theory IntEx2

imports "Quotient_Int"

begin

subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}

(*

context ring_1

begin

definition 

  of_int :: "int \<Rightarrow> 'a" 

where

  "of_int 

*)

subsection {* Binary representation *}

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/Tools/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"}.

  This two's complement binary representation derives from the paper 

  "An Efficient Representation of Arithmetic for Term Rewriting" by

  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,

  Springer LNCS 488 (240-251), 1991.

*}

subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}

definition

  Pls :: int where

  [code del]: "Pls = 0"

definition

  Min :: int where

  [code del]: "Min = - 1"

definition

  Bit0 :: "int \<Rightarrow> int" where

  [code del]: "Bit0 k = k + k"

definition

  Bit1 :: "int \<Rightarrow> int" where

  [code del]: "Bit1 k = 1 + k + k"

class number = -- {* for numeric types: nat, int, real, \dots *}

  fixes number_of :: "int \<Rightarrow> 'a"

(*use "~~/src/HOL/Tools/numeral.ML"

syntax

  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")

use "~~/src/HOL/Tools/numeral_syntax.ML"

setup NumeralSyntax.setup

abbreviation

  "Numeral0 \<equiv> number_of Pls"

abbreviation

  "Numeral1 \<equiv> number_of (Bit1 Pls)"

lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"

  -- {* Unfold all @{text let}s involving constants *}

  unfolding Let_def ..

definition

  succ :: "int \<Rightarrow> int" where

  [code del]: "succ k = k + 1"

definition

  pred :: "int \<Rightarrow> int" where

  [code del]: "pred k = k - 1"

lemmas

  max_number_of [simp] = max_def

    [of "number_of u" "number_of v", standard, simp]

and

  min_number_of [simp] = min_def 

    [of "number_of u" "number_of v", standard, simp]

  -- {* unfolding @{text minx} and @{text max} on numerals *}

lemmas numeral_simps = 

  succ_def pred_def Pls_def Min_def Bit0_def Bit1_def

text {* Removal of leading zeroes *}

lemma Bit0_Pls [simp, code_post]:

  "Bit0 Pls = Pls"

  unfolding numeral_simps by simp

lemma Bit1_Min [simp, code_post]:

  "Bit1 Min = Min"

  unfolding numeral_simps by simp

lemmas normalize_bin_simps =

  Bit0_Pls Bit1_Min

*)

end