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