src/HOL/Decision_Procs/cooper_tac.ML
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(*  Title:      HOL/Decision_Procs/cooper_tac.ML
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    Author:     Amine Chaieb, TU Muenchen
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*)
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structure Cooper_Tac =
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struct
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val trace = ref false;
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fun trace_msg s = if !trace then tracing s else ();
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val cooper_ss = @{simpset};
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val nT = HOLogic.natT;
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val binarith = @{thms normalize_bin_simps};
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val comp_arith = binarith @ simp_thms
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val zdvd_int = @{thm zdvd_int};
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val zdiff_int_split = @{thm zdiff_int_split};
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val all_nat = @{thm all_nat};
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val ex_nat = @{thm ex_nat};
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val number_of1 = @{thm number_of1};
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val number_of2 = @{thm number_of2};
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val split_zdiv = @{thm split_zdiv};
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val split_zmod = @{thm split_zmod};
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val mod_div_equality' = @{thm mod_div_equality'};
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val split_div' = @{thm split_div'};
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val Suc_plus1 = @{thm Suc_plus1};
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val imp_le_cong = @{thm imp_le_cong};
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val conj_le_cong = @{thm conj_le_cong};
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val mod_add_left_eq = @{thm mod_add_left_eq} RS sym;
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val mod_add_right_eq = @{thm mod_add_right_eq} RS sym;
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val mod_add_eq = @{thm mod_add_eq} RS sym;
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val nat_div_add_eq = @{thm div_add1_eq} RS sym;
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val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym;
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fun prepare_for_linz q fm = 
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  let
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    val ps = Logic.strip_params fm
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    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
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    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
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    fun mk_all ((s, T), (P,n)) =
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      if 0 mem loose_bnos P then
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        (HOLogic.all_const T $ Abs (s, T, P), n)
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      else (incr_boundvars ~1 P, n-1)
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    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
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    val rhs = hs
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    val np = length ps
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    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
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      (foldr HOLogic.mk_imp c rhs, np) ps
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    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
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      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');
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    val fm2 = foldr mk_all2 fm' vs
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  in (fm2, np + length vs, length rhs) end;
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(*Object quantifier to meta --*)
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fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
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(* object implication to meta---*)
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fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
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fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st =>
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  let
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    val g = List.nth (prems_of st, i - 1)
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    val thy = ProofContext.theory_of ctxt
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    (* Transform the term*)
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    val (t,np,nh) = prepare_for_linz q g
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    (* Some simpsets for dealing with mod div abs and nat*)
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    val mod_div_simpset = HOL_basic_ss 
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			addsimps [refl,mod_add_eq, mod_add_left_eq, 
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				  mod_add_right_eq,
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				  nat_div_add_eq, int_div_add_eq,
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				  @{thm mod_self}, @{thm "zmod_self"},
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				  @{thm mod_by_0}, @{thm div_by_0},
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				  @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},
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				  @{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"},
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				  Suc_plus1]
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			addsimps @{thms add_ac}
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			addsimprocs [cancel_div_mod_proc]
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    val simpset0 = HOL_basic_ss
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      addsimps [mod_div_equality', Suc_plus1]
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      addsimps comp_arith
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      addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]
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    (* Simp rules for changing (n::int) to int n *)
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    val simpset1 = HOL_basic_ss
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      addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)
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        [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}]
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      addsplits [zdiff_int_split]
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    (*simp rules for elimination of int n*)
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    val simpset2 = HOL_basic_ss
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      addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}]
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      addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}]
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    (* simp rules for elimination of abs *)
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    val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}]
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    val ct = cterm_of thy (HOLogic.mk_Trueprop t)
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    (* Theorem for the nat --> int transformation *)
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    val pre_thm = Seq.hd (EVERY
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      [simp_tac mod_div_simpset 1, simp_tac simpset0 1,
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       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),
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       TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]
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      (trivial ct))
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    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
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    (* The result of the quantifier elimination *)
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    val (th, tac) = case (prop_of pre_thm) of
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        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
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    let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1))
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    in 
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          ((pth RS iffD2) RS pre_thm,
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            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))
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    end
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      | _ => (pre_thm, assm_tac i)
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  in (rtac (((mp_step nh) o (spec_step np)) th) i 
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      THEN tac) st
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  end handle Subscript => no_tac st);
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fun linz_args meth =
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 let val parse_flag = 
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         Args.$$$ "no_quantify" >> (K (K false));
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 in
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   Method.simple_args 
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  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
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    curry (Library.foldl op |>) true)
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    (fn q => fn ctxt => meth ctxt q 1)
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  end;
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fun linz_method ctxt q i = Method.METHOD (fn facts =>
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  Method.insert_tac facts 1 THEN linz_tac ctxt q i);
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val setup =
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  Method.add_method ("cooper",
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     linz_args linz_method,
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     "decision procedure for linear integer arithmetic");
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end