src/HOL/ex/coopertac.ML
author chaieb
Wed Jun 06 16:12:08 2007 +0200 (2007-06-06)
changeset 23274 f997514ad8f4
child 23318 6d68b07ab5cf
permissions -rw-r--r--
New Reflected Presburger added to HOL/ex
     1 structure LinZTac =
     2 struct
     3 
     4 val trace = ref false;
     5 fun trace_msg s = if !trace then tracing s else ();
     6 
     7 val cooper_ss = @{simpset};
     8 
     9 val nT = HOLogic.natT;
    10 val binarith = map thm
    11   ["Pls_0_eq", "Min_1_eq"];
    12  val intarithrel = 
    13      (map thm ["int_eq_number_of_eq","int_neg_number_of_BIT", 
    14 		"int_le_number_of_eq","int_iszero_number_of_0",
    15 		"int_less_number_of_eq_neg"]) @
    16      (map (fn s => thm s RS thm "lift_bool") 
    17 	  ["int_iszero_number_of_Pls","int_iszero_number_of_1",
    18 	   "int_neg_number_of_Min"])@
    19      (map (fn s => thm s RS thm "nlift_bool") 
    20 	  ["int_nonzero_number_of_Min","int_not_neg_number_of_Pls"]);
    21      
    22 val intarith = map thm ["int_number_of_add_sym", "int_number_of_minus_sym",
    23 			"int_number_of_diff_sym", "int_number_of_mult_sym"];
    24 val natarith = map thm ["add_nat_number_of", "diff_nat_number_of",
    25 			"mult_nat_number_of", "eq_nat_number_of",
    26 			"less_nat_number_of"]
    27 val powerarith = 
    28     (map thm ["nat_number_of", "zpower_number_of_even", 
    29 	      "zpower_Pls", "zpower_Min"]) @ 
    30     [simplify (HOL_basic_ss addsimps [thm "zero_eq_Numeral0_nring", 
    31 			   thm "one_eq_Numeral1_nring"])
    32   (thm "zpower_number_of_odd")]
    33 
    34 val comp_arith = binarith @ intarith @ intarithrel @ natarith 
    35 	    @ powerarith @[thm"not_false_eq_true", thm "not_true_eq_false"];
    36 
    37 
    38 val zdvd_int = thm "zdvd_int";
    39 val zdiff_int_split = thm "zdiff_int_split";
    40 val all_nat = thm "all_nat";
    41 val ex_nat = thm "ex_nat";
    42 val number_of1 = thm "number_of1";
    43 val number_of2 = thm "number_of2";
    44 val split_zdiv = thm "split_zdiv";
    45 val split_zmod = thm "split_zmod";
    46 val mod_div_equality' = thm "mod_div_equality'";
    47 val split_div' = thm "split_div'";
    48 val Suc_plus1 = thm "Suc_plus1";
    49 val imp_le_cong = thm "imp_le_cong";
    50 val conj_le_cong = thm "conj_le_cong";
    51 val nat_mod_add_eq = mod_add1_eq RS sym;
    52 val nat_mod_add_left_eq = mod_add_left_eq RS sym;
    53 val nat_mod_add_right_eq = mod_add_right_eq RS sym;
    54 val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym;
    55 val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym;
    56 val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym;
    57 val nat_div_add_eq = @{thm "div_add1_eq"} RS sym;
    58 val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym;
    59 val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2;
    60 val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1;
    61 
    62 (*
    63 val fn_rews = List.concat (map thms ["allpairs.simps","iupt.simps","decr.simps", "decrnum.simps","disjuncts.simps","simpnum.simps", "simpfm.simps","numadd.simps","nummul.simps","numneg_def","numsub","simp_num_pair_def","not.simps","prep.simps","qelim.simps","minusinf.simps","plusinf.simps","rsplit0.simps","rlfm.simps","\\<Upsilon>.simps","\\<upsilon>.simps","linrqe_def", "ferrack_def", "Let_def", "numsub_def", "numneg_def","DJ_def", "imp_def", "evaldjf_def", "djf_def", "split_def", "eq_def", "disj_def", "simp_num_pair_def", "conj_def", "lt_def", "neq_def","gt_def"]);
    64 *)
    65 fun prepare_for_linz q fm = 
    66   let
    67     val ps = Logic.strip_params fm
    68     val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
    69     val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
    70     fun mk_all ((s, T), (P,n)) =
    71       if 0 mem loose_bnos P then
    72         (HOLogic.all_const T $ Abs (s, T, P), n)
    73       else (incr_boundvars ~1 P, n-1)
    74     fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
    75       val rhs = hs
    76 (*    val (rhs,irhs) = List.partition (relevant (rev ps)) hs *)
    77     val np = length ps
    78     val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
    79       (foldr HOLogic.mk_imp c rhs, np) ps
    80     val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
    81       (term_frees fm' @ term_vars fm');
    82     val fm2 = foldr mk_all2 fm' vs
    83   in (fm2, np + length vs, length rhs) end;
    84 
    85 (*Object quantifier to meta --*)
    86 fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
    87 
    88 (* object implication to meta---*)
    89 fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
    90 
    91 
    92 fun linz_tac ctxt q i = ObjectLogic.atomize_tac i THEN (fn st =>
    93   let
    94     val g = List.nth (prems_of st, i - 1)
    95     val thy = ProofContext.theory_of ctxt
    96     (* Transform the term*)
    97     val (t,np,nh) = prepare_for_linz q g
    98     (* Some simpsets for dealing with mod div abs and nat*)
    99     val mod_div_simpset = HOL_basic_ss 
   100 			addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, 
   101 				  nat_mod_add_right_eq, int_mod_add_eq, 
   102 				  int_mod_add_right_eq, int_mod_add_left_eq,
   103 				  nat_div_add_eq, int_div_add_eq,
   104 				  mod_self, @{thm "zmod_self"},
   105 				  DIVISION_BY_ZERO_MOD,DIVISION_BY_ZERO_DIV,
   106 				  ZDIVISION_BY_ZERO_MOD,ZDIVISION_BY_ZERO_DIV,
   107 				  @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},
   108 				  @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"},
   109 				  Suc_plus1]
   110 			addsimps add_ac
   111 			addsimprocs [cancel_div_mod_proc]
   112     val simpset0 = HOL_basic_ss
   113       addsimps [mod_div_equality', Suc_plus1]
   114       addsimps comp_arith
   115       addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]
   116     (* Simp rules for changing (n::int) to int n *)
   117     val simpset1 = HOL_basic_ss
   118       addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)
   119         [int_int_eq, zle_int, zless_int, zadd_int, zmult_int]
   120       addsplits [zdiff_int_split]
   121     (*simp rules for elimination of int n*)
   122 
   123     val simpset2 = HOL_basic_ss
   124       addsimps [nat_0_le, all_nat, ex_nat, number_of1, number_of2, int_0, int_1]
   125       addcongs [conj_le_cong, imp_le_cong]
   126     (* simp rules for elimination of abs *)
   127     val simpset3 = HOL_basic_ss addsplits [abs_split]
   128     val ct = cterm_of thy (HOLogic.mk_Trueprop t)
   129     (* Theorem for the nat --> int transformation *)
   130     val pre_thm = Seq.hd (EVERY
   131       [simp_tac mod_div_simpset 1, simp_tac simpset0 1,
   132        TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),
   133        TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]
   134       (trivial ct))
   135     fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
   136     (* The result of the quantifier elimination *)
   137     val (th, tac) = case (prop_of pre_thm) of
   138         Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
   139     let val pth = linzqe_oracle thy (Pattern.eta_long [] t1)
   140     in 
   141           ((pth RS iffD2) RS pre_thm,
   142             assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))
   143     end
   144       | _ => (pre_thm, assm_tac i)
   145   in (rtac (((mp_step nh) o (spec_step np)) th) i 
   146       THEN tac) st
   147   end handle Subscript => no_tac st);
   148 
   149 fun linz_args meth =
   150  let val parse_flag = 
   151          Args.$$$ "no_quantify" >> (K (K false));
   152  in
   153    Method.simple_args 
   154   (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
   155     curry (Library.foldl op |>) true)
   156     (fn q => fn ctxt => meth ctxt q 1)
   157   end;
   158 
   159 fun linz_method ctxt q i = Method.METHOD (fn facts =>
   160   Method.insert_tac facts 1 THEN linz_tac ctxt q i);
   161 
   162 val setup =
   163   Method.add_method ("cooper",
   164      linz_args linz_method,
   165      "decision procedure for linear integer arithmetic");
   166 
   167 end