(*  Title:      HOL/Quotient_Examples/Quotient_Int.thy

    Author:     Cezary Kaliszyk

    Author:     Christian Urban

Integers based on Quotients.

*)

theory Quotient_Int

imports Quotient_Product Nat

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

  by (auto simp add: equivp_def expand_fun_eq)

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

begin

quotient_definition

  "0 \<Colon> int" is "(0\<Colon>nat, 0\<Colon>nat)"

quotient_definition

  "1 \<Colon> int" is "(1\<Colon>nat, 0\<Colon>nat)"

fun

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

where

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

quotient_definition

  "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)" is "plus_int_raw"

fun

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

where

  "uminus_int_raw (x, y) = (y, x)"

quotient_definition

  "(uminus \<Colon> (int \<Rightarrow> int))" is "uminus_int_raw"

definition

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

fun

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

where

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

quotient_definition

  "(op *) :: (int \<Rightarrow> int \<Rightarrow> int)" is "times_int_raw"

fun

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

where

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

quotient_definition

  le_int_def: "(op \<le>) :: int \<Rightarrow> int \<Rightarrow> bool" is "le_int_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 [quot_respect]:

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

  and   "(op \<approx> ===> op \<approx>) uminus_int_raw uminus_int_raw"

  and   "(op \<approx> ===> op \<approx> ===> op =) le_int_raw le_int_raw"

  by auto

lemma times_int_raw_fst:

  assumes a: "x \<approx> z"

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

  using a

  apply(cases x, cases y, cases z)

  apply(auto simp add: times_int_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_int_raw_snd:

  assumes a: "x \<approx> z"

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

  using a

  apply(cases x, cases y, cases z)

  apply(auto simp add: times_int_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 [quot_respect]:

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

  apply(simp only: fun_rel_def)

  apply(rule allI | rule impI)+

  apply(rule equivp_transp[OF int_equivp])

  apply(rule times_int_raw_fst)

  apply(assumption)

  apply(rule times_int_raw_snd)

  apply(assumption)

  done

lemma plus_assoc_raw:

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

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

lemma plus_sym_raw:

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

  by (cases i, cases j) (simp)

lemma plus_zero_raw:

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

  by (cases i) (simp)

lemma plus_minus_zero_raw:

  shows "plus_int_raw (uminus_int_raw i) i \<approx> (0, 0)"

  by (cases i) (simp)

lemma times_assoc_raw:

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

  by (cases i, cases j, cases k)

     (simp add: algebra_simps)

lemma times_sym_raw:

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

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

lemma times_one_raw:

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

  by (cases i) (simp)

lemma times_plus_comm_raw:

  shows "times_int_raw (plus_int_raw i j) k \<approx> plus_int_raw (times_int_raw i k) (times_int_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_int_raw_rsp_aux:

  assumes a: "a \<approx> b" "c \<approx> d"

  shows "plus_int_raw a c \<approx> plus_int_raw b d"

  using a

  by (cases a, cases b, cases c, cases d)

     (simp)

lemma add_abs_int:

  "(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_int_raw.simps)

  apply(rule Quotient_rel_abs[OF Quotient_int])

  apply(rule plus_int_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" is "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"

  by (induct m)

     (simp_all add: zero_int_def one_int_def int_of_nat_def int_of_nat_raw add_abs_int)

lemma le_antisym_raw:

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

  by (cases i, cases j) (simp)

lemma le_refl_raw:

  shows "le_int_raw i i"

  by (cases i) (simp)

lemma le_trans_raw:

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

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

lemma le_cases_raw:

  shows "le_int_raw i j \<or> le_int_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_int_raw:

  shows "le_int_raw i j \<Longrightarrow> le_int_raw (plus_int_raw k i) (plus_int_raw k j)"

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

instance int :: ordered_cancel_ab_semigroup_add

proof

  fix i j k :: int

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

    by (lifting le_plus_int_raw)

qed

abbreviation

  "less_int_raw i j \<equiv> le_int_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_int_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

  by (drule_tac zero_le_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)

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

instance int :: linordered_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

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]

  minus_add_distrib[of "z1::int" "z2", standard]

lemma int_induct_raw:

  assumes a: "P (0::nat, 0)"

  and     b: "\<And>i. P i \<Longrightarrow> P (plus_int_raw i (1, 0))"

  and     c: "\<And>i. P i \<Longrightarrow> P (plus_int_raw i (uminus_int_raw (1, 0)))"

  shows      "P x"

proof (cases x, clarify)

  fix a b

  show "P (a, b)"

  proof (induct a arbitrary: b rule: Nat.induct)

    case zero

    show "P (0, b)" using assms by (induct b) auto

  next

    case (Suc n)

    then show ?case using assms by auto

  qed

qed

lemma int_induct:

  fixes x :: int

  assumes a: "P 0"

  and     b: "\<And>i. P i \<Longrightarrow> P (i + 1)"

  and     c: "\<And>i. P i \<Longrightarrow> P (i - 1)"

  shows      "P x"

  using a b c unfolding minus_int_def

  by (lifting int_induct_raw)

text {* Magnitide of an Integer, as a Natural Number: @{term nat} *}

definition

  "int_to_nat_raw \<equiv> \<lambda>(x, y).x - (y::nat)"

quotient_definition

  "int_to_nat::int \<Rightarrow> nat"

is

  "int_to_nat_raw"

lemma [quot_respect]:

  shows "(intrel ===> op =) int_to_nat_raw int_to_nat_raw"

  by (auto iff: int_to_nat_raw_def)

lemma nat_le_eq_zle_raw:

  assumes a: "less_int_raw (0, 0) w \<or> le_int_raw (0, 0) z"

  shows "(int_to_nat_raw w \<le> int_to_nat_raw z) = (le_int_raw w z)"

  using a

  by (cases w, cases z) (auto simp add: int_to_nat_raw_def)

lemma nat_le_eq_zle:

  fixes w z::"int"

  shows "0 < w \<or> 0 \<le> z \<Longrightarrow> (int_to_nat w \<le> int_to_nat z) = (w \<le> z)"

  unfolding less_int_def

  by (lifting nat_le_eq_zle_raw)

end