1 (* Title: HOL/Complex/ex/mirtac.ML |
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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 |
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6 structure MirTac = |
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7 struct |
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8 |
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9 val trace = ref false; |
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10 fun trace_msg s = if !trace then tracing s else (); |
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11 |
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12 val mir_ss = |
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13 let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff", "real_of_int_le_iff"] |
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14 in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) |
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15 end; |
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16 |
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17 val nT = HOLogic.natT; |
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18 val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", |
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19 "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]; |
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20 |
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21 val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", |
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22 "add_Suc", "add_number_of_left", "mult_number_of_left", |
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23 "Suc_eq_add_numeral_1"])@ |
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24 (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) |
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25 @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} |
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26 val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, |
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27 @{thm "real_of_nat_number_of"}, |
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28 @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"}, |
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29 @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"}, |
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30 @{thm "Ring_and_Field.divide_zero"}, |
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31 @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, |
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32 @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, |
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33 @{thm "diff_def"}, @{thm "minus_divide_left"}] |
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34 val comp_ths = ths @ comp_arith @ simp_thms |
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35 |
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36 |
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37 val zdvd_int = @{thm "zdvd_int"}; |
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38 val zdiff_int_split = @{thm "zdiff_int_split"}; |
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39 val all_nat = @{thm "all_nat"}; |
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40 val ex_nat = @{thm "ex_nat"}; |
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41 val number_of1 = @{thm "number_of1"}; |
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42 val number_of2 = @{thm "number_of2"}; |
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43 val split_zdiv = @{thm "split_zdiv"}; |
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44 val split_zmod = @{thm "split_zmod"}; |
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45 val mod_div_equality' = @{thm "mod_div_equality'"}; |
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46 val split_div' = @{thm "split_div'"}; |
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47 val Suc_plus1 = @{thm "Suc_plus1"}; |
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48 val imp_le_cong = @{thm "imp_le_cong"}; |
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49 val conj_le_cong = @{thm "conj_le_cong"}; |
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50 val nat_mod_add_eq = @{thm "mod_add1_eq"} RS sym; |
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51 val nat_mod_add_left_eq = @{thm "mod_add_left_eq"} RS sym; |
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52 val nat_mod_add_right_eq = @{thm "mod_add_right_eq"} RS sym; |
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53 val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym; |
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54 val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym; |
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55 val int_mod_add_right_eq = @{thm "zmod_zadd_right_eq"} RS sym; |
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56 val nat_div_add_eq = @{thm "div_add1_eq"} RS sym; |
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57 val int_div_add_eq = @{thm "zdiv_zadd1_eq"} RS sym; |
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58 val ZDIVISION_BY_ZERO_MOD = @{thm "DIVISION_BY_ZERO"} RS conjunct2; |
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59 val ZDIVISION_BY_ZERO_DIV = @{thm "DIVISION_BY_ZERO"} RS conjunct1; |
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60 |
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61 fun prepare_for_mir thy q fm = |
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62 let |
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63 val ps = Logic.strip_params fm |
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64 val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) |
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65 val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) |
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66 fun mk_all ((s, T), (P,n)) = |
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67 if 0 mem loose_bnos P then |
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68 (HOLogic.all_const T $ Abs (s, T, P), n) |
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69 else (incr_boundvars ~1 P, n-1) |
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70 fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; |
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71 val rhs = hs |
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72 (* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) |
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73 val np = length ps |
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74 val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) |
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75 (foldr HOLogic.mk_imp c rhs, np) ps |
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76 val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT) |
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77 (term_frees fm' @ term_vars fm'); |
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78 val fm2 = foldr mk_all2 fm' vs |
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79 in (fm2, np + length vs, length rhs) end; |
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80 |
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81 (*Object quantifier to meta --*) |
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82 fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; |
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83 |
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84 (* object implication to meta---*) |
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85 fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; |
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86 |
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87 |
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88 fun mir_tac ctxt q i = |
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89 (ObjectLogic.atomize_prems_tac i) |
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90 THEN (simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) i) |
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91 THEN (REPEAT_DETERM (split_tac [@{thm "split_min"}, @{thm "split_max"},@{thm "abs_split"}] i)) |
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92 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_mir thy 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, |
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101 @{thm "mod_self"}, @{thm "zmod_self"}, |
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102 @{thm "zdiv_zero"},@{thm "zmod_zero"},@{thm "div_0"}, @{thm "mod_0"}, |
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103 @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"}, |
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104 @{thm "Suc_plus1"}] |
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105 addsimps @{thms add_ac} |
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106 addsimprocs [cancel_div_mod_proc] |
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107 val simpset0 = HOL_basic_ss |
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108 addsimps [mod_div_equality', Suc_plus1] |
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109 addsimps comp_ths |
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110 addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, @{thm "split_min"}, @{thm "split_max"}] |
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111 (* Simp rules for changing (n::int) to int n *) |
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112 val simpset1 = HOL_basic_ss |
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113 addsimps [@{thm "nat_number_of_def"}, @{thm "zdvd_int"}] @ map (fn r => r RS sym) |
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114 [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"}, |
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115 @{thm "zmult_int"}] |
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116 addsplits [@{thm "zdiff_int_split"}] |
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117 (*simp rules for elimination of int n*) |
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118 |
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119 val simpset2 = HOL_basic_ss |
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120 addsimps [@{thm "nat_0_le"}, @{thm "all_nat"}, @{thm "ex_nat"}, @{thm "number_of1"}, |
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121 @{thm "number_of2"}, @{thm "int_0"}, @{thm "int_1"}] |
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122 addcongs [@{thm "conj_le_cong"}, @{thm "imp_le_cong"}] |
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123 (* simp rules for elimination of abs *) |
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124 val ct = cterm_of thy (HOLogic.mk_Trueprop t) |
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125 (* Theorem for the nat --> int transformation *) |
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126 val pre_thm = Seq.hd (EVERY |
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127 [simp_tac mod_div_simpset 1, simp_tac simpset0 1, |
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128 TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), TRY (simp_tac mir_ss 1)] |
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129 (trivial ct)) |
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130 fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) |
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131 (* The result of the quantifier elimination *) |
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132 val (th, tac) = case (prop_of pre_thm) of |
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133 Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => |
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134 let val pth = |
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135 (* If quick_and_dirty then run without proof generation as oracle*) |
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136 if !quick_and_dirty |
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137 then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1)) |
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138 else mirfr_oracle (true, cterm_of thy (Pattern.eta_long [] t1)) |
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139 in |
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140 (trace_msg ("calling procedure with term:\n" ^ |
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141 Syntax.string_of_term ctxt t1); |
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142 ((pth RS iffD2) RS pre_thm, |
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143 assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) |
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144 end |
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145 | _ => (pre_thm, assm_tac i) |
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146 in (rtac (((mp_step nh) o (spec_step np)) th) i |
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147 THEN tac) st |
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148 end handle Subscript => no_tac st); |
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149 |
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150 fun mir_args meth = |
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151 let val parse_flag = |
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152 Args.$$$ "no_quantify" >> (K (K false)); |
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153 in |
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154 Method.simple_args |
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155 (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> |
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156 curry (Library.foldl op |>) true) |
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157 (fn q => fn ctxt => meth ctxt q 1) |
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158 end; |
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159 |
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160 fun mir_method ctxt q i = Method.METHOD (fn facts => |
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161 Method.insert_tac facts 1 THEN mir_tac ctxt q i); |
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162 |
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163 val setup = |
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164 Method.add_method ("mir", |
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165 mir_args mir_method, |
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166 "decision procedure for MIR arithmetic"); |
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167 |
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168 |
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169 end |
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