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1 (* |
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2 ID: $Id$ |
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3 Author: Amine Chaieb, TU Muenchen |
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4 |
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5 Ferrante and Rackoff Algorithm. |
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6 *) |
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7 |
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8 structure Ferrante_Rackoff: |
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9 sig |
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10 val trace : bool ref |
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11 val ferrack_tac : bool -> int -> tactic |
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12 val setup : theory -> theory |
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13 end = |
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14 struct |
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15 |
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16 val trace = ref false; |
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17 fun trace_msg s = if !trace then tracing s else (); |
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18 |
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19 val context_ss = simpset_of (the_context ()); |
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20 |
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21 val nT = HOLogic.natT; |
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22 val binarith = map thm |
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23 ["Pls_0_eq", "Min_1_eq", |
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24 "bin_pred_Pls","bin_pred_Min","bin_pred_1","bin_pred_0", |
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25 "bin_succ_Pls", "bin_succ_Min", "bin_succ_1", "bin_succ_0", |
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26 "bin_add_Pls", "bin_add_Min", "bin_add_BIT_0", "bin_add_BIT_10", |
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27 "bin_add_BIT_11", "bin_minus_Pls", "bin_minus_Min", "bin_minus_1", |
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28 "bin_minus_0", "bin_mult_Pls", "bin_mult_Min", "bin_mult_1", "bin_mult_0", |
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29 "bin_add_Pls_right", "bin_add_Min_right"]; |
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30 val intarithrel = |
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31 (map thm ["int_eq_number_of_eq","int_neg_number_of_BIT", |
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32 "int_le_number_of_eq","int_iszero_number_of_0", |
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33 "int_less_number_of_eq_neg"]) @ |
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34 (map (fn s => thm s RS thm "lift_bool") |
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35 ["int_iszero_number_of_Pls","int_iszero_number_of_1", |
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36 "int_neg_number_of_Min"])@ |
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37 (map (fn s => thm s RS thm "nlift_bool") |
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38 ["int_nonzero_number_of_Min","int_not_neg_number_of_Pls"]); |
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39 |
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40 val intarith = map thm ["int_number_of_add_sym", "int_number_of_minus_sym", |
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41 "int_number_of_diff_sym", "int_number_of_mult_sym"]; |
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42 val natarith = map thm ["add_nat_number_of", "diff_nat_number_of", |
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43 "mult_nat_number_of", "eq_nat_number_of", |
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44 "less_nat_number_of"] |
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45 val powerarith = |
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46 (map thm ["nat_number_of", "zpower_number_of_even", |
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47 "zpower_Pls", "zpower_Min"]) @ |
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48 [(Tactic.simplify true [thm "zero_eq_Numeral0_nring", |
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49 thm "one_eq_Numeral1_nring"] |
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50 (thm "zpower_number_of_odd"))] |
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51 |
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52 val comp_arith = binarith @ intarith @ intarithrel @ natarith |
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53 @ powerarith @[thm"not_false_eq_true", thm "not_true_eq_false"]; |
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54 |
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55 fun prepare_for_linr sg q fm = |
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56 let |
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57 val ps = Logic.strip_params fm |
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58 val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) |
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59 val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) |
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60 fun mk_all ((s, T), (P,n)) = |
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61 if 0 mem loose_bnos P then |
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62 (HOLogic.all_const T $ Abs (s, T, P), n) |
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63 else (incr_boundvars ~1 P, n-1) |
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64 fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; |
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65 val rhs = hs |
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66 (* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) |
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67 val np = length ps |
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68 val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) |
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69 (foldr HOLogic.mk_imp c rhs, np) ps |
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70 val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT) |
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71 (term_frees fm' @ term_vars fm'); |
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72 val fm2 = foldr mk_all2 fm' vs |
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73 in (fm2, np + length vs, length rhs) end; |
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74 |
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75 (*Object quantifier to meta --*) |
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76 fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; |
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77 |
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78 (* object implication to meta---*) |
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79 fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; |
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80 |
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81 |
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82 fun ferrack_tac q i = ObjectLogic.atomize_tac i THEN (fn st => |
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83 let |
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84 val g = List.nth (prems_of st, i - 1) |
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85 val sg = sign_of_thm st |
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86 (* Transform the term*) |
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87 val (t,np,nh) = prepare_for_linr sg q g |
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88 (* Some simpsets for dealing with mod div abs and nat*) |
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89 val simpset0 = HOL_basic_ss addsimps comp_arith addsplits [split_min, split_max] |
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90 (* simp rules for elimination of abs *) |
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91 val simpset3 = HOL_basic_ss addsplits [abs_split] |
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92 val ct = cterm_of sg (HOLogic.mk_Trueprop t) |
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93 (* Theorem for the nat --> int transformation *) |
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94 val pre_thm = Seq.hd (EVERY |
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95 [simp_tac simpset0 1, |
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96 TRY (simp_tac simpset3 1), TRY (simp_tac context_ss 1)] |
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97 (trivial ct)) |
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98 fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) |
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99 (* The result of the quantifier elimination *) |
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100 val (th, tac) = case (prop_of pre_thm) of |
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101 Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => |
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102 let val pth = Ferrante_Rackoff_Proof.qelim (cterm_of sg (Pattern.eta_long [] t1)) |
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103 in |
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104 (trace_msg ("calling procedure with term:\n" ^ |
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105 Sign.string_of_term sg t1); |
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106 ((pth RS iffD2) RS pre_thm, |
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107 assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) |
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108 end |
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109 | _ => (pre_thm, assm_tac i) |
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110 in (rtac (((mp_step nh) o (spec_step np)) th) i |
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111 THEN tac) st |
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112 end handle Subscript => no_tac st | Ferrante_Rackoff_Proof.FAILURE _ => no_tac st); |
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113 |
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114 fun ferrack_args meth = |
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115 let val parse_flag = |
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116 Args.$$$ "no_quantify" >> (K (K false)); |
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117 in |
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118 Method.simple_args |
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119 (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> |
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120 curry (Library.foldl op |>) true) |
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121 (fn q => fn _ => meth q 1) |
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122 end; |
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123 |
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124 val setup = |
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125 Method.add_method ("ferrack", |
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126 ferrack_args (Method.SIMPLE_METHOD oo ferrack_tac), |
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127 "LCF-proof-producing decision procedure for linear real arithmetic"); |
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128 |
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129 end |