1 | |
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2 | include "RTLabs/CostSpec.ma". |
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3 | include "utilities/bool.ma". |
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4 | include "utilities/listb.ma". |
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5 | include "RTLabs/CostMisc.ma". |
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6 | |
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7 | definition check_well_cost_fn : internal_function → bool ≝ |
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8 | λf. |
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9 | idmap_all … (f_graph f) (λl,s,PR. well_cost_labelled_statement (f_graph f) s (f_closed f l s PR)) ∧ |
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10 | is_cost_label (lookup_present … (f_graph f) (f_entry f) ?). |
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11 | cases (f_entry f) // |
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12 | qed. |
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13 | |
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14 | lemma check_well_cost_fn_ok : ∀fn. bool_to_Prop (check_well_cost_fn fn) ↔ well_cost_labelled_fn fn. |
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15 | #fn % |
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16 | [ #H cases (andb_Prop_true … H) #ST #EN |
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17 | % [ lapply (proj1 … (idmap_all_ok …) ST) // | @EN ] |
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18 | | * #ST #EN @andb_Prop |
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19 | [ @(proj2 … (idmap_all_ok …)) #l #st #L |
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20 | cut (present ?? (f_graph fn) l) [ whd >L % #E destruct ] #PR |
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21 | lapply (ST l PR) generalize in ⊢ (?(???%) → ?); |
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22 | >(lookup_present_eq ????? L PR) // |
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23 | | @eq_true_to_b @EN |
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24 | ] |
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25 | ] qed. |
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26 | |
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27 | include alias "utilities/deqsets.ma". |
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28 | |
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29 | lemma successors_present : ∀g,st. |
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30 | labels_present g st → |
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31 | ∀l. l ∈ successors st → |
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32 | present ?? g l. |
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33 | #g * |
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34 | [ #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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35 | | #cs #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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36 | | #ty #r #c #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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37 | | #ty1 #ty2 #op #r1 #r2 #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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38 | | #ty1 #ty2 #ty3 #op #r1 #r2 #r3 #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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39 | | #ty #r1 #r2 #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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40 | | #ty #r1 #r2 #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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41 | | #id #args #dst #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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42 | | #r #args #dst #l1 #PR #l2 #IN >(memb_single … IN) @PR |
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43 | | #r #l1 #l2 * #PR1 #PR2 #l3 whd in ⊢ (?% → ?); @eqb_elim [ // | #_ #IN >(memb_single … IN) // ] |
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44 | | #_ #l * |
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45 | ] qed. |
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46 | |
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47 | include alias "common/Identifiers.ma". |
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48 | |
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49 | (* Check that from [checking] we reach a cost label without going through |
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50 | [checking_tail], which would form a loop in the CFG. We also have a set of |
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51 | labels that we have still [to_check], and return an updated set of labels |
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52 | to check if the check for the current label is successful. *) |
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53 | let rec check_label_bounded |
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54 | (g : graph statement) |
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55 | (CL : graph_closed g) |
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56 | (checking : label) |
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57 | (PR : present ?? g checking) |
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58 | (checking_tail : list label) |
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59 | (to_check : identifier_set LabelTag) |
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60 | (term_check : nat) |
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61 | on term_check : gt term_check (id_map_size … to_check) → option (identifier_set LabelTag) ≝ |
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62 | let stop_now ≝ Some ? to_check in |
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63 | match term_check return λx. ge x ? → ? with |
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64 | [ O ⇒ λH.⊥ |
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65 | | S term_check' ⇒ λH. |
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66 | let st ≝ lookup_present … g checking PR in |
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67 | let succs ≝ successors st in |
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68 | match succs return λsc. (∀l.l∈sc → ?) → ? with |
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69 | [ nil ⇒ λ_. stop_now |
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70 | | cons h t ⇒ |
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71 | match t with |
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72 | [ nil ⇒ λSC. (* single successor *) |
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73 | let PR' ≝ ? in |
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74 | let st' ≝ lookup_present … g h PR' in |
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75 | if is_cost_label st' then stop_now else |
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76 | match try_remove … to_check h return λx. (∀a,m'. x = ? → ?) → ? with |
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77 | [ Some to_check' ⇒ λH'. |
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78 | check_label_bounded g CL h PR' (checking::checking_tail) (\snd to_check') term_check' ? |
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79 | | None ⇒ λ_. |
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80 | if h == checking ∨ h ∈ checking_tail then None ? else |
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81 | stop_now (* already checked successor *) |
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82 | ] (try_remove_some_card … to_check h) |
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83 | | cons _ _ ⇒ λ_. stop_now (* all branches are followed by a cost label *) |
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84 | ] |
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85 | ] (successors_present g st (CL … checking … (lookup_lookup_present …))) |
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86 | ]. |
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87 | [ /2 by absurd/ |
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88 | | lapply (H' (\fst to_check') (\snd to_check') ?) [ cases to_check' // ] |
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89 | #E -PR' >E in H; #H' /2/ |
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90 | | @SC >memb_hd // |
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91 | ] qed. |
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92 | |
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93 | (* An inductive specification of the above function that's easier to work with. *) |
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94 | |
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95 | inductive check_label_bounded_spec (g:graph statement) : label → list label → identifier_set LabelTag → identifier_set LabelTag → Prop ≝ |
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96 | | clbs_ret : ∀l,PR,tl,toch. |
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97 | successors (lookup_present … g l PR) = [ ] → |
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98 | check_label_bounded_spec g l tl toch toch |
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99 | | clbs_checked : ∀l,PR,l',tl,toch. (* this case overlaps with clbs_cost *) |
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100 | successors (lookup_present … g l PR) = [l'] → |
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101 | ¬ l' ∈ toch → |
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102 | l' ≠ l → |
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103 | ¬ l' ∈ tl → |
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104 | check_label_bounded_spec g l tl toch toch |
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105 | | clbs_cost : ∀l,PR,l',PR',tl,toch. |
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106 | successors (lookup_present … g l PR) = [l'] → |
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107 | is_cost_label (lookup_present … g l' PR') → |
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108 | check_label_bounded_spec g l tl toch toch |
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109 | | clbs_step : ∀l,PR,l',PR',tl,toch,toch',toch''. |
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110 | successors (lookup_present … g l PR) = [l'] → |
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111 | (* l' ∈ toch → implied *) |
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112 | ¬ l' ∈ tl → |
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113 | ¬ is_cost_label (lookup_present … g l' PR') → |
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114 | try_remove … toch l' = Some ? 〈it,toch'〉 → |
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115 | check_label_bounded_spec g l' (l::tl) toch' toch'' → |
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116 | check_label_bounded_spec g l tl toch toch'' |
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117 | | clbs_branch : ∀l,PR,x,y,zs,tl,toch. (* the other check will show that these are cost labels *) |
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118 | successors (lookup_present … g l PR) = x::y::zs → |
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119 | check_label_bounded_spec g l tl toch toch. |
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120 | |
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121 | |
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122 | lemma check_label_bounded_s : ∀term_check,g,CL,checking,PR,checking_tail,to_check,TERM,to_check'. |
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123 | (∀l. l∈to_check → ¬ memb ? l (checking::checking_tail)) → |
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124 | check_label_bounded g CL checking PR checking_tail to_check term_check TERM = Some ? to_check' → |
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125 | check_label_bounded_spec g checking checking_tail to_check to_check'. |
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126 | #n elim n |
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127 | [ #a #b #c #d #d #e #g @⊥ /3 by n_plus_1_n_to_False, div_plus_times/ (* ! *) |
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128 | | #term_check #IH #g #CL #checking #PR #checking_tail #to_check #TERM #to_check' #REMOVING |
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129 | whd in ⊢ (??%? → ?); |
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130 | generalize in ⊢ (??(?%)? → ?); |
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131 | lapply (refl ? (successors (lookup_present … PR))) |
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132 | cases (successors (lookup_present … PR)) in ⊢ (???% → %); |
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133 | [ #SUCC whd in ⊢ (? → ??%? → ?); #_ #E destruct %1 // |
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134 | | #h * [ #SUCC whd in ⊢ (? → ??%? → ?); #PR' @if_elim |
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135 | [ #CS #E destruct @(clbs_cost … SUCC) // |
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136 | | #NCS generalize in ⊢ (??(?%)? → ?); |
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137 | lapply (refl ? (try_remove … to_check h)) |
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138 | cases (try_remove ????) in ⊢ (???% → %); |
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139 | [ #RM whd in ⊢ (? → ??%? → ?); #H @if_elim |
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140 | [ #H' #E destruct |
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141 | | #H' cases (not_orb ?? H') #H1 #H2 #E destruct @(clbs_checked … SUCC) |
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142 | [ whd in ⊢ (?(?%)); >(proj1 … (try_remove_empty …) RM) // |
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143 | | % #E >E in H1; >(proj2 … (eqb_true …) (refl ??)) * |
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144 | | assumption |
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145 | ] |
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146 | ] |
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147 | | * * #to_check'' #RM #H whd in ⊢ (??%? → ?); |
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148 | whd in ⊢ (??%? → ?); #H |
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149 | cases (try_remove_some ?????? RM) * #L1 #L2 #L3 |
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150 | @(clbs_step … SUCC ? NCS RM) |
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151 | [ @notb_Prop @(not_to_not … (Prop_notb … (REMOVING h ?))) /2/ |
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152 | whd in ⊢ (?%); >L1 // |
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153 | | @(IH … H) |
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154 | #l #IN'' @notb_Prop % whd in ⊢ (?% → ?); @if_elim |
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155 | [ #E #_ whd in IN'':(?%); >(proj1 ?? (eqb_true …) E) in IN''; |
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156 | >L2 * |
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157 | | #NE lapply (REMOVING l ?) |
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158 | [ whd in ⊢ (?%); cases (L3 l) [ #E destruct cases (Prop_notb … NE) #X @⊥ @X @eq_true_to_b @(proj2 ?? (eqb_true …)) % |
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159 | | #L >L @IN'' |
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160 | ] |
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161 | | cases (l∈checking::checking_tail) * * |
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162 | ] |
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163 | ] |
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164 | ] |
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165 | ] |
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166 | ] |
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167 | | #h2 #t #SUCCS whd in ⊢ (? → ??%? → ?); #PR' #E destruct |
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168 | @(clbs_branch … SUCCS) |
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169 | ] |
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170 | ] |
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171 | ] qed. |
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172 | |
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173 | lemma check_label_bounded_c : ∀g,CL,checking,checking_tail,to_check,to_check'. |
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174 | check_label_bounded_spec g checking checking_tail to_check to_check' → |
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175 | ∀term_check,TERM,PR. |
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176 | check_label_bounded g CL checking PR checking_tail to_check term_check TERM = Some ? to_check'. |
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177 | #g #CL #X1 #X2 #X3 #X4 #X elim X |
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178 | [ #l #PR #tl #toch #SUCC |
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179 | * [ #TERM @⊥ inversion TERM #E [2: #F #G #H] destruct ] #term_check' #TERM #PR |
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180 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >SUCC #H1 |
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181 | whd in ⊢ (??%?); % |
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182 | | #l #PR #l' #tl #toch #SUCC #NIN_toch #NEQ #NIN_tl |
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183 | * [ #TERM @⊥ inversion TERM #E [2: #F #G #H] destruct ] #term_check' #TERM #PR |
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184 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >SUCC #H1 |
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185 | whd in ⊢ (??%?); @if_elim [ // ] #NCS generalize in ⊢ (??(?%)?); |
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186 | cut (lookup … toch l' = None ?) [ whd in NIN_toch:(?(?%)); cases (lookup ????) in NIN_toch ⊢ %; [ // | * * ] ] |
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187 | #L >(proj2 … (try_remove_empty …) L) #H2 |
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188 | whd in ⊢ (??%?); >(proj2 … (eqb_false …) NEQ) >(Prop_notb … NIN_tl) |
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189 | % |
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190 | | #l #PR #l' #PR' #tl #toch #SUCC #CS |
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191 | * [ #TERM @⊥ inversion TERM #E [2: #F #G #H] destruct ] #term_check' #TERM #PR |
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192 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >SUCC #H1 |
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193 | whd in ⊢ (??%?); >CS whd in ⊢ (??%?); % |
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194 | | #l #PR #l' #PR' #tl #toch #toch' #toch'' #SUCC #NIN #NCS #RM #H #IH |
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195 | * [ #TERM @⊥ inversion TERM #E [2: #F #G #H] destruct ] #term_check' #TERM #PR |
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196 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >SUCC #H1 |
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197 | whd in ⊢ (??%?); >(not_b_to_eq_false ? (Prop_notb ? NCS)) |
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198 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >RM #H2 |
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199 | whd in ⊢ (??%?); >(not_b_to_eq_false ? (Prop_notb ? NIN)) |
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200 | whd in ⊢ (??%?); >(IH term_check' ??) % |
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201 | | #l #PR #x #y #zs #tl #toch #SUCC |
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202 | * [ #TERM @⊥ inversion TERM #E [2: #F #G #H] destruct ] #term_check' #TERM #PR |
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203 | whd in ⊢ (??%?); generalize in ⊢ (??(?%)?); >SUCC #H1 |
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204 | whd in ⊢ (??%?); % |
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205 | ] qed. |
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206 | |
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207 | lemma check_label_bounded_subset : ∀g,checking,checking_tail,to_check,to_check'. |
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208 | check_label_bounded_spec g checking checking_tail to_check to_check' → |
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209 | set_subset … to_check' to_check. |
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210 | #g #lX #lX' #tX #tX' #S elim S // |
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211 | #l #PR #l' #PR' #tl #toch #toch' #toch'' #SC #NI #CS #RM #H #IH |
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212 | cases (try_remove_some … toch' RM) * #L1 #L2 #L3 |
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213 | #id #IN cases (L3 id) |
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214 | [ #E destruct whd in ⊢ (?%); >L1 // |
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215 | | #L4 whd in ⊢ (?%); >L4 @IH @IN |
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216 | ] qed. |
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217 | |
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218 | lemma bound_on_instrs_to_cost_prime : ∀g,l,n. |
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219 | bound_on_instrs_to_cost g l n → |
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220 | bound_on_instrs_to_cost' g l n. |
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221 | #g #l #n #H inversion H |
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222 | #l' #n' #PR #H' #E1 #E2 #_ destruct |
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223 | lapply (refl ? (is_cost_label (lookup_present … g l' PR))) |
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224 | cases (is_cost_label ?) in ⊢ (???% → ?); #CS |
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225 | [ @boitc_here [ @PR | @eq_true_to_b @CS ] |
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226 | | @boitc_later [ @PR | @eq_false_to_notb @CS | @H ] |
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227 | ] qed. |
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228 | |
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229 | lemma successors_inv : ∀st,x,y,zs. |
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230 | successors st = x::y::zs → |
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231 | ∃r,l1,l2. st = St_cond r l1 l2. |
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232 | * normalize |
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233 | #a #b #c #d try #e try #f try #g try #h try #i try #j try #k try #l destruct |
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234 | /4/ |
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235 | qed. |
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236 | |
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237 | include alias "utilities/deqsets.ma". |
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238 | |
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239 | lemma after_branch_are_cost_labels : ∀g. ∀CL:graph_closed g. |
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240 | (∀l,PR. well_cost_labelled_statement g (lookup_present … g l PR) (CL l ? (lookup_lookup_present … PR))) → |
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241 | ∀l,PR,x,y,zs. successors (lookup_present … g l PR) = x::y::zs → |
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242 | ∀l'. ∀IN : l' ∈ successors (lookup_present … g l PR). |
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243 | is_cost_label (lookup_present … g l' ?). |
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244 | [2: @(successors_present … IN) @(CL l) @lookup_lookup_present ] |
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245 | #g #CL #WCL #l #PR #x #y #zs #SUCCS |
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246 | cases (successors_inv … SUCCS) |
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247 | #r * #l1 * #l2 #E |
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248 | #l' #IN generalize in ⊢ (?(?(?????%))); #OR' >E in IN; |
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249 | lapply (WCL l PR) generalize in ⊢ (?(???%) → ?); >E in ⊢ (% → % → ?); #LP #WCst |
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250 | cases (andb_Prop_true ?? WCst) |
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251 | #H1 #H2 |
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252 | whd in ⊢ (?% → ?); @eqb_elim |
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253 | [ 2: #_ #IN lapply (memb_single … IN) ] |
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254 | #E destruct // |
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255 | qed. |
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256 | |
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257 | (* Show that when we remove labels from to_check we do actually find a bound |
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258 | (if it exists). We need two extra facts: everything is "well" labelled, |
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259 | so that we know that branches are followed by labels and we don't need to |
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260 | follow them; and anything in the graph that isn't mentioned has already been |
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261 | successfully checked. *) |
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262 | |
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263 | lemma check_label_bounded_ok : ∀g,checking,checking_tail,to_check,to_check'. |
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264 | ∀CL:graph_closed g. |
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265 | (∀l,PR. well_cost_labelled_statement g (lookup_present … g l PR) (CL l ? (lookup_lookup_present … PR))) → |
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266 | check_label_bounded_spec g checking checking_tail to_check to_check' → |
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267 | (∀l. l∈g → ¬l ∈ to_check → l≠checking → ¬l∈checking_tail → ∃n. bound_on_instrs_to_cost g l n) → |
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268 | ∀l. l = checking ∨ bool_to_Prop (l∈to_check) → |
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269 | bool_to_Prop (l∈to_check') ∨ ∃n. bound_on_instrs_to_cost g l n. |
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270 | #g #X1 #X2 #X3 #X4 #CL #WCL #X elim X |
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271 | [ #l #PR #tl #toch #SUCC #INV #l' * |
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272 | [ #E destruct %2 %{1} % [ @PR | >SUCC #l' * ] |
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273 | | #IN %1 @IN |
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274 | ] |
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275 | | #l #PR #l' #tl #toch #SUCC #NIN_toch #NEQ #NIN_tl #INV #l'' * |
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276 | [ #E destruct %2 |
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277 | cases (INV l' ????) |
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278 | [ #n #B |
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279 | %{(S n)} % [ @PR | >SUCC #l'' * [2: *] #E destruct /2/ ] |
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280 | | @present_member @(successors_present … (CL l ? (lookup_lookup_present … PR))) |
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281 | >SUCC @eq_true_to_b @memb_hd |
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282 | | assumption |
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283 | | assumption |
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284 | | assumption |
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285 | ] |
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286 | | #IN %1 assumption |
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287 | ] |
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288 | | #l #PR #l' #PR' #tl #toch #SUCC #CS #INV #l'' * |
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289 | [ #E destruct %2 %{1} % [ // | #l'' >SUCC * [2: *] #E destruct @boitc_here // ] |
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290 | | /2/ |
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291 | ] |
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292 | | #l #PR #l' #PR' #tl #toch #toch' #toch'' #SUCC #NIN_tl #NCS #RM #H #IH #INV #l'' * |
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293 | [ #E destruct %2 cases (IH ? l' ?) |
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294 | [ #IN @⊥ lapply (check_label_bounded_subset … H l' IN) #IN' |
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295 | cases (try_remove_some ?????? RM) * #_ #L #_ whd in IN':(?%); |
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296 | cases (lookup ????) in IN' L; [ * | * #_ #E destruct ] |
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297 | | * #n #B %{(S n)} % [ // | #l'' >SUCC * [2: *] #E destruct /2/ ] |
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298 | | #l'' #INg #INch' #NEQ #NIN_ltl @(INV l'' INg ???) |
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299 | [ @notb_Prop % #INch @(absurd ?? (Prop_notb … INch')) |
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300 | cases (try_remove_some ?????? RM) * #_ #_ #L |
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301 | cases (L l'') [ #E destruct cases NEQ #X cases (X (refl ??)) ] |
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302 | #L' whd in ⊢ (?%); <L' @INch |
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303 | | % #E destruct cases (Prop_notb … NIN_ltl) #X @X @eq_true_to_b @memb_hd |
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304 | | @notb_Prop @(not_to_not … (Prop_notb … NIN_ltl)) #IN @eq_true_to_b @memb_cons @IN |
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305 | ] |
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306 | | %1 % |
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307 | ] |
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308 | | #IN @(IH ? l'' ?) |
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309 | [ #l''' #INg #NINch' #NEQ #NINtl @INV |
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310 | [ assumption |
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311 | | @notb_Prop @(not_to_not … (Prop_notb … NINch')) |
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312 | cases (try_remove_some ?????? RM) * #L1 #L2 #L3 |
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313 | cases (L3 l''') |
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314 | [ #E destruct @⊥ cases NEQ /2/ |
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315 | | #L whd in ⊢ (?% → ?%); >L // |
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316 | ] |
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317 | | % #E destruct cases (Prop_notb … NINtl) #X @X @eq_true_to_b @memb_hd |
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318 | | @notb_Prop @(not_to_not … (Prop_notb … NINtl)) #IN @eq_true_to_b @memb_cons @IN |
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319 | ] |
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320 | | cases (try_remove_some ?????? RM) * #L1 #L2 #L3 |
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321 | cases (L3 l'') [ #E destruct %1 % | #L %2 whd in ⊢ (?%); <L @IN ] |
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322 | ] |
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323 | ] |
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324 | | #l #PR #x #y #zs #tl #toch #SUCCS #INV #l' * |
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325 | [ #E destruct %2 %{1} % [ // ] |
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326 | #l' #EX @boitc_here |
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327 | [ @(successors_present … (CL l ? (lookup_lookup_present … PR))) |
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328 | @Exists_memb @EX |
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329 | | @(after_branch_are_cost_labels … WCL … SUCCS) |
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330 | ] |
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331 | | /2/ |
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332 | ] |
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333 | ] qed. |
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334 | |
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335 | |
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336 | (* When we reject a label it is because we found a loop without a cost label |
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337 | when we followed its successors. We need the invariant that only the |
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338 | initial label can be a cost label, and have two cases in the result: either |
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339 | we found the entire loop and return that, or we haven't come back out of the |
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340 | recursive calls to find the head again, so return part of the loop. *) |
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341 | |
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342 | lemma check_label_bounded_bad : ∀g,CL,term_check,checking,PR,checking_tail,to_check,TERM. |
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343 | (checking_tail ≠ [ ] → ¬ is_cost_label (lookup_present … g checking PR)) → |
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344 | check_label_bounded g CL checking PR checking_tail to_check term_check TERM = None ? → |
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345 | ∃head,PR'. bool_to_Prop (¬ is_cost_label (lookup_present … g head PR')) ∧ |
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346 | ((∃l. Exists ? (λl'.l'=l) (successors (lookup_present … g head PR')) ∧ |
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347 | bad_label_list g head l) ∨ |
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348 | (bool_to_Prop (head ∈ checking_tail) ∧ bad_label_list g head checking)). |
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349 | #g #CL #term_check elim term_check |
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350 | [ #A #B #C #D #TERM @⊥ inversion TERM #E try #F try #G try #H destruct |
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351 | | #term_check' #IH |
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352 | #checking #PR #checking_tail #to_check #TERM #PRECS |
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353 | whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); lapply (refl ? (successors (lookup_present … PR))) |
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354 | cases (successors ?) in ⊢ (???% → %); |
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355 | [ #H #H' whd in ⊢ (??%? → ?); #E destruct ] |
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356 | #h * [2: #y #zs #H #H' whd in ⊢ (??%? → ?); #E destruct ] |
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357 | #SUCC #PR' whd in ⊢ (??%? → ?); @if_elim |
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358 | [ #CS #E destruct ] #NCS generalize in ⊢ (??(?%)? → ?); |
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359 | cases (try_remove ????) |
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360 | [ #H whd in ⊢ (??%? → ?); @if_elim |
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361 | (* base case - we've found the head of the list *) |
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362 | [ #IN #_ cases (orb_Prop_true … IN) |
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363 | [ (* self-loop *) #E >(proj1 ?? (eqb_true …) E) in SUCC PR' NCS; #SUCC #PR' #NCS %{checking} %{PR} |
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364 | %{NCS} %1 %{checking} % [ >SUCC % % | // ] |
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365 | | #IN' %{h} % [ @PR' @eq_true_to_b @memb_hd ] %{NCS} %2 %{IN'} |
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366 | @(gl_step … (gl_end …)) |
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367 | [ @PR |
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368 | | >SUCC @eq_true_to_b @memb_hd |
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369 | | cases (orb_Prop_true … IN) |
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370 | [ #E <(proj1 ?? (eqb_true …) E) in PR ⊢ %; // |
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371 | | cases checking_tail in PRECS ⊢ %; |
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372 | [ #_ * | #h' #t #H #_ @H % #E destruct ] |
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373 | ] |
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374 | ] |
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375 | ] |
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376 | | #_ #E destruct |
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377 | ] |
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378 | | * * #to_check' #H1 whd in ⊢ (??%? → ?); |
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379 | #CHECK' cases (IH … CHECK') |
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380 | [ #head * #PRhead * #NCShead * |
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381 | [ #BLL %{head} %{PRhead} % /2/ |
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382 | | * whd in ⊢ (?% → ?); @if_elim |
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383 | [ #E #_ >(proj1 ?? (eqb_true …) E) in PRhead NCShead ⊢ %; #PRh #NCSh #BLL %{checking} %{PRh} %{NCSh} |
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384 | %1 %{h} >SUCC % [ % % | @BLL ] |
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385 | | #NE #INtl #BLL %{head} %{PRhead} %{NCShead} %2 % // |
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386 | @(gl_step … BLL) |
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387 | [ @PR |
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388 | | >SUCC @eq_true_to_b @memb_hd |
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389 | | cases checking_tail in PRECS INtl; |
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390 | [ #_ * |
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391 | | #h #t #H #_ @H % #E destruct |
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392 | ] |
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393 | ] |
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394 | ] |
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395 | ] |
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396 | | #_ // |
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397 | ] |
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398 | ] |
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399 | ] qed. |
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400 | |
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401 | |
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402 | (* Now keep checking as long as there's some instruction we haven't checked. *) |
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403 | |
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404 | let rec check_graph_bounded |
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405 | (g : graph statement) |
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406 | (CL : graph_closed g) |
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407 | (to_check : identifier_set LabelTag) |
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408 | (SUB : set_subset … to_check g) |
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409 | (start : label) |
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410 | (PR : present ?? g start) |
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411 | (REMOVED : notb (member ?? to_check start)) |
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412 | (SMALLER : gt (id_map_size … g) (id_map_size … to_check)) |
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413 | (term_check : nat) |
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414 | on term_check : gt term_check (id_map_size … to_check) → bool ≝ |
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415 | match term_check return λx. ge x ? → bool with |
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416 | [ O ⇒ λH.⊥ |
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417 | | S term_check' ⇒ λH. |
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418 | let TERM' ≝ ? in |
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419 | match check_label_bounded g CL start PR [ ] to_check (id_map_size … g) TERM' return λx. check_label_bounded ???????? = x → ? with |
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420 | [ None ⇒ λ_. false |
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421 | | Some to_check' ⇒ λH'. |
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422 | match choose … to_check' return λx. (∀id,a,m'. x = ? → ?) → (∀id,a,m'. x = ? → ?) → (∀id,a,m'. x = ? → ?) → ? with |
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423 | [ None ⇒ λ_.λ_.λ_. true |
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424 | | Some l_to_check'' ⇒ λL,SUB',C. |
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425 | check_graph_bounded g CL (\snd l_to_check'') ? (\fst (\fst l_to_check'')) ??? term_check' ? |
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426 | ] (choose_some … to_check') (choose_some_subset … to_check') (choose_some_card … to_check') |
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427 | ] (refl ??) |
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428 | ]. |
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429 | [ 2,3,4,5,6: cases l_to_check'' in C L SUB' ⊢ %; * #l * #to_check'' #C #L #SUB' |
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430 | lapply (check_label_bounded_subset … (check_label_bounded_s … H')) |
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431 | [ 1,3,5,7,9: #l' #IN @notb_Prop % #IS >(memb_single … IS) in IN; #IN' |
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432 | @(absurd … IN' (Prop_notb … REMOVED)) ] |
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433 | #SUB'' |
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434 | ] |
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435 | [ #id #IN @SUB @SUB'' @(SUB' ??? (refl ??)) @IN |
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436 | | @member_present @SUB @SUB'' cases (L ??? (refl ??)) * #L #_ #_ whd in ⊢ (?%); >L // |
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437 | | cases (L … (refl ??)) * #L1 #L2 #L3 whd in ⊢ (?(?%)); >L2 // |
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438 | | whd <(C ??? (refl ??)) lapply (subset_card … SUB'') #Z @(transitive_le … Z) /2/ |
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439 | | whd <(C ??? (refl ??)) lapply (subset_card … SUB'') #Z @(transitive_le … Z) /2/ |
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440 | | /2/ |
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441 | | @SMALLER |
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442 | ] qed. |
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443 | |
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444 | lemma check_graph_bounded_ok : ∀g.∀CL:graph_closed g. ∀term_check,to_check,SUB,start,PR,REMOVED,SMALLER,TERM. |
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445 | (∀l,PR. well_cost_labelled_statement g (lookup_present … g l PR) (CL l ? (lookup_lookup_present … PR))) → |
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446 | check_graph_bounded g CL to_check SUB start PR REMOVED SMALLER term_check TERM = true → |
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447 | (∀l. l∈g → ¬l∈to_check → l≠start → ∃n. bound_on_instrs_to_cost g l n) → |
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448 | ∀l. l∈g → ∃n. bound_on_instrs_to_cost g l n. |
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449 | #g #CL #term_check elim term_check |
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450 | [ #to_check #SUB #start #PR #REMOVED #SMALLER #TERM @⊥ inversion TERM #A try #B try #C try #D destruct |
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451 | | #term_check' #IH #to_check #SUB #start #PR #REMOVED #SMALLER #TERM #WCL |
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452 | whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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453 | cases (check_label_bounded ????????) in ⊢ (???% → ??(match % with [_⇒?|_⇒?]?)? → ?); |
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454 | [2:#to_check'] #CHECK_LABEL whd in ⊢ (??%? → ?); [2: #E destruct ] |
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455 | generalize in ⊢ (??(?%??)? → ?); generalize in ⊢ (? → ??(??%?)? → ?); generalize in ⊢ (? → ? → ??(???%)? → ?); |
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456 | lapply (choose_empty … to_check') |
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457 | cases (choose LabelTag unit to_check') |
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458 | [ * #H1 #H1' #H2 #H3 #H4 whd in ⊢ (??%? → ?); #_ #H #l #IN |
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459 | cases (true_or_false_Prop (l∈to_check ∨ l == start)) |
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460 | [ #CHECKED cases (check_label_bounded_ok … (check_label_bounded_s … CHECK_LABEL) ? l ?) |
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461 | [ #IN' @⊥ lapply (H1 (refl ??) l) #E whd in IN':(?%); >E in IN'; * |
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462 | | // |
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463 | | // |
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464 | | @WCL |
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465 | | #l' #IN @notb_Prop @(not_to_not … (Prop_notb … REMOVED)) #IN' |
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466 | >(memb_single … IN') in IN; // |
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467 | | #l' #IN_g #NIN_to_check #NEQ #_ @H // |
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468 | | cases (orb_Prop_true … CHECKED) /2/ #E %1 @(proj1 … (eqb_true …)) @E |
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469 | ] |
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470 | | #NOT @H // [ @notb_Prop @(not_to_not … NOT) /2/ | % #E @(absurd ?? NOT) @orb_Prop_r >(proj2 … (eqb_true …)) // ] |
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471 | ] |
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472 | | * * #next * #to_check'' * #H1 #H1' #H2 #H3 #H4 whd in ⊢ (??%? → ?); |
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473 | #CHECK #H @(IH … CHECK) |
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474 | [ @WCL |
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475 | | #l #INg #NIN'' #NEQ |
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476 | cut (¬l∈to_check') [ @notb_Prop @(not_to_not … (Prop_notb … NIN'')) |
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477 | cases (H4 … (refl ??)) * #L1 #L2 #L3 |
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478 | cases (L3 l) [ #EQ @⊥ @(absurd ?? NEQ) >EQ % | #E whd in ⊢ (?% → ?%); >E // ] ] |
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479 | #NIN' |
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480 | cases (true_or_false_Prop (l∈to_check ∨ l == start)) |
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481 | [ #CHECKED cases (check_label_bounded_ok … (check_label_bounded_s … CHECK_LABEL) ? l ?) |
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482 | [ #IN' @⊥ @(absurd … IN' (Prop_notb … NIN')) |
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483 | | // |
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484 | | // |
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485 | | @WCL |
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486 | | #l' #IN @notb_Prop @(not_to_not … (Prop_notb … REMOVED)) #IN' |
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487 | >(memb_single … IN') in IN; // |
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488 | | #l' #IN_g #NIN_to_check #NEQ #_ @H // |
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489 | | cases (orb_Prop_true … CHECKED) /2/ #E %1 @(proj1 … (eqb_true …)) @E |
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490 | ] |
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491 | | #NOT @H // [ @notb_Prop @(not_to_not … NOT) /2/ | % #E @(absurd ?? NOT) @orb_Prop_r >(proj2 … (eqb_true …)) // ] |
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492 | ] |
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493 | ] |
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494 | ] |
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495 | ] qed. |
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496 | |
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497 | (* The check_label_bounded_bad result gives us the loop, so we can use the |
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498 | loop_soundness_contradiction result to show that there's no bound for the |
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499 | head of the loop. *) |
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500 | |
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501 | lemma check_graph_bounded_bad : ∀g.∀CL:graph_closed g. ∀term_check,to_check,SUB,start,PR,REMOVED,SMALLER,TERM. |
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502 | check_graph_bounded g CL to_check SUB start PR REMOVED SMALLER term_check TERM = false → |
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503 | ∃l. present ?? g l ∧ ∀n. ¬ bound_on_instrs_to_cost g l n. |
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504 | #g #CL #term_check elim term_check |
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505 | [ #a #b #c #d #e #f #TERM @⊥ inversion TERM #A try #B try #C try #D destruct |
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506 | | #term_check' #IH #to_check #SUB #start #PR #REMOVED #SMALLER #TERM |
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507 | whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); |
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508 | cases (check_label_bounded ????????) in ⊢ (???% → ??(match % with [_⇒?|_⇒?]?)? → ?); |
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509 | [ #CHECK_LABEL whd in ⊢ (??%? → ?); #_ |
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510 | cases (check_label_bounded_bad … CHECK_LABEL) |
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511 | [ #head * #PR_head * #NCS_head * |
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512 | [ * #next * #NEXT #BLL |
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513 | %{head} %{PR_head} #n % #BOUND |
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514 | cases (bound_step1 … BOUND … NEXT) #m #BOUND' |
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515 | @(absurd ? BOUND' (loop_soundness_contradiction … NEXT NCS_head BLL m)) |
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516 | | * * |
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517 | ] |
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518 | | * #H cases (H (refl ??)) |
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519 | ] |
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520 | | #to_check' #CHECK_LABEL whd in ⊢ (??%? → ?); |
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521 | generalize in ⊢ (??(?%??)? → ?); generalize in ⊢ (? → ??(??%?)? → ?); generalize in ⊢ (? → ? → ??(???%)? → ?); |
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522 | cases (choose LabelTag unit to_check') |
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523 | [ #H1 #H2 #H3 whd in ⊢ (??%? → ?); #E destruct |
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524 | | * * #next * #to_check'' #H2 #H3 #H4 whd in ⊢ (??%? → ?); |
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525 | @IH |
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526 | ] |
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527 | ] |
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528 | ] qed. |
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529 | |
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530 | |
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531 | definition check_sound_cost_fn : internal_function → bool ≝ |
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532 | λfn. |
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533 | match try_remove … (id_set_of_map … (f_graph fn)) (f_entry fn) return λx. (x = ? → ?) → (∀a,m'. x = ? → ?) → (∀a,m'. x = ? → ?) → ? with |
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534 | [ None ⇒ λEMP. ⊥ |
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535 | | Some to_check ⇒ λ_.λCARD,L. |
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536 | check_graph_bounded (f_graph fn) (f_closed fn) (\snd to_check) ? (f_entry fn) ??? (|f_graph fn|) ? |
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537 | ] (proj1 ?? (try_remove_empty LabelTag unit (id_set_of_map … (f_graph fn)) (f_entry fn))) |
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538 | (try_remove_some_card ????) |
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539 | (try_remove_some ????). |
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540 | [ cases (f_entry fn) in EMP; #l #PR cases (proj1 … (id_set_of_map_present …) PR) #PR' #H |
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541 | @PR' @H % |
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542 | | cases to_check in L ⊢ %; * #m' #L cases (L … (refl ??)) * #L1 #L2 #L3 #id #IN |
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543 | cases (L3 id) |
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544 | [ #E destruct whd in ⊢ (?%); cases (f_entry fn) #l * cases (lookup ????) /2/ |
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545 | | #E whd in ⊢ (?%); lapply (id_set_of_map_present … (f_graph fn) id) * |
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546 | whd in match (present ????); whd in match (present ? unit ??); |
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547 | lapply (member_present … IN) whd in match (present ? unit ??); <E |
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548 | cases (lookup … (f_graph fn) id) // #X #_ #Y @⊥ cases (Y X) /2/ |
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549 | ] |
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550 | | cases (f_entry fn) // |
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551 | | cases to_check in L ⊢ %; * #to_check0 #L cases (L … (refl ??)) * #L1 #L2 #L3 |
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552 | whd in ⊢ (?(?%)); >L2 // |
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553 | | 5,6: <id_set_of_map_card in CARD; cases to_check * #m' #CARD >(CARD ?? (refl ??)) // |
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554 | ] qed. |
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555 | |
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556 | lemma check_sound_cost_fn_ok : ∀fn. |
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557 | well_cost_labelled_fn fn → |
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558 | bool_to_Prop (check_sound_cost_fn fn) ↔ soundly_labelled_fn fn. |
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559 | #fn #WCL % |
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560 | [ whd in ⊢ (?% → %); |
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561 | generalize in ⊢ (?(?%??) → ?); generalize in ⊢ (? → ?(??%?) → ?); generalize in ⊢ (? → ? → ?(???%) → ?); |
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562 | cases (try_remove ????) |
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563 | [ #H1 #H2 #H3 @⊥ cases (f_entry fn) in H3; #l #PR #E |
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564 | lapply (proj1 … (id_set_of_map_present …) PR) whd in ⊢ (% → ?); |
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565 | >(E (refl ??)) * /3/ |
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566 | | * * #to_check #H1 #H2 #H3 |
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567 | whd in ⊢ (?% → ?); #CHECK |
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568 | #l #PR @(check_graph_bounded_ok … CHECK) |
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569 | [ #l' #PR' cases WCL // |
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570 | | #l' #IN #NIN #NEQ @⊥ |
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571 | cases (H1 … (refl ??)) * #L1 #L2 #L3 |
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572 | @(absurd ?? (Prop_notb … NIN)) |
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573 | cases (L3 l') |
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574 | [ #E >E in NEQ; * #H cases (H (refl ??)) |
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575 | | #L whd in ⊢ (?%); <L @present_member @(proj1 … (id_set_of_map_present …)) |
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576 | @member_present @IN |
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577 | ] |
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578 | | /2/ |
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579 | ] |
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580 | ] |
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581 | | whd in ⊢ (% → %); #SOUND |
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582 | lapply (refl ? (check_sound_cost_fn fn)) |
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583 | cases (check_sound_cost_fn fn) in ⊢ (???% → %); |
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584 | [ // |
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585 | | whd in ⊢ (??%? → %); |
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586 | generalize in ⊢ (??(?%??)? → ?); generalize in ⊢ (? → ??(??%?)? → ?); generalize in ⊢ (? → ? → ??(???%)? → ?); |
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587 | cases (try_remove ????) |
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588 | [ #H1 #H2 #H3 @⊥ cases (f_entry fn) in H3; #l #PR #E |
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589 | lapply (proj1 … (id_set_of_map_present …) PR) whd in ⊢ (% → ?); |
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590 | >(E (refl ??)) * /3/ |
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591 | | * * #to_check #H1 #H2 #H3 |
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592 | whd in ⊢ (??%? → ?); #CHECK |
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593 | lapply (check_graph_bounded_bad … CHECK) -CHECK -H1 -H2 -H3 |
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594 | * #l * #PR #NOT cases (SOUND l PR) #n #B |
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595 | @(absurd ? B) @NOT |
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596 | ] |
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597 | ] |
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598 | ] qed. |
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599 | |
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600 | |
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601 | definition check_cost_program : RTLabs_program → bool ≝ |
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602 | λp. all ? (λfn. match \snd fn with |
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603 | [ Internal fn ⇒ check_well_cost_fn fn ∧ check_sound_cost_fn fn |
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604 | | External _ ⇒ true |
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605 | ]) (prog_funct … p). |
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606 | |
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607 | theorem check_cost_program_ok : ∀p. bool_to_Prop (check_cost_program p) ↔ well_cost_labelled_program p. |
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608 | #p whd in ⊢ (?(?%)%); % |
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609 | [ #H lapply ((proj1 ?? (all_All ???)) H) @All_mp |
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610 | * #id * #fd |
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611 | [ #H cases (andb_Prop_true … H) #W #S |
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612 | lapply (proj1 … (check_well_cost_fn_ok …) W) #W' |
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613 | % |
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614 | [ // |
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615 | | @(proj1 … (check_sound_cost_fn_ok …)) // |
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616 | ] |
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617 | | // |
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618 | ] |
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619 | | #H @(proj2 … (all_All …)) @(All_mp … H) |
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620 | * #id * #fd |
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621 | [ * #W #S |
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622 | lapply ((proj2 … (check_well_cost_fn_ok …)) W) |
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623 | lapply ((proj2 … (check_sound_cost_fn_ok …)) S) /2/ |
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624 | | // |
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625 | ] |
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626 | ] qed. |
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627 | |
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