(*  Title:      HOL/Decision_Procs/cooper_tac.ML

    Author:     Amine Chaieb, TU Muenchen

*)

structure Cooper_Tac =

struct

val trace = ref false;

fun trace_msg s = if !trace then tracing s else ();

val cooper_ss = @{simpset};

val nT = HOLogic.natT;

val binarith = @{thms normalize_bin_simps};

val comp_arith = binarith @ simp_thms

val zdvd_int = @{thm zdvd_int};

val zdiff_int_split = @{thm zdiff_int_split};

val all_nat = @{thm all_nat};

val ex_nat = @{thm ex_nat};

val number_of1 = @{thm number_of1};

val number_of2 = @{thm number_of2};

val split_zdiv = @{thm split_zdiv};

val split_zmod = @{thm split_zmod};

val mod_div_equality' = @{thm mod_div_equality'};

val split_div' = @{thm split_div'};

val Suc_plus1 = @{thm Suc_plus1};

val imp_le_cong = @{thm imp_le_cong};

val conj_le_cong = @{thm conj_le_cong};

val mod_add_left_eq = @{thm mod_add_left_eq} RS sym;

val mod_add_right_eq = @{thm mod_add_right_eq} RS sym;

val mod_add_eq = @{thm mod_add_eq} RS sym;

val nat_div_add_eq = @{thm div_add1_eq} RS sym;

val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym;

fun prepare_for_linz q fm = 

  let

    val ps = Logic.strip_params fm

    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)

    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)

    fun mk_all ((s, T), (P,n)) =

      if 0 mem loose_bnos P then

        (HOLogic.all_const T $ Abs (s, T, P), n)

      else (incr_boundvars ~1 P, n-1)

    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;

    val rhs = hs

    val np = length ps

    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))

      (foldr HOLogic.mk_imp c rhs, np) ps

    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)

      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');

    val fm2 = foldr mk_all2 fm' vs

  in (fm2, np + length vs, length rhs) end;

(*Object quantifier to meta --*)

fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;

(* object implication to meta---*)

fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;

fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st =>

  let

    val g = List.nth (prems_of st, i - 1)

    val thy = ProofContext.theory_of ctxt

    (* Transform the term*)

    val (t,np,nh) = prepare_for_linz q g

    (* Some simpsets for dealing with mod div abs and nat*)

    val mod_div_simpset = HOL_basic_ss 

			addsimps [refl,mod_add_eq, mod_add_left_eq, 

				  mod_add_right_eq,

				  nat_div_add_eq, int_div_add_eq,

				  @{thm mod_self}, @{thm "zmod_self"},

				  @{thm mod_by_0}, @{thm div_by_0},

				  @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},

				  @{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"},

				  Suc_plus1]

			addsimps @{thms add_ac}

			addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]

    val simpset0 = HOL_basic_ss

      addsimps [mod_div_equality', Suc_plus1]

      addsimps comp_arith

      addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]

    (* Simp rules for changing (n::int) to int n *)

    val simpset1 = HOL_basic_ss

      addsimps [@{thm nat_number_of_def}, zdvd_int] @ map (fn r => r RS sym)

        [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}]

      addsplits [zdiff_int_split]

    (*simp rules for elimination of int n*)

    val simpset2 = HOL_basic_ss

      addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}]

      addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}]

    (* simp rules for elimination of abs *)

    val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}]

    val ct = cterm_of thy (HOLogic.mk_Trueprop t)

    (* Theorem for the nat --> int transformation *)

    val pre_thm = Seq.hd (EVERY

      [simp_tac mod_div_simpset 1, simp_tac simpset0 1,

       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),

       TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]

      (trivial ct))

    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)

    (* The result of the quantifier elimination *)

    val (th, tac) = case (prop_of pre_thm) of

        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>

    let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1))

    in 

          ((pth RS iffD2) RS pre_thm,

            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))

    end

      | _ => (pre_thm, assm_tac i)

  in (rtac (((mp_step nh) o (spec_step np)) th) i 

      THEN tac) st

  end handle Subscript => no_tac st);

fun linz_args meth =

 let val parse_flag = 

         Args.$$$ "no_quantify" >> (K (K false));

 in

   Method.simple_args 

  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>

    curry (Library.foldl op |>) true)

    (fn q => fn ctxt => meth ctxt q)

  end;

fun linz_method ctxt q = SIMPLE_METHOD' (linz_tac ctxt q);

val setup =

  Method.add_method ("cooper",

     linz_args linz_method,

     "decision procedure for linear integer arithmetic");

end