theory IntEx
imports QuotMain
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 EQUIV_def)
  apply(auto simp add: mem_def expand_fun_eq)
  done

thm my_int_equiv

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 ho_plus_rsp[quot_rsp]:
  shows "(intrel ===> intrel ===> intrel) my_plus my_plus"
by (simp)

ML {* val qty = @{typ "my_int"} *}
ML {* val ty_name = "my_int" *}
ML {* val rsp_thms = @{thms ho_plus_rsp} *}
ML {* val defs = @{thms PLUS_def} *}
ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "my_int"; *}
ML {* val consts = lookup_quot_consts defs *}

ML {* fun lift_tac_intex lthy t = lift_tac lthy t [rel_eqv] rty [quot] defs *}

ML {* fun inj_repabs_tac_intex lthy = inj_repabs_tac lthy rty [quot] [rel_refl] [trans2] *}
ML {* fun all_inj_repabs_tac_intex lthy = all_inj_repabs_tac lthy rty [quot] [rel_refl] [trans2] *}


lemma "PLUS a b = PLUS b a"
apply(tactic {* procedure_tac @{context} @{thm plus_sym_pre} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
prefer 2
apply(tactic {* clean_tac @{context} [quot] defs 1 *})
apply(simp add: id_def)
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{context} 1*})
apply(tactic {* inj_repabs_tac_intex @{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} [rel_eqv] 1 *})
apply(simp)
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(tactic {* clean_tac @{context} [quot] defs 1 *})
done

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

(* FIXME/TODO: is this a respectfulness theorem? Does not look like one. *)
lemma map_prs: 
  "map REP_my_int (map ABS_my_int x) = x"
sorry

lemma foldl_prs[quot_rsp]: 
  "((op \<approx> ===> op \<approx> ===> op \<approx>) ===> op \<approx> ===> op = ===> op \<approx>) foldl foldl"
sorry

lemma "foldl PLUS x [] = x"
apply(tactic {* procedure_tac @{context} @{thm ho_tst} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*})
apply(simp add: map_prs)
apply(simp only: map_prs)
apply(tactic {* clean_tac @{context} [quot] defs 1 *})
sorry

(*
  FIXME: All below is your construction code; mostly commented out as it
  does not work.
*)

lemma "PLUS (PLUS i j) k = PLUS i (PLUS j k)"
apply(tactic {* procedure_tac @{context} @{thm plus_assoc_pre} 1 *})
apply(tactic {* regularize_tac @{context} [rel_eqv] 1 *})
apply(simp)
apply(tactic {* all_inj_repabs_tac_intex @{context} 1*}) 
apply(tactic {* clean_tac @{context} [quot] defs 1 *})
done