1 | include "joint/TranslateUtils_paolo.ma". |
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2 | include "utilities/hide.ma". |
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3 | |
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4 | definition graph_to_lin_statement : |
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5 | ∀p : unserialized_params.∀globals. |
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6 | joint_statement (mk_graph_params p) globals → joint_statement (mk_lin_params p) globals ≝ |
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7 | λp,globals,stmt.match stmt return λ_.joint_statement (mk_lin_params ?) ? with |
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8 | [ sequential c _ ⇒ sequential … c it |
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9 | | final c ⇒ final … c |
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10 | ]. |
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11 | |
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12 | lemma graph_to_lin_labels : |
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13 | ∀p : unserialized_params.∀globals,s. |
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14 | stmt_labels … (graph_to_lin_statement p globals s) = |
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15 | stmt_explicit_labels … s. |
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16 | #p#globals * [//] * // |
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17 | qed. |
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18 | |
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19 | (* discard all elements passing test, return first element failing it *) |
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20 | (* and the rest of the list, if any. *) |
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21 | let rec chop A (test : A → bool) (l : list A) on l : option (A × (list A)) ≝ |
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22 | match l with |
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23 | [ nil ⇒ None ? |
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24 | | cons x l' ⇒ if test x then chop A test l' else return 〈x, l'〉 |
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25 | ]. |
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26 | |
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27 | lemma chop_ok : ∀A,f,l. |
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28 | match chop A f l with |
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29 | [ Some pr ⇒ |
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30 | let x ≝ \fst pr in |
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31 | let post ≝ \snd pr in |
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32 | ∃pre.All ? (λx.bool_to_Prop (f x)) pre ∧ |
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33 | l = pre @ x :: post ∧ ¬bool_to_Prop (f x) |
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34 | | None ⇒ All A (λx.bool_to_Prop (f x)) l |
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35 | ]. |
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36 | #A #f #l elim l |
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37 | [ % |
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38 | | #hd #tl #IH whd in match (chop ???); |
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39 | elim (true_or_false_Prop (f hd)) #H >H normalize nodelta |
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40 | [ lapply IH elim (chop ???) normalize nodelta |
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41 | [ #G %{H G} |
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42 | | #pr * #pre ** #H1 #H2 #H3 %{(hd :: pre)} %{H3} % |
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43 | [ %{H H1} |
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44 | | >H2 % |
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45 | ] |
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46 | ] |
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47 | | %{[ ]} %{H} %% |
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48 | ] |
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49 | ] |
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50 | qed. |
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51 | |
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52 | unification hint 0 ≔ p, globals; |
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53 | lp ≟ lin_params_to_params p, |
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54 | sp ≟ stmt_pars lp, |
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55 | js ≟ joint_statement sp globals, |
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56 | lo ≟ labelled_obj LabelTag js |
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57 | (*----------------------------*)⊢ |
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58 | list lo ≡ codeT lp globals. |
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59 | |
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60 | |
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61 | definition graph_visit_ret_type ≝ λp,globals.λg : codeT (mk_graph_params p) globals. |
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62 | λentry : label. |
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63 | (Σ〈visited' : identifier_map LabelTag ℕ, |
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64 | required' : identifier_set LabelTag, |
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65 | generated' : codeT (mk_lin_params p) globals〉.'hide ( |
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66 | And (And (And (And |
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67 | (lookup ?? visited' entry = Some ? 0) |
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68 | (required' ⊆ visited')) |
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69 | (∃last.stmt_at … generated' 0 = Some ? (final … last))) |
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70 | (code_forall_labels … (λl.bool_to_Prop (l∈required')) (rev … generated'))) |
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71 | (∀l,n.lookup ?? visited' l = Some ? n → |
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72 | And (bool_to_Prop (code_has_label … (rev ? generated') l)) |
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73 | (∃s.And (And |
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74 | (lookup … g l = Some ? s) |
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75 | (nth_opt … n (rev … generated') = Some ? 〈Some ? l, graph_to_lin_statement … s〉)) |
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76 | (opt_All ? |
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77 | (λnext.Or (lookup … visited' next = Some ? (S n)) |
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78 | (nth_opt … (S n) (rev … generated') = Some ? 〈None ?, GOTO … next〉)) |
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79 | (stmt_implicit_label … s)))))). |
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80 | |
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81 | unification hint 0 ≔ tag ⊢ identifier_set tag ≡ identifier_map tag unit. |
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82 | |
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83 | include alias "common/Identifiers.ma". |
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84 | |
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85 | lemma lookup_safe_elim : ∀tag,A.∀P : A → Prop.∀m,i,prf. |
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86 | (∀x.lookup tag A m i = Some ? x → P x) → P (lookup_safe tag A m i prf). |
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87 | #tag #A #P #m #i #prf #H @H @lookup_eq_safe qed. |
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88 | |
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89 | let rec graph_visit |
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90 | (p : unserialized_params) |
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91 | (globals: list ident) |
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92 | (g : codeT (mk_graph_params p) globals) |
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93 | (* = graph (joint_statement (mk_graph_params p) globals *) |
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94 | (required: identifier_set LabelTag) |
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95 | (visited: identifier_map LabelTag ℕ) (* the reversed index of labels in the new code *) |
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96 | (generated: codeT (mk_lin_params p) globals) |
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97 | (* ≝ list ((option label)×(joint_statement (mk_lin_params p) globals)) *) |
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98 | (visiting: list label) |
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99 | (gen_length : ℕ) |
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100 | (n: nat) |
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101 | (entry : label) |
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102 | (g_prf : code_closed … g) |
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103 | (required_prf1 : ∀i.i∈required → Or (In ? visiting i) (bool_to_Prop (i∈visited))) |
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104 | (required_prf2 : code_forall_labels … (λl.bool_to_Prop (l ∈ required)) (rev … generated)) |
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105 | (generated_prf1 : ∀l,n.lookup … visited l = Some ? n → hide_Prop ( |
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106 | And (bool_to_Prop (code_has_label … (rev ? generated) l)) |
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107 | (∃s.And (And |
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108 | (lookup … g l = Some ? s) |
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109 | (nth_opt ? n (rev … generated) = Some ? 〈Some ? l, graph_to_lin_statement … s〉)) |
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110 | (opt_All ? (λnext.match lookup … visited next with |
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111 | [ Some n' ⇒ Or (n' = S n) (nth_opt ? (S n) (rev ? generated) = Some ? 〈None ?, GOTO … next〉) |
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112 | | None ⇒ And (nth_opt ? 0 visiting = Some ? next) (S n = gen_length) ]) (stmt_implicit_label … s))))) |
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113 | (generated_prf2 : ∀l.lookup … visited l = None ? → does_not_occur … l (rev ? generated)) |
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114 | (generated_prf3 : (∃last.stmt_at … generated 0 = Some ? (final … last)) ∨ |
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115 | (¬All ? (λx.bool_to_Prop (x∈visited)) visiting)) |
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116 | (visiting_prf : All … (λl.bool_to_Prop (l∈g)) visiting) |
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117 | (gen_length_prf : gen_length = length ? generated) |
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118 | (entry_prf : Or (And (And (visiting = [entry]) (gen_length = 0)) (Not (bool_to_Prop (entry∈visited)))) |
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119 | (lookup … visited entry = Some ? 0)) |
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120 | (n_prf : le (id_map_size … g) (plus n (id_map_size … visited))) |
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121 | on n |
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122 | : graph_visit_ret_type … g entry ≝ |
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123 | match chop ? (λx.x∈visited) visiting |
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124 | return λx.? → graph_visit_ret_type … g entry with |
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125 | [ None ⇒ λH. |
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126 | «〈visited, required, generated〉, ?» |
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127 | | Some pr ⇒ λH. |
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128 | let vis_hd ≝ \fst pr in |
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129 | let vis_tl ≝ \snd pr in |
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130 | match n return λx.? → graph_visit_ret_type … g entry with |
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131 | [ O ⇒ λn_prf'.⊥ |
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132 | | S n' ⇒ λn_prf'. |
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133 | (* add the label to the visited ones *) |
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134 | let visited' ≝ add … visited vis_hd gen_length in |
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135 | (* take l's statement *) |
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136 | let hd_vis_in_g ≝ (hide_prf ? ?) in |
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137 | let statement ≝ lookup_safe ?? g vis_hd hd_vis_in_g in |
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138 | (* translate it to its linear version *) |
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139 | let translated_statement ≝ graph_to_lin_statement p globals statement in |
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140 | (* add the translated statement to the code (which will be reversed later) *) |
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141 | let generated' ≝ 〈Some … vis_hd, translated_statement〉 :: generated in |
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142 | let required' ≝ union_set ??? (set_from_list … (stmt_explicit_labels … statement)) required in |
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143 | (* put successors in the visiting worklist *) |
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144 | let visiting' ≝ stmt_labels … statement @ vis_tl in |
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145 | (* finally, check the implicit successor *) |
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146 | (* if it has been already visited, we need to add a GOTO *) |
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147 | let add_req_gen ≝ |
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148 | match stmt_implicit_label … statement |
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149 | with |
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150 | [ Some next ⇒ |
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151 | if next ∈ visited' then |
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152 | 〈1, {(next)}, [〈None label, (GOTO … next : joint_statement ??)〉]〉 |
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153 | else 〈0, ∅, [ ]〉 |
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154 | | None ⇒ 〈0, ∅, [ ]〉 |
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155 | ] in |
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156 | (* prepare a common utility to deal with add_req_gen *) |
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157 | let add_req_gen_prf : |
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158 | ∀P : (ℕ × (identifier_set LabelTag) × (codeT (mk_lin_params p) globals)) → Prop. |
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159 | (opt_All ? (λnext.¬bool_to_Prop (next ∈ visited')) (stmt_implicit_label … statement) → |
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160 | P 〈0,∅,[ ]〉) → |
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161 | (∀next.stmt_implicit_label … statement = Some ? next → |
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162 | next ∈ visited' → |
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163 | P 〈1, {(next)}, [〈None ?, GOTO (mk_lin_params p) next〉]〉) → |
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164 | P add_req_gen ≝ hide_prf ?? in |
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165 | graph_visit ??? |
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166 | (union_set ??? (\snd (\fst add_req_gen)) required') |
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167 | visited' |
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168 | (\snd add_req_gen @ generated') |
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169 | visiting' |
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170 | (plus (\fst (\fst add_req_gen)) (S gen_length)) |
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171 | n' entry g_prf ????????? |
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172 | ] n_prf |
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173 | ] (chop_ok ? (λx.x∈visited) visiting). |
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174 | whd |
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175 | [ (* base case, visiting is all visited *) |
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176 | %[%[%[%]]] |
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177 | [ elim entry_prf |
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178 | [ ** #eq_visiting #gen_length_O #entry_vis >eq_visiting in H; * >entry_vis * |
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179 | | // |
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180 | ] |
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181 | | #l #l_req |
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182 | elim (required_prf1 … l_req) #G |
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183 | [ @(All_In … H G) |
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184 | | assumption |
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185 | ] |
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186 | | cases generated_prf3 [//] |
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187 | * #ABS @⊥ @ABS assumption |
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188 | | assumption |
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189 | | #l #n #H elim (generated_prf1 … H) |
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190 | #H1 * #s ** #H2 #H3 #H4 |
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191 | % [assumption] %{s} % |
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192 | [% assumption |
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193 | | @(opt_All_mp … H4) -l #l |
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194 | lapply (in_map_domain … visited l) |
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195 | elim (true_or_false_Prop (l∈visited)) #l_vis >l_vis |
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196 | normalize nodelta [ * #n' ] #EQlookup >EQlookup |
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197 | normalize nodelta * |
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198 | [ #EQn' % >EQn' % |
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199 | | #H %2{H} |
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200 | | #H' lapply (All_nth … H … H') >l_vis * |
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201 | ] |
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202 | ] |
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203 | ] |
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204 | (* first, close goals where add_gen_req plays no role *) |
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205 | |13: (* vis_hd is in g *) |
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206 | elim H #pre ** #_ #H2 #_ |
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207 | @(All_split … visiting_prf … H2) |
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208 | |2: (* n = 0 absrud *) |
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209 | elim H #pre ** #_ #H2 #H3 |
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210 | @(absurd … n_prf') |
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211 | @lt_to_not_le |
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212 | lapply (add_size … visited vis_hd 0 (* dummy value *)) |
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213 | >H3 normalize nodelta |
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214 | whd in match (lt ? ?); |
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215 | whd in match (1 + ?); |
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216 | #EQ <EQ @subset_card @add_subset |
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217 | [ @(All_split ? (λx.bool_to_Prop (x∈g)) ????? H2) @(All_mp … visiting_prf) |
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218 | #a elim g #gm whd in ⊢ (?→?%); #H >(lookup_eq_safe … H) % |
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219 | | #l #H |
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220 | elim (generated_prf1 … (lookup_eq_safe … H)) #_ * #s ** #s_in_g #_ #_ |
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221 | whd in ⊢ (?%); >s_in_g % |
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222 | ] |
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223 | |8: |
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224 | elim H #pre ** #_ #H2 #_ |
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225 | @All_append |
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226 | [ elim(g_prf … vis_hd statement ?) [2:@lookup_eq_safe] #G1 #G2 |
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227 | @(All_append … G1) |
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228 | whd in G2; lapply G2 elim statement normalize nodelta #s [2: #_ %] |
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229 | #l #G' %{G'} % |
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230 | | >H2 in visiting_prf; |
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231 | #H' lapply (All_append_r … H') -H' * #_ // |
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232 | ] |
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233 | |10: |
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234 | elim H #pre ** #_ #H2 #H3 -add_req_gen_prf |
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235 | %2 elim entry_prf |
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236 | [ ** >H2 cases pre |
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237 | [2: #x' #pre' #ABS normalize in ABS; destruct(ABS) |
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238 | cases pre' in e0; [2: #x'' #pre''] #ABS' normalize in ABS'; destruct(ABS') |
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239 | |1: #EQ normalize in EQ; destruct(EQ) |
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240 | #eq_gen_length #_ |
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241 | >lookup_add_hit >eq_gen_length % |
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242 | ] |
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243 | | #lookup_entry cut (entry ≠ vis_hd) |
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244 | [ % whd in match vis_hd; #entry_vis_hd <entry_vis_hd in H3; |
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245 | whd in ⊢ (?(?%)→?); >lookup_entry * #ABS @ABS % ] |
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246 | #entry_not_vis_hd >(lookup_add_miss ?????? entry_not_vis_hd) assumption |
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247 | ] |
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248 | |11: |
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249 | elim H #pre ** #_ #_ #H3 |
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250 | >commutative_plus |
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251 | >add_size >H3 normalize nodelta |
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252 | whd in match (1 + ?); |
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253 | >commutative_plus |
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254 | assumption |
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255 | |12: (* add_req_gen utility *) |
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256 | #P whd in match add_req_gen; |
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257 | elim (stmt_implicit_label ???) |
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258 | [ #H #_ @H % ] |
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259 | #next normalize nodelta elim (true_or_false_Prop (next ∈ visited')) #next_vis |
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260 | >next_vis normalize nodelta |
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261 | [ #_ #H @H [% | assumption] |
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262 | | #H #_ @H whd >next_vis % * |
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263 | ] |
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264 | | elim H #pre ** #H1 #H2 #_ |
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265 | #i >mem_set_union |
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266 | #H elim (orb_Prop_true … H) -H |
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267 | [ @add_req_gen_prf [ #_ >mem_set_empty * ] |
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268 | #next #_ #next_vis #H >(mem_set_singl_to_eq … H) %2 assumption |
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269 | | >mem_set_union |
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270 | #H elim (orb_Prop_true … H) -H |
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271 | [ #i_expl %1 @Exists_append_l @Exists_append_r |
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272 | @mem_list_as_set |
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273 | @i_expl |
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274 | | (* i was already required *) |
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275 | #i_req |
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276 | elim (required_prf1 … i_req) |
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277 | [ >H2 #H elim (Exists_append … H) -H |
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278 | [ (* i in preamble → i∈visited *) |
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279 | #i_pre %2 >mem_set_add @orb_Prop_r |
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280 | lapply (All_In … H1 i_pre) #H @H |
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281 | | * |
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282 | [ (* i is vis_hd *) |
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283 | #eq_i >eq_i %2 @mem_set_add_id |
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284 | | (* i in vis_tl → i∈visiting' *) |
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285 | #i_post % @Exists_append_r assumption |
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286 | ] |
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287 | ] |
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288 | | (* i in visited *) |
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289 | #i_vis %2 >mem_set_add @orb_Prop_r assumption |
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290 | ] |
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291 | ] |
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292 | ] |
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293 | |4,5,6: change with reverse in match rev; |
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294 | >reverse_append whd in match (reverse ??); >rev_append_def |
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295 | >associative_append |
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296 | [ #pt #s |
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297 | @(leb_elim (S pt) (|generated|)) #cmp |
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298 | whd in match (stmt_at ????); |
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299 | [ >nth_opt_append_l [2: >length_reverse assumption ] |
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300 | change with (stmt_at ???? = ? → ?) |
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301 | #EQ lapply(required_prf2 … EQ) @All_mp |
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302 | #l #l_req >mem_set_union @orb_Prop_r |
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303 | >mem_set_union @orb_Prop_r @l_req |
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304 | | >nth_opt_append_r [2: >length_reverse @not_lt_to_le assumption ] |
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305 | cases (pt - ?) |
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306 | [ whd in match (nth_opt ???); whd in ⊢ (??%?→?); |
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307 | #EQ destruct(EQ) whd |
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308 | >graph_to_lin_labels in ⊢ (???%); |
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309 | whd in match required'; |
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310 | generalize in match (stmt_explicit_labels … statement); |
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311 | #l @list_as_set_All |
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312 | #i >mem_set_union >mem_set_union |
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313 | #i_l @orb_Prop_r @orb_Prop_l @i_l |
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314 | | @add_req_gen_prf |
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315 | [ #_ | #next #_ #next_vis * |
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316 | [ whd in ⊢ (??%?→?); |
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317 | #EQ' destruct(EQ') whd %{I} >mem_set_union |
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318 | @orb_Prop_l @mem_set_add_id ]] |
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319 | #n whd in ⊢ (??%?→?); #ABS destruct(ABS) |
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320 | ] |
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321 | ] |
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322 | | elim H #pre ** #H1 #H2 #H3 |
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323 | #i whd in match visited'; |
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324 | @(eq_identifier_elim … i vis_hd) |
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325 | [ #EQi >EQi -i #pos |
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326 | >lookup_add_hit #EQpos (* too slow: destruct(EQpos) *) |
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327 | cut (gen_length = pos) |
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328 | [1,3,5: (* BUG: -graph_visit *) -visited destruct(EQpos) %] |
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329 | -EQpos #EQpos <EQpos -pos |
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330 | % |
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331 | [ >lin_code_has_label |
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332 | @add_req_gen_prf |
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333 | [ #_ |
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334 | | #next #_ #next_vis |
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335 | change with (? @ ([?] @ [?])) in match (? @ [? ; ?]); |
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336 | <associative_append >occurs_exactly_once_None |
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337 | ] |
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338 | >occurs_exactly_once_Some_eq >eq_identifier_refl |
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339 | normalize nodelta |
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340 | @generated_prf2 |
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341 | lapply (in_map_domain … visited vis_hd) |
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342 | >H3 normalize nodelta // |
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343 | | %{statement} |
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344 | % [ % ] |
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345 | [ @lookup_eq_safe |
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346 | | <associative_append @nth_opt_append_hit_l |
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347 | >nth_opt_append_r |
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348 | >rev_length |
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349 | <gen_length_prf |
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350 | [<minus_n_n] % |
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351 | | @add_req_gen_prf |
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352 | [ lapply (refl … (stmt_implicit_label ?? statement)) |
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353 | cases (stmt_implicit_label ???) in ⊢ (???%→%); |
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354 | [#_ #_ %] |
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355 | #next #K change with (¬bool_to_Prop (?∈?)→?) #next_vis |
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356 | | #next #K >K #next_vis |
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357 | ] whd |
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358 | >mem_set_add in next_vis; |
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359 | @eq_identifier_elim |
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360 | [1,3: #EQnext >EQnext * [#ABS elim(ABS I)] |
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361 | >lookup_add_hit |
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362 | |*: #NEQ >(lookup_add_miss … visited … NEQ) |
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363 | change with (?∈?) in match (false∨?); |
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364 | #next_vis lapply(in_map_domain … visited next) >next_vis |
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365 | whd in ⊢ (% → ?); [2: * #s ] |
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366 | #EQlookup >EQlookup |
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367 | ] whd |
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368 | [1,2: %2 <associative_append |
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369 | >nth_opt_append_r >append_length >rev_length >commutative_plus |
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370 | <gen_length_prf |
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371 | [1,3: <minus_n_n ] % |
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372 | | % [2: %] @nth_opt_append_hit_l whd in match (stmt_labels … statement); |
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373 | >K % |
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374 | ] |
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375 | ] |
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376 | ] |
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377 | | #NEQ #n_i >(lookup_add_miss … visited … NEQ) |
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378 | #Hlookup elim (generated_prf1 … Hlookup) |
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379 | #G1 * #s ** #G2 #G3 #G4 |
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380 | % |
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381 | [ >lin_code_has_label <associative_append |
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382 | >occurs_exactly_once_append |
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383 | @orb_Prop_l @andb_Prop |
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384 | [ >occurs_exactly_once_Some_eq |
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385 | >eq_identifier_false [2: % #ABS >ABS in NEQ; * #ABS' @ABS' % ] |
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386 | normalize nodelta >lin_code_has_label in G1; #K @K |
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387 | | @add_req_gen_prf |
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388 | [ #_ % | #next #_ #_ % ] |
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389 | ] |
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390 | | %{s} |
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391 | % [ % ] |
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392 | [ assumption |
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393 | | @nth_opt_append_hit_l assumption |
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394 | | @(opt_All_mp … G4) |
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395 | #x |
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396 | @(eq_identifier_elim … x vis_hd) #Heq |
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397 | [ >Heq |
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398 | lapply (in_map_domain … visited vis_hd) |
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399 | >H3 normalize nodelta in ⊢ (%→?); |
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400 | #EQlookup >EQlookup * #nth_opt_visiting #gen_length_eq |
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401 | >lookup_add_hit %1 >gen_length_eq % |
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402 | | >(lookup_add_miss ?????? Heq) |
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403 | lapply (in_map_domain … visited x) |
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404 | elim (true_or_false_Prop (x∈visited)) #x_vis >x_vis |
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405 | normalize nodelta in ⊢ (%→?); [ * #n' ] |
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406 | #EQlookupx >EQlookupx whd in ⊢ (%→%); * |
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407 | [ #G %1{G} |
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408 | | #G %2 @nth_opt_append_hit_l |
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409 | assumption |
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410 | | #G elim(absurd ?? Heq) |
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411 | (* BUG (but useless): -required -g -generated *) |
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412 | >H2 in G; lapply H1 cases pre |
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413 | [ #_ |
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414 | | #hd #tl * #hd_vis #_ |
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415 | ] normalize #EQ' destruct(EQ') |
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416 | [ % |
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417 | | >x_vis in hd_vis; * |
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418 | ] |
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419 | ] |
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420 | ] |
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421 | ] |
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422 | ] |
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423 | ] |
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424 | | #i whd in match visited'; |
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425 | @(eq_identifier_elim … i vis_hd) #Heq |
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426 | [ >Heq >lookup_add_hit #ABS destruct(ABS) ] |
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427 | >(lookup_add_miss ?????? Heq) |
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428 | #i_n_vis |
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429 | >does_not_occur_append @andb_Prop |
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430 | [ @generated_prf2 assumption |
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431 | | change with (bool_to_Prop (¬eq_identifier ??? ∧ ?)) |
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432 | >eq_identifier_false [2: % #ABS <ABS in Heq; * #ABS' @ABS' % ] |
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433 | @add_req_gen_prf [ #_ | #next #_ #_ ] % |
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434 | ] |
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435 | ] |
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436 | | @add_req_gen_prf |
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437 | [ #K | #next #K #next_vis %1 %{(GOTO … next)} % ] |
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438 | whd in match generated'; whd in match translated_statement; |
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439 | lapply K whd in match visiting'; |
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440 | whd in match (stmt_labels ???); cases statement #last |
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441 | [2: #_ %1 %{last} % ] #next whd in ⊢ (%→?); #next_vis |
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442 | %2 % * >next_vis * |
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443 | | whd in match generated'; |
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444 | @add_req_gen_prf [ #_ | #next #_ #_ ] normalize >gen_length_prf % |
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445 | ] |
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446 | qed. |
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447 | |
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448 | (* CSC: The branch compression (aka tunneling) optimization is not implemented |
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449 | in Matita *) |
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450 | definition branch_compress |
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451 | ≝ λp: graph_params.λglobals.λg:codeT p globals. |
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452 | λentry : Σl.bool_to_Prop (code_has_label … g l).g. |
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453 | |
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454 | lemma branch_compress_closed : ∀p,globals,g,l.code_closed ?? g → |
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455 | code_closed … (branch_compress p globals g l). |
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456 | #p#globals#g#l#H @H qed. |
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457 | |
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458 | lemma branch_compress_has_entry : ∀p,globals,g,l. |
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459 | code_has_label … (branch_compress p globals g l) l. |
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460 | #p#globals#g*#l#l_prf @l_prf qed. |
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461 | |
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462 | definition filter_labels ≝ λtag,A.λtest : identifier tag → bool.λc : list (labelled_obj tag A). |
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463 | map ?? |
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464 | (λs. let 〈l_opt,x〉 ≝ s in |
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465 | 〈! l ← l_opt ; if test l then return l else None ?, x〉) c. |
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466 | |
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467 | lemma does_not_occur_filter_labels : |
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468 | ∀tag,A,test,id,c. |
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469 | does_not_occur ?? id (filter_labels tag A test c) = |
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470 | (does_not_occur ?? id c ∨ ¬ test id). |
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471 | #tag #A #test #id #c elim c |
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472 | [ // |
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473 | | ** [2: #lbl] #s #tl #IH |
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474 | whd in match (filter_labels ????); normalize nodelta |
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475 | whd in match (does_not_occur ????) in ⊢ (??%%); |
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476 | [2: @IH] |
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477 | normalize in match (! l ← ? ; ?); >IH |
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478 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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479 | elim (test lbl) normalize nodelta |
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480 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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481 | [1,2: >eq_identifier_refl [2: >commutative_orb] normalize % |
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482 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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483 | ] |
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484 | ] |
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485 | qed. |
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486 | |
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487 | lemma occurs_exactly_once_filter_labels : |
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488 | ∀tag,A,test,id,c. |
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489 | occurs_exactly_once ?? id (filter_labels tag A test c) = |
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490 | (occurs_exactly_once ?? id c ∧ test id). |
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491 | #tag #A #test #id #c elim c |
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492 | [ // |
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493 | | ** [2: #lbl] #s #tl #IH |
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494 | whd in match (filter_labels ????); normalize nodelta |
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495 | whd in match (occurs_exactly_once ????) in ⊢ (??%%); |
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496 | [2: @IH] |
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497 | normalize in match (! l ← ? ; ?); >IH |
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498 | >does_not_occur_filter_labels |
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499 | @(eq_identifier_elim ?? lbl id) #Heq [<Heq] |
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500 | elim (test lbl) normalize nodelta |
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501 | change with (eq_identifier ???) in match (instruction_matches_identifier ????); |
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502 | [1,2: >eq_identifier_refl >commutative_andb [ >(commutative_andb ? true) >commutative_orb | >(commutative_andb ? false)] normalize % |
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503 | |*: >(eq_identifier_false ??? Heq) normalize nodelta % |
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504 | ] |
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505 | ] |
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506 | qed. |
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507 | |
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508 | lemma nth_opt_filter_labels : ∀tag,A,test,instrs,n. |
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509 | nth_opt ? n (filter_labels tag A test instrs) = |
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510 | ! 〈lopt, s〉 ← nth_opt ? n instrs ; |
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511 | return 〈 ! lbl ← lopt; if test lbl then return lbl else None ?, s〉. |
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512 | #tag #A #test #instrs elim instrs |
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513 | [ * [2: #n'] % |
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514 | | * #lopt #s #tl #IH * [2: #n'] |
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515 | whd in match (filter_labels ????); normalize nodelta |
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516 | whd in match (nth_opt ???) in ⊢ (??%%); [>IH] % |
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517 | ] |
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518 | qed. |
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519 | |
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520 | lemma stmt_at_filter_labels : ∀p : lin_params.∀globals,test.∀c : codeT p globals. |
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521 | ∀i.stmt_at p globals (filter_labels ?? test c) i = stmt_at p globals c i. |
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522 | #p#globals#test #c#i |
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523 | whd in ⊢ (??%%); >nth_opt_filter_labels |
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524 | elim (nth_opt ???); // |
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525 | qed. |
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526 | |
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527 | lemma option_bind_inverse : ∀A,B.∀m : option A.∀f : A → option B.∀r. |
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528 | ! x ← m ; f x = return r → |
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529 | ∃x.m = return x ∧ f x = return r. |
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530 | #A #B * normalize [2:#x] #f #r #EQ destruct |
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531 | %{x} %{EQ} % |
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532 | qed. |
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533 | |
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534 | lemma nth_opt_reverse_hit : |
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535 | ∀A,l,n.n < |l| → nth_opt A n (reverse ? l) = nth_opt A (|l| - (S n)) l. |
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536 | #A #l elim l |
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537 | [ #n #ABS normalize in ABS; @⊥ -A /2 by absurd/ |
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538 | | #hd #tl #IH #n #lim whd in match (reverse ??); >rev_append_def |
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539 | @(leb_elim (S n) (|tl|)) #H |
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540 | [ >nth_opt_append_l [2: >length_reverse @H ] |
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541 | >(IH … H) >(minus_Sn_m … H) % |
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542 | | >(le_to_le_to_eq … (le_S_S_to_le … lim) (not_lt_to_le … H)) |
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543 | >nth_opt_append_r >length_reverse [2: % ] |
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544 | <minus_n_n <minus_n_n % |
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545 | ] |
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546 | ] |
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547 | qed. |
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548 | |
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549 | lemma nth_opt_reverse_hit_inv : |
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550 | ∀A,l,n.n < |l| → nth_opt A (|l| - (S n)) (reverse ? l) = nth_opt A n l. |
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551 | #A #l #n #H <(reverse_reverse ? l) in ⊢ (???%); @sym_eq |
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552 | <length_reverse @nth_opt_reverse_hit >length_reverse @H |
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553 | qed. |
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554 | |
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555 | definition linearise_code: |
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556 | ∀p : unserialized_params.∀globals. |
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557 | ∀g : codeT (mk_graph_params p) globals.code_closed … g → |
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558 | ∀entry : (Σl.bool_to_Prop (code_has_label … g l)). |
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559 | (Σc : codeT (mk_lin_params p) globals. |
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560 | code_closed … c ∧ |
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561 | ∃ sigma : identifier_map LabelTag ℕ. |
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562 | lookup … sigma entry = Some ? 0 ∧ |
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563 | ∀l,n.lookup … sigma l = Some ? n → |
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564 | ∃s. lookup … g l = Some ? s ∧ |
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565 | opt_Exists ? |
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566 | (λls.let 〈lopt, ts〉 ≝ ls in |
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567 | opt_All ? (eq ? l) lopt ∧ |
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568 | ts = graph_to_lin_statement … s ∧ |
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569 | opt_All ? |
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570 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
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571 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
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572 | (stmt_implicit_label … s)) (nth_opt … n c)) |
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573 | ≝ |
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574 | λp,globals,g,g_prf,entry_sig. |
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575 | let g ≝ branch_compress (mk_graph_params p) ? g entry_sig in |
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576 | let g_prf ≝ branch_compress_closed … g entry_sig g_prf in |
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577 | match graph_visit p globals g ∅ (empty_map …) [ ] [entry_sig] 0 (|g|) |
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578 | (entry_sig) g_prf ????????? |
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579 | with |
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580 | [ mk_Sig triple H ⇒ |
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581 | let sigma ≝ \fst (\fst triple) in |
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582 | let required ≝ \snd (\fst triple) in |
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583 | let crev ≝ \snd triple in |
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584 | let lbld_code ≝ rev ? crev in |
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585 | mk_Sig ?? (filter_labels … (λl.l∈required) lbld_code) ? ]. |
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586 | [ (* done later *) |
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587 | | #i >mem_set_empty * |
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588 | |3,4: #a #b whd in ⊢ (??%?→?); #ABS destruct(ABS) |
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589 | | #l #_ % |
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590 | | %2 % * >mem_set_empty * |
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591 | | % [2: %] @(branch_compress_has_entry … g entry_sig) |
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592 | | % |
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593 | | % % [% %] cases (pi1 … entry_sig) normalize #_ % // |
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594 | | >commutative_plus change with (? ≤ |g|) % |
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595 | ] |
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596 | lapply (refl … triple) |
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597 | generalize in match triple in ⊢ (???%→%); ** |
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598 | #visited #required #generated #EQtriple |
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599 | >EQtriple in H ⊢ %; normalize nodelta **** |
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600 | #entry_O #req_vis #last_fin #labels_in_req #sigma_prop |
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601 | % >EQtriple |
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602 | [ #i #s |
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603 | >stmt_at_filter_labels #EQ |
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604 | % |
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605 | [ @stmt_forall_labels_explicit |
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606 | @(All_mp … (labels_in_req … EQ)) |
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607 | #l #l_req >lin_code_has_label |
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608 | >occurs_exactly_once_filter_labels >l_req |
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609 | >commutative_andb whd in ⊢ (?%); |
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610 | elim (sigma_prop ?? (lookup_eq_safe … (req_vis … l_req))) |
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611 | >lin_code_has_label #H #_ @H |
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612 | | lapply EQ cases s #s' [2: #_ % ] |
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613 | * -EQ #EQ change with (bool_to_Prop (code_has_point ????)) |
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614 | whd in match (point_of_succ ???); |
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615 | >lin_code_has_point @leb_elim [ #_ % ] >length_map >length_reverse |
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616 | #INEQ |
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617 | cut (|generated| = S i) |
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618 | [ @(le_to_le_to_eq … (not_lt_to_le … INEQ) ) |
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619 | elim (option_bind_inverse ????? EQ) #x * #EQ1 #EQ2 |
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620 | <length_reverse |
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621 | @(nth_opt_hit_length … EQ1) |
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622 | ] #EQ_length |
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623 | elim last_fin #fin #EQ' |
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624 | lapply EQ whd in match (stmt_at ????); |
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625 | >nth_opt_reverse_hit >EQ_length [2: % ] <minus_n_n |
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626 | change with (stmt_at ???? = ? → ?) |
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627 | >EQ' #ABS destruct(ABS) |
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628 | ] |
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629 | ] |
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630 | %{visited} % [assumption] |
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631 | #lbl #n #eq_lookup elim (sigma_prop ?? eq_lookup) |
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632 | #lbl_in_gen * #stmt ** #stmt_in_g #nth_opt_is_stmt #succ_is_in |
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633 | % [2: % [ assumption ] |] |
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634 | >nth_opt_filter_labels in ⊢ (???%); |
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635 | >nth_opt_is_stmt >m_return_bind whd >m_return_bind |
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636 | % [ % ] |
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637 | [ elim (lbl ∈ required) % |
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638 | | % |
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639 | | lapply succ_is_in |
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640 | lapply (refl … (stmt_implicit_label … stmt)) |
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641 | cases (stmt_implicit_label … stmt) in ⊢ (???%→%); [#_ #_ %] |
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642 | #next #EQ_next * |
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643 | [ #H %1{H} ] |
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644 | #H %2 |
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645 | >nth_opt_filter_labels >H % |
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646 | ] |
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647 | qed. |
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648 | |
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649 | (* Paolo: for now I'm leaving the closed graph stuff out, waiting for it to |
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650 | be done on all passes *) |
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651 | definition linearise_int_fun : |
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652 | ∀p : unserialized_params. |
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653 | ∀globals. |
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654 | joint_internal_function (mk_graph_params p) globals → |
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655 | joint_internal_function (mk_lin_params p) globals |
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656 | (* ∃sigma : identifier_map LabelTag ℕ. |
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657 | let g ≝ joint_if_code ?? (pi1 … fin) in |
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658 | let c ≝ joint_if_code ?? (pi1 … fout) in |
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659 | let entry ≝ joint_if_entry ?? (pi1 … fin) in |
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660 | lookup … sigma entry = Some ? 0 ∧ |
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661 | ∀l,n.lookup … sigma l = Some ? n → |
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662 | ∃s. lookup … g l = Some ? s ∧ |
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663 | opt_Exists ? |
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664 | (λls.let 〈lopt, ts〉 ≝ ls in |
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665 | opt_All ? (eq ? l) lopt ∧ |
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666 | ts = graph_to_lin_statement … s ∧ |
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667 | opt_All ? |
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668 | (λnext.Or (lookup … sigma next = Some ? (S n)) |
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669 | (nth_opt … (S n) c = Some ? 〈None ?, GOTO … next〉)) |
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670 | (stmt_implicit_label … s)) (nth_opt … n c)*) ≝ |
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671 | λp,globals,f_in. |
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672 | mk_joint_internal_function (mk_lin_params p) globals |
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673 | (joint_if_luniverse ?? f_in) (joint_if_runiverse ?? f_in) |
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674 | (joint_if_result ?? f_in) (joint_if_params ?? f_in) (joint_if_locals ?? f_in) |
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675 | (joint_if_stacksize ?? f_in) |
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676 | (linearise_code p globals (joint_if_code … f_in) |
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677 | (match daemon in False with [ ]) |
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678 | (joint_if_entry … f_in)) |
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679 | 0 0 (* exit is dummy! *). |
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680 | elim (linearise_code ?????) #c * #code_closed |
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681 | @hide_prf |
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682 | * #sigma * #O_in #sigma_prop |
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683 | >lin_code_has_point |
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684 | elim (sigma_prop … O_in) #s * #_ |
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685 | cases c |
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686 | [1,3: * |
---|
687 | |*: #hd #tl #_ % |
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688 | ] |
---|
689 | qed. |
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