1 | |
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2 | include "Clight/label.ma". |
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3 | include "Clight/Cexec.ma". |
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4 | |
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5 | (* for the induction principle *) |
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6 | include "Clight/CexecSound.ma". |
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7 | |
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8 | lemma commute_fst_pair : ∀A,B,C,D:Type[0].∀e:A×B.∀F:A → B → C×D. |
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9 | \fst (let 〈x,y〉 ≝ e in F x y) = let 〈x,y〉 ≝ e in \fst (F x y). |
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10 | #A #B #C #D * // |
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11 | qed. |
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12 | |
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13 | lemma shift_fst : ∀A,B,C,D:Type[0].∀e:A×B.∀F:A → C.∀G:B → D. |
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14 | \fst (let 〈x,y〉 ≝ e in 〈F x, G y〉) = F (\fst e). |
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15 | #A #B #C #D * // |
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16 | qed. |
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17 | |
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18 | lemma label_expr_elim : ∀e,u.∀T:Type[0].∀F:expr → universe CostTag → T. ∀P:T → Type[0]. |
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19 | (∀u'. P (F (\fst (label_expr e u)) u')) → |
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20 | P (let 〈e',u'〉 ≝ label_expr e u in F e' u'). |
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21 | #e #u #T #F #P |
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22 | cases (label_expr e u) |
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23 | #e' #u' #H @H |
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24 | qed. |
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25 | |
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26 | lemma add_cost_expr_elim : ∀e,u.∀T:Type[0].∀F:expr → universe CostTag → T. ∀P:T → Type[0]. |
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27 | (∀u',l. P (F (Expr (Ecost l e) (typeof e)) u')) → |
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28 | P (let 〈e',u'〉 ≝ add_cost_expr e u in F e' u'). |
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29 | #e #u #T #F #P #H whd in match (add_cost_expr ??); |
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30 | cases (fresh ??) #l #u' @H |
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31 | qed. |
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32 | |
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33 | |
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34 | |
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35 | lemma label_expr_type : ∀e,u. |
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36 | typeof (\fst (label_expr e u)) = typeof e. |
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37 | * #e #ty #u >shift_fst // |
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38 | qed. |
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39 | |
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40 | lemma label_expr_ok : ∀ge,en,m,e,v,tr. |
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41 | exec_expr ge en m e = OK … 〈v,tr〉 → |
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42 | ∀u. ∃tr'. exec_expr ge en m (\fst (label_expr e u)) = OK … 〈v,tr'〉. |
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43 | #ge #en #m #e |
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44 | @(expr_lvalue_ind ? |
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45 | (λe,ty. ∀b,o,tr. exec_lvalue' ge en m e ty = OK … 〈〈b,o〉,tr〉 → |
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46 | ∀u. ∃tr'. exec_lvalue' ge en m (\fst (label_expr_descr e u ty)) ty = OK … 〈〈b,o〉,tr'〉) … e) |
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47 | [ 1,2: normalize /2/ |
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48 | | * // |
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49 | [ normalize /2 by ex_intro/ |
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50 | | #e1 #ty #IH #v #tr #EX #u |
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51 | cases (bind_inversion ????? EX) * * #b #o #tr1 * #EX1 #EX1r |
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52 | cases (bind_inversion ????? EX1r) #v * #EXl #EXrem |
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53 | normalize in EXrem; destruct |
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54 | cases (IH … EX1 u) #tr1' #EX1' |
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55 | %{tr1'} >shift_fst in EX1'; >shift_fst >shift_fst #EX1' whd in ⊢ (??%?); >EX1' |
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56 | whd in ⊢ (??%?); >EXl @refl |
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57 | | #e1 #id #ty #IH #v #tr #EX #u |
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58 | cases (bind_inversion ????? EX) * * #b #o #tr1 * #EX1 #EX1r |
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59 | cases (bind_inversion ????? EX1r) #v * #EXl #EXrem |
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60 | normalize in EXrem; destruct |
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61 | cases (IH … EX1 u) #tr1' #EX1' |
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62 | %{tr1'} >shift_fst in EX1'; >shift_fst >shift_fst #EX1' whd in ⊢ (??%?); >EX1' |
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63 | whd in ⊢ (??%?); >EXl @refl |
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64 | ] |
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65 | | #v #ty #b #o #tr #EX #u %{tr} @EX |
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66 | | #e1 #ty #IH #b #o #tr #EX #u |
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67 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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68 | cases (IH … EX1 u) #tr1' #EX1' |
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69 | %{tr1'} >shift_fst whd in ⊢ (??%?); >EX1' |
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70 | whd in ⊢ (??%?); |
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71 | cases v1 in EX1rem ⊢ %; normalize #A try #B try #C destruct @refl |
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72 | | #ty' #e1 #ty #IH #v #tr #EX #u |
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73 | cases (bind_inversion ????? EX) * * #b1 #o1 #tr1 * #EX1 #EX1rem |
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74 | cases (IH … EX1 u) #tr1' #EX1' |
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75 | %{tr1'} >shift_fst >shift_fst >shift_fst whd in ⊢ (??%?); |
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76 | whd in ⊢ (??(match % with [_⇒?|_⇒?])?); >EX1' |
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77 | whd in ⊢ (??%?); |
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78 | cases ty' in EX1rem ⊢ %; |
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79 | [ 4: #r #ty' whd in ⊢ (??%? → ??%?); cases (pointer_compat_dec b1 r) |
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80 | #pc normalize #E destruct @refl |
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81 | | *: normalize #A try #B try #C try #D destruct |
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82 | ] |
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83 | | #ty #op #e1 #IH #v #tr #EX #u |
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84 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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85 | cases (bind_inversion ????? EX1rem) #v2 * #EX2 #EXrem |
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86 | normalize in EXrem; destruct |
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87 | cases (IH … EX1 u) #tr1' #EX1' |
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88 | %{tr1'} >shift_fst |
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89 | whd in ⊢ (??(????(?(???%)?))?); |
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90 | @label_expr_elim #u2 |
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91 | whd in ⊢ (??%?); >EX1' |
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92 | whd in ⊢ (??%?); >label_expr_type >EX2 @refl |
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93 | | #ty #op #e1 #e2 #IH1 #IH2 #v #tr #EX |
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94 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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95 | cases (bind_inversion ????? EX1rem) * #v2 #tr2 * #EX2 #EX2rem |
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96 | cases (bind_inversion ????? EX2rem) #v' * #EXop #EXop' |
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97 | #u |
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98 | cases (IH1 … EX1 u) #tr1' #EX1' |
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99 | >shift_fst |
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100 | whd in ⊢ (??(λ_.??(????(?(???%)?))?)); |
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101 | @label_expr_elim #u2 |
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102 | cases (IH2 … EX2 u2) #tr2' #EX2' |
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103 | @label_expr_elim #u3 %{(tr1'⧺tr2')} |
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104 | whd in ⊢ (??%?); >EX1' |
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105 | whd in ⊢ (??%?); >EX2' |
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106 | whd in ⊢ (??%?); >label_expr_type >label_expr_type >EXop |
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107 | normalize in EXop'; destruct @refl |
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108 | | #ty #ty' #e1 #IH #v #tr #EX #u |
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109 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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110 | cases (bind_inversion ????? EX1rem) #v2 * #EX2 #EX2rem |
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111 | normalize in EX2rem; destruct |
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112 | cases (IH … EX1 u) #tr1' #EX1' |
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113 | %{tr1'} >shift_fst >shift_fst |
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114 | whd in ⊢ (??%?); >EX1' |
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115 | whd in ⊢ (??%?); >label_expr_type >EX2 @refl |
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116 | | #ty #e1 #e2 #e3 #IH1 #IH2 #IH3 #v #tr #EX #u |
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117 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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118 | cases (bind_inversion ????? EX1rem) * * #EXguard #EXguardrem |
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119 | cases (bind_inversion ????? EXguardrem) * #v2 #tr2 * #EX2 #EX2rem |
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120 | normalize in EX2rem; destruct |
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121 | cases (IH1 … EX1 u) #tr1' #EX1' |
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122 | >shift_fst whd in ⊢ (??(λ_.??(????(?(???%)?))?)); |
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123 | @label_expr_elim #u2 |
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124 | @label_expr_elim #u3 |
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125 | @add_cost_expr_elim #u4 #l4 |
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126 | @label_expr_elim #u5 |
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127 | @add_cost_expr_elim #u6 #l6 |
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128 | [ cases (IH2 … EX2 u2) | cases (IH3 … EX2 u4) ] #trx' #EXx' |
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129 | [ %{(tr1'⧺trx'⧺(Echarge l4))} | %{(tr1'⧺trx'⧺(Echarge l6))} ] |
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130 | whd in ⊢ (??%?); >EX1' |
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131 | whd in ⊢ (??%?); >label_expr_type >EXguard |
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132 | whd in ⊢ (??%?); whd in ⊢ (??(match % with [_⇒?|_⇒?])?); >EXx' |
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133 | normalize @refl |
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134 | | #ty #e1 #e2 #IH1 #IH2 #v #tr #EX #u |
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135 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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136 | cases (bind_inversion ????? EX1rem) * * #EXguard #EXguardrem |
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137 | >shift_fst whd in ⊢ (??(λ_.??(????(?(???%)?))?)); |
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138 | @label_expr_elim #u2 |
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139 | @label_expr_elim #u3 |
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140 | @add_cost_expr_elim #u4 #l4 |
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141 | @add_cost_expr_elim #u5 #l5 |
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142 | [ cases (bind_inversion ????? EXguardrem) * #v2 #tr2 * #EX2 #EX2rem |
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143 | cases (bind_inversion ????? EX2rem) * * #EX3 #EX3rem |
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144 | cases (IH1 … EX1 u) #tr1' #EX1' |
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145 | cases (IH2 … EX2 u2) #tr2' #EX2' |
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146 | %{(tr1'⧺(tr2'⧺E0)⧺Echarge l4)} |
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147 | whd in ⊢ (??%?); >EX1' |
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148 | whd in ⊢ (??%?); >label_expr_type >EX2' >EXguard |
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149 | whd in ⊢ (??%?); whd in ⊢ (??(match % with [_⇒?|_⇒?])?); |
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150 | whd in ⊢ (??(match (match % with [_⇒?|_⇒?]) with [_⇒?|_⇒?])?); >EX2' |
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151 | whd in ⊢ (??(match (match % with [_⇒?|_⇒?]) with [_⇒?|_⇒?])?); >label_expr_type >EX3 |
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152 | normalize in EX3rem ⊢ %; destruct @refl |
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153 | | cases (IH1 … EX1 u) #tr1' #EX1' |
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154 | %{(tr1'⧺E0⧺Echarge l5)} |
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155 | whd in ⊢ (??%?); >EX1' |
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156 | whd in ⊢ (??%?); >label_expr_type >EXguard |
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157 | normalize in EXguardrem; destruct @refl |
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158 | ] |
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159 | | #ty #e1 #e2 #IH1 #IH2 #v #tr #EX #u |
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160 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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161 | cases (bind_inversion ????? EX1rem) * * #EXguard #EXguardrem |
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162 | >shift_fst whd in ⊢ (??(λ_.??(????(?(???%)?))?)); |
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163 | @label_expr_elim #u2 |
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164 | @add_cost_expr_elim #u4 #l4 |
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165 | @label_expr_elim #u3 |
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166 | @add_cost_expr_elim #u5 #l5 |
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167 | [ cases (IH1 … EX1 u) #tr1' #EX1' |
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168 | %{(tr1'⧺Echarge l4)} |
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169 | whd in ⊢ (??%?); >EX1' |
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170 | whd in ⊢ (??%?); >label_expr_type >EXguard |
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171 | normalize in EXguardrem ⊢ %; destruct @refl |
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172 | | cases (bind_inversion ????? EXguardrem) * #v2 #tr2 * #EX2 #EX2rem |
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173 | cases (bind_inversion ????? EX2rem) * * #EX3 #EX3rem |
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174 | cases (IH1 … EX1 u) #tr1' #EX1' |
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175 | cases (IH2 … EX2 u4) #tr2' #EX2' |
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176 | %{(tr1'⧺(tr2'⧺E0)⧺Echarge l5)} |
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177 | whd in ⊢ (??%?); >EX1' |
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178 | whd in ⊢ (??%?); >label_expr_type >EX2' >EXguard |
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179 | whd in ⊢ (??%?); whd in ⊢ (??(match % with [_⇒?|_⇒?])?); |
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180 | whd in ⊢ (??(match (match % with [_⇒?|_⇒?]) with [_⇒?|_⇒?])?); >EX2' |
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181 | whd in ⊢ (??(match (match % with [_⇒?|_⇒?]) with [_⇒?|_⇒?])?); >label_expr_type >EX3 |
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182 | normalize in EX3rem ⊢ %; destruct @refl |
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183 | ] |
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184 | | #ty #ty' #v #tr normalize /2/ |
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185 | | #ty #e1 #ty' #id #IH #b #o #tr #EX #u cases ty' in IH EX ⊢ %; |
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186 | [ 7: #id #fl #IH #EX |
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187 | cases (bind_inversion ????? EX) * * #b1 #o1 #tr1 * #EX1 #EX1rem |
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188 | cases (bind_inversion ????? EX1rem) #n * #EXoff #EXoffrem |
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189 | whd in EXoffrem:(???%); destruct >shift_fst >shift_fst |
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190 | cases (IH … EX1 u) #tr1' #EX1' |
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191 | %{tr1'} |
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192 | whd in ⊢ (??%?); whd in ⊢ (??(match % with [_⇒?|_⇒?])?); >EX1' |
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193 | whd in ⊢ (??%?); >EXoff whd in ⊢ (??%?); @refl |
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194 | | 8: #id #fl #IH #EX |
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195 | cases (bind_inversion ????? EX) * * #b1 #o1 #tr1 * #EX1 #EX1rem |
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196 | whd in EX1rem:(???%); destruct >shift_fst >shift_fst |
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197 | cases (IH … EX1 u) #tr1' #EX1' |
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198 | %{tr1'} |
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199 | whd in ⊢ (??%?); whd in ⊢ (??(match % with [_⇒?|_⇒?])?); >EX1' |
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200 | whd in ⊢ (??%?); @refl |
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201 | | *: normalize #A #B try #C try #D try #F destruct |
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202 | ] |
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203 | | #ty #l #e1 #IH #v #tr #EX #u |
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204 | cases (bind_inversion ????? EX) * #v1 #tr1 * #EX1 #EX1rem |
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205 | whd in EX1rem:(???%); destruct |
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206 | cases (IH … EX1 u) #tr1' #EX1' %{(tr1'@Echarge l)} |
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207 | >shift_fst >shift_fst whd in ⊢ (??%?); >EX1' |
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208 | whd in ⊢ (??%?); @refl |
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209 | | #e1 #ty #NLV #b #o #tr #EX @⊥ |
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210 | cases e1 in NLV EX; normalize // |
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211 | #A #B #C try #D try #F destruct |
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212 | ] qed. |
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