(*  Title:      HOL/Tools/Qelim/presburger.ML

     Author:     Amine Chaieb, TU Muenchen

 *)

 signature PRESBURGER =

 sig

   val cooper_tac: bool -> thm list -> thm list -> Proof.context -> int -> tactic

 end;

 structure Presburger : PRESBURGER = 

 struct

 open Conv;

 val comp_ss = HOL_ss addsimps @{thms "Groebner_Basis.comp_arith"};

 fun strip_objimp ct =

   (case Thm.term_of ct of

     Const ("op -->", _) $ _ $ _ =>

       let val (A, B) = Thm.dest_binop ct

       in A :: strip_objimp B end

   | _ => [ct]);

 fun strip_objall ct = 

  case term_of ct of 

   Const ("All", _) $ Abs (xn,xT,p) => 

    let val (a,(v,t')) = (apsnd (Thm.dest_abs (SOME xn)) o Thm.dest_comb) ct

    in apfst (cons (a,v)) (strip_objall t')

    end

 | _ => ([],ct);

 local

   val all_maxscope_ss = 

      HOL_basic_ss addsimps map (fn th => th RS sym) @{thms "all_simps"}

 in

 fun thin_prems_tac P = simp_tac all_maxscope_ss THEN'

   CSUBGOAL (fn (p', i) =>

     let

      val (qvs, p) = strip_objall (Thm.dest_arg p')

      val (ps, c) = split_last (strip_objimp p)

      val qs = filter P ps

      val q = if P c then c else @{cterm "False"}

      val ng = fold_rev (fn (a,v) => fn t => Thm.capply a (Thm.cabs v t)) qvs 

          (fold_rev (fn p => fn q => Thm.capply (Thm.capply @{cterm "op -->"} p) q) qs q)

      val g = Thm.capply (Thm.capply @{cterm "op ==>"} (Thm.capply @{cterm "Trueprop"} ng)) p'

      val ntac = (case qs of [] => q aconvc @{cterm "False"}

                          | _ => false)

     in 

     if ntac then no_tac

       else rtac (Goal.prove_internal [] g (K (blast_tac HOL_cs 1))) i

     end)

 end;

 local

  fun isnum t = case t of 

    Const(@{const_name Groups.zero},_) => true

  | Const(@{const_name Groups.one},_) => true

  | @{term "Suc"}$s => isnum s

  | @{term "nat"}$s => isnum s

  | @{term "int"}$s => isnum s

  | Const(@{const_name Groups.uminus},_)$s => isnum s

  | Const(@{const_name Groups.plus},_)$l$r => isnum l andalso isnum r

  | Const(@{const_name Groups.times},_)$l$r => isnum l andalso isnum r

  | Const(@{const_name Groups.minus},_)$l$r => isnum l andalso isnum r

  | Const(@{const_name Power.power},_)$l$r => isnum l andalso isnum r

  | Const(@{const_name Divides.mod},_)$l$r => isnum l andalso isnum r

  | Const(@{const_name Divides.div},_)$l$r => isnum l andalso isnum r

  | _ => can HOLogic.dest_number t orelse can HOLogic.dest_nat t

  fun ty cts t = 

  if not (typ_of (ctyp_of_term t) mem [HOLogic.intT, HOLogic.natT, HOLogic.boolT]) then false 

     else case term_of t of 

       c$l$r => if c mem [@{term"op *::int => _"}, @{term"op *::nat => _"}] 

                then not (isnum l orelse isnum r)

                else not (member (op aconv) cts c)

     | c$_ => not (member (op aconv) cts c)

     | c => not (member (op aconv) cts c)

  val term_constants =

   let fun h acc t = case t of

     Const _ => insert (op aconv) t acc

   | a$b => h (h acc a) b

   | Abs (_,_,t) => h acc t

   | _ => acc

  in h [] end;

 in 

 fun is_relevant ctxt ct = 

  subset (op aconv) (term_constants (term_of ct) , snd (CooperData.get ctxt))

  andalso forall (fn Free (_,T) => T mem [@{typ "int"}, @{typ nat}]) (OldTerm.term_frees (term_of ct))

  andalso forall (fn Var (_,T) => T mem [@{typ "int"}, @{typ nat}]) (OldTerm.term_vars (term_of ct));

 fun int_nat_terms ctxt ct =

  let 

   val cts = snd (CooperData.get ctxt)

   fun h acc t = if ty cts t then insert (op aconvc) t acc else

    case (term_of t) of

     _$_ => h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)

   | Abs(_,_,_) => Thm.dest_abs NONE t ||> h acc |> uncurry (remove (op aconvc))

   | _ => acc

  in h [] ct end

 end;

 fun generalize_tac f = CSUBGOAL (fn (p, i) => PRIMITIVE (fn st =>

  let 

    fun all T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "all"}

    fun gen x t = Thm.capply (all (ctyp_of_term x)) (Thm.cabs x t)

    val ts = sort (fn (a,b) => Term_Ord.fast_term_ord (term_of a, term_of b)) (f p)

    val p' = fold_rev gen ts p

  in implies_intr p' (implies_elim st (fold forall_elim ts (assume p'))) end));

 local

 val ss1 = comp_ss

   addsimps @{thms simp_thms} @ [@{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"}]

 val ss2 = HOL_basic_ss

   addsimps [@{thm "nat_0_le"}, @{thm "int_nat_number_of"},

             @{thm "all_nat"}, @{thm "ex_nat"}, @{thm "number_of1"}, 

             @{thm "number_of2"}, @{thm "int_0"}, @{thm "int_1"}, @{thm "Suc_eq_plus1"}]

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

 val div_mod_ss = HOL_basic_ss addsimps @{thms simp_thms}

   @ map (symmetric o mk_meta_eq) 

     [@{thm "dvd_eq_mod_eq_0"},

      @{thm "mod_add_left_eq"}, @{thm "mod_add_right_eq"}, 

      @{thm "mod_add_eq"}, @{thm "div_add1_eq"}, @{thm "zdiv_zadd1_eq"}]

   @ [@{thm "mod_self"}, @{thm "zmod_self"}, @{thm "mod_by_0"}, 

      @{thm "div_by_0"}, @{thm "DIVISION_BY_ZERO"} RS conjunct1, 

      @{thm "DIVISION_BY_ZERO"} RS conjunct2, @{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_eq_plus1"}]

   @ @{thms add_ac}

  addsimprocs [cancel_div_mod_nat_proc, cancel_div_mod_int_proc]

  val splits_ss = comp_ss addsimps [@{thm "mod_div_equality'"}] addsplits 

      [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, 

       @{thm "split_min"}, @{thm "split_max"}, @{thm "abs_split"}]

 in

 fun nat_to_int_tac ctxt = 

   simp_tac (Simplifier.context ctxt ss1) THEN_ALL_NEW

   simp_tac (Simplifier.context ctxt ss2) THEN_ALL_NEW

   simp_tac (Simplifier.context ctxt comp_ss);

 fun div_mod_tac ctxt i = simp_tac (Simplifier.context ctxt div_mod_ss) i;

 fun splits_tac ctxt i = simp_tac (Simplifier.context ctxt splits_ss) i;

 end;

 fun core_cooper_tac ctxt = CSUBGOAL (fn (p, i) =>

    let 

     val cpth = 

        if !quick_and_dirty 

        then linzqe_oracle (Thm.cterm_of (ProofContext.theory_of ctxt)

              (Envir.beta_norm (Pattern.eta_long [] (term_of (Thm.dest_arg p)))))

        else arg_conv (Cooper.cooper_conv ctxt) p

     val p' = Thm.rhs_of cpth

     val th = implies_intr p' (equal_elim (symmetric cpth) (assume p'))

    in rtac th i end

    handle Cooper.COOPER _ => no_tac);

 fun finish_tac q = SUBGOAL (fn (_, i) =>

   (if q then I else TRY) (rtac TrueI i));

 fun cooper_tac elim add_ths del_ths ctxt =

 let val ss = Simplifier.context ctxt (fst (CooperData.get ctxt)) delsimps del_ths addsimps add_ths

     val aprems = Arith_Data.get_arith_facts ctxt

 in

   Method.insert_tac aprems

   THEN_ALL_NEW Object_Logic.full_atomize_tac

   THEN_ALL_NEW CONVERSION Thm.eta_long_conversion

   THEN_ALL_NEW simp_tac ss

   THEN_ALL_NEW (TRY o generalize_tac (int_nat_terms ctxt))

   THEN_ALL_NEW Object_Logic.full_atomize_tac

   THEN_ALL_NEW (thin_prems_tac (is_relevant ctxt))

   THEN_ALL_NEW Object_Logic.full_atomize_tac

   THEN_ALL_NEW div_mod_tac ctxt

   THEN_ALL_NEW splits_tac ctxt

   THEN_ALL_NEW simp_tac ss

   THEN_ALL_NEW CONVERSION Thm.eta_long_conversion

   THEN_ALL_NEW nat_to_int_tac ctxt

   THEN_ALL_NEW (core_cooper_tac ctxt)

   THEN_ALL_NEW finish_tac elim

 end;

 end;