header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}

theory LarryInt 

imports Nat "../QuotMain"

begin

fun

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

where

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

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

  by (auto simp add: equivp_def expand_fun_eq)

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

begin

quotient_def 

  Zero_int_def: "0::int" as "(0::nat, 0::nat)"

quotient_def

  One_int_def: "1::int" as "(1::nat, 0::nat)"

quotient_def

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

as 

  "\<lambda>(x, y) (u, v). (x + (u::nat), y + (v::nat))"

quotient_def

  "uminus :: int \<Rightarrow> int" 

as 

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

fun

  mult_aux::"nat \<times> nat \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"

where

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

quotient_def

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

as 

  "mult_aux"

quotient_def 

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

as 

  "\<lambda>(x, y) (u, v). (x+v \<le> u+(y::nat))"

definition

  less_int_def: "z < (w::int) \<equiv> (z \<le> w & z \<noteq> w)"

definition

  diff_int_def:  "z - (w::int) \<equiv> z + (-w)"

instance ..

end

subsection{*Construction of the Integers*}

abbreviation

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

lemma zminus_zminus_aux:

  "uminus_aux (uminus_aux z) = z"

  by (cases z) (simp)

lemma [quot_respect]:

  shows "(intrel ===> intrel) uminus_aux uminus_aux"

  by simp

lemma zminus_zminus: 

  shows "- (- z) = (z::int)"

apply(lifting zminus_zminus_aux)

apply(injection)

apply(rule quot_respect)

apply(rule quot_respect)

done

(* PROBLEM *)

lemma zminus_0_aux:

  shows "uminus_aux (0, 0) = (0, 0::nat)"

by simp

lemma zminus_0: "- 0 = (0::int)"

apply(lifting zminus_0_aux)

apply(injection)

apply(rule quot_respect)

done

(* PROBLEM *)

subsection{*Integer Addition*}

definition

  "add_aux \<equiv> \<lambda>(x, y) (u, v). (x + (u::nat), y + (v::nat))"

lemma zminus_zadd_distrib_aux:

  shows "uminus_aux (add_aux z w) = add_aux (uminus_aux z) (uminus_aux w)"

by (cases z, cases w)

   (auto simp add: add_aux_def)

lemma [quot_respect]:

  shows "(intrel ===> intrel ===> intrel) 

            (\<lambda>(x, y) (u, v). (x + u, y + (v::nat)))  (\<lambda>(x, y) (u, v). (x + u, y + (v::nat)))"

by simp

lemma zminus_zadd_distrib: 

  shows "- (z + w) = (- z) + (- w::int)"

apply(lifting zminus_zadd_distrib_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

lemma zadd_commute_aux:

  shows "add_aux z w = add_aux w z"

by (cases z, cases w)

   (simp add: add_aux_def)

lemma zadd_commute: 

  shows "(z::int) + w = w + z"

apply(lifting zadd_commute_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

lemma zadd_assoc_aux:

  shows "add_aux (add_aux z1 z2) z3 = add_aux z1 (add_aux z2 z3)"

by (cases z1, cases z2, cases z3) (simp add: add_aux_def)

lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"

apply(lifting zadd_assoc_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

lemma zadd_0_aux:

  fixes z::"nat \<times> nat"

  shows "add_aux (0, 0) z = z"

by (simp add: add_aux_def)

(*also for the instance declaration int :: plus_ac0*)

lemma zadd_0: "(0::int) + z = z"

apply(lifting zadd_0_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

lemma zadd_zminus_inverse_aux:

  shows "intrel (add_aux (uminus_aux z) z) (0, 0)"

by (cases z) (simp add: add_aux_def)

lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"

apply(lifting zadd_zminus_inverse_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

subsection{*Integer Multiplication*}

lemma zmult_zminus_aux:

  shows "mult_aux (uminus_aux z) w = uminus_aux (mult_aux z w)"

apply(cases z, cases w)

apply(simp)

done

lemma mult_aux_fst:

  assumes a: "intrel x z"

  shows "intrel (mult_aux x y) (mult_aux z y)"

using a

apply(cases x, cases y, cases z)

apply(auto simp add: mult_aux.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_aux_snd:

  assumes a: "intrel x z"

  shows "intrel (mult_aux y x) (mult_aux y z)"

using a

apply(cases x, cases y, cases z)

apply(auto simp add: mult_aux.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 "(intrel ===> intrel ===> intrel) mult_aux mult_aux"

apply(simp only: fun_rel.simps)

apply(rule allI | rule impI)+

apply(rule equivp_transp[OF int_equivp])

apply(rule mult_aux_fst)

apply(assumption)

apply(rule mult_aux_snd)

apply(assumption)

done

lemma zmult_zminus: "(- z) * w = - (z * (w::int))"

apply(lifting zmult_zminus_aux)

apply(injection)

apply(rule quot_respect)

apply(rule quot_respect)

done

lemma zmult_commute_aux: 

  shows "mult_aux z w = mult_aux w z"

apply(cases z, cases w)

apply(simp add: add_ac mult_ac)

done

lemma zmult_commute: "(z::int) * w = w * z"

by (lifting zmult_commute_aux)

lemma zmult_assoc_aux:

  shows "mult_aux (mult_aux z1 z2) z3 = mult_aux z1 (mult_aux z2 z3)"

apply(cases z1, cases z2, cases z3)

apply(simp add: add_mult_distrib2 mult_ac)

done

lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"

by (lifting zmult_assoc_aux)

lemma zadd_mult_distrib_aux:

  shows "mult_aux (add_aux z1 z2) w = add_aux (mult_aux z1 w) (mult_aux z2 w)"

apply(cases z1, cases z2, cases w)

apply(simp add: add_mult_distrib2 mult_ac add_aux_def)

done

lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"

apply(lifting zadd_mult_distrib_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"

by (simp add: zmult_commute [of w] zadd_zmult_distrib)

lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"

by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)

lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"

by (simp add: zmult_commute [of w] zdiff_zmult_distrib)

lemmas int_distrib =

  zadd_zmult_distrib zadd_zmult_distrib2

  zdiff_zmult_distrib zdiff_zmult_distrib2

lemma zmult_1_aux:

  shows "mult_aux (1, 0) z = z"

apply(cases z)

apply(auto)

done

lemma zmult_1: "(1::int) * z = z"

apply(lifting zmult_1_aux)

done

lemma zmult_1_right: "z * (1::int) = z"

by (rule trans [OF zmult_commute zmult_1])

lemma zero_not_one:

  shows "(0, 0) \<noteq> (1::nat, 0::nat)"

by simp

text{*The Integers Form A Ring*}

instance int :: comm_ring_1

proof

  fix i j k :: int

  show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)

  show "i + j = j + i" by (simp add: zadd_commute)

  show "0 + i = i" by (rule zadd_0)

  show "- i + i = 0" by (rule zadd_zminus_inverse2)

  show "i - j = i + (-j)" by (simp add: diff_int_def)

  show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)

  show "i * j = j * i" by (rule zmult_commute)

  show "1 * i = i" by (rule zmult_1) 

  show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)

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

    by (lifting zero_not_one) (auto) (* PROBLEM? regularize failed *)

qed

subsection{*The @{text "\<le>"} Ordering*}

abbreviation

  "le_aux \<equiv> \<lambda>(x, y) (u, v). (x+v \<le> u+(y::nat))"

lemma zle_refl_aux:

  "le_aux w w"

apply(cases w)

apply(simp)

done

lemma [quot_respect]:

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

by auto

lemma zle_refl: "w \<le> (w::int)"

apply(lifting zle_refl_aux)

apply(injection)

apply(rule quot_respect)

done

(* PROBLEM *)

lemma zle_trans_aux:

  shows "\<lbrakk>le_aux i j; le_aux j k\<rbrakk> \<Longrightarrow> le_aux i k"

apply(cases i, cases j, cases k)

apply(auto)

done

lemma zle_trans: "\<lbrakk>i \<le> j; j \<le> k\<rbrakk> \<Longrightarrow> i \<le> (k::int)"

apply(lifting zle_trans_aux)

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

lemma zle_anti_sym_aux:

  shows "\<lbrakk>le_aux z w; le_aux w z\<rbrakk> \<Longrightarrow> intrel z w"

apply(cases z, cases w)

apply(auto)

done

lemma zle_anti_sym: "\<lbrakk>z \<le> w; w \<le> z\<rbrakk> \<Longrightarrow> z = (w::int)"

apply(lifting zle_anti_sym_aux)

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

(* Axiom 'order_less_le' of class 'order': *)

lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"

by (simp add: less_int_def)

instance int :: order

apply(intro_classes)

apply(auto intro: zle_refl zle_trans zle_anti_sym zless_le simp add: less_int_def)

done

(* Axiom 'linorder_linear' of class 'linorder': *)

lemma zle_linear_aux:

  "le_aux z w \<or> le_aux w z"

apply(cases w, cases z)

apply(auto)

done

lemma zle_linear: "(z::int) \<le> w \<or> w \<le> z"

apply(lifting zle_linear_aux)

apply(injection)

apply(rule quot_respect)+

done

instance int :: linorder

proof qed (rule zle_linear)

lemma zadd_left_mono_aux:

  shows "le_aux i j \<Longrightarrow> le_aux (add_aux k i) (add_aux k j)"

apply(cases k)

apply(auto simp add: add_aux_def)

done

lemma zadd_left_mono: "i \<le> j \<Longrightarrow> k + i \<le> k + (j::int)"

apply(lifting zadd_left_mono_aux[simplified add_aux_def])

apply(injection)

apply(rule quot_respect)+

done

(* PROBLEM *)

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

(* PROBLEM: this has to be a definition, not an abbreviation *)

(* otherwise the lemma nat_le_eq_zle cannot be lifted        *)

fun

  nat_aux

where

  "nat_aux (x, y) =  x - (y::nat)"

quotient_def

  "nat2::int\<Rightarrow>nat"

as

  "nat_aux"

abbreviation

  "less_aux x y \<equiv> (le_aux x y \<and> \<not>(x = y))"

lemma nat_le_eq_zle_aux:

  shows "less_aux (0, 0) w \<or> le_aux (0, 0) z \<Longrightarrow> (nat_aux w \<le> nat_aux z) = (le_aux w z)"

apply(auto)

sorry

lemma [quot_respect]:

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

apply(auto)

done

lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> (nat2 w \<le> nat2 z) = (w\<le>z)"

unfolding less_int_def

apply(lifting nat_le_eq_zle_aux)

apply(injection)

apply(simp_all only: quot_respect)

done

(* PROBLEM *)

end