(*  Title:      HOL/Reflection/mir_tac.ML

     Author:     Amine Chaieb, TU Muenchen

 *)

 structure Mir_Tac =

 struct

 val trace = ref false;

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

 val mir_ss = 

 let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff", "real_of_int_le_iff"]

 in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths)

 end;

 val nT = HOLogic.natT;

   val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", 

                        "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"];

   val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", 

                  "add_Suc", "add_number_of_left", "mult_number_of_left", 

                  "Suc_eq_add_numeral_1"])@

                  (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])

                  @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} 

   val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 

              @{thm "real_of_nat_number_of"},

              @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"},

              @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"},

              @{thm "Ring_and_Field.divide_zero"}, 

              @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, 

              @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},

              @{thm "diff_def"}, @{thm "minus_divide_left"}]

 val comp_ths = ths @ comp_arith @ 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_eq = @{thm "mod_add_eq"} RS sym;

 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 nat_div_add_eq = @{thm "div_add1_eq"} RS sym;

 val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym;

 val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2;

 val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1;

 fun prepare_for_mir thy 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 (rhs,irhs) = List.partition (relevant (rev ps)) 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 mir_tac ctxt q i = 

     (ObjectLogic.atomize_prems_tac i)

         THEN (simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) i)

         THEN (REPEAT_DETERM (split_tac [@{thm "split_min"}, @{thm "split_max"},@{thm "abs_split"}] 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_mir thy q g

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

     val mod_div_simpset = HOL_basic_ss 

                         addsimps [refl, mod_add_eq, 

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

                                   @{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"},

                                   @{thm "Suc_plus1"}]

                         addsimps @{thms add_ac}

                         addsimprocs [cancel_div_mod_proc]

     val simpset0 = HOL_basic_ss

       addsimps [mod_div_equality', Suc_plus1]

       addsimps comp_ths

       addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "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"}, @{thm "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 [@{thm "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 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 mir_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 =

           (* If quick_and_dirty then run without proof generation as oracle*)

              if !quick_and_dirty

              then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1))

              else mirfr_oracle (true, cterm_of thy (Pattern.eta_long [] t1))

     in 

           (trace_msg ("calling procedure with term:\n" ^

              Syntax.string_of_term ctxt t1);

            ((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 mir_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 1)

   end;

 fun mir_method ctxt q i = Method.METHOD (fn facts =>

   Method.insert_tac facts 1 THEN mir_tac ctxt q i);

 val setup =

   Method.add_method ("mir",

      mir_args mir_method,

      "decision procedure for MIR arithmetic");

 end