src/HOL/Integ/Numeral.thy

import -> imports

(*  Title:	HOL/Integ/Numeral.thy
    ID:         $Id$
    Author:	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright	1994  University of Cambridge
*)

header{*Arithmetic on Binary Integers*}

theory Numeral
imports IntDef
files "Tools/numeral_syntax.ML"
begin

text{* The file @{text numeral_syntax.ML} hides the constructors Pls and Min.
   Only qualified access Numeral.Pls and Numeral.Min is allowed.
   We do not hide Bit because we need the BIT infix syntax.*}

text{*This formalization defines binary arithmetic in terms of the integers
rather than using a datatype. This avoids multiple representations (leading
zeroes, etc.)  See @{text "ZF/Integ/twos-compl.ML"}, function @{text
int_of_binary}, for the numerical interpretation.

The representation expects that @{text "(m mod 2)"} is 0 or 1,
even if m is negative;
For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
@{text "-5 = (-3)*2 + 1"}.
*}


typedef (Bin)
  bin = "UNIV::int set"
    by (auto)

constdefs
  Pls :: "bin"
   "Pls == Abs_Bin 0"

  Min :: "bin"
   "Min == Abs_Bin (- 1)"

  Bit :: "[bin,bool] => bin"    (infixl "BIT" 90)
   --{*That is, 2w+b*}
   "w BIT b == Abs_Bin ((if b then 1 else 0) + Rep_Bin w + Rep_Bin w)"


axclass
  number < type  -- {* for numeric types: nat, int, real, \dots *}

consts
  number_of :: "bin => 'a::number"

syntax
  "_Numeral" :: "num_const => 'a"    ("_")
  Numeral0 :: 'a
  Numeral1 :: 'a

translations
  "Numeral0" == "number_of Numeral.Pls"
  "Numeral1" == "number_of (Numeral.Pls BIT True)"


setup NumeralSyntax.setup

syntax (xsymbols)
  "_square" :: "'a => 'a"  ("(_\<twosuperior>)" [1000] 999)
syntax (HTML output)
  "_square" :: "'a => 'a"  ("(_\<twosuperior>)" [1000] 999)
syntax (output)
  "_square" :: "'a => 'a"  ("(_ ^/ 2)" [81] 80)
translations
  "x\<twosuperior>" == "x^2"
  "x\<twosuperior>" <= "x^(2::nat)"


lemma Let_number_of [simp]: "Let (number_of v) f == f (number_of v)"
  -- {* Unfold all @{text let}s involving constants *}
  by (simp add: Let_def)

lemma Let_0 [simp]: "Let 0 f == f 0"
  by (simp add: Let_def)

lemma Let_1 [simp]: "Let 1 f == f 1"
  by (simp add: Let_def)


constdefs
  bin_succ  :: "bin=>bin"
   "bin_succ w == Abs_Bin(Rep_Bin w + 1)"

  bin_pred  :: "bin=>bin"
   "bin_pred w == Abs_Bin(Rep_Bin w - 1)"

  bin_minus  :: "bin=>bin"
   "bin_minus w == Abs_Bin(- (Rep_Bin w))"

  bin_add  :: "[bin,bin]=>bin"
   "bin_add v w == Abs_Bin(Rep_Bin v + Rep_Bin w)"

  bin_mult  :: "[bin,bin]=>bin"
   "bin_mult v w == Abs_Bin(Rep_Bin v * Rep_Bin w)"


lemmas Bin_simps = 
       bin_succ_def bin_pred_def bin_minus_def bin_add_def bin_mult_def
       Pls_def Min_def Bit_def Abs_Bin_inverse Rep_Bin_inverse Bin_def

text{*Removal of leading zeroes*}
lemma Pls_0_eq [simp]: "Numeral.Pls BIT False = Numeral.Pls"
by (simp add: Bin_simps)

lemma Min_1_eq [simp]: "Numeral.Min BIT True = Numeral.Min"
by (simp add: Bin_simps)

subsection{*The Functions @{term bin_succ},  @{term bin_pred} and @{term bin_minus}*}

lemma bin_succ_Pls [simp]: "bin_succ Numeral.Pls = Numeral.Pls BIT True"
by (simp add: Bin_simps) 

lemma bin_succ_Min [simp]: "bin_succ Numeral.Min = Numeral.Pls"
by (simp add: Bin_simps) 

lemma bin_succ_1 [simp]: "bin_succ(w BIT True) = (bin_succ w) BIT False"
by (simp add: Bin_simps add_ac) 

lemma bin_succ_0 [simp]: "bin_succ(w BIT False) = w BIT True"
by (simp add: Bin_simps add_ac) 

lemma bin_pred_Pls [simp]: "bin_pred Numeral.Pls = Numeral.Min"
by (simp add: Bin_simps) 

lemma bin_pred_Min [simp]: "bin_pred Numeral.Min = Numeral.Min BIT False"
by (simp add: Bin_simps diff_minus) 

lemma bin_pred_1 [simp]: "bin_pred(w BIT True) = w BIT False"
by (simp add: Bin_simps) 

lemma bin_pred_0 [simp]: "bin_pred(w BIT False) = (bin_pred w) BIT True"
by (simp add: Bin_simps diff_minus add_ac) 

lemma bin_minus_Pls [simp]: "bin_minus Numeral.Pls = Numeral.Pls"
by (simp add: Bin_simps) 

lemma bin_minus_Min [simp]: "bin_minus Numeral.Min = Numeral.Pls BIT True"
by (simp add: Bin_simps) 

lemma bin_minus_1 [simp]:
     "bin_minus (w BIT True) = bin_pred (bin_minus w) BIT True"
by (simp add: Bin_simps add_ac diff_minus) 

 lemma bin_minus_0 [simp]: "bin_minus(w BIT False) = (bin_minus w) BIT False"
by (simp add: Bin_simps) 


subsection{*Binary Addition and Multiplication:
         @{term bin_add} and @{term bin_mult}*}

lemma bin_add_Pls [simp]: "bin_add Numeral.Pls w = w"
by (simp add: Bin_simps) 

lemma bin_add_Min [simp]: "bin_add Numeral.Min w = bin_pred w"
by (simp add: Bin_simps diff_minus add_ac) 

lemma bin_add_BIT_11 [simp]:
     "bin_add (v BIT True) (w BIT True) = bin_add v (bin_succ w) BIT False"
by (simp add: Bin_simps add_ac)

lemma bin_add_BIT_10 [simp]:
     "bin_add (v BIT True) (w BIT False) = (bin_add v w) BIT True"
by (simp add: Bin_simps add_ac)

lemma bin_add_BIT_0 [simp]:
     "bin_add (v BIT False) (w BIT y) = bin_add v w BIT y"
by (simp add: Bin_simps add_ac)

lemma bin_add_Pls_right [simp]: "bin_add w Numeral.Pls = w"
by (simp add: Bin_simps) 

lemma bin_add_Min_right [simp]: "bin_add w Numeral.Min = bin_pred w"
by (simp add: Bin_simps diff_minus) 

lemma bin_mult_Pls [simp]: "bin_mult Numeral.Pls w = Numeral.Pls"
by (simp add: Bin_simps) 

lemma bin_mult_Min [simp]: "bin_mult Numeral.Min w = bin_minus w"
by (simp add: Bin_simps) 

lemma bin_mult_1 [simp]:
     "bin_mult (v BIT True) w = bin_add ((bin_mult v w) BIT False) w"
by (simp add: Bin_simps add_ac left_distrib)

lemma bin_mult_0 [simp]: "bin_mult (v BIT False) w = (bin_mult v w) BIT False"
by (simp add: Bin_simps left_distrib)



subsection{*Converting Numerals to Rings: @{term number_of}*}

axclass number_ring \<subseteq> number, comm_ring_1
  number_of_eq: "number_of w = of_int (Rep_Bin w)"

lemma number_of_succ:
     "number_of(bin_succ w) = (1 + number_of w ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)

lemma number_of_pred:
     "number_of(bin_pred w) = (- 1 + number_of w ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps)

lemma number_of_minus:
     "number_of(bin_minus w) = (- (number_of w)::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

lemma number_of_add:
     "number_of(bin_add v w) = (number_of v + number_of w::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

lemma number_of_mult:
     "number_of(bin_mult v w) = (number_of v * number_of w::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

text{*The correctness of shifting.  But it doesn't seem to give a measurable
  speed-up.*}
lemma double_number_of_BIT:
     "(1+1) * number_of w = (number_of (w BIT False) ::'a::number_ring)"
by (simp add: number_of_eq Bin_simps left_distrib) 

text{*Converting numerals 0 and 1 to their abstract versions*}
lemma numeral_0_eq_0 [simp]: "Numeral0 = (0::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

lemma numeral_1_eq_1 [simp]: "Numeral1 = (1::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

text{*Special-case simplification for small constants*}

text{*Unary minus for the abstract constant 1. Cannot be inserted
  as a simprule until later: it is @{text number_of_Min} re-oriented!*}
lemma numeral_m1_eq_minus_1: "(-1::'a::number_ring) = - 1"
by (simp add: number_of_eq Bin_simps) 


lemma mult_minus1 [simp]: "-1 * z = -(z::'a::number_ring)"
by (simp add: numeral_m1_eq_minus_1)

lemma mult_minus1_right [simp]: "z * -1 = -(z::'a::number_ring)"
by (simp add: numeral_m1_eq_minus_1)

(*Negation of a coefficient*)
lemma minus_number_of_mult [simp]:
     "- (number_of w) * z = number_of(bin_minus w) * (z::'a::number_ring)"
by (simp add: number_of_minus)

text{*Subtraction*}
lemma diff_number_of_eq:
     "number_of v - number_of w =
      (number_of(bin_add v (bin_minus w))::'a::number_ring)"
by (simp add: diff_minus number_of_add number_of_minus)


lemma number_of_Pls: "number_of Numeral.Pls = (0::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

lemma number_of_Min: "number_of Numeral.Min = (- 1::'a::number_ring)"
by (simp add: number_of_eq Bin_simps) 

lemma number_of_BIT:
     "number_of(w BIT x) = (if x then 1 else (0::'a::number_ring)) +
	                   (number_of w) + (number_of w)"
by (simp add: number_of_eq Bin_simps) 



subsection{*Equality of Binary Numbers*}

text{*First version by Norbert Voelker*}

lemma eq_number_of_eq:
  "((number_of x::'a::number_ring) = number_of y) =
   iszero (number_of (bin_add x (bin_minus y)) :: 'a)"
by (simp add: iszero_def compare_rls number_of_add number_of_minus)

lemma iszero_number_of_Pls: "iszero ((number_of Numeral.Pls)::'a::number_ring)"
by (simp add: iszero_def numeral_0_eq_0)

lemma nonzero_number_of_Min: "~ iszero ((number_of Numeral.Min)::'a::number_ring)"
by (simp add: iszero_def numeral_m1_eq_minus_1 eq_commute)


subsection{*Comparisons, for Ordered Rings*}

lemma double_eq_0_iff: "(a + a = 0) = (a = (0::'a::ordered_idom))"
proof -
  have "a + a = (1+1)*a" by (simp add: left_distrib)
  with zero_less_two [where 'a = 'a]
  show ?thesis by force
qed

lemma le_imp_0_less: 
  assumes le: "0 \<le> z" shows "(0::int) < 1 + z"
proof -
  have "0 \<le> z" .
  also have "... < z + 1" by (rule less_add_one) 
  also have "... = 1 + z" by (simp add: add_ac)
  finally show "0 < 1 + z" .
qed

lemma odd_nonzero: "1 + z + z \<noteq> (0::int)";
proof (cases z rule: int_cases)
  case (nonneg n)
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
  thus ?thesis using  le_imp_0_less [OF le]
    by (auto simp add: add_assoc) 
next
  case (neg n)
  show ?thesis
  proof
    assume eq: "1 + z + z = 0"
    have "0 < 1 + (int n + int n)"
      by (simp add: le_imp_0_less add_increasing) 
    also have "... = - (1 + z + z)" 
      by (simp add: neg add_assoc [symmetric]) 
    also have "... = 0" by (simp add: eq) 
    finally have "0<0" ..
    thus False by blast
  qed
qed


text{*The premise involving @{term Ints} prevents @{term "a = 1/2"}.*}
lemma Ints_odd_nonzero: "a \<in> Ints ==> 1 + a + a \<noteq> (0::'a::ordered_idom)"
proof (unfold Ints_def) 
  assume "a \<in> range of_int"
  then obtain z where a: "a = of_int z" ..
  show ?thesis
  proof
    assume eq: "1 + a + a = 0"
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
    with odd_nonzero show False by blast
  qed
qed 

lemma Ints_number_of: "(number_of w :: 'a::number_ring) \<in> Ints"
by (simp add: number_of_eq Ints_def) 


lemma iszero_number_of_BIT:
     "iszero (number_of (w BIT x)::'a) = 
      (~x & iszero (number_of w::'a::{ordered_idom,number_ring}))"
by (simp add: iszero_def number_of_eq Bin_simps double_eq_0_iff 
              Ints_odd_nonzero Ints_def)

lemma iszero_number_of_0:
     "iszero (number_of (w BIT False) :: 'a::{ordered_idom,number_ring}) = 
      iszero (number_of w :: 'a)"
by (simp only: iszero_number_of_BIT simp_thms)

lemma iszero_number_of_1:
     "~ iszero (number_of (w BIT True)::'a::{ordered_idom,number_ring})"
by (simp only: iszero_number_of_BIT simp_thms)



subsection{*The Less-Than Relation*}

lemma less_number_of_eq_neg:
    "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
     = neg (number_of (bin_add x (bin_minus y)) :: 'a)"
apply (subst less_iff_diff_less_0) 
apply (simp add: neg_def diff_minus number_of_add number_of_minus)
done

text{*If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
  @{term Numeral0} IS @{term "number_of Numeral.Pls"} *}
lemma not_neg_number_of_Pls:
     "~ neg (number_of Numeral.Pls ::'a::{ordered_idom,number_ring})"
by (simp add: neg_def numeral_0_eq_0)

lemma neg_number_of_Min:
     "neg (number_of Numeral.Min ::'a::{ordered_idom,number_ring})"
by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)

lemma double_less_0_iff: "(a + a < 0) = (a < (0::'a::ordered_idom))"
proof -
  have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
  also have "... = (a < 0)"
    by (simp add: mult_less_0_iff zero_less_two 
                  order_less_not_sym [OF zero_less_two]) 
  finally show ?thesis .
qed

lemma odd_less_0: "(1 + z + z < 0) = (z < (0::int))";
proof (cases z rule: int_cases)
  case (nonneg n)
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
                             le_imp_0_less [THEN order_less_imp_le])  
next
  case (neg n)
  thus ?thesis by (simp del: int_Suc
			add: int_Suc0_eq_1 [symmetric] zadd_int compare_rls)
qed

text{*The premise involving @{term Ints} prevents @{term "a = 1/2"}.*}
lemma Ints_odd_less_0: 
     "a \<in> Ints ==> (1 + a + a < 0) = (a < (0::'a::ordered_idom))";
proof (unfold Ints_def) 
  assume "a \<in> range of_int"
  then obtain z where a: "a = of_int z" ..
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
    by (simp add: a)
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
  also have "... = (a < 0)" by (simp add: a)
  finally show ?thesis .
qed

lemma neg_number_of_BIT:
     "neg (number_of (w BIT x)::'a) = 
      neg (number_of w :: 'a::{ordered_idom,number_ring})"
by (simp add: neg_def number_of_eq Bin_simps double_less_0_iff
              Ints_odd_less_0 Ints_def)


text{*Less-Than or Equals*}

text{*Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals*}
lemmas le_number_of_eq_not_less =
       linorder_not_less [of "number_of w" "number_of v", symmetric, 
                          standard]

lemma le_number_of_eq:
    "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
     = (~ (neg (number_of (bin_add y (bin_minus x)) :: 'a)))"
by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)


text{*Absolute value (@{term abs})*}

lemma abs_number_of:
     "abs(number_of x::'a::{ordered_idom,number_ring}) =
      (if number_of x < (0::'a) then -number_of x else number_of x)"
by (simp add: abs_if)


text{*Re-orientation of the equation nnn=x*}
lemma number_of_reorient: "(number_of w = x) = (x = number_of w)"
by auto




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

lemmas bin_arith_extra_simps = 
       number_of_add [symmetric]
       number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
       number_of_mult [symmetric]
       diff_number_of_eq abs_number_of 

text{*For making a minimal simpset, one must include these default simprules.
  Also include @{text simp_thms} or at least @{term "(~False)=True"} *}
lemmas bin_arith_simps = 
       Pls_0_eq Min_1_eq
       bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0
       bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
       bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10 bin_add_BIT_11
       bin_minus_Pls bin_minus_Min bin_minus_1 bin_minus_0
       bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0 
       bin_add_Pls_right bin_add_Min_right
       abs_zero abs_one bin_arith_extra_simps

text{*Simplification of relational operations*}
lemmas bin_rel_simps = 
       eq_number_of_eq iszero_number_of_Pls nonzero_number_of_Min
       iszero_number_of_0 iszero_number_of_1
       less_number_of_eq_neg
       not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
       neg_number_of_Min neg_number_of_BIT
       le_number_of_eq

declare bin_arith_extra_simps [simp]
declare bin_rel_simps [simp]


subsection{*Simplification of arithmetic when nested to the right*}

lemma add_number_of_left [simp]:
     "number_of v + (number_of w + z) =
      (number_of(bin_add v w) + z::'a::number_ring)"
by (simp add: add_assoc [symmetric])

lemma mult_number_of_left [simp]:
    "number_of v * (number_of w * z) =
     (number_of(bin_mult v w) * z::'a::number_ring)"
by (simp add: mult_assoc [symmetric])

lemma add_number_of_diff1:
    "number_of v + (number_of w - c) = 
     number_of(bin_add v w) - (c::'a::number_ring)"
by (simp add: diff_minus add_number_of_left)

lemma add_number_of_diff2 [simp]: "number_of v + (c - number_of w) =
     number_of (bin_add v (bin_minus w)) + (c::'a::number_ring)"
apply (subst diff_number_of_eq [symmetric])
apply (simp only: compare_rls)
done

end