theory IntEx2
imports "../QuotMain" "../QuotProd" Nat
(*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_type int = "nat \<times> nat" / intrel
  unfolding equivp_def
  by (auto simp add: mem_def expand_fun_eq)

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

ML {* @{term "0 \<Colon> int"} *}

quotient_definition
  "0 \<Colon> int" as "(0\<Colon>nat, 0\<Colon>nat)"

quotient_definition
  "1 \<Colon> int" as "(1\<Colon>nat, 0\<Colon>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_definition
  "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)" as "plus_raw"

fun
  uminus_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
where
  "uminus_raw (x, y) = (y, x)"

quotient_definition
  "(uminus \<Colon> (int \<Rightarrow> int))" as "uminus_raw"

definition
  minus_int_def [code del]:  "z - w = z + (-w\<Colon>int)"

fun
  mult_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> (nat \<times> nat)"
where
  "mult_raw (x, y) (u, v) = (x*u + y*v, x*v + y*u)"

quotient_definition
  mult_int_def: "(op *) :: (int \<Rightarrow> int \<Rightarrow> int)" as "mult_raw"

fun
  le_raw :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
where
  "le_raw (x, y) (u, v) = (x+v \<le> u+y)"

quotient_definition
  le_int_def: "(op \<le>) :: int \<Rightarrow> int \<Rightarrow> bool" as "le_raw"

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

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

definition
  zsgn_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 uminus_raw_rsp[quot_respect]:
  shows "(op \<approx> ===> op \<approx>) uminus_raw uminus_raw"
  by auto

lemma mult_raw_fst:
  assumes a: "x \<approx> z"
  shows "mult_raw x y \<approx> mult_raw z y"
using a
apply(cases x, cases y, cases z)
apply(auto simp add: mult_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_snd:
  assumes a: "x \<approx> z"
  shows "mult_raw y x \<approx> mult_raw y z"
using a
apply(cases x, cases y, cases z)
apply(auto simp add: mult_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>) mult_raw mult_raw"
apply(simp only: fun_rel.simps)
apply(rule allI | rule impI)+
apply(rule equivp_transp[OF int_equivp])
apply(rule mult_raw_fst)
apply(assumption)
apply(rule mult_raw_snd)
apply(assumption)
done

lemma le_raw_rsp[quot_respect]:
  shows "(op \<approx> ===> op \<approx> ===> op =) le_raw le_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 (uminus_raw i) i \<approx> (0, 0)"
by (cases i) (simp)

lemma times_assoc_raw:
  shows "mult_raw (mult_raw i j) k \<approx> mult_raw i (mult_raw j k)"
by (cases i, cases j, cases k) 
   (simp add: algebra_simps)

lemma times_sym_raw:
  shows "mult_raw i j \<approx> mult_raw j i"
by (cases i, cases j) (simp add: algebra_simps)

lemma times_one_raw:
  shows "mult_raw  (1, 0) i \<approx> i"
by (cases i) (simp)

lemma times_plus_comm_raw:
  shows "mult_raw (plus_raw i j) k \<approx> plus_raw (mult_raw i k) (mult_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}*}

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 plus_raw_rsp_aux:
  assumes a: "a \<approx> b" "c \<approx> d"
  shows "plus_raw a c \<approx> plus_raw b d"
using a
by (cases a, cases b, cases c, cases d) 
   (simp)

lemma add:
     "(abs_int (x,y)) + (abs_int (u,v)) =
      (abs_int (x + u, y + v))"
apply(simp add: plus_int_def id_simps)
apply(fold plus_raw.simps)
apply(rule Quotient_rel_abs[OF Quotient_int])
apply(rule plus_raw_rsp_aux)
apply(simp_all add: rep_abs_rsp_left[OF Quotient_int])
done

definition int_of_nat_raw: 
  "int_of_nat_raw m = (m :: nat, 0 :: nat)"

quotient_definition
  "int_of_nat :: nat \<Rightarrow> int" as "int_of_nat_raw"

lemma[quot_respect]: 
  shows "(op = ===> op \<approx>) int_of_nat_raw int_of_nat_raw"
by (simp add: equivp_reflp[OF int_equivp])

lemma int_of_nat:
  shows "of_nat m = int_of_nat m"
apply (induct m)
apply (simp_all add: zero_int_def one_int_def int_of_nat_def int_of_nat_raw add)
done

lemma le_antisym_raw:
  shows "le_raw i j \<Longrightarrow> le_raw j i \<Longrightarrow> i \<approx> j"
by (cases i, cases j) (simp)

lemma le_refl_raw:
  shows "le_raw i i"
by (cases i) (simp)

lemma le_trans_raw:
  shows "le_raw i j \<Longrightarrow> le_raw j k \<Longrightarrow> le_raw i k"
by (cases i, cases j, cases k) (simp)

lemma le_cases_raw:
  shows "le_raw i j \<or> le_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 "le_raw i j \<Longrightarrow> le_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> le_raw i j \<and> \<not>(i \<approx> j)"

lemma zmult_zless_mono2_lemma:
  fixes i j::int
  and   k::nat
  shows "i < j \<Longrightarrow> 0 < k \<Longrightarrow> of_nat k * i < of_nat k * j"
apply(induct "k")
apply(simp)
apply(case_tac "k = 0")
apply(simp_all add: left_distrib add_strict_mono)
done

lemma zero_le_imp_eq_int_raw:
  fixes k::"(nat \<times> nat)"
  shows "less_raw (0, 0) k \<Longrightarrow> (\<exists>n > 0. k \<approx> int_of_nat_raw n)"
apply(cases k)
apply(simp add:int_of_nat_raw)
apply(auto)
apply(rule_tac i="b" and j="a" in less_Suc_induct)
apply(auto)
done

lemma zero_le_imp_eq_int:
  fixes k::int
  shows "0 < k \<Longrightarrow> \<exists>n > 0. k = of_nat n"
  unfolding less_int_def int_of_nat
  by (lifting zero_le_imp_eq_int_raw)

lemma zmult_zless_mono2: 
  fixes i j k::int
  assumes a: "i < j" "0 < k"
  shows "k * i < k * j"
using a
using a
apply(drule_tac zero_le_imp_eq_int)
apply(auto simp add: zmult_zless_mono2_lemma)
done

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"
    by (rule zmult_zless_mono2)
  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
    by (simp only: zabs_def)
  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
    by (simp only: zsgn_def)
qed

instance int :: lordered_ring
proof  
  fix k :: int
  show "abs k = sup k (- k)"
    by (auto simp add: sup_int_def zabs_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)"

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