theory IntEx
imports "../QuotList" Nitpick
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 my_int = "nat \<times> nat" / intrel
  apply(unfold equivp_def)
  apply(auto simp add: mem_def expand_fun_eq)
  done

thm quot_equiv

thm quot_thm

thm my_int_equivp

print_theorems
print_quotients

quotient_def 
  ZERO::"my_int"
where
  "ZERO \<equiv> (0::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ZERO
thm ZERO_def

ML {* prop_of @{thm ZERO_def} *}

ML {* separate *}

quotient_def 
  ONE::"my_int"
where
  "ONE \<equiv> (1::nat, 0::nat)"

ML {* print_qconstinfo @{context} *}

term ONE
thm ONE_def

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_def 
  PLUS::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"
where
  "PLUS \<equiv> my_plus"

term my_plus
term PLUS
thm PLUS_def

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

quotient_def 
  NEG::"my_int \<Rightarrow> my_int"
where
  "NEG \<equiv> my_neg"

term NEG
thm NEG_def

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_def 
  MULT::"my_int \<Rightarrow> my_int \<Rightarrow> my_int"
where
  "MULT \<equiv> my_mult"

term MULT
thm MULT_def

(* 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_def 
  LE :: "my_int \<Rightarrow> my_int \<Rightarrow> bool"
where
  "LE \<equiv> 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))"

ML {* print_qconstinfo @{context} *}

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 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(tactic {* procedure_tac @{context} @{thm test2} 1 *})
apply(rule impI)
apply(rule ballI)
apply(rule apply_rsp[OF Quotient_my_int plus_rsp])
apply(simp only: in_respects)
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
done

lemma test3: "my_plus = my_plus"
apply(rule refl)
done

lemma "PLUS = PLUS"
apply(tactic {* procedure_tac @{context} @{thm test3} 1 *})
apply(rule impI)
apply(rule plus_rsp)
apply(injection)
apply(cleaning)
done


lemma "PLUS a b = PLUS b a"
apply(tactic {* procedure_tac @{context} @{thm plus_sym_pre} 1 *})
apply(tactic {* regularize_tac @{context} 1 *})
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
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(tactic {* procedure_tac @{context} @{thm plus_assoc_pre} 1 *})
apply(tactic {* regularize_tac @{context} 1 *})
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
done

lemma ho_tst: "foldl my_plus x [] = x"
apply simp
done

lemma "foldl PLUS x [] = x"
apply(lifting ho_tst)
apply(tactic {* clean_tac @{context} 1 *})
apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int] nil_prs[OF Quotient_my_int])
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(tactic {* procedure_tac @{context} @{thm ho_tst2} 1 *})
apply(tactic {* regularize_tac @{context} 1 *})
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int] cons_prs[OF Quotient_my_int])
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(tactic {* procedure_tac @{context} @{thm ho_tst3} 1 *})
apply(tactic {* regularize_tac @{context} 1 *})
apply(tactic {* all_inj_repabs_tac @{context} 1*})
(* TODO: does not work when this is added *)
(* apply(tactic {* lambda_prs_tac @{context} 1 *})*)
apply(tactic {* clean_tac @{context} 1 *})
apply(simp only: foldl_prs[OF Quotient_my_int Quotient_my_int] nil_prs[OF Quotient_my_int])
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(tactic {* procedure_tac @{context} @{thm lam_tst} 1 *})
apply(tactic {* regularize_tac @{context} 1 *})
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac @{context} 1 *})
apply(simp add: pair_prs)
done

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

(* test about lifting identity equations *)

ML {*
(* helper code from QuotMain *)
val pat_ball = @{term "Ball (Respects (R1 ===> R2)) P"}
val pat_bex  = @{term "Bex (Respects (R1 ===> R2)) P"}
val simproc = Simplifier.simproc_i @{theory} "" [pat_ball, pat_bex] (K (ball_bex_range_simproc))
val simpset = (mk_minimal_ss @{context}) 
                       addsimps @{thms ball_reg_eqv bex_reg_eqv babs_reg_eqv}
                       addsimprocs [simproc] addSolver equiv_solver
*}

(* What regularising does *)
(*========================*)

(* 0. preliminary simplification step *)
thm ball_reg_eqv bex_reg_eqv (* of no use: babs_reg_eqv *)
    ball_reg_eqv_range bex_reg_eqv_range
(* 1. first two rtacs *)
thm ball_reg_right bex_reg_left
(* 2. monos *)
(* 3. commutation rules *)
thm ball_all_comm bex_ex_comm
(* 4. then rel-equality *)
thm eq_imp_rel
(* 5. then simplification like 0 *)
(* finally jump to 1 *)


(* What cleaning does *)
(*====================*)

(* 1. conversion *)
thm lambda_prs
(* 2. simplification with *)
thm all_prs ex_prs
(* 3. Simplification with *)
thm fun_map.simps id_simps Quotient_abs_rep Quotient_rel_rep
(* 4. Test for refl *)

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(tactic {* procedure_tac @{context} @{thm lam_tst3a} 1 *})
apply(rule impI)
apply(rule lam_tst3a_reg)
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac  @{context} 1 *})
apply(simp only: babs_prs[OF Quotient_my_int Quotient_my_int])
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(tactic {* procedure_tac @{context} @{thm lam_tst3b} 1 *})
apply(rule impI)
apply (rule babs_rsp[OF fun_quotient[OF Quotient_my_int Quotient_my_int]])
apply (simp add: id_rsp)
apply(tactic {* all_inj_repabs_tac @{context} 1*})
apply(tactic {* clean_tac  @{context} 1 *})
apply(subst babs_prs)
apply(tactic {* quotient_tac @{context} 1 *})
apply(tactic {* quotient_tac @{context} 1 *})
apply(subst babs_prs)
apply(tactic {* quotient_tac @{context} 1 *})
apply(tactic {* quotient_tac @{context} 1 *})
apply(rule refl)
done

term map

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)
apply(cleaning)
apply(simp only: babs_prs[OF Quotient_my_int Quotient_my_int])
apply(simp only: map_prs[OF Quotient_my_int Quotient_my_int, symmetric])
done

end