1 | |
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2 | include "RTLabs/semantics.ma". |
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3 | include "common/StructuredTraces.ma". |
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4 | |
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5 | discriminator status_class. |
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6 | |
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7 | (* NB: For RTLabs we only classify branching behaviour as jumps. Other jumps |
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8 | will be added later (LTL → LIN). *) |
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9 | |
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10 | definition RTLabs_classify : state → status_class ≝ |
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11 | λs. match s with |
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12 | [ State f _ _ ⇒ |
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13 | match lookup_present ?? (f_graph (func f)) (next f) (next_ok f) with |
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14 | [ St_cond _ _ _ ⇒ cl_jump |
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15 | | St_jumptable _ _ ⇒ cl_jump |
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16 | | _ ⇒ cl_other |
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17 | ] |
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18 | | Callstate _ _ _ _ _ ⇒ cl_call |
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19 | | Returnstate _ _ _ _ ⇒ cl_return |
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20 | ]. |
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21 | |
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22 | definition is_cost_label : statement → bool ≝ |
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23 | λs. match s with [ St_cost _ _ ⇒ true | _ ⇒ false ]. |
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24 | |
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25 | definition RTLabs_cost : state → bool ≝ |
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26 | λs. match s with |
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27 | [ State f fs m ⇒ |
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28 | is_cost_label (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) |
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29 | | _ ⇒ false |
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30 | ]. |
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31 | |
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32 | definition RTLabs_status : genv → abstract_status ≝ |
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33 | λge. |
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34 | mk_abstract_status |
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35 | state |
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36 | (λs,s'. ∃t. eval_statement ge s = Value ??? 〈t,s'〉) |
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37 | (λs,c. RTLabs_classify s = c) |
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38 | (λs. RTLabs_cost s = true) |
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39 | (λs,s'. match s with |
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40 | [ mk_Sig s p ⇒ |
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41 | match s return λs. RTLabs_classify s = cl_call → ? with |
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42 | [ Callstate fd args dst stk m ⇒ |
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43 | λ_. match s' with |
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44 | [ State f fs m ⇒ match stk with [ nil ⇒ False | cons h t ⇒ next h = next f ] |
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45 | | _ ⇒ False |
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46 | ] |
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47 | | State f fs m ⇒ λH.⊥ |
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48 | | _ ⇒ λH.⊥ |
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49 | ] p |
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50 | ]). |
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51 | [ normalize in H; destruct |
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52 | | whd in H:(??%?); |
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53 | cases (lookup_present LabelTag statement (f_graph (func f)) (next f) (next_ok f)) in H; |
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54 | normalize try #a try #b try #c try #d try #e try #g try #h destruct |
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55 | ] qed. |
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56 | |
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57 | lemma RTLabs_not_cost : ∀ge,s. |
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58 | RTLabs_cost s = false → |
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59 | ¬ as_costed (RTLabs_status ge) s. |
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60 | #ge #s #E % whd in ⊢ (% → ?); >E #E' destruct |
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61 | qed. |
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62 | |
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63 | (* Before attempting to construct a structured trace, let's show that we can |
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64 | form flat traces with evidence that they were constructed from an execution. |
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65 | |
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66 | For now we don't consider I/O. *) |
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67 | |
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68 | |
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69 | coinductive exec_no_io (o:Type[0]) (i:o → Type[0]) : execution state o i → Prop ≝ |
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70 | | noio_stop : ∀a,b,c. exec_no_io o i (e_stop … a b c) |
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71 | | noio_step : ∀a,b,e. exec_no_io o i e → exec_no_io o i (e_step … a b e) |
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72 | | noio_wrong : ∀m. exec_no_io o i (e_wrong … m). |
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73 | |
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74 | (* add I/O? *) |
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75 | coinductive flat_trace (o:Type[0]) (i:o → Type[0]) (ge:genv) : state → Type[0] ≝ |
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76 | | ft_stop : ∀s. RTLabs_is_final s ≠ None ? → flat_trace o i ge s |
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77 | | ft_step : ∀s,tr,s'. eval_statement ge s = Value ??? 〈tr,s'〉 → flat_trace o i ge s' → flat_trace o i ge s |
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78 | | ft_wrong : ∀s,m. eval_statement ge s = Wrong ??? m → flat_trace o i ge s. |
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79 | |
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80 | coinductive not_wrong (o:Type[0]) (i:o → Type[0]) (ge:genv) : ∀s. flat_trace o i ge s → Type[0] ≝ |
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81 | | nw_stop : ∀s,H. not_wrong o i ge s (ft_stop o i ge s H) |
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82 | | nw_step : ∀s,tr,s',H,tr'. not_wrong o i ge s' tr' → not_wrong o i ge s (ft_step o i ge s tr s' H tr'). |
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83 | |
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84 | lemma still_not_wrong : ∀o,i,ge,s,tr,s',H,tr'. |
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85 | not_wrong o i ge s (ft_step o i ge s tr s' H tr') → |
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86 | not_wrong o i ge s' tr'. |
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87 | #o #i #ge #s #tr #s' #H #tr' #NW inversion NW |
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88 | [ #H105 #H106 #H107 #H108 #H109 destruct |
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89 | | #H111 #H112 #H113 #H114 #H115 #H116 #H117 #H118 #H119 destruct // |
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90 | ] qed. |
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91 | |
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92 | inductive flat_trace_prefix (o:Type[0]) (i:o → Type[0]) (ge:genv) : ∀s,s'. flat_trace o i ge s → flat_trace o i ge s' → Prop ≝ |
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93 | | ftp_refl : ∀s,t. flat_trace_prefix o i ge s s t t |
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94 | | ftp_step : ∀s1,tr,s2,s3,t2,t3,EV. |
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95 | flat_trace_prefix o i ge s2 s3 t2 t3 → |
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96 | flat_trace_prefix o i ge s1 s3 (ft_step o i ge s1 tr s2 EV t2) t3 |
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97 | . |
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98 | |
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99 | lemma concatenate_flat_trace_prefix : ∀o,i,ge,s1,s2,s3,t1,t2,t3. |
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100 | flat_trace_prefix o i ge s1 s2 t1 t2 → |
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101 | flat_trace_prefix o i ge s2 s3 t2 t3 → |
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102 | flat_trace_prefix o i ge s1 s3 t1 t3. |
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103 | #o #i #ge #s1 #s2 #s3 #t1 #t2 #t3 #p1 elim p1 |
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104 | [ // |
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105 | | #s1' #tr #s2' #s3' #t2' #t3' #EV #p2 #IH #p3 |
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106 | %2 /2/ |
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107 | ] qed. |
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108 | |
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109 | let corec make_flat_trace ge s |
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110 | (H:exec_no_io … (exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s))) : |
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111 | flat_trace io_out io_in ge s ≝ |
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112 | let e ≝ exec_inf_aux … RTLabs_fullexec ge (eval_statement ge s) in |
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113 | match e return λx. e = x → ? with |
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114 | [ e_stop tr i s' ⇒ λE. ft_step … s tr s' ? (ft_stop … s' ?) |
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115 | | e_step tr s' e' ⇒ λE. ft_step … s tr s' ? (make_flat_trace ge s' ?) |
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116 | | e_wrong m ⇒ λE. ft_wrong … s m ? |
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117 | | e_interact o f ⇒ λE. ⊥ |
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118 | ] (refl ? e). |
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119 | [ 1,2: whd in E:(??%?); >exec_inf_aux_unfold in E; |
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120 | cases (eval_statement ge s) |
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121 | [ 1,4: #O #K whd in ⊢ (??%? → ?); #E destruct |
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122 | | 2,5: * #tr #s1 whd in ⊢ (??%? → ?); |
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123 | >(?:is_final ????? = RTLabs_is_final s1) // |
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124 | lapply (refl ? (RTLabs_is_final s1)) |
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125 | cases (RTLabs_is_final s1) in ⊢ (???% → %); |
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126 | [ 1,3: #_ whd in ⊢ (??%? → ?); #E destruct |
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127 | | #i #_ whd in ⊢ (??%? → ?); #E destruct /2/ @refl |
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128 | | #i #E whd in ⊢ (??%? → ?); #E2 destruct >E % #E' destruct |
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129 | ] |
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130 | | *: #m whd in ⊢ (??%? → ?); #E destruct |
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131 | ] |
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132 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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133 | cases (eval_statement ge s) |
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134 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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135 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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136 | cases (is_final ?????) |
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137 | [ whd in ⊢ (??%? → ?); #E destruct @refl |
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138 | | #i whd in ⊢ (??%? → ?); #E destruct |
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139 | ] |
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140 | | #m whd in ⊢ (??%? → ?); #E destruct |
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141 | ] |
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142 | | whd in E:(??%?); >E in H; #H >exec_inf_aux_unfold in E; |
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143 | cases (eval_statement ge s) |
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144 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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145 | | * #tr #s1 whd in ⊢ (??%? → ?); |
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146 | cases (is_final ?????) |
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147 | [ whd in ⊢ (??%? → ?); #E |
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148 | change with (eval_statement ge s1) in E:(??(??????(?????%))?); |
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149 | destruct |
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150 | inversion H |
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151 | [ #a #b #c #E1 destruct |
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152 | | #trx #sx #ex #H1 #E2 #E3 destruct @H1 |
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153 | | #m #E1 destruct |
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154 | ] |
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155 | | #i whd in ⊢ (??%? → ?); #E destruct |
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156 | ] |
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157 | | #m whd in ⊢ (??%? → ?); #E destruct |
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158 | ] |
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159 | | whd in E:(??%?); >exec_inf_aux_unfold in E; |
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160 | cases (eval_statement ge s) |
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161 | [ #O #K whd in ⊢ (??%? → ?); #E destruct |
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162 | | * #tr1 #s1 whd in ⊢ (??%? → ?); |
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163 | cases (is_final ?????) |
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164 | [ whd in ⊢ (??%? → ?); #E destruct |
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165 | | #i whd in ⊢ (??%? → ?); #E destruct |
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166 | ] |
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167 | | #m whd in ⊢ (??%? → ?); #E destruct @refl |
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168 | ] |
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169 | | whd in E:(??%?); >E in H; #H |
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170 | inversion H |
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171 | [ #a #b #c #E destruct |
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172 | | #a #b #c #d #E1 destruct |
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173 | | #m #E1 destruct |
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174 | ] |
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175 | ] qed. |
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176 | |
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177 | let corec make_whole_flat_trace p s |
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178 | (H:exec_no_io … (exec_inf … RTLabs_fullexec p)) |
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179 | (I:make_initial_state ??? p = OK ? s) : |
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180 | flat_trace io_out io_in (make_global … RTLabs_fullexec p) s ≝ |
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181 | let ge ≝ make_global … p in |
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182 | let e ≝ exec_inf_aux ?? RTLabs_fullexec ge (Value … 〈E0, s〉) in |
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183 | match e return λx. e = x → ? with |
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184 | [ e_stop tr i s' ⇒ λE. ft_stop ?? ge s ? |
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185 | | e_step _ _ e' ⇒ λE. make_flat_trace ge s ? |
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186 | | e_wrong m ⇒ λE. ⊥ |
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187 | | e_interact o f ⇒ λE. ⊥ |
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188 | ] (refl ? e). |
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189 | [ whd in E:(??%?); >exec_inf_aux_unfold in E; |
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190 | whd in ⊢ (??%? → ?); |
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191 | >(?:is_final ????? = RTLabs_is_final s) // |
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192 | lapply (refl ? (RTLabs_is_final s)) |
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193 | cases (RTLabs_is_final s) in ⊢ (???% → %); |
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194 | [ #_ whd in ⊢ (??%? → ?); #E destruct |
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195 | | #i #E whd in ⊢ (??%? → ?); #E2 % #E3 destruct |
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196 | ] |
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197 | | whd in H:(???%); >I in H; whd in ⊢ (???% → ?); whd in E:(??%?); |
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198 | >exec_inf_aux_unfold in E ⊢ %; whd in ⊢ (??%? → ???% → ?); cases (is_final ?????) |
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199 | [ whd in ⊢ (??%? → ???% → ?); #E #H inversion H |
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200 | [ #a #b #c #E1 destruct |
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201 | | #tr1 #s1 #e1 #H1 #E1 #E2 -E2 -I destruct (E1) |
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202 | @H1 |
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203 | | #m #E1 destruct |
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204 | ] |
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205 | | #i whd in ⊢ (??%? → ???% → ?); #E destruct |
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206 | ] |
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207 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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208 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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209 | | whd in E:(??%?); >exec_inf_aux_unfold in E; whd in ⊢ (??%? → ?); |
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210 | cases (is_final ?????) [2:#i] whd in ⊢ (??%? → ?); #E destruct |
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211 | ] qed. |
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212 | |
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213 | (* Need a way to choose whether a called function terminates. Then, |
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214 | if the initial function terminates we generate a purely inductive structured trace, |
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215 | otherwise we start generating the coinductive one, and on every function call |
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216 | use the choice method again to decide whether to step over or keep going. |
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217 | |
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218 | Not quite what we need - have to decide on seeing each label whether we will see |
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219 | another or hit a non-terminating call? |
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220 | |
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221 | Also - need the notion of well-labelled in order to break loops. |
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222 | |
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223 | |
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224 | |
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225 | outline: |
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226 | |
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227 | does function terminate? |
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228 | - yes, get (bound on the number of steps until return), generate finite |
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229 | structure using bound as termination witness |
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230 | - no, get (¬ bound on steps to return), start building infinite trace out of |
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231 | finite steps. At calls, check for termination, generate appr. form. |
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232 | |
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233 | generating the finite parts: |
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234 | |
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235 | We start with the status after the call has been executed; well-labelling tells |
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236 | us that this is a labelled state. Now we want to generate a trace_label_return |
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237 | and also return the remainder of the flat trace. |
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238 | |
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239 | *) |
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240 | |
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241 | (* [will_return ge depth s trace] says that after a finite number of steps of |
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242 | [trace] from [s] we reach the return state for the current function. [depth] |
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243 | performs the call/return counting necessary for handling deeper function |
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244 | calls. It should be zero at the top level. *) |
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245 | inductive will_return (ge:genv) : nat → ∀s. flat_trace io_out io_in ge s → Type[0] ≝ |
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246 | | wr_step : ∀s,tr,s',depth,EX,trace. |
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247 | RTLabs_classify s = cl_other ∨ RTLabs_classify s = cl_jump → |
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248 | will_return ge depth s' trace → |
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249 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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250 | | wr_call : ∀s,tr,s',depth,EX,trace. |
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251 | RTLabs_classify s = cl_call → |
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252 | will_return ge (S depth) s' trace → |
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253 | will_return ge depth s (ft_step ?? ge s tr s' EX trace) |
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254 | | wr_ret : ∀s,tr,s',depth,EX,trace. |
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255 | RTLabs_classify s = cl_return → |
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256 | will_return ge depth s' trace → |
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257 | will_return ge (S depth) s (ft_step ?? ge s tr s' EX trace) |
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258 | (* Note that we require the ability to make a step after the return (this |
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259 | corresponds to somewhere that will be guaranteed to be a label at the |
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260 | end of the compilation chain). *) |
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261 | | wr_base : ∀s,tr,s',EX,trace. |
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262 | RTLabs_classify s = cl_return → |
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263 | will_return ge O s (ft_step ?? ge s tr s' EX trace) |
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264 | . |
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265 | |
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266 | (* The way we will use [will_return] won't satisfy Matita's guardedness check, |
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267 | so we will measure the length of these termination proofs and use an upper |
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268 | bound to show termination of the finite structured trace construction |
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269 | functions. *) |
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270 | |
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271 | let rec will_return_length ge d s tr (T:will_return ge d s tr) on T : nat ≝ |
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272 | match T with |
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273 | [ wr_step _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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274 | | wr_call _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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275 | | wr_ret _ _ _ _ _ _ _ T' ⇒ S (will_return_length … T') |
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276 | | wr_base _ _ _ _ _ _ ⇒ S O |
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277 | ]. |
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278 | |
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279 | include alias "arithmetics/nat.ma". |
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280 | |
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281 | (* Specialised to the particular situation it is used in. *) |
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282 | lemma wrl_nonzero : ∀ge,d,s,tr,T. O ≥ 3 * (will_return_length ge d s tr T) → False. |
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283 | #ge #d #s #tr * #s1 #tr1 #s2 [ 1,2,3: #d ] #EX #tr' #CL [1,2,3:#IH] |
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284 | whd in ⊢ (??(??%) → ?); |
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285 | >commutative_times |
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286 | #H lapply (le_plus_b … H) |
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287 | #H lapply (le_to_leb_true … H) |
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288 | normalize #E destruct |
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289 | qed. |
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290 | |
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291 | (* Inversion lemmas on [will_return] that also note the effect on the length |
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292 | of the proof. *) |
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293 | lemma will_return_call : ∀ge,d,s,tr,s',EX,trace. |
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294 | RTLabs_classify s = cl_call → |
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295 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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296 | ΣTM':will_return ge (S d) s' trace. will_return_length … TM > will_return_length … TM'. |
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297 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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298 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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299 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 destruct % // |
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300 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 @⊥ destruct >CL in H53; #E destruct |
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301 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 @⊥ destruct >CL in H66; #E destruct |
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302 | ] qed. |
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303 | |
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304 | lemma will_return_return : ∀ge,d,s,tr,s',EX,trace. |
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305 | RTLabs_classify s = cl_return → |
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306 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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307 | match d with |
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308 | [ O ⇒ True |
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309 | | S d' ⇒ |
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310 | ΣTM':will_return ge d' s' trace. will_return_length … TM > will_return_length … TM' |
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311 | ]. |
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312 | #ge #d #s #tr #s' #EX #trace #CL #TERM inversion TERM |
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313 | [ #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 #H30 #H31 @⊥ destruct >CL in H25; * #E destruct |
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314 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 @⊥ destruct >CL in H39; #E destruct |
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315 | | #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 #H59 destruct % // |
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316 | | #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct @I |
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317 | ] qed. |
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318 | |
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319 | lemma will_return_notfn : ∀ge,d,s,tr,s',EX,trace. |
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320 | (RTLabs_classify s = cl_other) ⊎ (RTLabs_classify s = cl_jump) → |
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321 | ∀TM:will_return ge d s (ft_step ?? ge s tr s' EX trace). |
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322 | ΣTM':will_return ge d s' trace. will_return_length … TM > will_return_length … TM'. |
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323 | #ge #d #s #tr #s' #EX #trace * #CL #TERM inversion TERM |
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324 | [ #H290 #H291 #H292 #H293 #H294 #H295 #H296 #H297 #H298 #H299 #H300 #H301 #H302 destruct % // |
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325 | | #H304 #H305 #H306 #H307 #H308 #H309 #H310 #H311 #H312 #H313 #H314 #H315 #H316 @⊥ destruct >CL in H310; #E destruct |
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326 | | #H318 #H319 #H320 #H321 #H322 #H323 #H324 #H325 #H326 #H327 #H328 #H329 #H330 @⊥ destruct >CL in H324; #E destruct |
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327 | | #H332 #H333 #H334 #H335 #H336 #H337 #H338 #H339 #H340 #H341 @⊥ destruct >CL in H337; #E destruct |
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328 | | #H343 #H344 #H345 #H346 #H347 #H348 #H349 #H350 #H351 #H352 #H353 #H354 #H355 destruct % // |
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329 | | #H357 #H358 #H359 #H360 #H361 #H362 #H363 #H364 #H365 #H366 #H367 #H368 #H369 @⊥ destruct >CL in H363; #E destruct |
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330 | | #H371 #H372 #H373 #H374 #H375 #H376 #H377 #H378 #H379 #H380 #H381 #H382 #H383 @⊥ destruct >CL in H377; #E destruct |
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331 | | #H385 #H386 #H387 #H388 #H389 #H390 #H391 #H392 #H393 #H394 @⊥ destruct >CL in H390; #E destruct |
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332 | ] qed. |
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333 | |
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334 | (* We require that labels appear after branch instructions and at the start of |
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335 | functions. The first is required for preciseness, the latter for soundness. |
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336 | We will make a separate requirement for there to be a finite number of steps |
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337 | between labels to catch loops for soundness (is this sufficient?). *) |
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338 | |
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339 | definition well_cost_labelled_statement : ∀f:internal_function. ∀s. labels_present (f_graph f) s → Prop ≝ |
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340 | λf,s. match s return λs. labels_present ? s → Prop with |
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341 | [ St_cond _ l1 l2 ⇒ λH. |
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342 | is_cost_label (lookup_present … (f_graph f) l1 ?) = true ∧ |
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343 | is_cost_label (lookup_present … (f_graph f) l2 ?) = true |
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344 | | St_jumptable _ ls ⇒ λH. |
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345 | (* I did have a dependent version of All here, but it's a pain. *) |
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346 | All … (λl. ∃H. is_cost_label (lookup_present … (f_graph f) l H) = true) ls |
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347 | | _ ⇒ λ_. True |
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348 | ]. whd in H; |
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349 | [ @(proj1 … H) |
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350 | | @(proj2 … H) |
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351 | ] qed. |
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352 | |
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353 | definition well_cost_labelled_fn : internal_function → Prop ≝ |
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354 | λf. (∀l. ∀H:present … (f_graph f) l. |
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355 | well_cost_labelled_statement f (lookup_present … (f_graph f) l H) (f_closed f l …)) ∧ |
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356 | is_cost_label (lookup_present … (f_graph f) (f_entry f) ?) = true. |
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357 | [ @lookup_lookup_present | cases (f_entry f) // ] qed. |
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358 | |
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359 | (* We need to ensure that any code we come across is well-cost-labelled. We may |
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360 | get function code from either the global environment or the state. *) |
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361 | |
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362 | definition well_cost_labelled_ge : genv → Prop ≝ |
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363 | λge. ∀b,f. find_funct_ptr ?? ge b = Some ? (Internal ? f) → well_cost_labelled_fn f. |
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364 | |
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365 | definition well_cost_labelled_state : state → Prop ≝ |
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366 | λs. match s with |
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367 | [ State f fs m ⇒ well_cost_labelled_fn (func f) ∧ All ? (λf. well_cost_labelled_fn (func f)) fs |
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368 | | Callstate fd _ _ fs _ ⇒ match fd with [ Internal fn ⇒ well_cost_labelled_fn fn | External _ ⇒ True ] ∧ |
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369 | All ? (λf. well_cost_labelled_fn (func f)) fs |
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370 | | Returnstate _ _ fs _ ⇒ All ? (λf. well_cost_labelled_fn (func f)) fs |
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371 | ]. |
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372 | |
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373 | lemma well_cost_labelled_state_step : ∀ge,s,tr,s'. |
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374 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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375 | well_cost_labelled_ge ge → |
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376 | well_cost_labelled_state s → |
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377 | well_cost_labelled_state s'. |
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378 | #ge #s #tr' #s' #EV cases (eval_perserves … EV) |
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379 | [ #ge #f #f' #fs #m #m' * #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #Hge * #H1 #H2 % // |
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380 | | #ge #f #fs #m * #fn #args #f' #dst * #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #b #Hfn #Hge * #H1 #H2 % /2/ |
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381 | (* |
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382 | | #ge #f #fs #m * #fn #args #f' #dst #m' #b #Hge * #H1 #H2 % /2/ |
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383 | *) |
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384 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #Hge * #H1 #H2 % /2/ |
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385 | | #ge #f #fs #m #rtv #dst #m' #Hge * #H1 #H2 @H2 |
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386 | | #ge #f #fs #rtv #dst #f' #m * #func #locals #next #nok #sp #retdst #locals' #next' #nok' #Hge * #H1 #H2 % // |
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387 | ] qed. |
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388 | |
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389 | lemma rtlabs_jump_inv : ∀s. |
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390 | RTLabs_classify s = cl_jump → |
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391 | ∃f,fs,m. s = State f fs m ∧ |
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392 | let stmt ≝ lookup_present ?? (f_graph (func f)) (next f) (next_ok f) in |
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393 | (∃r,l1,l2. stmt = St_cond r l1 l2) ∨ (∃r,ls. stmt = St_jumptable r ls). |
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394 | * |
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395 | [ #f #fs #m #E |
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396 | %{f} %{fs} %{m} % |
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397 | [ @refl |
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398 | | whd in E:(??%?); cases (lookup_present ? statement ???) in E ⊢ %; |
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399 | try (normalize try #A try #B try #C try #D try #F try #G try #H destruct) |
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400 | [ %1 %{A} %{B} %{C} @refl |
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401 | | %2 %{A} %{B} @refl |
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402 | ] |
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403 | ] |
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404 | | normalize #H1 #H2 #H3 #H4 #H5 #H6 destruct |
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405 | | normalize #H8 #H9 #H10 #H11 #H12 destruct |
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406 | ] qed. |
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407 | |
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408 | lemma well_cost_labelled_jump : ∀ge,s,tr,s'. |
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409 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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410 | well_cost_labelled_state s → |
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411 | RTLabs_classify s = cl_jump → |
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412 | RTLabs_cost s' = true. |
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413 | #ge #s #tr #s' #EV #H #CL |
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414 | cases (rtlabs_jump_inv s CL) |
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415 | #fr * #fs * #m * #Es * |
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416 | [ * #r * #l1 * #l2 #Estmt |
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417 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
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418 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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419 | >Estmt #LP whd in ⊢ (??%? → ?); |
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420 | (* replace with lemma on successors? *) |
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421 | @bind_value #v #Ev @bind_ok * #Eb whd in ⊢ (??%? → ?); #E destruct |
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422 | lapply (Hbody (next fr) (next_ok fr)) |
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423 | generalize in ⊢ (???% → ?); |
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424 | >Estmt #LP' |
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425 | whd in ⊢ (% → ?); |
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426 | * #H1 #H2 [ @H1 | @H2 ] |
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427 | | * #r * #ls #Estmt |
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428 | >Es in H; whd in ⊢ (% → ?); * * #Hbody #_ #Hfs |
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429 | >Es in EV; whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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430 | >Estmt #LP whd in ⊢ (??%? → ?); |
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431 | (* replace with lemma on successors? *) |
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432 | @bind_value #a cases a [ | #sz #i | #f | #r | #ptr ] #Ea whd in ⊢ (??%? → ?); |
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433 | [ 2: (* later *) |
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434 | | *: #E destruct |
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435 | ] |
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436 | lapply (Hbody (next fr) (next_ok fr)) |
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437 | generalize in ⊢ (???% → ?); >Estmt #LP' whd in ⊢ (% → ?); #CP |
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438 | generalize in ⊢ (??(?%)? → ?); |
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439 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [_⇒?|_⇒?]?)? → ?); |
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440 | [ #E1 #E2 whd in E2:(??%?); destruct |
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441 | | #l' #E1 #E2 whd in E2:(??%?); destruct |
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442 | cases (All_nth ???? CP ? E1) |
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443 | #H1 #H2 @H2 |
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444 | ] |
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445 | ] qed. |
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446 | |
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447 | lemma rtlabs_call_inv : ∀s. |
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448 | RTLabs_classify s = cl_call → |
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449 | ∃fd,args,dst,stk,m. s = Callstate fd args dst stk m. |
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450 | * [ #f #fs #m whd in ⊢ (??%? → ?); |
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451 | cases (lookup_present … (next f) (next_ok f)) normalize |
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452 | try #A try #B try #C try #D try #E try #F try #G destruct |
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453 | | #fd #args #dst #stk #m #E %{fd} %{args} %{dst} %{stk} %{m} @refl |
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454 | | normalize #H411 #H412 #H413 #H414 #H415 destruct |
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455 | ] qed. |
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456 | |
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457 | lemma well_cost_labelled_call : ∀ge,s,tr,s'. |
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458 | eval_statement ge s = Value ??? 〈tr,s'〉 → |
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459 | well_cost_labelled_state s → |
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460 | RTLabs_classify s = cl_call → |
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461 | RTLabs_cost s' = true. |
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462 | #ge #s #tr #s' #EV #WCL #CL |
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463 | cases (rtlabs_call_inv s CL) |
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464 | #fd * #args * #dst * #stk * #m #E >E in EV WCL; |
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465 | whd in ⊢ (??%? → % → ?); |
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466 | cases fd |
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467 | [ #fn whd in ⊢ (??%? → % → ?); |
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468 | @bind_value #lcl #Elcl cases (alloc m O (f_stacksize fn) Any) |
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469 | #m' #b whd in ⊢ (??%? → ?); #E' destruct |
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470 | * whd in ⊢ (% → ?); * #WCL1 #WCL2 #WCL3 |
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471 | @WCL2 |
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472 | | #fn whd in ⊢ (??%? → % → ?); |
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473 | @bindIO_value #evargs #Eargs |
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474 | whd in ⊢ (??%? → ?); |
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475 | #E' destruct |
---|
476 | ] qed. |
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477 | |
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478 | |
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479 | (* The preservation of (most of) the stack is useful to show as_after_return. |
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480 | We do this by showing that during the execution of a function the lower stack |
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481 | frames never change, and that after returning from the function we preserve |
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482 | the identity of the next instruction to execute. |
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483 | *) |
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484 | |
---|
485 | inductive stack_of_state : list frame → state → Prop ≝ |
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486 | | sos_State : ∀f,fs,m. stack_of_state fs (State f fs m) |
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487 | | sos_Callstate : ∀fd,args,dst,f,fs,m. stack_of_state fs (Callstate fd args dst (f::fs) m) |
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488 | | sos_Returnstate : ∀rtv,dst,fs,m. stack_of_state fs (Returnstate rtv dst fs m) |
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489 | . |
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490 | |
---|
491 | inductive stack_preserved : trace_ends_with_ret → state → state → Prop ≝ |
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492 | | sp_normal : ∀fs,s1,s2. |
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493 | stack_of_state fs s1 → |
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494 | stack_of_state fs s2 → |
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495 | stack_preserved doesnt_end_with_ret s1 s2 |
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496 | | sp_finished : ∀s1,f,f',fs,m. |
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497 | next f = next f' → |
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498 | stack_of_state (f::fs) s1 → |
---|
499 | stack_preserved ends_with_ret s1 (State f' fs m). |
---|
500 | |
---|
501 | discriminator list. |
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502 | |
---|
503 | lemma stack_of_state_eq : ∀fs,fs',s. |
---|
504 | stack_of_state fs s → |
---|
505 | stack_of_state fs' s → |
---|
506 | fs = fs'. |
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507 | #fs0 #fs0' #s0 * |
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508 | [ #f #fs #m #H inversion H |
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509 | #a #b #c #d #e #g try #h try #i try #j destruct @refl |
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510 | | #fd #args #dst #f #fs #m #H inversion H |
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511 | #a #b #c #d #e #g try #h try #i try #j destruct @refl |
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512 | | #rtv #dst #fs #m #H inversion H |
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513 | #a #b #c #d #e #g try #h try #i try #j destruct @refl |
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514 | ] qed. |
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515 | |
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516 | lemma stack_preserved_join : ∀e,s1,s2,s3. |
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517 | stack_preserved doesnt_end_with_ret s1 s2 → |
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518 | stack_preserved e s2 s3 → |
---|
519 | stack_preserved e s1 s3. |
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520 | #e1 #s1 #s2 #s3 #H1 inversion H1 |
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521 | [ #fs #s1' #s2' #S1 #S2 #E1 #E2 #E3 #E4 destruct |
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522 | #H2 inversion H2 |
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523 | [ #fs' #s1'' #s2'' #S1' #S2' #E1 #E2 #E3 #E4 destruct |
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524 | @(sp_normal fs) // <(stack_of_state_eq … S1' S2) // |
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525 | | #s1'' #f #f' #fs' #m #N #S1' #E1 #E2 #E3 #E4 destruct |
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526 | @(sp_finished … N) >(stack_of_state_eq … S1' S2) // |
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527 | ] |
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528 | | #H25 #H26 #H27 #H28 #H29 #H30 #H31 #H32 #H33 #H34 #H35 #H36 destruct |
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529 | ] qed. |
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530 | |
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531 | lemma stack_preserved_return : ∀ge,s1,s2,tr. |
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532 | RTLabs_classify s1 = cl_return → |
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533 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
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534 | stack_preserved ends_with_ret s1 s2. |
---|
535 | #ge * |
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536 | [ #f #fs #m #s2 #tr #E @⊥ whd in E:(??%?); |
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537 | cases (lookup_present ??? (next f) (next_ok f)) in E; |
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538 | normalize #a try #b try #c try #d try #e try #f try #g destruct |
---|
539 | | #fd #args #dst #fs #m #s2 #tr #E normalize in E; destruct |
---|
540 | | #res #dst * |
---|
541 | [ #m #s2 #tr #_ #EV whd in EV:(??%?); destruct |
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542 | | #f #fs #m #s2 #tr #_ whd in ⊢ (??%? → ?); @bind_value #locals #El #EV |
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543 | whd in EV:(??%?); destruct @(sp_finished ? f) // |
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544 | ] |
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545 | ] qed. |
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546 | |
---|
547 | lemma stack_preserved_step : ∀ge,s1,s2,tr. |
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548 | RTLabs_classify s1 = cl_other ∨ RTLabs_classify s1 = cl_jump → |
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549 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
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550 | stack_preserved doesnt_end_with_ret s1 s2. |
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551 | #ge0 #s1 #s2 #tr #CL #EV inversion (eval_perserves … EV) |
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552 | [ #ge #f #f' #fs #m #m' #F #E1 #E2 #E3 #E4 destruct /2/ |
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553 | | #ge #f #fs #m #fd #args #f' #dst #F #b #FFP #E1 #E2 #E3 #E4 /2/ |
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554 | | #ge #fn #locals #next #nok #sp #fs #m #args #dst #m' #E1 #E2 #E3 #E4 destruct |
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555 | normalize in CL; cases CL #E destruct |
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556 | | #ge #f #fs #m #rtv #dst #m' #E1 #E2 #E3 #E4 destruct /2/ |
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557 | | #ge #f #fs #rtv #dst #f' #m #F #E1 #E2 #E3 #E4 destruct cases CL |
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558 | #E normalize in E; destruct |
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559 | ] qed. |
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560 | |
---|
561 | lemma stack_preserved_call : ∀ge,s1,s2,s3,tr. |
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562 | RTLabs_classify s1 = cl_call → |
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563 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
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564 | stack_preserved ends_with_ret s2 s3 → |
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565 | stack_preserved doesnt_end_with_ret s1 s3. |
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566 | #ge #s1 #s2 #s3 #tr #CL #EV #SP |
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567 | cases (rtlabs_call_inv … CL) |
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568 | #fd * #args * #dst * #stk * #m #E destruct |
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569 | inversion SP |
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570 | [ #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 #H46 destruct |
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571 | | #s2' #f #f' #fs #m' #N #S #E1 #E2 #E3 #E4 destruct |
---|
572 | inversion (eval_perserves … EV) |
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573 | [ #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 #H56 #H57 #H58 destruct |
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574 | | #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 #H71 #H72 #H73 #H74 destruct |
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575 | | #ge' #fn #locals #next #nok #sp #fs1 #m1 #args1 #dst1 #m2 #E1 #E2 #E3 #E4 destruct |
---|
576 | inversion S |
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577 | [ #fx #fsx #mx #E1 #E2 #E3 destruct /2/ |
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578 | | #H76 #H77 #H78 #H79 #H80 #H81 #H82 #H83 #H84 destruct |
---|
579 | | #H86 #H87 #H88 #H89 #H90 #H91 #H92 destruct |
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580 | ] |
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581 | | #H94 #H95 #H96 #H97 #H98 #H99 #H100 #H101 #H102 #H103 #H104 destruct |
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582 | | #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 #H115 #H116 #H117 destruct |
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583 | ] |
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584 | ] qed. |
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585 | |
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586 | lemma RTLabs_after_call : ∀ge,s1,s2,s3,tr. |
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587 | ∀CL : RTLabs_classify s1 = cl_call. |
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588 | eval_statement ge s1 = Value ??? 〈tr,s2〉 → |
---|
589 | stack_preserved ends_with_ret s2 s3 → |
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590 | as_after_return (RTLabs_status ge) «s1,CL» s3. |
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591 | #ge #s1 #s2 #s3 #tr #CL #EV #S23 |
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592 | cases (rtlabs_call_inv … CL) #fn * #args * #dst * #stk * #m #E destruct |
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593 | inversion S23 |
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594 | [ #H129 #H130 #H131 #H132 #H133 #H134 #H135 #H136 #H137 destruct |
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595 | | #s2' #f #f' #fs #m' #N #S #E1 #E2 #E3 #E4 destruct |
---|
596 | inversion (eval_perserves … EV) |
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597 | [ #H139 #H140 #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 #H149 destruct |
---|
598 | | #H151 #H152 #H153 #H154 #H155 #H156 #H157 #H158 #H159 #H160 #H161 #H162 #H163 #H164 #H165 destruct |
---|
599 | | #gex #fnx #locals #next #nok #sp #fsx #mx #argsx #dstx #mx' #E1 #E2 #E3 #E4 destruct |
---|
600 | inversion S |
---|
601 | [ #fy #fsy #my #E1 #E2 #E3 destruct @N |
---|
602 | | #H167 #H168 #H169 #H170 #H171 #H172 #H173 #H174 #H175 destruct |
---|
603 | | #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
604 | ] |
---|
605 | | #H185 #H186 #H187 #H188 #H189 #H190 #H191 #H192 #H193 #H194 #H195 destruct |
---|
606 | | #H197 #H198 #H199 #H200 #H201 #H202 #H203 #H204 #H205 #H206 #H207 #H208 destruct |
---|
607 | ] |
---|
608 | ] qed. |
---|
609 | |
---|
610 | (* Don't need to know that labels break loops because we have termination. *) |
---|
611 | |
---|
612 | (* A bit of mucking around with the depth to avoid proving termination after |
---|
613 | termination. Note that we keep a proof that our upper bound on the length |
---|
614 | of the termination proof is respected. *) |
---|
615 | record trace_result (ge:genv) (depth:nat) (ends:trace_ends_with_ret) (start:state) (full:flat_trace io_out io_in ge start) (T:state → Type[0]) (limit:nat) : Type[0] ≝ { |
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616 | new_state : state; |
---|
617 | remainder : flat_trace io_out io_in ge new_state; |
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618 | is_remainder : flat_trace_prefix … full remainder; |
---|
619 | cost_labelled : well_cost_labelled_state new_state; |
---|
620 | new_trace : T new_state; |
---|
621 | stack_ok : stack_preserved ends start new_state; |
---|
622 | terminates : match depth with |
---|
623 | [ O ⇒ True |
---|
624 | | S d ⇒ ΣTM:will_return ge d new_state remainder. limit > will_return_length … TM |
---|
625 | ] |
---|
626 | }. |
---|
627 | |
---|
628 | (* The same with a flag indicating whether the function returned, as opposed to |
---|
629 | encountering a label. *) |
---|
630 | record sub_trace_result (ge:genv) (depth:nat) (start:state) (full:flat_trace io_out io_in ge start) (T:trace_ends_with_ret → state → Type[0]) (limit:nat) : Type[0] ≝ { |
---|
631 | ends : trace_ends_with_ret; |
---|
632 | trace_res :> trace_result ge (match ends with [ doesnt_end_with_ret ⇒ S depth | _ ⇒ depth]) ends start full (T ends) limit |
---|
633 | }. |
---|
634 | |
---|
635 | (* We often return the result from a recursive call with an addition to the |
---|
636 | structured trace, so we define a couple of functions to help. The bound on |
---|
637 | the size of the termination proof might need to be relaxed, too. *) |
---|
638 | |
---|
639 | definition replace_trace : ∀ge,d,e1,e2,s1,s2,t1,t2,T1,T2,l1,l2. l2 ≥ l1 → |
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640 | ∀r:trace_result ge d e1 s1 t1 T1 l1. |
---|
641 | (∀s,t. flat_trace_prefix … s1 s t1 t → flat_trace_prefix … s2 s t2 t) → |
---|
642 | T2 (new_state … r) → |
---|
643 | stack_preserved e2 s2 (new_state … r) → |
---|
644 | trace_result ge d e2 s2 t2 T2 l2 ≝ |
---|
645 | λge,d,e1,e2,s1,s2,t1,t2,T1,T2,l1,l2,lGE,r,PRE,trace,SP. |
---|
646 | mk_trace_result ge d e2 s2 t2 T2 l2 |
---|
647 | (new_state … r) |
---|
648 | (remainder … r) |
---|
649 | (PRE ?? (is_remainder ??????? r)) |
---|
650 | (cost_labelled … r) |
---|
651 | trace |
---|
652 | SP |
---|
653 | (match d return λd'.match d' with [ O ⇒ True | S d'' ⇒ ΣTM.l1 > will_return_length ge d'' (new_state … r) (remainder … r) TM] → |
---|
654 | match d' with [ O ⇒ True | S d'' ⇒ ΣTM.l2 > will_return_length ge d'' (new_state … r) (remainder … r) TM] with |
---|
655 | [O ⇒ λ_. I | _ ⇒ λTM. «pi1 … TM, ?» ] (terminates ??????? r)) |
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656 | . |
---|
657 | @(transitive_le … lGE) @(pi2 … TM) qed. |
---|
658 | |
---|
659 | definition replace_sub_trace : ∀ge,d,s1,s2,t1,t2,T1,T2,l1,l2. l2 ≥ l1 → |
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660 | ∀r:sub_trace_result ge d s1 t1 T1 l1. |
---|
661 | (∀s,t. flat_trace_prefix … s1 s t1 t → flat_trace_prefix … s2 s t2 t) → |
---|
662 | T2 (ends … r) (new_state … r) → |
---|
663 | stack_preserved (ends … r) s2 (new_state … r) → |
---|
664 | sub_trace_result ge d s2 t2 T2 l2 ≝ |
---|
665 | λge,d,s1,s2,t1,t2,T1,T2,l1,l2,lGE,r,PRE,trace,SP. |
---|
666 | mk_sub_trace_result ge d s2 t2 T2 l2 |
---|
667 | (ends … r) |
---|
668 | (replace_trace … lGE … r PRE trace SP). |
---|
669 | |
---|
670 | (* Small syntax hack to avoid ambiguous input problems. *) |
---|
671 | definition myge : nat → nat → Prop ≝ ge. |
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672 | |
---|
673 | let rec make_label_return ge depth s |
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674 | (trace: flat_trace io_out io_in ge s) |
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675 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
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676 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
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677 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
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678 | (TERMINATES: will_return ge depth s trace) |
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679 | (TIME: nat) |
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680 | (TERMINATES_IN_TIME: myge TIME (plus 2 (times 3 (will_return_length … TERMINATES)))) |
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681 | on TIME : trace_result ge depth ends_with_ret s trace |
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682 | (trace_label_return (RTLabs_status ge) s) |
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683 | (will_return_length … TERMINATES) ≝ |
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684 | |
---|
685 | match TIME return λTIME. TIME ≥ ? → ? with |
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686 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
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687 | | S TIME ⇒ λTERMINATES_IN_TIME. |
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688 | |
---|
689 | let r ≝ make_label_label ge depth s |
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690 | trace |
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691 | ENV_COSTLABELLED |
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692 | STATE_COSTLABELLED |
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693 | STATEMENT_COSTLABEL |
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694 | TERMINATES |
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695 | TIME ? in |
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696 | match ends … r return λx. trace_result ge (match x with [ doesnt_end_with_ret ⇒ S depth | _ ⇒ depth]) x s trace (trace_label_label (RTLabs_status ge) x s) ? → trace_result ge depth ends_with_ret s trace (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) with |
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697 | [ ends_with_ret ⇒ λr. |
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698 | replace_trace … r ? (tlr_base (RTLabs_status ge) s (new_state … r) (new_trace … r)) (stack_ok … r) |
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699 | | doesnt_end_with_ret ⇒ λr. |
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700 | let r' ≝ make_label_return ge depth (new_state … r) |
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701 | (remainder … r) |
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702 | ENV_COSTLABELLED |
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703 | (cost_labelled … r) ? |
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704 | (pi1 … (terminates … r)) TIME ? in |
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705 | replace_trace … r' ? |
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706 | (tlr_step (RTLabs_status ge) s (new_state … r) |
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707 | (new_state … r') (new_trace … r) (new_trace … r')) ? |
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708 | ] (trace_res … r) |
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709 | |
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710 | ] TERMINATES_IN_TIME |
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711 | |
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712 | |
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713 | and make_label_label ge depth s |
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714 | (trace: flat_trace io_out io_in ge s) |
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715 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
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716 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
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717 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
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718 | (TERMINATES: will_return ge depth s trace) |
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719 | (TIME: nat) |
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720 | (TERMINATES_IN_TIME: myge TIME (plus 1 (times 3 (will_return_length … TERMINATES)))) |
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721 | on TIME : sub_trace_result ge depth s trace |
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722 | (λends. trace_label_label (RTLabs_status ge) ends s) |
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723 | (will_return_length … TERMINATES) ≝ |
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724 | |
---|
725 | match TIME return λTIME. TIME ≥ ? → ? with |
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726 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
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727 | | S TIME ⇒ λTERMINATES_IN_TIME. |
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728 | |
---|
729 | let r ≝ make_any_label ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED TERMINATES TIME ? in |
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730 | replace_sub_trace … r ? |
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731 | (tll_base (RTLabs_status ge) (ends … r) s (new_state … r) (new_trace … r) STATEMENT_COSTLABEL) (stack_ok … r) |
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732 | |
---|
733 | ] TERMINATES_IN_TIME |
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734 | |
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735 | |
---|
736 | and make_any_label ge depth s |
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737 | (trace: flat_trace io_out io_in ge s) |
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738 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
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739 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
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740 | (TERMINATES: will_return ge depth s trace) |
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741 | (TIME: nat) |
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742 | (TERMINATES_IN_TIME: myge TIME (times 3 (will_return_length … TERMINATES))) |
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743 | on TIME : sub_trace_result ge depth s trace |
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744 | (λends. trace_any_label (RTLabs_status ge) ends s) |
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745 | (will_return_length … TERMINATES) ≝ |
---|
746 | |
---|
747 | match TIME return λTIME. TIME ≥ ? → ? with |
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748 | [ O ⇒ λTERMINATES_IN_TIME. ⊥ |
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749 | | S TIME ⇒ λTERMINATES_IN_TIME. |
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750 | |
---|
751 | match trace return λs,trace. well_cost_labelled_state s → ∀TM:will_return ??? trace. myge ? (times 3 (will_return_length ??? trace TM)) → sub_trace_result ge depth s trace (λends. trace_any_label (RTLabs_status ge) ends s) (will_return_length … TM) with |
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752 | [ ft_stop st FINAL ⇒ |
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753 | λSTATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. ? |
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754 | |
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755 | | ft_step start tr next EV trace' ⇒ λSTATE_COSTLABELLED,TERMINATES,TERMINATES_IN_TIME. |
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756 | match RTLabs_classify start return λx. RTLabs_classify start = x → sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
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757 | [ cl_other ⇒ λCL. |
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758 | match RTLabs_cost next return λx. RTLabs_cost next = x → sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
759 | (* We're about to run into a label. *) |
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760 | [ true ⇒ λCS. |
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761 | mk_sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends start) ? |
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762 | doesnt_end_with_ret |
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763 | (mk_trace_result ge … next trace' ?? |
---|
764 | (tal_base_not_return (RTLabs_status ge) start next ?? CS) ??) |
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765 | (* An ordinary step, keep going. *) |
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766 | | false ⇒ λCS. |
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767 | let r ≝ make_any_label ge depth next trace' ENV_COSTLABELLED ? (will_return_notfn … TERMINATES) TIME ? in |
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768 | replace_sub_trace … r ? |
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769 | (tal_step_default (RTLabs_status ge) (ends … r) |
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770 | start next (new_state … r) ? (new_trace … r) ? (RTLabs_not_cost … CS)) ? |
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771 | ] (refl ??) |
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772 | |
---|
773 | | cl_jump ⇒ λCL. |
---|
774 | mk_sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends start) ? |
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775 | doesnt_end_with_ret |
---|
776 | (mk_trace_result ge ?????? next trace' ?? |
---|
777 | (tal_base_not_return (RTLabs_status ge) start next ???) ??) |
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778 | |
---|
779 | | cl_call ⇒ λCL. |
---|
780 | let r ≝ make_label_return ge (S depth) next trace' ENV_COSTLABELLED ?? (will_return_call … TERMINATES) TIME ? in |
---|
781 | match RTLabs_cost (new_state … r) return λx. RTLabs_cost (new_state … r) = x → sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends ?) (will_return_length … TERMINATES) with |
---|
782 | (* We're about to run into a label, use base case for call *) |
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783 | [ true ⇒ λCS. |
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784 | mk_sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends start) ? |
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785 | doesnt_end_with_ret |
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786 | (replace_trace … r ? |
---|
787 | (tal_base_call (RTLabs_status ge) start next (new_state … r) |
---|
788 | ? CL ? (new_trace … r) CS) ?) |
---|
789 | (* otherwise use step case *) |
---|
790 | | false ⇒ λCS. |
---|
791 | let r' ≝ make_any_label ge depth |
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792 | (new_state … r) (remainder … r) ENV_COSTLABELLED ? |
---|
793 | (pi1 … (terminates … r)) TIME ? in |
---|
794 | replace_sub_trace … r' ? |
---|
795 | (tal_step_call (RTLabs_status ge) (ends … r') |
---|
796 | start next (new_state … r) (new_state … r') ? CL ? |
---|
797 | (new_trace … r) (RTLabs_not_cost … CS) (new_trace … r')) ? |
---|
798 | ] (refl ??) |
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799 | |
---|
800 | | cl_return ⇒ λCL. |
---|
801 | mk_sub_trace_result ge depth start ? (λends. trace_any_label (RTLabs_status ge) ends start) ? |
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802 | ends_with_ret |
---|
803 | (mk_trace_result ge ?????? |
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804 | next |
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805 | trace' |
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806 | ? |
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807 | ? |
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808 | (tal_base_return (RTLabs_status ge) start next ? CL) |
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809 | ? |
---|
810 | ?) |
---|
811 | ] (refl ? (RTLabs_classify start)) |
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812 | |
---|
813 | | ft_wrong start m EV ⇒ λSTATE_COSTLABELLED,TERMINATES. ⊥ |
---|
814 | |
---|
815 | ] STATE_COSTLABELLED TERMINATES TERMINATES_IN_TIME |
---|
816 | ] TERMINATES_IN_TIME. |
---|
817 | |
---|
818 | [ cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
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819 | | // |
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820 | | // |
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821 | | cases r #H1 #H2 #H3 #H4 #H5 #H6 * #H7 @le_S_to_le |
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822 | | #s0 #t @concatenate_flat_trace_prefix @(is_remainder … r) |
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823 | | @(stack_preserved_join … (stack_ok … r)) // |
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824 | | @(trace_label_label_label … (new_trace … r)) |
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825 | | cases r #H1 #H2 #H3 #H4 #H5 #H6 * #H7 #LT |
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826 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
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827 | @(transitive_le … (3*(will_return_length … TERMINATES))) |
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828 | [ >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
829 | @(monotonic_le_times_r 3 … LT) |
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830 | | @le_S @le_S @le_n |
---|
831 | ] |
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832 | | @le_S_S_to_le @TERMINATES_IN_TIME |
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833 | | cases (not_le_Sn_O ?) [ #H @H @TERMINATES_IN_TIME ] |
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834 | | @le_n |
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835 | | // |
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836 | | @le_S_S_to_le @TERMINATES_IN_TIME |
---|
837 | | @(wrl_nonzero … TERMINATES_IN_TIME) |
---|
838 | | (* Bad - we've reached the end of the trace; need to fix semantics so that |
---|
839 | this can't happen *) |
---|
840 | | @(will_return_return … CL TERMINATES) |
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841 | | /2 by stack_preserved_return/ |
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842 | | %{tr} @EV |
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843 | | @(well_cost_labelled_state_step … EV) // |
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844 | | %2 %1 |
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845 | | whd @(will_return_notfn … TERMINATES) %2 @CL |
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846 | | @stack_preserved_step /2/ |
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847 | | %{tr} @EV |
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848 | | %1 whd @CL |
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849 | | @(well_cost_labelled_jump … EV) // |
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850 | | @(well_cost_labelled_state_step … EV) // |
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851 | | %2 %1 |
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852 | | @(stack_preserved_call … EV (stack_ok … r)) // |
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853 | | %{tr} @EV |
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854 | | @RTLabs_after_call // |
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855 | | #s0 #t #p %2 @p |
---|
856 | | cases (will_return_call … TERMINATES) #H @le_S_to_le |
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857 | | cases r #H1 #H2 #H3 #H4 #H5 #H6 * #H7 |
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858 | cases (will_return_call … CL TERMINATES) |
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859 | #TM #X #Y @le_S_to_le @(transitive_lt … Y X) |
---|
860 | | #s0 #t #p1 %2 @(concatenate_flat_trace_prefix … p1) @(is_remainder … r) |
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861 | | @RTLabs_after_call // |
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862 | | %{tr} @EV |
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863 | | @(stack_preserved_join … (stack_ok … r')) @(stack_preserved_call … EV (stack_ok … r)) // |
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864 | | @(cost_labelled … r) |
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865 | | cases r #H72 #H73 #H74 #H75 #HX #HY * #H76 #H78 |
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866 | @(le_plus_to_le … 1) @(transitive_le … TERMINATES_IN_TIME) |
---|
867 | cases (will_return_call … TERMINATES) in H78; |
---|
868 | #X #Y #Z |
---|
869 | @(transitive_le … (monotonic_lt_times_r 3 … Y)) |
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870 | [ @(transitive_le … (monotonic_lt_times_r 3 … Z)) // |
---|
871 | | // |
---|
872 | ] |
---|
873 | | @(well_cost_labelled_state_step … EV) // |
---|
874 | | @(well_cost_labelled_call … EV) // |
---|
875 | | skip |
---|
876 | | cases (will_return_call … TERMINATES) |
---|
877 | #TM #GT @le_S_S_to_le |
---|
878 | >commutative_times change with ((S ?) * 3 ≤ ?) >commutative_times |
---|
879 | @(transitive_le … TERMINATES_IN_TIME) |
---|
880 | @(monotonic_le_times_r 3 … GT) |
---|
881 | | whd @(will_return_notfn … TERMINATES) %1 @CL |
---|
882 | | @(stack_preserved_step … EV) /2/ |
---|
883 | | %{tr} @EV |
---|
884 | | %2 whd @CL |
---|
885 | | @(well_cost_labelled_state_step … EV) // |
---|
886 | | %2 %1 |
---|
887 | | cases (will_return_notfn … TERMINATES) #TM @le_S_to_le |
---|
888 | | #s0 #t #p %2 @p |
---|
889 | | @CL |
---|
890 | | %{tr} @EV |
---|
891 | | @(stack_preserved_join … (stack_ok … r)) @(stack_preserved_step … EV) /2/ |
---|
892 | | @(well_cost_labelled_state_step … EV) // |
---|
893 | | %1 @CL |
---|
894 | | cases (will_return_notfn … TERMINATES) #TM #GT |
---|
895 | @le_S_S_to_le |
---|
896 | @(transitive_le … (monotonic_lt_times_r … GT) TERMINATES_IN_TIME) |
---|
897 | // |
---|
898 | | inversion TERMINATES |
---|
899 | [ #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 #H13 -TERMINATES -TERMINATES destruct |
---|
900 | | #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 #H29 -TERMINATES -TERMINATES destruct |
---|
901 | | #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 #H42 #H43 #H44 #H45 -TERMINATES -TERMINATES destruct |
---|
902 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 -TERMINATES -TERMINATES destruct |
---|
903 | ] |
---|
904 | ] cases daemon qed. |
---|
905 | |
---|
906 | (* We can initialise TIME with a suitably large value based on the length of the |
---|
907 | termination proof. *) |
---|
908 | let rec make_label_return' ge depth s |
---|
909 | (trace: flat_trace io_out io_in ge s) |
---|
910 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
911 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
912 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
913 | (TERMINATES: will_return ge depth s trace) |
---|
914 | : trace_result ge depth ends_with_ret s trace (trace_label_return (RTLabs_status ge) s) (will_return_length … TERMINATES) ≝ |
---|
915 | make_label_return ge depth s trace ENV_COSTLABELLED STATE_COSTLABELLED STATEMENT_COSTLABEL TERMINATES |
---|
916 | (2 + 3 * will_return_length … TERMINATES) ?. |
---|
917 | @le_n |
---|
918 | qed. |
---|
919 | |
---|
920 | (* FIXME: there's trouble at the end of the program because we can't make a step |
---|
921 | away from the final return. |
---|
922 | |
---|
923 | We need to show that the "next pc" is preserved through a function call. |
---|
924 | |
---|
925 | Tail-calls are not handled properly (which means that if we try to show the |
---|
926 | full version with non-termination we'll fail because calls and returns aren't |
---|
927 | balanced. |
---|
928 | *) |
---|
929 | |
---|
930 | inductive inhabited (T:Type[0]) : Prop ≝ |
---|
931 | | witness : T → inhabited T. |
---|
932 | |
---|
933 | (* We also require that program's traces are soundly labelled: for any state |
---|
934 | in the execution, we can give a distance to a labelled state or termination. |
---|
935 | |
---|
936 | Note that this differs from the syntactic notions in earlier languages |
---|
937 | because it is a global property. In principle, we would have a loop broken |
---|
938 | only by a call to a function (which necessarily has a label) and no local |
---|
939 | cost label. |
---|
940 | *) |
---|
941 | |
---|
942 | let rec nth_state ge s |
---|
943 | (trace: flat_trace io_out io_in ge s) |
---|
944 | n |
---|
945 | on n : option state ≝ |
---|
946 | match n with |
---|
947 | [ O ⇒ Some ? s |
---|
948 | | S n' ⇒ |
---|
949 | match trace with |
---|
950 | [ ft_step _ _ s' _ trace' ⇒ nth_state ge s' trace' n' |
---|
951 | | _ ⇒ None ? |
---|
952 | ] |
---|
953 | ]. |
---|
954 | |
---|
955 | definition soundly_labelled_trace : ∀ge,s. flat_trace io_out io_in ge s → Prop ≝ |
---|
956 | λge,s,trace. ∀n.∃m. ∀s'. nth_state ge s trace (n+m) = Some ? s' → RTLabs_cost s' = true. |
---|
957 | |
---|
958 | lemma soundly_labelled_step : ∀ge,s,tr,s',EV,trace'. |
---|
959 | soundly_labelled_trace ge s (ft_step … ge s tr s' EV trace') → |
---|
960 | soundly_labelled_trace ge s' trace'. |
---|
961 | #ge #s #tr #s' #EV #trace' #H |
---|
962 | #n cases (H (S n)) #m #H' %{m} @H' |
---|
963 | qed. |
---|
964 | |
---|
965 | (* Define a notion of sound labellings of RTLabs programs. *) |
---|
966 | |
---|
967 | let rec successors (s : statement) : list label ≝ |
---|
968 | match s with |
---|
969 | [ St_skip l ⇒ [l] |
---|
970 | | St_cost _ l ⇒ [l] |
---|
971 | | St_const _ _ l ⇒ [l] |
---|
972 | | St_op1 _ _ _ _ _ l ⇒ [l] |
---|
973 | | St_op2 _ _ _ _ l ⇒ [l] |
---|
974 | | St_load _ _ _ l ⇒ [l] |
---|
975 | | St_store _ _ _ l ⇒ [l] |
---|
976 | | St_call_id _ _ _ l ⇒ [l] |
---|
977 | | St_call_ptr _ _ _ l ⇒ [l] |
---|
978 | (* |
---|
979 | | St_tailcall_id _ _ ⇒ [ ] |
---|
980 | | St_tailcall_ptr _ _ ⇒ [ ] |
---|
981 | *) |
---|
982 | | St_cond _ l1 l2 ⇒ [l1; l2] |
---|
983 | | St_jumptable _ ls ⇒ ls |
---|
984 | | St_return ⇒ [ ] |
---|
985 | ]. |
---|
986 | |
---|
987 | definition actual_successor : state → option label ≝ |
---|
988 | λs. match s with |
---|
989 | [ State f fs m ⇒ Some ? (next f) |
---|
990 | | Callstate _ _ _ fs _ ⇒ match fs with [ cons f _ ⇒ Some ? (next f) | _ ⇒ None ? ] |
---|
991 | | Returnstate _ _ _ _ ⇒ None ? |
---|
992 | ]. |
---|
993 | |
---|
994 | lemma nth_opt_Exists : ∀A,n,l,a. |
---|
995 | nth_opt A n l = Some A a → |
---|
996 | Exists A (λa'. a' = a) l. |
---|
997 | #A #n elim n |
---|
998 | [ * [ #a #E normalize in E; destruct | #a #l #a' #E normalize in E; destruct % // ] |
---|
999 | | #m #IH * |
---|
1000 | [ #a #E normalize in E; destruct |
---|
1001 | | #a #l #a' #E %2 @IH @E |
---|
1002 | ] |
---|
1003 | ] qed. |
---|
1004 | |
---|
1005 | lemma eval_successor : ∀ge,f,fs,m,tr,s'. |
---|
1006 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
1007 | RTLabs_classify s' = cl_return ∨ |
---|
1008 | ∃l. actual_successor s' = Some ? l ∧ Exists ? (λl0. l0 = l) (successors (lookup_present … (f_graph (func f)) (next f) (next_ok f))). |
---|
1009 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
1010 | whd in ⊢ (??%? → ?); |
---|
1011 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
1012 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1013 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1014 | | #r #c #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1015 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1016 | | #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1017 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1018 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1019 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1020 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // % // |
---|
1021 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct %2 cases b [ %{l1} | %{l2} ] % // [ % | %2 %] // |
---|
1022 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev |
---|
1023 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #r #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
1024 | whd in ⊢ (??%? → ?); |
---|
1025 | generalize in ⊢ (??(?%)? → ?); |
---|
1026 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
1027 | [ #e #E normalize in E; destruct |
---|
1028 | | #l #El whd in ⊢ (??%? → ?); #E destruct %2 %{l} % // @(nth_opt_Exists … El) |
---|
1029 | ] |
---|
1030 | | #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev whd in ⊢ (??%? → ?); #E destruct %1 @refl |
---|
1031 | ] qed. |
---|
1032 | |
---|
1033 | definition steps_for_statement : statement → nat ≝ |
---|
1034 | λs. S (match s with [ St_call_id _ _ _ _ ⇒ 1 | St_call_ptr _ _ _ _ ⇒ 1 | St_return ⇒ 1 | _ ⇒ 0 ]). |
---|
1035 | |
---|
1036 | inductive bound_on_steps_to_cost (g:graph statement) : label → nat → Prop ≝ |
---|
1037 | | bostc_here : ∀l,n,H. is_cost_label (lookup_present … g l H) → bound_on_steps_to_cost g l n |
---|
1038 | | bostc_later : ∀l,n. bound_on_steps_to_cost1 g l n → bound_on_steps_to_cost g l n |
---|
1039 | with bound_on_steps_to_cost1 : label → nat → Prop ≝ |
---|
1040 | | bostc_step : ∀l,n,H. |
---|
1041 | let stmt ≝ lookup_present … g l H in |
---|
1042 | (∀l'. Exists label (λl0. l0 = l') (successors stmt) → |
---|
1043 | bound_on_steps_to_cost g l' n) → |
---|
1044 | bound_on_steps_to_cost1 g l (steps_for_statement stmt + n). |
---|
1045 | |
---|
1046 | (* |
---|
1047 | lemma steps_to_label_bound_inv : ∀g,l,n. |
---|
1048 | steps_to_label_bound g l n → |
---|
1049 | ∀H. let stmt ≝ lookup_present … g l H in |
---|
1050 | ∃n'. n = steps_for_statement stmt + n' ∧ |
---|
1051 | (∀l'. Exists label (λl0. l0 = l') (successors stmt) → |
---|
1052 | (∃H'. bool_to_Prop (is_cost_label (lookup_present … g l' H'))) ∨ |
---|
1053 | steps_to_label_bound g l' n'). |
---|
1054 | #g #l0 #n0 #S inversion S #l #n #H #IH #E1 #E2 #_ destruct #H' |
---|
1055 | % [2: % [ @refl | #l' #EX cases (IH l' EX) /2/ ] | skip ] |
---|
1056 | qed. |
---|
1057 | *) |
---|
1058 | discriminator nat. |
---|
1059 | (* |
---|
1060 | definition soundly_labelled_pc ≝ λg,l. ∃n. steps_to_label_bound g l n. |
---|
1061 | |
---|
1062 | let rec soundly_labelled_fn (fn : internal_function) : Prop ≝ |
---|
1063 | soundly_labelled_pc (f_graph fn) (f_entry fn). |
---|
1064 | |
---|
1065 | |
---|
1066 | definition soundly_labelled_frame : frame → Prop ≝ |
---|
1067 | λf. soundly_labelled_pc (f_graph (func f)) (next f). |
---|
1068 | |
---|
1069 | definition soundly_labelled_state : state → Prop ≝ |
---|
1070 | λs. match s with |
---|
1071 | [ State f _ _ ⇒ soundly_labelled_frame f |
---|
1072 | | Callstate _ _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
1073 | | Returnstate _ _ stk _ ⇒ match stk with [ nil ⇒ False | cons f _ ⇒ soundly_labelled_frame f ] |
---|
1074 | ]. |
---|
1075 | *) |
---|
1076 | definition frame_bound_on_steps_to_cost : frame → nat → Prop ≝ |
---|
1077 | λf. bound_on_steps_to_cost (f_graph (func f)) (next f). |
---|
1078 | definition frame_bound_on_steps_to_cost1 : frame → nat → Prop ≝ |
---|
1079 | λf. bound_on_steps_to_cost1 (f_graph (func f)) (next f). |
---|
1080 | |
---|
1081 | inductive state_bound_on_steps_to_cost : state → nat → Prop ≝ |
---|
1082 | | sbostc_state : ∀f,fs,m,n. frame_bound_on_steps_to_cost1 f n → state_bound_on_steps_to_cost (State f fs m) n |
---|
1083 | | sbostc_call : ∀fd,args,dst,f,fs,m,n. frame_bound_on_steps_to_cost f n → state_bound_on_steps_to_cost (Callstate fd args dst (f::fs) m) (S n) |
---|
1084 | | sbostc_ret : ∀rtv,dst,f,fs,m,n. frame_bound_on_steps_to_cost f n → state_bound_on_steps_to_cost (Returnstate rtv dst (f::fs) m) (S n) |
---|
1085 | . |
---|
1086 | |
---|
1087 | lemma state_bound_on_steps_to_cost_zero : ∀s. |
---|
1088 | ¬ state_bound_on_steps_to_cost s O. |
---|
1089 | #s % #H inversion H |
---|
1090 | [ #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 destruct |
---|
1091 | whd in H50; @(bound_on_steps_to_cost1_inv_ind … H50) (* XXX inversion H50*) |
---|
1092 | #H55 #H56 #H57 #H58 #H59 #H60 #H61 normalize in H60; destruct |
---|
1093 | | #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 destruct |
---|
1094 | | #H13 #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 destruct |
---|
1095 | ] qed. |
---|
1096 | |
---|
1097 | lemma eval_steps : ∀ge,f,fs,m,tr,s'. |
---|
1098 | eval_statement ge (State f fs m) = Value ??? 〈tr,s'〉 → |
---|
1099 | steps_for_statement (lookup_present ?? (f_graph (func f)) (next f) (next_ok f)) = |
---|
1100 | match s' with [ State _ _ _ ⇒ 1 | Callstate _ _ _ _ _ ⇒ 2 | Returnstate _ _ _ _ ⇒ 2 ]. |
---|
1101 | #ge * #func #locals #next #next_ok #sp #dst #fs #m #tr #s' |
---|
1102 | whd in ⊢ (??%? → ?); |
---|
1103 | generalize in ⊢ (??(?%)? → ?); cases (lookup_present ??? next next_ok) |
---|
1104 | [ #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1105 | | #cl #l #LP whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1106 | | #r #c #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1107 | | #ty #ty' #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1108 | | #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev' @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1109 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #locals' #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1110 | | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #m' #Em whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1111 | | #id #rs #r #l #LP whd in ⊢ (??%? → ?); @bind_value #b #Eb @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1112 | | #r #rs #r' #l #LP whd in ⊢ (??%? → ?); @bind_value #fv #Efv @bind_ok #fd #Efd @bind_ok #vs #Evs whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1113 | | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev @bind_ok #b #Eb whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1114 | | #r #ls #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev |
---|
1115 | cases v [ #E normalize in E; destruct | #sz #i | #f #E normalize in E; destruct | #r #E normalize in E; destruct | #p #E normalize in E; destruct ] |
---|
1116 | whd in ⊢ (??%? → ?); |
---|
1117 | generalize in ⊢ (??(?%)? → ?); |
---|
1118 | cases (nth_opt label (nat_of_bitvector (bitsize_of_intsize sz) i) ls) in ⊢ (???% → ??(match % with [ _ ⇒ ? | _ ⇒ ? ] ?)? → ?); |
---|
1119 | [ #e #E normalize in E; destruct |
---|
1120 | | #l #El whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1121 | ] |
---|
1122 | | #LP whd in ⊢ (??%? → ?); @bind_value #v #Ev whd in ⊢ (??%? → ?); #E destruct @refl |
---|
1123 | ] qed. |
---|
1124 | |
---|
1125 | lemma bound_after_step : ∀ge,s,tr,s',n. |
---|
1126 | state_bound_on_steps_to_cost s (S n) → |
---|
1127 | eval_statement ge s = Value ??? 〈tr, s'〉 → |
---|
1128 | RTLabs_cost s' = false → |
---|
1129 | (RTLabs_classify s' = cl_return ∨ RTLabs_classify s = cl_call) ∨ |
---|
1130 | state_bound_on_steps_to_cost s' n. |
---|
1131 | #ge #s #tr #s' #n #BOUND1 inversion BOUND1 |
---|
1132 | [ #f #fs #m #m #FS #E1 #E2 #_ destruct |
---|
1133 | #EVAL cases (eval_successor … EVAL) |
---|
1134 | [ /3/ |
---|
1135 | | * #l * #S1 #S2 #NC %2 |
---|
1136 | (* |
---|
1137 | cases (bound_on_steps_to_cost1_inv … FS ?) [2: @(next_ok f) ] |
---|
1138 | *) |
---|
1139 | @(bound_on_steps_to_cost1_inv_ind … FS) #next #n' #next_ok #IH #E1 #E2 #E3 destruct |
---|
1140 | inversion (eval_perserves … EVAL) |
---|
1141 | [ #ge0 #f0 #f' #fs' #m0 #m' #F #E4 #E5 #E6 #_ destruct |
---|
1142 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
1143 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
1144 | %1 inversion (IH … S2) |
---|
1145 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
1146 | change with (RTLabs_cost (State (mk_frame H1 H7 lx LPx H5 H6) fs' m')) in CSx:(?%); |
---|
1147 | whd in S1:(??%?); destruct >NC in CSx; * |
---|
1148 | | whd in S1:(??%?); destruct #H71 #H72 #H73 #H74 #H75 #H76 destruct @H73 |
---|
1149 | ] |
---|
1150 | | #ge0 #f0 #fs' #m0 #fd #args #f' #dst #F #b #FFP #E4 #E5 #E6 #_ destruct |
---|
1151 | >(eval_steps … EVAL) in E2; #En normalize in En; |
---|
1152 | inversion F #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9 #H10 #H11 #H12 destruct |
---|
1153 | %2 @IH normalize in S1; destruct @S2 |
---|
1154 | | #H14 #H15 #H16 #H17 #H18 #H19 #H20 #H21 #H22 #H23 #H24 #H25 #H26 #H27 #H28 |
---|
1155 | destruct |
---|
1156 | | #H31 #H32 #H33 #H34 #H35 #H36 #H37 #H38 #H39 #H40 #H41 destruct |
---|
1157 | normalize in S1; destruct |
---|
1158 | | #H44 #H45 #H46 #H47 #H48 #H49 #H50 #H51 #H52 #H53 #H54 #H55 destruct |
---|
1159 | ] |
---|
1160 | ] |
---|
1161 | | #H58 #H59 #H60 #H61 #H62 #H63 #H64 #H65 #H66 #H67 #H68 #H69 #H70 destruct |
---|
1162 | /3/ |
---|
1163 | | #rtv #dst #f #fs #m #n' #FS #E1 #E2 #_ destruct |
---|
1164 | #EVAL #NC %2 inversion (eval_perserves … EVAL) |
---|
1165 | [ #H72 #H73 #H74 #H75 #H76 #H77 #H78 #H79 #H80 #H81 #H82 destruct |
---|
1166 | | #H84 #H85 #H86 #H87 #H88 #H89 #H90 #H91 #H92 #H93 #H94 #H95 #H96 #H97 #H98 destruct |
---|
1167 | | #H100 #H101 #H102 #H103 #H104 #H105 #H106 #H107 #H108 #H109 #H110 #H111 #H112 #H113 #H114 destruct |
---|
1168 | | #H116 #H117 #H118 #H119 #H120 #H121 #H122 #H123 #H124 #H125 #H126 destruct |
---|
1169 | | #ge' #f' #fs' #rtv' #dst' #f'' #m' #F #E1 #E2 #E3 #_ destruct |
---|
1170 | %1 whd in FS ⊢ %; |
---|
1171 | inversion (stack_preserved_return … EVAL) [ @refl | #H141 #H142 #H143 #H144 #H145 #H146 #H147 #H148 #H149 destruct ] |
---|
1172 | #s1 #f1 #f2 #fs #m #FE #SS1 #_ #E1 #E2 #_ destruct <FE |
---|
1173 | inversion SS1 [ #H163 #H164 #H165 #H166 #H167 #H168 destruct | #H170 #H171 #H172 #H173 #H174 #H175 #H176 #H177 #H178 destruct | #rtv #dst #fs0 #m0 #E1 #E2 #_ destruct ] |
---|
1174 | inversion F #func #locals #next #next_ok #sp #retdst #locals' #next' #next_ok' #E1 #E2 #_ destruct |
---|
1175 | inversion FS |
---|
1176 | [ #lx #nx #LPx #CSx #E1x #E2x @⊥ destruct |
---|
1177 | change with (RTLabs_cost (State (mk_frame func locals' lx ? sp retdst) fs m0)) in CSx:(?%); |
---|
1178 | >NC in CSx; * |
---|
1179 | | #lx #nx #H #E1x #E2x #_ destruct @H |
---|
1180 | ] |
---|
1181 | ] |
---|
1182 | ] qed. |
---|
1183 | |
---|
1184 | (* When constructing an infinite trace, we need to be able to grab the finite |
---|
1185 | portion of the trace for the next [trace_label_diverges] constructor. We |
---|
1186 | use the fact that the trace is soundly labelled to achieve this. *) |
---|
1187 | |
---|
1188 | inductive finite_prefix (ge:genv) : state → Prop ≝ |
---|
1189 | | fp_tal : ∀s,s'. |
---|
1190 | trace_any_label (RTLabs_status ge) doesnt_end_with_ret s s' → |
---|
1191 | flat_trace io_out io_in ge s' → |
---|
1192 | finite_prefix ge s |
---|
1193 | | fp_tac : ∀s,s'. |
---|
1194 | trace_any_call (RTLabs_status ge) s s' → |
---|
1195 | flat_trace io_out io_in ge s' → |
---|
1196 | finite_prefix ge s |
---|
1197 | . |
---|
1198 | |
---|
1199 | definition fp_add_default : ∀ge,s,s'. |
---|
1200 | RTLabs_classify s = cl_other → |
---|
1201 | finite_prefix ge s' → |
---|
1202 | (∃t. eval_statement ge s = Value ??? 〈t,s'〉) → |
---|
1203 | RTLabs_cost s' = false → |
---|
1204 | finite_prefix ge s ≝ |
---|
1205 | λge,s,s',OTHER,fp. |
---|
1206 | match fp return λs'.λ_. (∃t. eval_statement ge ? = Value ??? 〈t,s'〉) → RTLabs_cost s' = false → finite_prefix ge s with |
---|
1207 | [ fp_tal s' sf TAL rem ⇒ λEVAL, NOT_COST. fp_tal ge s sf |
---|
1208 | (tal_step_default (RTLabs_status ge) doesnt_end_with_ret s s' sf EVAL TAL OTHER (RTLabs_not_cost … NOT_COST)) |
---|
1209 | rem |
---|
1210 | | fp_tac s' sf TAC rem ⇒ λEVAL, NOT_COST. fp_tac ge s sf |
---|
1211 | (tac_step_default (RTLabs_status ge) s sf s' EVAL TAC OTHER (RTLabs_not_cost … NOT_COST)) rem |
---|
1212 | ]. |
---|
1213 | |
---|
1214 | definition fp_add_terminating_call : ∀ge,s,s1,s'. |
---|
1215 | (∃t. eval_statement ge s = Value ??? 〈t,s1〉) → |
---|
1216 | ∀CALL:RTLabs_classify s = cl_call. |
---|
1217 | finite_prefix ge s' → |
---|
1218 | trace_label_return (RTLabs_status ge) s1 s' → |
---|
1219 | as_after_return (RTLabs_status ge) (mk_Sig ?? s CALL) s' → |
---|
1220 | RTLabs_cost s' = false → |
---|
1221 | finite_prefix ge s ≝ |
---|
1222 | λge,s,s1,s',EVAL,CALL,fp. |
---|
1223 | match fp return λs'.λ_. trace_label_return (RTLabs_status ge) ? s' → as_after_return (RTLabs_status ge) ? s' → RTLabs_cost s' = false → finite_prefix ge s with |
---|
1224 | [ fp_tal s' sf TAL rem ⇒ λTLR,RET,NOT_COST. fp_tal ge s sf |
---|
1225 | (tal_step_call (RTLabs_status ge) doesnt_end_with_ret s s1 s' sf EVAL CALL RET TLR (RTLabs_not_cost … NOT_COST) TAL) |
---|
1226 | rem |
---|
1227 | | fp_tac s' sf TAC rem ⇒ λTLR,RET,NOT_COST. fp_tac ge s sf |
---|
1228 | (tac_step_call (RTLabs_status ge) s s' sf s1 EVAL CALL RET TLR (RTLabs_not_cost … NOT_COST) TAC) |
---|
1229 | rem |
---|
1230 | ]. |
---|
1231 | |
---|
1232 | definition termination_oracle ≝ ∀ge,depth,s,trace. |
---|
1233 | inhabited (will_return ge depth s trace) ∨ ¬ inhabited (will_return ge depth s trace). |
---|
1234 | |
---|
1235 | let rec finite_segment ge s n trace |
---|
1236 | (ORACLE: termination_oracle) |
---|
1237 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
1238 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
1239 | (NO_TERMINATION: Not (∃depth. inhabited (will_return ge depth s trace))) |
---|
1240 | (NOT_UNDEFINED: not_wrong … trace) |
---|
1241 | (LABEL_LIMIT: state_bound_on_steps_to_cost s n) |
---|
1242 | on n : finite_prefix ge s ≝ |
---|
1243 | match n return λn. state_bound_on_steps_to_cost s n → finite_prefix ge s with |
---|
1244 | [ O ⇒ λLABEL_LIMIT. ⊥ |
---|
1245 | | S n' ⇒ |
---|
1246 | match trace return λs,trace. not_wrong ??? s trace → well_cost_labelled_state s → (Not (∃depth. inhabited (will_return ge depth s trace))) → state_bound_on_steps_to_cost s (S n') → finite_prefix ge s with |
---|
1247 | [ ft_stop st FINAL ⇒ λNOT_UNDEFINED,STATE_COSTLABELLED,NO_TERMINATION,LABEL_LIMIT. ⊥ |
---|
1248 | | ft_step start tr next EV trace' ⇒ λNOT_UNDEFINED,STATE_COSTLABELLED,NO_TERMINATION,LABEL_LIMIT. |
---|
1249 | match RTLabs_classify start return λx. RTLabs_classify start = x → ? with |
---|
1250 | [ cl_other ⇒ λCL. |
---|
1251 | match RTLabs_cost next return λx. RTLabs_cost next = x → ? with |
---|
1252 | [ true ⇒ λCS. |
---|
1253 | fp_tal ge start next (tal_base_not_return (RTLabs_status ge) start next ?? CS) trace' |
---|
1254 | | false ⇒ λCS. |
---|
1255 | let fs ≝ finite_segment ge next n' trace' ORACLE ENV_COSTLABELLED ???? in |
---|
1256 | fp_add_default ge ?? CL fs ? CS |
---|
1257 | ] (refl ??) |
---|
1258 | | cl_jump ⇒ λCL. |
---|
1259 | fp_tal ge start next (tal_base_not_return (RTLabs_status ge) start next ?? ?) trace' |
---|
1260 | | cl_call ⇒ λCL. |
---|
1261 | match ORACLE ge O next trace' return λ_. finite_prefix ge start with |
---|
1262 | [ or_introl TERMINATES ⇒ |
---|
1263 | match TERMINATES with [ witness TERMINATES ⇒ |
---|
1264 | let tlr ≝ make_label_return' ge O next trace' ENV_COSTLABELLED ?? TERMINATES in |
---|
1265 | match RTLabs_cost (new_state … tlr) return λx. RTLabs_cost (new_state … tlr) = x → finite_prefix ge start with |
---|
1266 | [ true ⇒ λCS. fp_tal ge start (new_state … tlr) (tal_base_call (RTLabs_status ge) start next (new_state … tlr) ? CL ? (new_trace … tlr) CS) (remainder … tlr) |
---|
1267 | | false ⇒ λCS. |
---|
1268 | let fs ≝ finite_segment ge (new_state … tlr) n' (remainder … tlr) ORACLE ENV_COSTLABELLED ???? in |
---|
1269 | fp_add_terminating_call … fs (new_trace … tlr) ? CS |
---|
1270 | ] (refl ??) |
---|
1271 | ] |
---|
1272 | | or_intror NO_TERMINATION ⇒ |
---|
1273 | fp_tac ??? (tac_base (RTLabs_status ge) start CL) (ft_step io_out io_in ge start tr next EV trace') |
---|
1274 | ] |
---|
1275 | | cl_return ⇒ λCL. ⊥ |
---|
1276 | ] (refl ??) |
---|
1277 | | ft_wrong start m EV ⇒ λNOT_UNDEFINED,STATE_COSTLABELLED,NO_TERMINATION,LABEL_LIMIT. ⊥ |
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1278 | ] NOT_UNDEFINED STATE_COSTLABELLED NO_TERMINATION |
---|
1279 | ] LABEL_LIMIT. |
---|
1280 | [ cases (state_bound_on_steps_to_cost_zero s) /2/ |
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1281 | | (* TODO: how do we know that we're not at the final state? *) |
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1282 | | @(absurd ?? NO_TERMINATION) |
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1283 | %{0} % @wr_base // |
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1284 | | @(well_cost_labelled_jump … EV) // |
---|
1285 | | 5,6,7,8,9,10: /2/ |
---|
1286 | | /2/ |
---|
1287 | | // |
---|
1288 | | (* FIXME: post-call non-termination *) |
---|
1289 | | (* FIXME: post-call not-wrong *) |
---|
1290 | | (* FIXME: bound on steps after call *) |
---|
1291 | | @(well_cost_labelled_state_step … EV) // |
---|
1292 | | @(well_cost_labelled_call … EV) // |
---|
1293 | | 18,19,20: /2/ |
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1294 | | @(well_cost_labelled_state_step … EV) // |
---|
1295 | | @(not_to_not … NO_TERMINATION) |
---|
1296 | * #depth * #TERM %{depth} % @wr_step /2/ |
---|
1297 | | @(still_not_wrong … NOT_UNDEFINED) |
---|
1298 | | cases (bound_after_step … LABEL_LIMIT EV ?) |
---|
1299 | [ * [ #TERMINATES @⊥ @(absurd ?? NO_TERMINATION) %{0} % @wr_step [ %1 // | |
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1300 | inversion trace' |
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1301 | [ cases daemon (* TODO again *) | #s1 #tr1 #s2 #EVAL' #trace'' #E1 #E2 destruct |
---|
1302 | @wr_base // |
---|
1303 | | #H99 #H100 #H101 #H102 #H103 destruct |
---|
1304 | inversion NOT_UNDEFINED |
---|
1305 | [ #H137 #H138 #H139 #H140 #H141 destruct |
---|
1306 | | #H143 #H144 #H145 #H146 #H147 #H148 #H149 #H150 #H151 destruct |
---|
1307 | inversion H148 |
---|
1308 | [ #H153 #H154 #H155 #H156 #H157 destruct |
---|
1309 | | #H159 #H160 #H161 #H162 #H163 #H164 #H165 #H166 #H167 destruct |
---|
1310 | ] |
---|
1311 | ] |
---|
1312 | ] |
---|
1313 | ] |
---|
1314 | | >CL #E destruct |
---|
1315 | ] |
---|
1316 | | // |
---|
1317 | | // |
---|
1318 | ] |
---|
1319 | | inversion NOT_UNDEFINED |
---|
1320 | [ #H169 #H170 #H171 #H172 #H173 destruct |
---|
1321 | | #H175 #H176 #H177 #H178 #H179 #H180 #H181 #H182 #H183 destruct |
---|
1322 | ] |
---|
1323 | ] cases daemon qed. |
---|
1324 | |
---|
1325 | (* |
---|
1326 | let corec make_label_diverges ge s |
---|
1327 | (trace: flat_trace io_out io_in ge s) |
---|
1328 | (ENV_COSTLABELLED: well_cost_labelled_ge ge) |
---|
1329 | (STATE_COSTLABELLED: well_cost_labelled_state s) (* functions in the state *) |
---|
1330 | (STATEMENT_COSTLABEL: RTLabs_cost s = true) (* current statement is a cost label *) |
---|
1331 | (SOUNDLY_COSTLABELLED: soundly_labelled_trace … trace) |
---|
1332 | (NO_TERMINATION: Not (∃depth. inhabited (will_return ge depth s trace))) |
---|
1333 | : trace_label_diverges (RTLabs_status ge) s ≝ ? |
---|
1334 | . |
---|
1335 | *) |
---|