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1 (* Title: HOL/Reflection/cooper_tac.ML |
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2 Author: Amine Chaieb, TU Muenchen |
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3 *) |
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4 |
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5 structure Cooper_Tac = |
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6 struct |
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7 |
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8 val trace = ref false; |
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9 fun trace_msg s = if !trace then tracing s else (); |
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10 |
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11 val cooper_ss = @{simpset}; |
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12 |
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13 val nT = HOLogic.natT; |
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14 val binarith = @{thms normalize_bin_simps}; |
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15 val comp_arith = binarith @ simp_thms |
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16 |
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17 val zdvd_int = @{thm zdvd_int}; |
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18 val zdiff_int_split = @{thm zdiff_int_split}; |
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19 val all_nat = @{thm all_nat}; |
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20 val ex_nat = @{thm ex_nat}; |
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21 val number_of1 = @{thm number_of1}; |
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22 val number_of2 = @{thm number_of2}; |
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23 val split_zdiv = @{thm split_zdiv}; |
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24 val split_zmod = @{thm split_zmod}; |
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25 val mod_div_equality' = @{thm mod_div_equality'}; |
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26 val split_div' = @{thm split_div'}; |
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27 val Suc_plus1 = @{thm Suc_plus1}; |
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28 val imp_le_cong = @{thm imp_le_cong}; |
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29 val conj_le_cong = @{thm conj_le_cong}; |
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30 val nat_mod_add_eq = @{thm mod_add1_eq} RS sym; |
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31 val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym; |
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32 val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym; |
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33 val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym; |
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34 val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym; |
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35 val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym; |
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36 val nat_div_add_eq = @{thm div_add1_eq} RS sym; |
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37 val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; |
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38 |
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39 fun prepare_for_linz q fm = |
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40 let |
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41 val ps = Logic.strip_params fm |
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42 val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) |
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43 val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) |
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44 fun mk_all ((s, T), (P,n)) = |
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45 if 0 mem loose_bnos P then |
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46 (HOLogic.all_const T $ Abs (s, T, P), n) |
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47 else (incr_boundvars ~1 P, n-1) |
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48 fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; |
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49 val rhs = hs |
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50 val np = length ps |
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51 val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) |
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52 (foldr HOLogic.mk_imp c rhs, np) ps |
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53 val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT) |
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54 (OldTerm.term_frees fm' @ OldTerm.term_vars fm'); |
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55 val fm2 = foldr mk_all2 fm' vs |
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56 in (fm2, np + length vs, length rhs) end; |
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57 |
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58 (*Object quantifier to meta --*) |
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59 fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; |
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60 |
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61 (* object implication to meta---*) |
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62 fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; |
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63 |
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64 |
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65 fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st => |
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66 let |
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67 val g = List.nth (prems_of st, i - 1) |
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68 val thy = ProofContext.theory_of ctxt |
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69 (* Transform the term*) |
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70 val (t,np,nh) = prepare_for_linz q g |
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71 (* Some simpsets for dealing with mod div abs and nat*) |
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72 val mod_div_simpset = HOL_basic_ss |
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73 addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, |
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74 nat_mod_add_right_eq, int_mod_add_eq, |
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75 int_mod_add_right_eq, int_mod_add_left_eq, |
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76 nat_div_add_eq, int_div_add_eq, |
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77 @{thm mod_self}, @{thm "zmod_self"}, |
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78 @{thm mod_by_0}, @{thm div_by_0}, |
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79 @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"}, |
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80 @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"}, |
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81 Suc_plus1] |
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82 addsimps @{thms add_ac} |
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83 addsimprocs [cancel_div_mod_proc] |
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84 val simpset0 = HOL_basic_ss |
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85 addsimps [mod_div_equality', Suc_plus1] |
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86 addsimps comp_arith |
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87 addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}] |
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88 (* Simp rules for changing (n::int) to int n *) |
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89 val simpset1 = HOL_basic_ss |
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90 addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym) |
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91 [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}] |
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92 addsplits [zdiff_int_split] |
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93 (*simp rules for elimination of int n*) |
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94 |
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95 val simpset2 = HOL_basic_ss |
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96 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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97 addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}] |
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98 (* simp rules for elimination of abs *) |
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99 val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}] |
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100 val ct = cterm_of thy (HOLogic.mk_Trueprop t) |
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101 (* Theorem for the nat --> int transformation *) |
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102 val pre_thm = Seq.hd (EVERY |
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103 [simp_tac mod_div_simpset 1, simp_tac simpset0 1, |
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104 TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), |
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105 TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)] |
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106 (trivial ct)) |
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107 fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) |
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108 (* The result of the quantifier elimination *) |
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109 val (th, tac) = case (prop_of pre_thm) of |
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110 Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => |
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111 let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1)) |
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112 in |
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113 ((pth RS iffD2) RS pre_thm, |
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114 assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i)) |
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115 end |
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116 | _ => (pre_thm, assm_tac i) |
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117 in (rtac (((mp_step nh) o (spec_step np)) th) i |
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118 THEN tac) st |
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119 end handle Subscript => no_tac st); |
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120 |
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121 fun linz_args meth = |
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122 let val parse_flag = |
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123 Args.$$$ "no_quantify" >> (K (K false)); |
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124 in |
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125 Method.simple_args |
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126 (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> |
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127 curry (Library.foldl op |>) true) |
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128 (fn q => fn ctxt => meth ctxt q 1) |
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129 end; |
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130 |
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131 fun linz_method ctxt q i = Method.METHOD (fn facts => |
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132 Method.insert_tac facts 1 THEN linz_tac ctxt q i); |
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133 |
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134 val setup = |
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135 Method.add_method ("cooper", |
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136 linz_args linz_method, |
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137 "decision procedure for linear integer arithmetic"); |
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138 |
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139 end |