff divide_le_0_iff 

    mult_less_0_iff divide_less_0_iff 

    zero_le_power2 power2_less_0

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

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 (uminus v) :: int)"

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

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  @{term Suc} *}

lemma eq_number_of_Suc [simp]:

     "(number_of v = Suc n) =  

        (let pv = number_of (Int.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 (Int.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 (Int.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 (Int.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 (Int.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 (Int.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])

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))"

apply (simp only: number_of_BIT lemma1 lemma2 eq_commute

               OrderedGroup.add_left_cancel add_assoc OrderedGroup.add_0_left

            split add: bit.split)

apply simp

done

lemma eq_number_of_BIT_Pls:

     "((number_of (v BIT x) ::int) = Numeral0) =  

      (x=bit.B0 & (((number_of v) ::int) = Numeral0))"

apply (simp only: simp_thms  add: number_of_BIT number_of_Pls eq_commute

            split add: bit.split 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 Int.Min) =  

      (x=bit.B1 & (((number_of v) ::int) = number_of Int.Min))"

apply (simp only: simp_thms  add: number_of_BIT number_of_Min eq_commute

            split add: bit.split 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: "(Numeral0 ::int) ~= number_of Int.Min"

by auto

subsection{*Max and Min Combined with @{term Suc} *}

lemma max_number_of_Suc [simp]:

     "max (Suc n) (number_of v) =  

        (let pv = number_of (Int.pred v) in  

         if neg pv then Suc n else Suc(max n (nat pv)))"

apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 

            split add: split_if nat.split)

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

apply auto

done

lemma max_Suc_number_of [simp]:

     "max (number_of v) (Suc n) =  

        (let pv = number_of (Int.pred v) in  

         if neg pv then Suc n else Suc(max (nat pv) n))"

apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 

            split add: split_if nat.split)

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

apply auto

done

lemma min_number_of_Suc [simp]:

     "min (Suc n) (number_of v) =  

        (let pv = number_of (Int.pred v) in  

         if neg pv then 0 else Suc(min n (nat pv)))"

apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 

            split add: split_if nat.split)

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

apply auto

done

lemma min_Suc_number_of [simp]:

     "min (number_of v) (Suc n) =  

        (let pv = number_of (Int.pred v) in  

         if neg pv then 0 else Suc(min (nat pv) n))"

apply (simp only: Let_def neg_eq_less_0 number_of_pred nat_number_of_def 

            split add: split_if nat.split)

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

apply auto

done

subsection{*Literal arithmetic involving powers*}

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

apply (induct "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)

lemmas power_nat_number_of_number_of = power_nat_number_of [of _ "number_of w", standard]

declare power_nat_number_of_number_of [simp]

text{*For arbitrary rings*}

lemma power_number_of_even:

  fixes z :: "'a::{number_ring,recpower}"

  shows "z ^ number_of (w BIT bit.B0) = (let w = z ^ (number_of w) in w * w)"

unfolding Let_def nat_number_of_def number_of_BIT bit.cases

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 power_number_of_odd:

  fixes z :: "'a::{number_ring,recpower}"

  shows "z ^ number_of (w BIT bit.B1) = (if (0::int) <= number_of w

     then (let w = z ^ (number_of w) in z * w * w) else 1)"

unfolding Let_def nat_number_of_def number_of_BIT bit.cases

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

apply (simp only: nat_add_distrib nat_mult_distrib)

apply simp

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

done

lemmas zpower_number_of_even = power_number_of_even [where 'a=int]

lemmas zpower_number_of_odd = power_number_of_odd [where 'a=int]

lemmas power_number_of_even_number_of [simp] =

    power_number_of_even [of "number_of v", standard]

lemmas power_number_of_odd_number_of [simp] =

    power_number_of_odd [of "number_of v", standard]

ML

{*

val numeral_ss = simpset() addsimps @{thms numerals};

val nat_bin_arith_setup =

 LinArith.map_data

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

     {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms,

      inj_thms = inj_thms,

      lessD = lessD, neqE = neqE,

      simpset = simpset addsimps [Suc_nat_number_of, int_nat_number_of,

        @{thm not_neg_number_of_Pls}, @{thm neg_number_of_Min},

        @{thm neg_number_of_BIT}]})

*}

declaration {* K 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: "Numeral0 = (0::nat)"

  by (simp add: number_of_Pls nat_number_of_def)

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

  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)

  done

lemma nat_number_of_BIT_1:

  "number_of (w BIT bit.B1) =

    (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 zadd_assoc split: bit.split)

  apply simp

  done

lemma nat_number_of_BIT_0:

    "number_of (w BIT bit.B0) = (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 zadd_assoc)

  apply simp

  done

lemmas nat_number =

  nat_number_of_Pls nat_number_of_Min

  nat_number_of_BIT_1 nat_number_of_BIT_0

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,recpower})"

by (simp add: power_mult power_Suc); 

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

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)

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)"

by (simp add: int_number_of_def nat_number_of_def number_of_eq of_nat_nat);

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 (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 (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

lemma nat_mult_dvd_cancel_disj[simp]:

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

by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])

lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd 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[simp]:

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

by (simp add: nat_mult_div_cancel1)

end