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

print_theorems
print_quotients

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

term ZERO
thm ZERO_def

ML {* prop_of @{thm ZERO_def} *}

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

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 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} *}

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:
  "(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} @ @{thms ho_all_prs ho_ex_prs} *}
ML {* val defs = @{thms PLUS_def} *}

ML {*
fun lift_thm_my_int lthy t =
  lift_thm lthy qty ty_name rsp_thms defs t
*}

print_quotients

ML {* lift_thm_my_int @{context} @{thm plus_sym_pre} *}

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

ML {* lift_thm_my_int @{context} @{thm plus_assoc_pre} *}








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

text {* Below is the construction site code used if things do not work *}
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 = [@{const_name my_plus}] *}*)
ML {* val consts = lookup_quot_consts defs *}
ML {* val t_a = atomize_thm @{thm ho_tst} *}

(*
prove t_r: {* build_regularize_goal t_a rty rel @{context} *}
ML_prf {*   fun tac ctxt =
      (ObjectLogic.full_atomize_tac) THEN'
     REPEAT_ALL_NEW (FIRST' [
      rtac rel_refl,
      atac,
      rtac @{thm universal_twice},
      (rtac @{thm impI} THEN' atac),
      (*rtac @{thm equality_twice},*)
      EqSubst.eqsubst_tac ctxt [0]
        [(@{thm equiv_res_forall} OF [rel_eqv]),
         (@{thm equiv_res_exists} OF [rel_eqv])],
      (rtac @{thm impI} THEN' (asm_full_simp_tac (Simplifier.context ctxt HOL_ss)) THEN' rtac rel_refl),
      (rtac @{thm RIGHT_RES_FORALL_REGULAR})
     ]);*}
apply (atomize(full))
apply (tactic {* tac @{context} 1 *})
apply (auto)
done
ML {* val t_r = @{thm t_r} OF [t_a] *}*)
ML {* val t_r = regularize t_a rty rel rel_eqv rel_refl @{context} *}
ML {*
  val rt = build_repabs_term @{context} t_r consts rty qty
  val rg = Logic.mk_equals ((Thm.prop_of t_r), rt);
*}
*)

lemma foldl_rsp:
  "((IntEx.intrel ===> IntEx.intrel ===> IntEx.intrel) ===>
           IntEx.intrel ===> op = ===> IntEx.intrel)
           foldl foldl"
  apply (simp only:FUN_REL.simps)
  apply (rule allI)
  apply (rule allI)
  apply (rule impI)
  apply (rule allI)
  apply (rule allI)
  apply (rule impI)
  apply (rule allI)
  apply (rule allI)
  apply (rule impI)
  apply (simp only:)
  apply (rule list.induct)
  apply (simp)
  apply (simp only: foldl.simps)
  sorry

ML {* val rsp_thms = @{thm foldl_rsp} :: rsp_thms *}
prove t_t: rg
apply(atomize(full))
ML_prf {* fun r_mk_comb_tac_int lthy = r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms *}
apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_int @{context}) 1 *})
done
ML {* val t_t = @{thm Pure.equal_elim_rule1} OF [@{thm t_t},t_r] *}
ML {* val abs = findabs rty (prop_of t_a) *}
ML {* val aps = findaps rty (prop_of t_a); *}
ML {* val simp_lam_prs_thms = map (make_simp_prs_thm @{context} quot @{thm LAMBDA_PRS}) abs *}

(*ML {* val t_t = repabs @{context} @{thm t_r} consts rty qty quot rel_refl trans2 rsp_thms *}*)
ML {* val (alls, exs) = findallex rty qty (prop_of t_a) *}
ML {* val allthms = map (make_allex_prs_thm @{context} quot @{thm FORALL_PRS}) alls *}
ML {* val t_a = MetaSimplifier.rewrite_rule (allthms) t_t *}
ML {* val t_l = repeat_eqsubst_thm @{context} simp_lam_prs_thms t_a *}
ML {* val defs_sym = add_lower_defs @{context} defs *}
ML {* val t_d = repeat_eqsubst_thm @{context} defs_sym t_l *}
ML {* val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d *}
ML {* ObjectLogic.rulify t_r *}