theory IntEx

imports "../QuotProd" "../QuotList"

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 

  my_int = "nat \<times> nat" / intrel

  apply(unfold equivp_def)

  apply(auto simp add: mem_def expand_fun_eq)

  done

quotient_definition

  "ZERO :: my_int"

as

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

quotient_definition

  "ONE :: my_int"

as

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

fun

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

where

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

quotient_definition

  "PLUS :: my_int \<Rightarrow> my_int \<Rightarrow> my_int"

as

  "my_plus"

fun

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

where

  "my_neg (x, y) = (y, x)"

quotient_definition

  "NEG :: my_int \<Rightarrow> my_int"

as

  "my_neg"

definition

  MINUS :: "my_int \<Rightarrow> my_int \<Rightarrow> my_int"

where

  "MINUS z w = PLUS z (NEG w)"

fun

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

where

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

quotient_definition

  "MULT :: my_int \<Rightarrow> my_int \<Rightarrow> my_int"

as

  "my_mult"

(* NOT SURE WETHER THIS DEFINITION IS CORRECT *)

fun

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

where

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

quotient_definition

   "LE :: my_int \<Rightarrow> my_int \<Rightarrow> bool"

as

  "my_le"

term LE

thm LE_def

definition

  LESS :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"

where

  "LESS z w = (LE z w \<and> z \<noteq> w)"

term LESS

thm LESS_def

definition

  ABS :: "my_int \<Rightarrow> my_int"

where

  "ABS i = (if (LESS i ZERO) then (NEG i) else i)"

definition

  SIGN :: "my_int \<Rightarrow> my_int"

where

 "SIGN i = (if i = ZERO then ZERO else if (LESS ZERO i) then ONE else (NEG ONE))"

print_quotconsts

lemma plus_sym_pre:

  shows "my_plus a b \<approx> my_plus b a"

  apply(cases a)

  apply(cases b)

  apply(auto)

  done

lemma plus_rsp[quot_respect]:

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

by (simp)

lemma neg_rsp[quot_respect]:

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

by simp

lemma test1: "my_plus a b = my_plus a b"

apply(rule refl)

done

lemma "PLUS a b = PLUS a b"

apply(lifting_setup test1)

apply(regularize)

apply(injection)

apply(cleaning)

done

thm lambda_prs

lemma test2: "my_plus a = my_plus a"

apply(rule refl)

done

lemma "PLUS a = PLUS a"

apply(lifting_setup test2)

apply(rule impI)

apply(rule ballI)

apply(rule apply_rsp[OF Quotient_my_int plus_rsp])

apply(simp only: in_respects)

apply(injection)

apply(cleaning)

done

lemma test3: "my_plus = my_plus"

apply(rule refl)

done

lemma "PLUS = PLUS"

apply(lifting_setup test3)

apply(rule impI)

apply(rule plus_rsp)

apply(injection)

apply(cleaning)

done

lemma "PLUS a b = PLUS b a"

apply(lifting plus_sym_pre)

done

lemma plus_assoc_pre:

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

  apply (cases i)

  apply (cases j)

  apply (cases k)

  apply (simp)

  done

lemma plus_assoc: "PLUS (PLUS x xa) xb = PLUS x (PLUS xa xb)"

apply(lifting plus_assoc_pre)

done

lemma int_induct_raw:

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

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

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

  shows      "P x"

  apply(case_tac x) apply(simp)

  apply(rule_tac x="b" in spec)

  apply(rule_tac Nat.induct)

  apply(rule allI)

  apply(rule_tac Nat.induct)

  using a b c apply(auto)

  done

lemma int_induct:

  assumes a: "P ZERO"

  and     b: "\<And>i. P i \<Longrightarrow> P (PLUS i ONE)"

  and     c: "\<And>i. P i \<Longrightarrow> P (PLUS i (NEG ONE))"

  shows      "P x"

  using a b c

  by (lifting int_induct_raw)

lemma ex1tst: "Bexeq (op \<approx>) (\<lambda>x :: nat \<times> nat. x \<approx> x)"

sorry

lemma ex1tst': "\<exists>! (x :: my_int). x = x"

apply(lifting ex1tst)

done

lemma ho_tst: "foldl my_plus x [] = x"

apply simp

done

term foldl

lemma "foldl PLUS x [] = x"

apply(lifting ho_tst)

done

lemma ho_tst2: "foldl my_plus x (h # t) \<approx> my_plus h (foldl my_plus x t)"

sorry

lemma "foldl PLUS x (h # t) = PLUS h (foldl PLUS x t)"

apply(lifting ho_tst2)

done

lemma ho_tst3: "foldl f (s::nat \<times> nat) ([]::(nat \<times> nat) list) = s"

by simp

lemma "foldl f (x::my_int) ([]::my_int list) = x"

apply(lifting ho_tst3)

done

lemma lam_tst: "(\<lambda>x. (x, x)) y = (y, (y :: nat \<times> nat))"

by simp

(* Don't know how to keep the goal non-contracted... *)

lemma "(\<lambda>x. (x, x)) (y::my_int) = (y, y)"

apply(lifting lam_tst)

done

lemma lam_tst2: "(\<lambda>(y :: nat \<times> nat). y) = (\<lambda>(x :: nat \<times> nat). x)"

by simp

lemma

  shows "equivp (op \<approx>)"

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

(* Nitpick finds a counterexample! *)

oops

lemma lam_tst3a: "(\<lambda>(y :: nat \<times> nat). y) = (\<lambda>(x :: nat \<times> nat). x)"

by auto

lemma id_rsp:

  shows "(R ===> R) id id"

by simp

lemma lam_tst3a_reg: "(op \<approx> ===> op \<approx>) (Babs (Respects op \<approx>) (\<lambda>y. y)) (Babs (Respects op \<approx>) (\<lambda>x. x))"

apply (rule babs_rsp[OF Quotient_my_int])

apply (simp add: id_rsp)

done

lemma "(\<lambda>(y :: my_int). y) = (\<lambda>(x :: my_int). x)"

apply(lifting lam_tst3a)

apply(rule impI)

apply(rule lam_tst3a_reg)

done

lemma lam_tst3b: "(\<lambda>(y :: nat \<times> nat \<Rightarrow> nat \<times> nat). y) = (\<lambda>(x :: nat \<times> nat \<Rightarrow> nat \<times> nat). x)"

by auto

lemma "(\<lambda>(y :: my_int => my_int). y) = (\<lambda>(x :: my_int => my_int). x)"

apply(lifting lam_tst3b)

apply(rule impI)

apply(rule babs_rsp[OF fun_quotient[OF Quotient_my_int Quotient_my_int]])

apply(simp add: id_rsp)

done

lemma lam_tst4: "map (\<lambda>x. my_plus x (0,0)) l = l"

apply (induct l)

apply simp

apply (case_tac a)

apply simp

done

lemma "map (\<lambda>x. PLUS x ZERO) l = l"

apply(lifting lam_tst4)

done

end