1 | include "Clight/Cexec.ma". |
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2 | |
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3 | (*include "Plogic/connectives.ma".*) |
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4 | |
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5 | (* Is rather careful about using destruct because it currently uses excessive |
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6 | normalization. *) |
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7 | lemma exec_bool_of_val_sound: ∀v,ty,r. |
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8 | exec_bool_of_val v ty = OK ? r → bool_of_val v ty (of_bool r). |
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9 | #v #ty #r |
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10 | cases v; [ | #sz #i | #f | #r1 | #r' #b #pc #of ] |
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11 | cases ty; [ 2,11,20,29,38: #sz' #sg | 3,12,21,30,39: #sz' | 4,13,22,31,40: #rg #ty | 5,14,23,32,41: #r #ty #n | 6,15,24,33,42: #args #rty | 7,8,16,17,25,26,34,35,43,44: #id #fs | 9,18,27,36,45: #r #id ] |
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12 | whd in ⊢ (??%? → ?) |
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13 | [ 2: @intsize_eq_elim_elim |
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14 | [ #NE #H destruct |
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15 | | *; whd @eq_bv_elim |
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16 | [ #E1 #E2 destruct @bool_of_val_false @is_false_int |
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17 | | #E1 #E2 >(?:r=¬false) [ @bool_of_val_true @is_true_int_int @E1 | destruct @refl ] |
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18 | ] |
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19 | ] |
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20 | | 8: #H cases (eq_dec f Fzero) |
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21 | [ #e >e in H ⊢ % >Feq_zero_true #E destruct @bool_of_val_false @is_false_float |
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22 | | #ne >Feq_zero_false in H // #E >(?:r=¬false) [ @bool_of_val_true @is_true_float @ne | destruct @refl ] |
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23 | ] |
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24 | | 14: #H >(?:r=false) [ @bool_of_val_false @is_false_pointer | destruct @refl ] |
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25 | | 15: #H >(?:r=true) [ @bool_of_val_true @is_true_pointer_pointer | destruct @refl ] |
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26 | | *: #H destruct |
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27 | ] qed. |
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28 | |
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29 | lemma bool_val_distinct: Vtrue ≠ Vfalse. |
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30 | % #H whd in H:(??%%); (*Wilmer: XXX*) cases daemon (*destruct; @(absurd ? e0 one_not_zero)*) |
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31 | qed. |
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32 | |
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33 | lemma bool_of: ∀v,ty,b. bool_of_val v ty (of_bool b) → |
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34 | if b then is_true v ty else is_false v ty. |
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35 | #v #ty #b cases b; #H inversion H; #v' #ty' #H' #ev #et #ev' //; |
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36 | @⊥ cases (bool_val_distinct) #X @X [2: @sym_eq] @ev' |
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37 | qed. |
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38 | |
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39 | lemma try_cast_null_sound: ∀m,sz,i,ty,ty',v'. try_cast_null m sz i ty ty' = OK ? v' → cast m (Vint sz i) ty ty' v'. |
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40 | #m #sz #i #ty #ty' #v' |
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41 | whd in ⊢ (??%? → ?) |
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42 | @eq_bv_elim |
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43 | [ #e >e |
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44 | cases ty; [ | #sz' #sg | #fs | #sp #ty | #sp #ty #n | #args #rty | #id #fs | #id #fs | #r #id ] |
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45 | whd in ⊢ (??%? → ?); #H [ 2: | *: destruct ] |
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46 | cases ty' in H ⊢ %; [ | #sz'' #sg | #fs | #sp #ty | #sp #ty #n | #args #rty | #id #fs | #id #fs | #r #id ] |
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47 | try (@eq_intsize_elim #E) whd in ⊢ (??%? → ?); #H destruct @cast_ip_z // |
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48 | | #_ whd in ⊢ (??%? → ?); #H destruct |
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49 | ] |
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50 | qed. |
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51 | |
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52 | axiom exec_cast_sound : ∀m:mem. ∀v:val. ∀ty:type. ∀ty':type. ∀v':val. exec_cast m v ty ty' = OK ? v' → cast m v ty ty' v'. |
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53 | (*Wilmer: XXXX #m #v #ty #ty' #v' |
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54 | cases v; |
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55 | [ #H whd in H:(??%?); destruct; |
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56 | | #sz #i cases ty; |
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57 | [ #H whd in H:(??%?); destruct; |
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58 | | 3: #a #H whd in H:(??%?); destruct; |
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59 | | 7,8,9: #a #b #H whd in H:(??%?); destruct; |
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60 | | #sz1 #si1 cases ty'; |
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61 | [ whd in ⊢ (??%? → ?) @intsize_eq_elim_elim |
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62 | [ #NE #H destruct |
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63 | | *; whd #H whd in H:(??%?); destruct; |
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64 | ] |
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65 | | 3: #a whd in ⊢ (??%? → ?) @intsize_eq_elim_elim |
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66 | [ #E #H whd in H:(??%?); destruct |
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67 | | *; whd #H whd in H:(??%?); destruct; @cast_if |
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68 | ] |
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69 | | 2,7,8,9: #a #b whd in ⊢ (??%? → ?) @intsize_eq_elim_elim |
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70 | [ 1,3,5,7: #NE #H destruct |
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71 | | *: *; whd #H whd in H:(??%?); destruct; // |
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72 | ] |
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73 | | 4,5,6: [ #sp #ty'' letin t ≝ (Tpointer sp ty'') |
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74 | | #sp #ty'' #n letin t ≝ (Tarray sp ty'' n) |
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75 | | #args #rty letin t ≝ (Tfunction args rty) ] |
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76 | whd in ⊢ (??%? → ?) @intsize_eq_elim_elim |
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77 | [ 1,3,5: #NE #H destruct |
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78 | | *: *; whd |
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79 | lapply (try_cast_null_sound m sz i (Tint sz si1) t v'); |
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80 | cases (try_cast_null m sz i (Tint sz si1) t); |
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81 | [ 1,3,5: #v'' #H' #e @H' @e |
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82 | | *: #m #_ whd in ⊢ (??%? → ?); #H destruct (H); |
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83 | ] |
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84 | ] |
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85 | ] |
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86 | | *: [ #sp #ty'' letin t ≝ (Tpointer sp ty'') |
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87 | | #sp #ty'' #n letin t ≝ (Tarray sp ty'' n) |
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88 | | #args #rty letin t ≝ (Tfunction args rty) ] |
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89 | whd in ⊢ (??%? → ?); |
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90 | lapply (try_cast_null_sound m sz i t ty' v'); |
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91 | cases (try_cast_null m sz i t ty'); |
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92 | [ 1,3,5: #v'' #H' #e @H' @e |
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93 | | *: #m #_ whd in ⊢ (??%? → ?); #H destruct (H); |
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94 | ] |
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95 | ] |
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96 | | #f cases ty; [ 3: #x | 2,4,6,7,8,9: #x #y | 5: #x #y #z ] |
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97 | [ cases ty'; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ] |
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98 | whd in e:(??%?); destruct; //; |
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99 | | *: #e whd in e:(??%?); destruct |
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100 | ] |
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101 | | #r cases ty; [ 3: #x | 2,4,6,7,8,9: #x #y | 5: #x #y #z ] |
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102 | whd in ⊢ (??%? → ?); #H destruct; cases (eq_region_dec r ?) in H; |
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103 | whd in ⊢ (? → ??%? → ?); #H1 #H2 destruct; |
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104 | cases ty' in H2; normalize; try #a try #b try #c try #d destruct; |
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105 | @cast_pp_z //; |
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106 | | #r #b #pc #of whd in ⊢ (??%? → ?) #e |
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107 | elim (bind_inversion ????? e) #s * #es #e' |
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108 | elim (bind_inversion ????? e') #u * #eu #e'' -e'; |
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109 | elim (bind_inversion ????? e'') #s' * #es' #e'''; -e''; |
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110 | cut (type_region ty s); |
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111 | [ cases ty in es:(??%?) ⊢ %; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ] |
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112 | whd in e:(??%?); destruct; //; |
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113 | | #Hty |
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114 | cut (type_region ty' s'); |
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115 | [ cases ty' in es' ⊢ %; [ #e | 3: #a #e | 2,4,6,7,8,9: #a #b #e | #a #b #c #e ] |
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116 | whd in e:(??%?); destruct; //; |
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117 | | #Hty' |
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118 | cut (s = r). elim (eq_region_dec r s) in eu; //; normalize; #_ #e destruct. |
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119 | #e >e in Hty #Hty |
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120 | cases (pointer_compat_dec b s') in e''' |
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121 | #Hcompat #e''' whd in e''':(??%?); destruct (e'''); /2/ |
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122 | ] |
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123 | ] |
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124 | ] qed.*) |
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125 | |
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126 | let rec expr_lvalue_ind |
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127 | (P:expr → Prop) |
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128 | (Q:expr_descr → type → Prop) |
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129 | (ci:∀sz,ty,i.P (Expr (Econst_int sz i) ty)) |
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130 | (cf:∀ty,f.P (Expr (Econst_float f) ty)) |
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131 | (lv:∀e,ty. Q e ty → Plvalue P e ty) |
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132 | (vr:∀v,ty.Q (Evar v) ty) |
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133 | (dr:∀e,ty.P e → Q (Ederef e) ty) |
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134 | (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty)) |
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135 | (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty)) |
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136 | (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty)) |
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137 | (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty)) |
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138 | (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty)) |
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139 | (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty)) |
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140 | (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty)) |
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141 | (sz:∀ty,ty'. P (Expr (Esizeof ty') ty)) |
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142 | (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty) |
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143 | (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty)) |
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144 | (xx:∀e,ty. is_not_lvalue e → Q e ty) |
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145 | (e:expr) on e : P e ≝ |
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146 | match e with |
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147 | [ Expr e' ty ⇒ |
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148 | match e' with |
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149 | [ Econst_int sz i ⇒ ci sz ty i |
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150 | | Econst_float f ⇒ cf ty f |
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151 | | Evar v ⇒ lv (Evar v) ty (vr v ty) |
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152 | | Ederef e'' ⇒ lv (Ederef e'') ty (dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'')) |
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153 | | Eaddrof e'' ⇒ match e'' with [ Expr e0 ty0 ⇒ ao ty e0 ty0 (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e0 ty0) ] |
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154 | | Eunop op e'' ⇒ uo ty op e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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155 | | Ebinop op e1 e2 ⇒ bo ty op e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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156 | | Ecast ty' e'' ⇒ ca ty ty' e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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157 | | Econdition e1 e2 e3 ⇒ cd ty e1 e2 e3 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e3) |
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158 | | Eandbool e1 e2 ⇒ ab ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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159 | | Eorbool e1 e2 ⇒ ob ty e1 e2 (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e1) (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e2) |
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160 | | Esizeof ty' ⇒ sz ty ty' |
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161 | | Efield e'' i ⇒ match e'' with [ Expr ef tyf ⇒ lv (Efield (Expr ef tyf) i) ty (fl ty ef tyf i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx ef tyf)) ] |
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162 | | Ecost l e'' ⇒ co ty l e'' (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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163 | ] |
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164 | ] |
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165 | and lvalue_expr_ind |
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166 | (P:expr → Prop) |
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167 | (Q:expr_descr → type → Prop) |
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168 | (ci:∀sz,ty,i.P (Expr (Econst_int sz i) ty)) |
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169 | (cf:∀ty,f.P (Expr (Econst_float f) ty)) |
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170 | (lv:∀e,ty. Q e ty → Plvalue P e ty) |
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171 | (vr:∀v,ty.Q (Evar v) ty) |
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172 | (dr:∀e,ty.P e → Q (Ederef e) ty) |
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173 | (ao:∀ty,e,ty'.Q e ty' → P (Expr (Eaddrof (Expr e ty')) ty)) |
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174 | (uo:∀ty,op,e.P e → P (Expr (Eunop op e) ty)) |
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175 | (bo:∀ty,op,e1,e2.P e1 → P e2 → P (Expr (Ebinop op e1 e2) ty)) |
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176 | (ca:∀ty,ty',e.P e → P (Expr (Ecast ty' e) ty)) |
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177 | (cd:∀ty,e1,e2,e3.P e1 → P e2 → P e3 → P (Expr (Econdition e1 e2 e3) ty)) |
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178 | (ab:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eandbool e1 e2) ty)) |
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179 | (ob:∀ty,e1,e2.P e1 → P e2 → P (Expr (Eorbool e1 e2) ty)) |
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180 | (sz:∀ty,ty'. P (Expr (Esizeof ty') ty)) |
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181 | (fl:∀ty,e,ty',i. Q e ty' → Q (Efield (Expr e ty') i) ty) |
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182 | (co:∀ty,l,e. P e → P (Expr (Ecost l e) ty)) |
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183 | (xx:∀e,ty. is_not_lvalue e → Q e ty) |
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184 | (e:expr_descr) (ty:type) on e : Q e ty ≝ |
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185 | match e return λe0. Q e0 ty with |
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186 | [ Evar v ⇒ vr v ty |
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187 | | Ederef e'' ⇒ dr e'' ty (expr_lvalue_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'') |
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188 | | Efield e' i ⇒ match e' return λe1.Q (Efield e1 i) ty with [ Expr e'' ty'' ⇒ fl ty e'' ty'' i (lvalue_expr_ind P Q ci cf lv vr dr ao uo bo ca cd ab ob sz fl co xx e'' ty'') ] |
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189 | | _ ⇒ xx ? ty ? |
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190 | ]. whd; @I qed. |
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191 | |
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192 | |
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193 | theorem exec_expr_sound: ∀ge:genv. ∀en:env. ∀m:mem. ∀e:expr. |
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194 | (P_res ? (λx.eval_expr ge en m e (\fst x) (\snd x)) (exec_expr ge en m e)). |
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195 | #ge #en #m #e @(expr_lvalue_ind ? (λe',ty.P_res ? (λr.eval_lvalue ge en m (Expr e' ty) (\fst (\fst r)) (\snd (\fst r)) (\snd r)) (exec_lvalue' ge en m e' ty)) … e) |
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196 | (* XXX // fails [ 1,2: #ty #c whd // *) |
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197 | [ #sz #ty #c whd in ⊢ (???%) cases ty try #sz' try #sg try #x try @I whd in ⊢ (???%) |
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198 | @eq_intsize_elim #E try @I <E whd % |
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199 | | #ty #c whd %2 |
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200 | (* expressions that are lvalues *) |
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201 | | #e' #ty cases e'; //; [ #i #He' | #e #He' | #e #i #He' ] whd in He' ⊢ %; |
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202 | @bind2_OK #x cases x; #y cases y; #sp #loc #ofs #tr #H |
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203 | @opt_bind_OK #vl #evl whd in evl:(??%?); @(eval_Elvalue … evl) |
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204 | >H in He' #He' @He' |
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205 | | #v #ty |
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206 | whd in ⊢ (???%); |
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207 | lapply (refl ? (lookup SymbolTag block en v)) |
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208 | cases (lookup SymbolTag block en v) in ⊢ (???% → %) |
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209 | [ #eget @opt_bind_OK #sploc cases sploc; #sp #loc #efind |
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210 | whd; @(eval_Evar_global … eget efind) |
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211 | | #loc #eget @(eval_Evar_local … eget) |
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212 | ] |
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213 | | #ty #e #He whd in ⊢ (???%) |
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214 | @bind2_OK #v cases v // #r #l #pc #ofs #tr #Hv whd |
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215 | >Hv in He #He |
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216 | @eval_Ederef [ 3: @He | *: skip ] |
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217 | | #ty #e'' #ty'' #He'' @bind2_OK * #loc #ofs #tr #H |
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218 | cases ty // * #pty |
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219 | cases loc in H ⊢ % * #loc' #H |
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220 | whd try @I |
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221 | @eval_Eaddrof whd in H:(??%?) >H in He'' #He'' @He'' |
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222 | | #ty #op #e1 #He1 @bind2_OK #v1 #tr #Hv1 |
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223 | @opt_bind_OK #v #ev |
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224 | @(eval_Eunop … ev) |
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225 | >Hv1 in He1 #He1 @He1 |
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226 | | #ty #op #e1 #e2 #He1 #He2 |
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227 | @bind2_OK #v1 #tr1 #ev1 >ev1 in He1 #He1 |
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228 | @bind2_OK #v2 #tr2 #ev2 >ev2 in He2 #He2 |
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229 | @opt_bind_OK #v #ev whd in He1 He2; whd; |
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230 | @(eval_Ebinop … He1 He2 ev) |
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231 | | #ty #ty' #e' #He' |
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232 | @bind2_OK #v #tr #Hv >Hv in He' #He' |
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233 | @bind_OK #v' #ev' |
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234 | @(eval_Ecast … He' ?) |
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235 | (* XXX /2/; *) |
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236 | @(exec_cast_sound … ev') |
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237 | | #ty #e1 #e2 #e3 #He1 #He2 #He3 |
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238 | @bind2_OK #vb #tr1 #Hvb >Hvb in He1 #He1 |
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239 | @bind_OK #b |
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240 | cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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241 | @bind2_OK #v #tr #Hv whd in Hv:(??%?) |
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242 | [ >Hv in He2 #He2 whd in He2 Hv:(??%?) ⊢%; |
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243 | @(eval_Econdition_true … He1 ? He2) @(bool_of ??? Hb) |
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244 | | >Hv in He3 #He3 whd in He3 Hv:(??%?) ⊢%; |
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245 | @(eval_Econdition_false … He1 ? He3) @(bool_of ??? Hb) |
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246 | ] |
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247 | | #ty #e1 #e2 #He1 #He2 |
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248 | @bind2_OK #v1 #tr1 #Hv1 >Hv1 in He1 #He1 |
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249 | @bind_OK #b1 cases b1; #eb1 lapply (exec_bool_of_val_sound … eb1); #Hb1 |
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250 | [ @bind2_OK #v2 #tr2 #Hv2 >Hv2 in He2 #He2 |
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251 | @bind_OK #b2 #eb2 |
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252 | @(eval_Eandbool_2 … He1 … He2) |
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253 | [ @(bool_of … Hb1) | @(exec_bool_of_val_sound … eb2) ] |
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254 | | @(eval_Eandbool_1 … He1) @(bool_of … Hb1) |
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255 | ] |
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256 | | #ty #e1 #e2 #He1 #He2 |
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257 | @bind2_OK #v1 #tr1 #Hv1 >Hv1 in He1 #He1 |
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258 | @bind_OK #b1 cases b1; #eb1 lapply (exec_bool_of_val_sound … eb1); #Hb1 |
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259 | [ @(eval_Eorbool_1 … He1) @(bool_of … Hb1) |
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260 | | @bind2_OK #v2 #tr2 #Hv2 >Hv2 in He2 #He2 |
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261 | @bind_OK #b2 #eb2 |
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262 | @(eval_Eorbool_2 … He1 … He2) |
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263 | [ @(bool_of … Hb1) | @(exec_bool_of_val_sound … eb2) ] |
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264 | ] |
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265 | | #ty #ty' cases ty try #sz try #sg try #x try @I whd; (* XXX //*) @eval_Esizeof |
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266 | | #ty #e' #ty' #i cases ty'; //; |
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267 | [ #id #fs #He' @bind2_OK #x cases x; #sp #l #ofs #H |
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268 | @bind_OK #delta #Hdelta whd in H:(??%?) >H in He' #He' |
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269 | @(eval_Efield_struct … He' (refl ??) Hdelta) |
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270 | | #id #fs #He' @bind2_OK #x cases x; #sp #l #ofs #H whd in H:(??%?) |
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271 | >H in He' #He' |
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272 | @(eval_Efield_union … He' (refl ??)) |
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273 | ] |
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274 | | #ty #l #e' #He' @bind2_OK #v #tr1 #H >H in He' #He' |
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275 | @(eval_Ecost … He') |
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276 | (* exec_lvalue fails on non-lvalues. *) |
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277 | | #e' #ty cases e'; |
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278 | [ 2,5,12: #a #H | 3,4: #a * | 13,14: #a #b * | 1,6,8,10,11: #a #b #H | 7,9: #a #b #c #H ] |
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279 | @I |
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280 | ] qed. |
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281 | |
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282 | lemma addrof_eval_lvalue: ∀ge,en,m,e,r,loc,pc,off,tr,ty. |
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283 | eval_expr ge en m (Expr (Eaddrof e) ty) (Vptr r loc pc off) tr → |
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284 | eval_lvalue ge en m e loc off tr. |
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285 | #ge #en #m #e #r #loc #pc #off #tr #ty #H inversion H; |
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286 | [ 1,2,5: #a #b #c #H @False_ind destruct (H); |
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287 | | #a #b #c #d #e #f #H1 #g #H2 <H2 in H1 #H1 @False_ind |
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288 | @(eval_lvalue_inv_ind … H1) |
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289 | [ #a #b #c #d #bad destruct (bad); |
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290 | | #a #b #c #d #e #bad destruct (bad); |
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291 | | #a #b #c #d #e #f #g #h #bad destruct (bad); |
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292 | | #a #b #c #d #e #f #g #h #i #j #k #l #m #bad @False_ind destruct (bad); |
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293 | | #a #b #c #d #e #f #g #h #i #j #k #bad destruct (bad); |
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294 | ] |
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295 | | #e' #ty' #r' #loc' #ofs' #tr' #H' #pc' #e1 #e2 #e3 destruct (e1 e2 e3); @H' |
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296 | | #a #b #c #d #e #f #g #h #i #bad destruct (bad); |
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297 | | #a #b #c #d #e #f #g #h #i #j #k #l #k #l #bad destruct (bad); |
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298 | | #a #b #c #d #e #f #g #h #i #j #k #l #m #bad destruct (bad); |
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299 | | #a #b #c #d #e #f #g #h #i #j #k #l #m #bad destruct (bad); |
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300 | | #a #b #c #d #e #f #g #h #bad destruct (bad); |
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301 | | #a #b #c #d #e #f #g #h #i #j #k #l #m #n #bad destruct (bad); |
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302 | | #a #b #c #d #e #f #g #h #bad destruct (bad); |
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303 | | #a #b #c #d #e #f #g #h #i #j #k #l #m #n #bad destruct (bad); |
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304 | | #a #b #c #d #e #f #g #h #i #bad destruct (bad); |
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305 | | #a #b #c #d #e #f #g #bad destruct (bad); |
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306 | ] qed. |
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307 | |
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308 | lemma addrof_exec_lvalue: ∀ge,en,m,e,r,loc,off,tr. |
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309 | exec_lvalue ge en m e = OK ? 〈〈mk_block r loc,off〉,tr〉 → |
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310 | exec_expr ge en m (Expr (Eaddrof e) (Tpointer r Tvoid)) = OK ? 〈Vptr r (mk_block r loc) (same_compat ??) off, tr〉. |
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311 | #ge #en #m #e #r #loc #off #tr #H whd in ⊢ (??%?) |
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312 | >H whd in ⊢ (??%?) cases r @refl |
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313 | qed. |
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314 | |
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315 | theorem exec_lvalue_sound: ∀ge,en,m,e. |
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316 | P_res ? (λr.eval_lvalue ge en m e (\fst (\fst r)) (\snd (\fst r)) (\snd r)) (exec_lvalue ge en m e). |
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317 | #ge #en #m #e lapply (refl ? (exec_lvalue ge en m e)); |
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318 | cases (exec_lvalue ge en m e) in ⊢ (???% → %); |
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319 | [ #x cases x #y cases y #loc #off #tr #H whd cases loc in H ⊢ % #locr #loci #H |
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320 | @(addrof_eval_lvalue … locr … (Tpointer locr Tvoid)) [ @same_compat ] |
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321 | lapply (addrof_exec_lvalue … H) #H' |
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322 | lapply (exec_expr_sound ge en m (Expr (Eaddrof e) (Tpointer locr Tvoid))) |
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323 | >H' #H'' @H'' |
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324 | | #msg #_ whd @I |
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325 | ] qed. |
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326 | |
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327 | (* Plain equality versions of the above *) |
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328 | |
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329 | definition exec_expr_sound' ≝ λge,en,m,e,v. |
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330 | λH:exec_expr ge en m e = OK ? v. |
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331 | P_res_to_P ???? (exec_expr_sound ge en m e) H. |
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332 | |
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333 | definition exec_lvalue_sound' ≝ λge,en,m,e,loc,off,tr. |
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334 | λH:exec_lvalue ge en m e = OK ? 〈〈loc,off〉,tr〉. |
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335 | P_res_to_P ???? (exec_lvalue_sound ge en m e) H. |
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336 | |
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337 | lemma exec_exprlist_sound: ∀ge,e,m,l. P_res ? (λvltr:list val×trace. eval_exprlist ge e m l (\fst vltr) (\snd vltr)) (exec_exprlist ge e m l). |
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338 | #ge #e #m #l elim l; |
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339 | whd; (* XXX //; *) |
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340 | [ % |
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341 | | #e1 #es #IH |
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342 | @bind2_OK #v #tr1 #Hv |
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343 | @bind2_OK #vs #tr2 #Hvs |
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344 | whd; @eval_Econs |
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345 | [ @(P_res_to_P … (exec_expr_sound ge e m e1) Hv) |
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346 | | @(P_res_to_P … IH Hvs) |
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347 | ] |
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348 | ] qed. |
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349 | |
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350 | lemma exec_alloc_variables_sound : ∀l,en,m,en',m'. |
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351 | exec_alloc_variables en m l = 〈en',m'〉 → |
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352 | alloc_variables en m l en' m'. |
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353 | #l elim l |
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354 | [ #en #m #en' #m' #EXEC whd in EXEC:(??%?); destruct % |
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355 | | * #id #ty #t #IH #en #m #en' #m' |
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356 | lapply (refl ? (alloc m O (sizeof ty) Any)) #ALLOC #EXEC |
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357 | whd in EXEC:(??%?) ALLOC:(???%) |
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358 | @(alloc_variables_cons … ALLOC) |
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359 | @IH @EXEC |
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360 | qed. |
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361 | |
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362 | lemma exec_bind_parameters_sound : ∀ps,vs,en,m. |
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363 | P_res ? (λm'. bind_parameters en m ps vs m') (exec_bind_parameters en m ps vs). |
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364 | #ps elim ps |
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365 | [ * // |
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366 | | * #id #ty #ps' #IH * |
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367 | [ // |
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368 | | #v #vs #en #m |
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369 | @opt_bind_OK #b #GET |
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370 | @opt_bind_OK #m' #STORE |
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371 | lapply (refl ? (exec_bind_parameters en m' ps' vs)) |
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372 | cases (exec_bind_parameters en m' ps' vs) in ⊢ (???% → %) [2: #msg #_ %] |
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373 | #m'' #BIND |
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374 | @(bind_parameters_cons … GET STORE) |
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375 | lapply (IH vs en m') whd in ⊢ (% → ?) >BIND // |
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376 | ] |
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377 | ] qed. |
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378 | |
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379 | lemma check_eventval_list_sound : ∀vs,tys. |
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380 | P_res ? (λevs. eventval_list_match evs tys vs) (check_eventval_list vs tys). |
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381 | #vs0 elim vs0 |
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382 | [ * // |
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383 | | #v #vs #IH * |
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384 | [ // |
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385 | | #ty #tys whd in ⊢ (???%) |
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386 | cases ty [ #sz #sg | #sz | #r ] cases v // |
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387 | [ #sz' #v @bind_OK #ev whd in ⊢ (??%? → ?) |
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388 | @eq_intsize_elim #E #CHECKev whd in CHECKev:(??%?); destruct |
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389 | | #v ] @bind_OK #evs #CHECKevs |
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390 | @(evl_match_cons ??????? (P_res_to_P ???? (IH ?) CHECKevs)) |
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391 | // |
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392 | ] |
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393 | ] qed. |
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394 | |
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395 | theorem exec_step_sound: ∀ge,st. |
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396 | P_io ??? (λr. step ge st (\fst r) (\snd r)) (exec_step ge st). |
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397 | #ge #st cases st; |
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398 | [ #f #s #k #e #m cases s; |
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399 | [ cases k; |
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400 | [ whd in ⊢ (?????%); |
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401 | lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %); |
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402 | //; #H whd; |
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403 | @step_skip_call //; |
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404 | | #s' #k' whd; (* XXX //; *) @step_skip_seq |
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405 | | #ex #s' #k' @step_skip_or_continue_while % //; |
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406 | | #ex #s' #k' |
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407 | @res_bindIO2_OK #v #tr #Hv |
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408 | letin bexpr ≝ (exec_bool_of_val v (typeof ex)); |
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409 | lapply (refl ? bexpr); |
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410 | cases bexpr in ⊢ (???% → %); |
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411 | [ #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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412 | whd in ⊢ (?????%); |
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413 | [ @(step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv)) |
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414 | [ % // | @(bool_of … Hb) ] |
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415 | | @(step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv)) |
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416 | [ % // | @(bool_of … Hb) ] |
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417 | ] |
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418 | | #msg #_ //; |
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419 | ] |
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420 | | #ex #s1 #s2 #k' @step_skip_or_continue_for2 % //; |
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421 | | #ex #s1 #s2 #k' @step_skip_for3 |
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422 | | #k' @step_skip_break_switch % //; |
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423 | | #r #f' #e' #k' whd in ⊢ (?????%); |
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424 | lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %); |
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425 | //; #H whd; |
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426 | @step_skip_call //; |
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427 | ] |
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428 | | #ex1 #ex2 |
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429 | @res_bindIO2_OK #x cases x; #y cases y; #pcl #loc #ofs #tr1 #Hlval |
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430 | @res_bindIO2_OK #v2 #tr2 #Hv2 |
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431 | @opt_bindIO_OK #m' #em' |
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432 | whd; @(step_assign … (exec_lvalue_sound' … Hlval) (exec_expr_sound' … Hv2) em') |
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433 | | #lex #fex #args |
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434 | @res_bindIO2_OK #vf #tr1 #Hvf0 lapply (exec_expr_sound' … Hvf0); #Hvf |
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435 | @res_bindIO2_OK #vargs #tr2 #Hvargs0 lapply (P_res_to_P ???? (exec_exprlist_sound …) Hvargs0); #Hvargs |
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436 | @opt_bindIO_OK #fd #efd |
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437 | @bindIO_OK #ety |
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438 | cases lex; whd; |
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439 | [ @(step_call_none … Hvf Hvargs efd ety) |
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440 | | #lhs' |
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441 | @res_bindIO2_OK #x cases x; #y cases y; #pcl #loc #ofs #tr3 #Hlocofs |
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442 | whd; @(step_call_some … (exec_lvalue_sound' … Hlocofs) Hvf Hvargs efd ety) |
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443 | ] |
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444 | | #s1 #s2 whd; (* XXX //; *) @step_seq |
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445 | | #ex #s1 #s2 |
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446 | @res_bindIO2_OK #v #tr #Hv |
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447 | letin bexpr ≝ (exec_bool_of_val v (typeof ex)); |
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448 | lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //; |
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449 | #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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450 | [ @(step_ifthenelse_true … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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451 | | @(step_ifthenelse_false … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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452 | ] |
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453 | | #ex #s' |
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454 | @res_bindIO2_OK #v #tr #Hv |
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455 | letin bexpr ≝ (exec_bool_of_val v (typeof ex)); |
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456 | lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //; |
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457 | #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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458 | [ @(step_while_true … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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459 | | @(step_while_false … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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460 | ] |
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461 | | #ex #s' whd; (* XXX //; *) @step_dowhile |
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462 | | #s1 #ex #s2 #s3 whd in ⊢ (?????%); elim (is_Sskip s1); #Hs1 whd in ⊢ (?????%); |
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463 | [ >Hs1 |
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464 | @res_bindIO2_OK #v #tr #Hv |
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465 | letin bexpr ≝ (exec_bool_of_val v (typeof ex)); |
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466 | lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //; |
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467 | #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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468 | [ @(step_for_true … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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469 | | @(step_for_false … (exec_expr_sound' … Hv)) @(bool_of … Hb) |
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470 | ] |
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471 | | @step_for_start //; |
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472 | ] |
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473 | | whd in ⊢ (?????%); cases k; // #k' @step_skip_break_switch %2 // |
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474 | | whd in ⊢ (?????%); cases k; //; |
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475 | [ #ex #s' #k' whd; @step_skip_or_continue_while %2 ; //; |
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476 | | #ex #s' #k' whd; |
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477 | @res_bindIO2_OK #v #tr #Hv |
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478 | letin bexpr ≝ (exec_bool_of_val v (typeof ex)); |
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479 | lapply (refl ? bexpr); cases bexpr in ⊢ (???% → %); //; |
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480 | #b cases b; #eb lapply (exec_bool_of_val_sound … eb); #Hb |
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481 | [ @(step_skip_or_continue_dowhile_true … (exec_expr_sound' … Hv)) |
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482 | [ %2 ; // | @(bool_of … Hb) ] |
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483 | | @(step_skip_or_continue_dowhile_false … (exec_expr_sound' … Hv)) |
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484 | [ %2 ; // | @(bool_of … Hb) ] |
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485 | ] |
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486 | | #ex #s1 #s2 #k' whd; @step_skip_or_continue_for2 %2 ; // |
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487 | ] |
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488 | | #r whd in ⊢ (?????%); cases r; |
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489 | [ whd; lapply (refl ? (fn_return f)); cases (fn_return f) in ⊢ (???% → %); //; #H |
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490 | @step_return_0 @H |
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491 | | #ex cases (type_eq_dec (fn_return f) Tvoid); //; whd in ⊢ (% → ?????%); #Hnotvoid |
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492 | @res_bindIO2_OK #v #tr #Hv |
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493 | whd; @(step_return_1 … Hnotvoid (exec_expr_sound' … Hv)) |
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494 | ] |
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495 | | #ex #ls @res_bindIO2_OK * // #sz #n #tr #Hv |
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496 | @step_switch @(exec_expr_sound' … Hv) |
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497 | | #l #s' whd; @step_label (* XXX //; *) |
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498 | | #l whd in ⊢ (?????%); lapply (refl ? (find_label l (fn_body f) (call_cont k))); |
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499 | cases (find_label l (fn_body f) (call_cont k)) in ⊢ (???% → %); //; |
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500 | #sk cases sk; #s' #k' #H |
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501 | @(step_goto … H) |
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502 | | #l #s' whd; (* XXX //; *) @step_cost |
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503 | ] |
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504 | | #f0 #vargs #k #m whd in ⊢ (?????%); cases f0; |
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505 | [ #fn whd in ⊢ (?????%) |
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506 | lapply (refl ? (exec_alloc_variables empty_env m (fn_params fn@fn_vars fn))) |
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507 | cases (exec_alloc_variables empty_env m (fn_params fn@fn_vars fn)) in ⊢ (???% → %) |
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508 | #en' #m' #ALLOC whd in ⊢ (?????%) |
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509 | @res_bindIO_OK #m2 #BIND |
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510 | whd; @(step_internal_function … (exec_alloc_variables_sound … ALLOC)) |
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511 | @(P_res_to_P … (exec_bind_parameters_sound …) BIND) |
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512 | | #id #argtys #rty @res_bindIO_OK #evs #Hevs |
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513 | @bindIO_OK #eres whd; |
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514 | @step_external_function % [ @(P_res_to_P … (check_eventval_list_sound …) Hevs) |
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515 | | @mk_val_correct ] |
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516 | ] |
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517 | | #v #k' #m' whd in ⊢ (?????%); cases k'; //; |
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518 | #r #f #e #k whd in ⊢ (?????%); cases r; |
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519 | [ whd; @step_returnstate_0 |
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520 | | #x cases x; #y cases y; #z cases z; #pcl #b #ofs #ty |
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521 | @opt_bindIO_OK #m' #em' @step_returnstate_1 whd in em':(??%?); //; |
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522 | ] |
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523 | ] |
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524 | qed. |
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525 | |
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526 | lemma make_initial_state_sound : ∀p. P_res ? (initial_state p) (make_initial_state p). |
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527 | #p cases p; #fns #main #vars |
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528 | whd in ⊢ (???%); |
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529 | @bind_OK #m #Em |
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530 | @opt_bind_OK #x cases x; #sp #b #esb |
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531 | @opt_bind_OK #f #ef |
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532 | @(initial_state_intro … Em esb ef) @refl |
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533 | qed. |
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534 | |
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535 | theorem exec_steps_sound: ∀ge,n,st. |
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536 | P_io ??? (λts:trace×state. star (mk_transrel ?? step) ge st (\fst ts) (\snd ts)) |
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537 | (exec_steps n ge st). |
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538 | #ge #n elim n; |
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539 | [ #st whd in ⊢ (?????%); elim (is_final_state st); #H whd; % |
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540 | | #n' #IH #st whd in ⊢ (?????%); elim (is_final_state st); #H |
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541 | [ whd; % |
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542 | | @(P_bindIO2_OK ????????? (exec_step_sound …)) #t #s' cases s'; |
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543 | [ #f #s #k #e #m | #fd #args #k #m | #r #k #m ] |
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544 | whd in ⊢ (? → ?????(??????%?)); |
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545 | cases m; #mc #mn #mp #Hstep |
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546 | whd in ⊢ (?????(??????%?)); |
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547 | @(P_bindIO2_OK ????????? (IH …)) #t' #s'' #Hsteps |
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548 | whd; @(star_step ????????? Hsteps) [ 2,5,8: @Hstep | 3,6,9: // ] |
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549 | ] |
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550 | qed. |
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551 | |
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552 | lemma is_final_sound: ∀s,r. is_final s = Some ? r → final_state s r. |
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553 | * [ 3: #v * [ #m #r cases v [ 2: * #r' #E normalize in E; destruct % |
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554 | | *: normalize #x1 try #x2 try #x3 try #x4 try #x5 destruct |
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555 | ] |
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556 | | *: normalize #x1 try #x2 try #x3 try #x4 try #x5 try #x6 try #x7 destruct |
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557 | ] |
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558 | | *: normalize #x1 #x2 #x3 #x4 #x5 #x6 try #x7 destruct |
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559 | ] qed. |
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