theory IntEx2

imports "../QuotMain" Nat Presburger

(*uses

  ("Tools/numeral.ML")

  ("Tools/numeral_syntax.ML")

  ("Tools/int_arith.ML")*)

begin

fun

  intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infix "\<approx>" 50)

where

  "intrel (x, y) (u, v) = (x + v = u + y)"

quotient int = "nat \<times> nat" / intrel

  apply(unfold equivp_def)

  apply(auto simp add: mem_def expand_fun_eq)

  done

instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"

begin

quotient_def

  zero_int::"0 :: int"

where

  "(0::nat, 0::nat)"

quotient_def

  one_int::"1 :: int"

where

  "(1::nat, 0::nat)"

fun

  plus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "plus_raw (x, y) (u, v) = (x + u, y + v)"

quotient_def

  plus_int::"(op +) :: (int \<Rightarrow> int \<Rightarrow> int)"

where

  "plus_raw"

fun

  minus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "minus_raw (x, y) = (y, x)"

quotient_def

  uminus_int::"(uminus :: (int \<Rightarrow> int))"

where

  "minus_raw"

definition

  minus_int_def [code del]:  "z - w = z + (-w::int)"

fun

  times_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"

where

  "times_raw (x, y) (u, v) = (x*u + y*v, x*v + y*u)"

quotient_def

  times_int::"(op *) :: (int \<Rightarrow> int \<Rightarrow> int)"

where

  "times_raw"

fun

  less_eq_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"

where

  "less_eq_raw (x, y) (u, v) = (x+v \<le> u+y)"

quotient_def

  less_eq_int :: "(op \<le>) :: int \<Rightarrow> int \<Rightarrow> bool"

where

  "less_eq_raw"

definition

  less_int_def [code del]: "(z\<Colon>int) < w = (z \<le> w \<and> z \<noteq> w)"

definition

  abs_int_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"

definition

  sgn_int_def: "sgn (i\<Colon>int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"

instance ..

end

lemma plus_raw_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) plus_raw plus_raw"

by auto

lemma minus_raw_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx>) minus_raw minus_raw"

  by auto

lemma times_raw_fst:

  assumes a: "x \<approx> z"

  shows "times_raw x y \<approx> times_raw z y"

using a

apply(cases x, cases y, cases z)

apply(auto simp add: times_raw.simps intrel.simps)

apply(rename_tac u v w x y z)

apply(subgoal_tac "u*w + z*w = y*w + v*w  &  u*x + z*x = y*x + v*x")

apply(simp add: mult_ac)

apply(simp add: add_mult_distrib [symmetric])

done

lemma times_raw_snd:

  assumes a: "x \<approx> z"

  shows "times_raw y x \<approx> times_raw y z"

using a

apply(cases x, cases y, cases z)

apply(auto simp add: times_raw.simps intrel.simps)

apply(rename_tac u v w x y z)

apply(subgoal_tac "u*w + z*w = y*w + v*w  &  u*x + z*x = y*x + v*x")

apply(simp add: mult_ac)

apply(simp add: add_mult_distrib [symmetric])

done

lemma mult_raw_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) times_raw times_raw"

apply(simp only: fun_rel.simps)

apply(rule allI | rule impI)+

apply(rule equivp_transp[OF int_equivp])

apply(rule times_raw_fst)

apply(assumption)

apply(rule times_raw_snd)

apply(assumption)

done

lemma less_eq_raw_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx> ===> op =) less_eq_raw less_eq_raw"

by auto

lemma plus_assoc_raw:

  shows "plus_raw (plus_raw i j) k \<approx> plus_raw i (plus_raw j k)"

by (cases i, cases j, cases k) (simp)

lemma plus_sym_raw:

  shows "plus_raw i j \<approx> plus_raw j i"

by (cases i, cases j) (simp)

lemma plus_zero_raw:

  shows "plus_raw  (0, 0) i \<approx> i"

by (cases i) (simp)

lemma plus_minus_zero_raw:

  shows "plus_raw (minus_raw i) i \<approx> (0, 0)"

by (cases i) (simp)

lemma times_assoc_raw:

  shows "times_raw (times_raw i j) k \<approx> times_raw i (times_raw j k)"

by (cases i, cases j, cases k) 

   (simp add: algebra_simps)

lemma times_sym_raw:

  shows "times_raw i j \<approx> times_raw j i"

by (cases i, cases j) (simp add: algebra_simps)

lemma times_one_raw:

  shows "times_raw  (1, 0) i \<approx> i"

by (cases i) (simp)

lemma times_plus_comm_raw:

  shows "times_raw (plus_raw i j) k \<approx> plus_raw (times_raw i k) (times_raw j k)"

by (cases i, cases j, cases k) 

   (simp add: algebra_simps)

lemma one_zero_distinct:

  shows "\<not> (0, 0) \<approx> ((1::nat), (0::nat))"

  by simp

text{* The integers form a @{text comm_ring_1}*}

ML {* qconsts_lookup @{theory} @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} *}

ML {* dest_Type (snd (dest_Const @{term "0 :: int"})) *}

ML {* @{term "0 :: int"} *}

instance int :: comm_ring_1

proof

  fix i j k :: int

  show "(i + j) + k = i + (j + k)"

     by (lifting plus_assoc_raw)

  show "i + j = j + i" 

    by (lifting plus_sym_raw)

  show "0 + i = (i::int)"

    by (lifting plus_zero_raw)

  show "- i + i = 0"

    by (lifting plus_minus_zero_raw)

  show "i - j = i + - j"

    by (simp add: minus_int_def)

  show "(i * j) * k = i * (j * k)"

    by (lifting times_assoc_raw)

  show "i * j = j * i"

    by (lifting times_sym_raw)

  show "1 * i = i"

    by (lifting times_one_raw)

  show "(i + j) * k = i * k + j * k"

    by (lifting times_plus_comm_raw)

  show "0 \<noteq> (1::int)"

    by (lifting one_zero_distinct)

qed

lemma add:

     "(ABS_int (x,y)) + (ABS_int (u,v)) =

      (ABS_int (x+u, y+v))"

apply(simp add: plus_int_def)

lemma int_def: "of_nat m = ABS_int (m, 0)"

apply(induct m)

apply(simp add: zero_int_def)

apply(simp add: one_int_def add)

done

lemma le_antisym_raw:

  shows "less_eq_raw i j \<Longrightarrow> less_eq_raw j i \<Longrightarrow> i \<approx> j"

by (cases i, cases j) (simp)

lemma le_refl_raw:

  shows "less_eq_raw i i"

by (cases i) (simp)

lemma le_trans_raw:

  shows "less_eq_raw i j \<Longrightarrow> less_eq_raw j k \<Longrightarrow> less_eq_raw i k"

by (cases i, cases j, cases k) (simp)

lemma le_cases_raw:

  shows "less_eq_raw i j \<or> less_eq_raw j i"

by (cases i, cases j) 

   (simp add: linorder_linear)

instance int :: linorder

proof

  fix i j k :: int

  show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"

    by (lifting le_antisym_raw)

  show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"

    by (auto simp add: less_int_def dest: antisym) 

  show "i \<le> i"

    by (lifting le_refl_raw)

  show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"

    by (lifting le_trans_raw)

  show "i \<le> j \<or> j \<le> i"

    by (lifting le_cases_raw)

qed

instantiation int :: distrib_lattice

begin

definition

  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"

definition

  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"

instance

  by intro_classes

    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)

end

lemma le_plus_raw:

  shows "less_eq_raw i j \<Longrightarrow> less_eq_raw (plus_raw k i) (plus_raw k j)"

by (cases i, cases j, cases k) (simp)

instance int :: pordered_cancel_ab_semigroup_add

proof

  fix i j k :: int

  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"

    by (lifting le_plus_raw)

qed

abbreviation

  "less_raw i j \<equiv> less_eq_raw i j \<and> \<not>(i \<approx> j)"

lemma test:

  "\<lbrakk>less_raw i j; less_raw (0, 0) k\<rbrakk>

    \<Longrightarrow> less_raw (times_raw k i) (times_raw k j)"

apply(cases i, cases j, cases k)

apply(simp only: less_eq_raw.simps times_raw.simps)

apply(simp)

apply(rename_tac u v w x y z)

apply(rule conjI)

apply(subgoal_tac "y*u + y*x \<le> y*w + y*v  &  z*v \<le> y*v")

apply(simp add: )

sorry

text{*The integers form an ordered integral domain*}

instance int :: ordered_idom

proof

  fix i j k :: int

  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"

    unfolding less_int_def by (lifting test)

  show "\<bar>i\<bar> = (if i < 0 then -i else i)"

    by (simp only: abs_int_def)

  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"

    by (simp only: sgn_int_def)

qed

instance int :: lordered_ring

proof  

  fix k :: int

  show "abs k = sup k (- k)"

    by (auto simp add: sup_int_def abs_int_def less_minus_self_iff [symmetric])

qed

lemmas int_distrib =

  left_distrib [of "z1::int" "z2" "w", standard]

  right_distrib [of "w::int" "z1" "z2", standard]

  left_diff_distrib [of "z1::int" "z2" "w", standard]

  right_diff_distrib [of "w::int" "z1" "z2", standard]

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)"