src/HOL/Nat_Numeral.thy

hiding constants and facts in the Quickcheck_Exhaustive and Quickcheck_Narrowing theory;

(*  Title:      HOL/Nat_Numeral.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1999  University of Cambridge
*)

header {* Binary numerals for the natural numbers *}

theory Nat_Numeral
imports Int
begin

subsection {* Numerals for natural numbers *}

text {*
  Arithmetic for naturals is reduced to that for the non-negative integers.
*}

instantiation nat :: number_semiring
begin

definition
  nat_number_of_def [code_unfold, code del]: "number_of v = nat (number_of v)"

instance proof
  fix n show "number_of (int n) = (of_nat n :: nat)"
    unfolding nat_number_of_def number_of_eq by simp
qed
 
end

lemma [code_post]:
  "nat (number_of v) = number_of v"
  unfolding nat_number_of_def ..


subsection {* Special case: squares and cubes *}

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

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

context power
begin

abbreviation (xsymbols)
  power2 :: "'a \<Rightarrow> 'a"  ("(_\<twosuperior>)" [1000] 999) where
  "x\<twosuperior> \<equiv> x ^ 2"

notation (latex output)
  power2  ("(_\<twosuperior>)" [1000] 999)

notation (HTML output)
  power2  ("(_\<twosuperior>)" [1000] 999)

end

context monoid_mult
begin

lemma power2_eq_square: "a\<twosuperior> = a * a"
  by (simp add: numeral_2_eq_2)

lemma power3_eq_cube: "a ^ 3 = a * a * a"
  by (simp add: numeral_3_eq_3 mult_assoc)

lemma power_even_eq:
  "a ^ (2*n) = (a ^ n) ^ 2"
  by (subst mult_commute) (simp add: power_mult)

lemma power_odd_eq:
  "a ^ Suc (2*n) = a * (a ^ n) ^ 2"
  by (simp add: power_even_eq)

end

context semiring_1
begin

lemma zero_power2 [simp]: "0\<twosuperior> = 0"
  by (simp add: power2_eq_square)

lemma one_power2 [simp]: "1\<twosuperior> = 1"
  by (simp add: power2_eq_square)

end

context ring_1
begin

lemma power2_minus [simp]:
  "(- a)\<twosuperior> = a\<twosuperior>"
  by (simp add: power2_eq_square)

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

lemma power_minus1_even [simp]:
  "(- 1) ^ (2*n) = 1"
proof (induct n)
  case 0 show ?case by simp
next
  case (Suc n) then show ?case by (simp add: power_add)
qed

lemma power_minus1_odd:
  "(- 1) ^ Suc (2*n) = - 1"
  by simp

lemma power_minus_even [simp]:
  "(-a) ^ (2*n) = a ^ (2*n)"
  by (simp add: power_minus [of a]) 

end

context ring_1_no_zero_divisors
begin

lemma zero_eq_power2 [simp]:
  "a\<twosuperior> = 0 \<longleftrightarrow> a = 0"
  unfolding power2_eq_square by simp

lemma power2_eq_1_iff:
  "a\<twosuperior> = 1 \<longleftrightarrow> a = 1 \<or> a = - 1"
  unfolding power2_eq_square by (rule square_eq_1_iff)

end

context idom
begin

lemma power2_eq_iff: "x\<twosuperior> = y\<twosuperior> \<longleftrightarrow> x = y \<or> x = - y"
  unfolding power2_eq_square by (rule square_eq_iff)

end

context linordered_ring
begin

lemma sum_squares_ge_zero:
  "0 \<le> x * x + y * y"
  by (intro add_nonneg_nonneg zero_le_square)

lemma not_sum_squares_lt_zero:
  "\<not> x * x + y * y < 0"
  by (simp add: not_less sum_squares_ge_zero)

end

context linordered_ring_strict
begin

lemma sum_squares_eq_zero_iff:
  "x * x + y * y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: add_nonneg_eq_0_iff)

lemma sum_squares_le_zero_iff:
  "x * x + y * y \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  by (simp add: le_less not_sum_squares_lt_zero sum_squares_eq_zero_iff)

lemma sum_squares_gt_zero_iff:
  "0 < x * x + y * y \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
  by (simp add: not_le [symmetric] sum_squares_le_zero_iff)

end

context linordered_semidom
begin

lemma power2_le_imp_le:
  "x\<twosuperior> \<le> y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y"
  unfolding numeral_2_eq_2 by (rule power_le_imp_le_base)

lemma power2_less_imp_less:
  "x\<twosuperior> < y\<twosuperior> \<Longrightarrow> 0 \<le> y \<Longrightarrow> x < y"
  by (rule power_less_imp_less_base)

lemma power2_eq_imp_eq:
  "x\<twosuperior> = y\<twosuperior> \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x = y"
  unfolding numeral_2_eq_2 by (erule (2) power_eq_imp_eq_base) simp

end

context linordered_idom
begin

lemma zero_le_power2 [simp]:
  "0 \<le> a\<twosuperior>"
  by (simp add: power2_eq_square)

lemma zero_less_power2 [simp]:
  "0 < a\<twosuperior> \<longleftrightarrow> a \<noteq> 0"
  by (force simp add: power2_eq_square zero_less_mult_iff linorder_neq_iff)

lemma power2_less_0 [simp]:
  "\<not> a\<twosuperior> < 0"
  by (force simp add: power2_eq_square mult_less_0_iff) 

lemma abs_power2 [simp]:
  "abs (a\<twosuperior>) = a\<twosuperior>"
  by (simp add: power2_eq_square abs_mult abs_mult_self)

lemma power2_abs [simp]:
  "(abs a)\<twosuperior> = a\<twosuperior>"
  by (simp add: power2_eq_square abs_mult_self)

lemma odd_power_less_zero:
  "a < 0 \<Longrightarrow> a ^ Suc (2*n) < 0"
proof (induct n)
  case 0
  then show ?case by simp
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)
  thus ?case
    by (simp del: power_Suc add: Suc mult_less_0_iff mult_neg_neg)
qed

lemma odd_0_le_power_imp_0_le:
  "0 \<le> a ^ Suc (2*n) \<Longrightarrow> 0 \<le> a"
  using odd_power_less_zero [of a n]
    by (force simp add: linorder_not_less [symmetric]) 

lemma zero_le_even_power'[simp]:
  "0 \<le> a ^ (2*n)"
proof (induct n)
  case 0
    show ?case by simp
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: Suc zero_le_mult_iff)
qed

lemma sum_power2_ge_zero:
  "0 \<le> x\<twosuperior> + y\<twosuperior>"
  unfolding power2_eq_square by (rule sum_squares_ge_zero)

lemma not_sum_power2_lt_zero:
  "\<not> x\<twosuperior> + y\<twosuperior> < 0"
  unfolding power2_eq_square by (rule not_sum_squares_lt_zero)

lemma sum_power2_eq_zero_iff:
  "x\<twosuperior> + y\<twosuperior> = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  unfolding power2_eq_square by (rule sum_squares_eq_zero_iff)

lemma sum_power2_le_zero_iff:
  "x\<twosuperior> + y\<twosuperior> \<le> 0 \<longleftrightarrow> x = 0 \<and> y = 0"
  unfolding power2_eq_square by (rule sum_squares_le_zero_iff)

lemma sum_power2_gt_zero_iff:
  "0 < x\<twosuperior> + y\<twosuperior> \<longleftrightarrow> x \<noteq> 0 \<or> y \<noteq> 0"
  unfolding power2_eq_square by (rule sum_squares_gt_zero_iff)

end

lemma power2_sum:
  fixes x y :: "'a::number_semiring"
  shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"
  by (simp add: algebra_simps power2_eq_square semiring_mult_2_right)

lemma power2_diff:
  fixes x y :: "'a::number_ring"
  shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"
  by (simp add: ring_distribs power2_eq_square mult_2) (rule mult_commute)


subsection {* Predicate for negative binary numbers *}

definition neg  :: "int \<Rightarrow> bool" where
  "neg Z \<longleftrightarrow> Z < 0"

lemma not_neg_int [simp]: "~ neg (of_nat n)"
by (simp add: neg_def)

lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
by (simp add: neg_def del: of_nat_Suc)

lemmas neg_eq_less_0 = neg_def

lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
by (simp add: neg_def linorder_not_less)

text{*To simplify inequalities when Numeral1 can get simplified to 1*}

lemma not_neg_0: "~ neg 0"
by (simp add: One_int_def neg_def)

lemma not_neg_1: "~ neg 1"
by (simp add: neg_def linorder_not_less)

lemma neg_nat: "neg z ==> nat z = 0"
by (simp add: neg_def order_less_imp_le) 

lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
by (simp add: linorder_not_less neg_def)

text {*
  If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
  @{term Numeral0} IS @{term "number_of Pls"}
*}

lemma not_neg_number_of_Pls: "~ neg (number_of Int.Pls)"
  by (simp add: neg_def)

lemma neg_number_of_Min: "neg (number_of Int.Min)"
  by (simp add: neg_def)

lemma neg_number_of_Bit0:
  "neg (number_of (Int.Bit0 w)) = neg (number_of w)"
  by (simp add: neg_def)

lemma neg_number_of_Bit1:
  "neg (number_of (Int.Bit1 w)) = neg (number_of w)"
  by (simp add: neg_def)

lemmas neg_simps [simp] =
  not_neg_0 not_neg_1
  not_neg_number_of_Pls neg_number_of_Min
  neg_number_of_Bit0 neg_number_of_Bit1


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

declare 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, code_post]: "Numeral0 = (0::nat)"
  by (rule semiring_numeral_0_eq_0)

lemma nat_numeral_1_eq_1 [simp]: "Numeral1 = (1::nat)"
  by (rule semiring_numeral_1_eq_1)

lemma Numeral1_eq1_nat:
  "(1::nat) = Numeral1"
  by simp

lemma numeral_1_eq_Suc_0 [code_post]: "Numeral1 = Suc 0"
by (simp only: nat_numeral_1_eq_1 One_nat_def)


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

lemma int_nat_number_of [simp]:
     "int (number_of v) =  
         (if neg (number_of v :: int) then 0  
          else (number_of v :: int))"
  unfolding nat_number_of_def number_of_is_id neg_def
  by simp (* FIXME: redundant with of_nat_number_of_eq *)

lemma nonneg_int_cases:
  fixes k :: int assumes "0 \<le> k" obtains n where "k = of_nat n"
  using assms by (cases k, simp, simp)

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)
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)"
  unfolding nat_number_of_def number_of_is_id neg_def numeral_simps
  by (simp add: Suc_nat_eq_nat_zadd1 add_ac)

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 *}

lemma add_nat_number_of [simp]:
     "(number_of v :: nat) + number_of v' =  
         (if v < Int.Pls then number_of v'  
          else if v' < Int.Pls then number_of v  
          else number_of (v + v'))"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by (simp add: nat_add_distrib)

lemma nat_number_of_add_1 [simp]:
  "number_of v + (1::nat) =
    (if v < Int.Pls then 1 else number_of (Int.succ v))"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by (simp add: nat_add_distrib)

lemma nat_1_add_number_of [simp]:
  "(1::nat) + number_of v =
    (if v < Int.Pls then 1 else number_of (Int.succ v))"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by (simp add: nat_add_distrib)

lemma nat_1_add_1 [simp]: "1 + 1 = (2::nat)"
  by (rule semiring_one_add_one_is_two)

text {* TODO: replace simp rules above with these generic ones: *}

lemma semiring_add_number_of:
  "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
    (number_of v :: 'a::number_semiring) + number_of v' = number_of (v + v')"
  unfolding Int.Pls_def
  by (elim nonneg_int_cases,
    simp only: number_of_int of_nat_add [symmetric])

lemma semiring_number_of_add_1:
  "Int.Pls \<le> v \<Longrightarrow>
    number_of v + (1::'a::number_semiring) = number_of (Int.succ v)"
  unfolding Int.Pls_def Int.succ_def
  by (elim nonneg_int_cases,
    simp only: number_of_int add_commute [where b=1] of_nat_Suc [symmetric])

lemma semiring_1_add_number_of:
  "Int.Pls \<le> v \<Longrightarrow>
    (1::'a::number_semiring) + number_of v = number_of (Int.succ v)"
  unfolding Int.Pls_def Int.succ_def
  by (elim nonneg_int_cases,
    simp only: number_of_int add_commute [where b=1] of_nat_Suc [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)"
by (simp add: Let_def nat_diff_distrib [symmetric] neg_eq_less_0 not_neg_eq_ge_0)


lemma diff_nat_number_of [simp]: 
     "(number_of v :: nat) - number_of v' =  
        (if v' < Int.Pls then number_of v  
         else let d = number_of (v + uminus v') in     
              if neg d then 0 else nat d)"
  unfolding nat_number_of_def number_of_is_id numeral_simps neg_def
  by auto

lemma nat_number_of_diff_1 [simp]:
  "number_of v - (1::nat) =
    (if v \<le> Int.Pls then 0 else number_of (Int.pred v))"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by auto


subsubsection{*Multiplication *}

lemma mult_nat_number_of [simp]:
     "(number_of v :: nat) * number_of v' =  
       (if v < Int.Pls then 0 else number_of (v * v'))"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by (simp add: nat_mult_distrib)

(* TODO: replace mult_nat_number_of with this next rule *)
lemma semiring_mult_number_of:
  "\<lbrakk>Int.Pls \<le> v; Int.Pls \<le> v'\<rbrakk> \<Longrightarrow>
    (number_of v :: 'a::number_semiring) * number_of v' = number_of (v * v')"
  unfolding Int.Pls_def
  by (elim nonneg_int_cases,
    simp only: number_of_int of_nat_mult [symmetric])


subsection{*Comparisons*}

subsubsection{*Equals (=) *}

lemma eq_nat_number_of [simp]:
     "((number_of v :: nat) = number_of v') =  
      (if neg (number_of v :: int) then (number_of v' :: int) \<le> 0
       else if neg (number_of v' :: int) then (number_of v :: int) = 0
       else v = v')"
  unfolding nat_number_of_def number_of_is_id neg_def
  by auto


subsubsection{*Less-than (<) *}

lemma less_nat_number_of [simp]:
  "(number_of v :: nat) < number_of v' \<longleftrightarrow>
    (if v < v' then Int.Pls < v' else False)"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by auto


subsubsection{*Less-than-or-equal *}

lemma le_nat_number_of [simp]:
  "(number_of v :: nat) \<le> number_of v' \<longleftrightarrow>
    (if v \<le> v' then True else v \<le> Int.Pls)"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by auto

(*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{*Squares of literal numerals will be evaluated.*}
lemmas power2_eq_square_number_of [simp] =
  power2_eq_square [of "number_of w"] for w


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

(*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"] for v

subsubsection{*Arith *}

lemma Suc_eq_plus1: "Suc n = n + 1"
  unfolding One_nat_def by simp

lemma Suc_eq_plus1_left: "Suc n = 1 + n"
  unfolding One_nat_def by simp

(* 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))"
  unfolding One_nat_def by (cases m) simp_all

lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
  unfolding One_nat_def by (cases m) simp_all

lemma power_eq_if: "(p ^ m :: nat) = (if m=0 then 1 else p * (p ^ (m - 1)))"
  unfolding One_nat_def by (cases m) simp_all


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) \<longleftrightarrow> v \<le> Int.Pls"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by auto

lemma eq_0_number_of [simp]:
  "(0::nat) = number_of v \<longleftrightarrow> v \<le> Int.Pls"
by (rule trans [OF eq_sym_conv eq_number_of_0])

lemma less_0_number_of [simp]:
   "(0::nat) < number_of v \<longleftrightarrow> Int.Pls < v"
  unfolding nat_number_of_def number_of_is_id numeral_simps
  by simp

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])



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 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 linorder_not_less [symmetric])


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 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"] for w
declare power_nat_number_of_number_of [simp]



text{*For arbitrary rings*}

lemma power_number_of_even:
  fixes z :: "'a::monoid_mult"
  shows "z ^ number_of (Int.Bit0 w) = (let w = z ^ (number_of w) in w * w)"
by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
  nat_add_distrib power_add simp del: nat_number_of)

lemma power_number_of_odd:
  fixes z :: "'a::monoid_mult"
  shows "z ^ number_of (Int.Bit1 w) = (if (0::int) <= number_of w
     then (let w = z ^ (number_of w) in z * w * w) else 1)"
unfolding Let_def Bit1_def nat_number_of_def number_of_is_id
apply (cases "0 <= w")
apply (simp only: mult_assoc nat_add_distrib power_add, simp)
apply (simp add: not_le mult_2 [symmetric] add_assoc)
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"] for v

lemmas power_number_of_odd_number_of [simp] =
    power_number_of_odd [of "number_of v"] for v

lemma nat_number_of_Pls: "Numeral0 = (0::nat)"
  by (simp add: nat_number_of_def)

lemma nat_number_of_Min [no_atp]: "number_of Int.Min = (0::nat)"
  apply (simp only: number_of_Min nat_number_of_def nat_zminus_int)
  done

lemma nat_number_of_Bit0:
    "number_of (Int.Bit0 w) = (let n::nat = number_of w in n + n)"
by (cases "w \<ge> 0") (auto simp add: Let_def Bit0_def nat_number_of_def number_of_is_id
  nat_add_distrib simp del: nat_number_of)

lemma nat_number_of_Bit1:
  "number_of (Int.Bit1 w) =
    (if neg (number_of w :: int) then 0
     else let n = number_of w in Suc (n + n))"
unfolding Let_def Bit1_def nat_number_of_def number_of_is_id neg_def
apply (cases "w < 0")
apply (simp add: mult_2 [symmetric] add_assoc)
apply (simp only: nat_add_distrib, simp)
done

lemmas eval_nat_numeral =
  nat_number_of_Bit0 nat_number_of_Bit1

lemmas nat_arith =
  add_nat_number_of
  diff_nat_number_of
  mult_nat_number_of
  eq_nat_number_of
  less_nat_number_of

lemmas semiring_norm =
  Let_def arith_simps nat_arith rel_simps neg_simps if_False
  if_True add_0 add_Suc add_number_of_left mult_number_of_left
  numeral_1_eq_1 [symmetric] Suc_eq_plus1
  numeral_0_eq_0 [symmetric] numerals [symmetric]
  not_iszero_Numeral1

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

lemma power_m1_even: "(-1) ^ (2*n) = (1::'a::{number_ring})"
  by (simp only: number_of_Min power_minus1_even)

lemma power_m1_odd: "(-1) ^ Suc(2*n) = (-1::'a::{number_ring})"
  by (simp only: number_of_Min power_minus1_odd)

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 (auto simp add: neg_def)

lemma nat_number_of_mult_left:
     "number_of v * (number_of v' * (k::nat)) =  
         (if v < Int.Pls then 0
          else number_of (v * v') * k)"
by (auto simp add: not_less Pls_def nat_number_of_def number_of_is_id
  nat_mult_distrib simp del: nat_number_of)


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_semiring)
          else 0)"
  by (auto simp add: int_number_of_def nat_number_of_def number_of_int
    elim!: nonneg_int_cases)

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_semiring))"
  by (simp only: of_nat_number_of_lemma neg_def, simp)


subsubsection{*For simplifying @{term "Suc m - K"} and  @{term "K - Suc m"}*}

text{*Where K above is a literal*}

lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)"
by (simp split: nat_diff_split)

text {*Now just instantiating @{text n} to @{text "number_of v"} does
  the right simplification, but with some redundant inequality
  tests.*}
lemma neg_number_of_pred_iff_0:
  "neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))"
apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ")
apply (simp only: less_Suc_eq_le le_0_eq)
apply (subst less_number_of_Suc, simp)
done

text{*No longer required as a simprule because of the @{text inverse_fold}
   simproc*}
lemma Suc_diff_number_of:
     "Int.Pls < v ==>
      Suc m - (number_of v) = m - (number_of (Int.pred v))"
apply (subst Suc_diff_eq_diff_pred)
apply simp
apply (simp del: nat_numeral_1_eq_1)
apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
                        neg_number_of_pred_iff_0)
done

lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
by (simp split: nat_diff_split)


subsubsection{*For @{term nat_case} and @{term nat_rec}*}

lemma nat_case_number_of [simp]:
     "nat_case a f (number_of v) =
        (let pv = number_of (Int.pred v) in
         if neg pv then a else f (nat pv))"
by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)

lemma nat_case_add_eq_if [simp]:
     "nat_case a f ((number_of v) + n) =
       (let pv = number_of (Int.pred v) in
         if neg pv then nat_case a f n else f (nat pv + n))"
apply (subst add_eq_if)
apply (simp split add: nat.split
            del: nat_numeral_1_eq_1
            add: nat_numeral_1_eq_1 [symmetric]
                 numeral_1_eq_Suc_0 [symmetric]
                 neg_number_of_pred_iff_0)
done

lemma nat_rec_number_of [simp]:
     "nat_rec a f (number_of v) =
        (let pv = number_of (Int.pred v) in
         if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
apply (case_tac " (number_of v) ::nat")
apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
apply (simp split add: split_if_asm)
done

lemma nat_rec_add_eq_if [simp]:
     "nat_rec a f (number_of v + n) =
        (let pv = number_of (Int.pred v) in
         if neg pv then nat_rec a f n
                   else f (nat pv + n) (nat_rec a f (nat pv + n)))"
apply (subst add_eq_if)
apply (simp split add: nat.split
            del: nat_numeral_1_eq_1
            add: nat_numeral_1_eq_1 [symmetric]
                 numeral_1_eq_Suc_0 [symmetric]
                 neg_number_of_pred_iff_0)
done


subsubsection{*Various Other Lemmas*}

lemma card_UNIV_bool[simp]: "card (UNIV :: bool set) = 2"
by(simp add: UNIV_bool)

text {*Evens and Odds, for Mutilated Chess Board*}

text{*Lemmas for specialist use, NOT as default simprules*}
lemma nat_mult_2: "2 * z = (z+z::nat)"
by (rule semiring_mult_2)

lemma nat_mult_2_right: "z * 2 = (z+z::nat)"
by (rule semiring_mult_2_right)

text{*Case analysis on @{term "n<2"}*}
lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0"
by (auto simp add: nat_1_add_1 [symmetric])

text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}

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

end