1 | (* Various small results used in at least two files in the front-end. *) |
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2 | |
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3 | include "Clight/TypeComparison.ma". |
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4 | include "common/Pointers.ma". |
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5 | |
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6 | lemma eq_intsize_identity : ∀id. eq_intsize id id = true. |
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7 | * normalize // |
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8 | qed. |
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9 | |
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10 | lemma sz_eq_identity : ∀s. sz_eq_dec s s = inl ?? (refl ? s). |
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11 | * normalize // |
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12 | qed. |
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13 | |
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14 | lemma type_eq_identity : ∀t. type_eq t t = true. |
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15 | #t normalize elim (type_eq_dec t t) |
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16 | [ 1: #Heq normalize // |
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17 | | 2: #H destruct elim H #Hcontr elim (Hcontr (refl ? t)) ] qed. |
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18 | |
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19 | lemma type_neq_not_identity : ∀t1, t2. t1 ≠ t2 → type_eq t1 t2 = false. |
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20 | #t1 #t2 #Hneq normalize elim (type_eq_dec t1 t2) |
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21 | [ 1: #Heq destruct elim Hneq #Hcontr elim (Hcontr (refl ? t2)) |
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22 | | 2: #Hneq' normalize // ] qed. |
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23 | |
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24 | (* useful facts on various boolean operations *) |
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25 | lemma andb_lsimpl_true : ∀x. andb true x = x. // qed. |
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26 | lemma andb_lsimpl_false : ∀x. andb false x = false. normalize // qed. |
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27 | lemma andb_comm : ∀x,y. andb x y = andb y x. // qed. |
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28 | lemma notb_true : notb true = false. // qed. |
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29 | lemma notb_false : notb false = true. % #H destruct qed. |
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30 | lemma notb_fold : ∀x. if x then false else true = (¬x). // qed. |
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31 | |
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32 | (* useful facts on block *) |
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33 | lemma not_eq_block : ∀b1,b2. b1 ≠ b2 → eq_block b1 b2 = false. |
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34 | #b1 #b2 #Hneq |
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35 | @(eq_block_elim … b1 b2) |
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36 | [ 1: #Heq destruct elim Hneq #H @(False_ind … (H (refl ??))) |
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37 | | 2: #_ @refl ] qed. |
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38 | |
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39 | lemma not_eq_block_rev : ∀b1,b2. b2 ≠ b1 → eq_block b1 b2 = false. |
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40 | #b1 #b2 #Hneq |
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41 | @(eq_block_elim … b1 b2) |
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42 | [ 1: #Heq destruct elim Hneq #H @(False_ind … (H (refl ??))) |
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43 | | 2: #_ @refl ] qed. |
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44 | |
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45 | (* useful facts on Z *) |
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46 | lemma Zltb_to_Zleb_true : ∀a,b. Zltb a b = true → Zleb a b = true. |
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47 | #a #b #Hltb lapply (Zltb_true_to_Zlt … Hltb) #Hlt @Zle_to_Zleb_true |
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48 | /3 by Zlt_to_Zle, transitive_Zle/ qed. |
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49 | |
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50 | lemma Zle_to_Zle_to_eq : ∀a,b. Zle a b → Zle b a → a = b. |
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51 | #a #b elim b |
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52 | [ 1: elim a // #a' normalize [ 1: @False_ind | 2: #_ @False_ind ] ] |
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53 | #b' elim a normalize |
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54 | [ 1: #_ @False_ind |
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55 | | 2: #a' #H1 #H2 >(le_to_le_to_eq … H1 H2) @refl |
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56 | | 3: #a' #_ @False_ind |
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57 | | 4: @False_ind |
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58 | | 5: #a' @False_ind |
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59 | | 6: #a' #H2 #H1 >(le_to_le_to_eq … H1 H2) @refl |
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60 | ] qed. |
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61 | |
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62 | lemma Zleb_true_to_Zleb_true_to_eq : ∀a,b. (Zleb a b = true) → (Zleb b a = true) → a = b. |
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63 | #a #b #H1 #H2 |
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64 | /3 by Zle_to_Zle_to_eq, Zleb_true_to_Zle/ |
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65 | qed. |
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66 | |
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67 | lemma Zltb_dec : ∀a,b. (Zltb a b = true) ∨ (Zltb a b = false ∧ Zleb b a = true). |
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68 | #a #b |
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69 | lapply (Z_compare_to_Prop … a b) |
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70 | cases a |
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71 | [ 1: | 2,3: #a' ] |
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72 | cases b |
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73 | whd in match (Z_compare OZ OZ); normalize nodelta |
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74 | [ 2,3,5,6,8,9: #b' ] |
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75 | whd in match (Zleb ? ?); |
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76 | try /3 by or_introl, or_intror, conj, I, refl/ |
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77 | whd in match (Zltb ??); |
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78 | whd in match (Zleb ??); #_ |
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79 | [ 1: cases (decidable_le (succ a') b') |
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80 | [ 1: #H lapply (le_to_leb_true … H) #H %1 assumption |
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81 | | 2: #Hnotle lapply (not_le_to_lt … Hnotle) #Hlt %2 @conj try @le_to_leb_true |
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82 | /2 by monotonic_pred/ @(not_le_to_leb_false … Hnotle) ] |
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83 | | 2: cases (decidable_le (succ b') a') |
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84 | [ 1: #H lapply (le_to_leb_true … H) #H %1 assumption |
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85 | | 2: #Hnotle lapply (not_le_to_lt … Hnotle) #Hlt %2 @conj try @le_to_leb_true |
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86 | /2 by monotonic_pred/ @(not_le_to_leb_false … Hnotle) ] |
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87 | ] qed. |
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88 | |
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89 | lemma Zleb_unsigned_OZ : ∀bv. Zleb OZ (Z_of_unsigned_bitvector 16 bv) = true. |
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90 | #bv elim bv try // #n' * #tl normalize /2/ qed. |
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91 | |
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92 | lemma Zltb_unsigned_OZ : ∀bv. Zltb (Z_of_unsigned_bitvector 16 bv) OZ = false. |
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93 | #bv elim bv try // #n' * #tl normalize /2/ qed. |
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94 | |
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95 | lemma Z_of_unsigned_not_neg : ∀bv. |
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96 | (Z_of_unsigned_bitvector 16 bv = OZ) ∨ (∃p. Z_of_unsigned_bitvector 16 bv = pos p). |
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97 | #bv elim bv |
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98 | [ 1: normalize %1 @refl |
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99 | | 2: #n #hd #tl #Hind cases hd |
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100 | [ 1: normalize %2 /2 by ex_intro/ |
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101 | | 2: cases Hind normalize [ 1: #H %1 @H | 2: * #p #H >H %2 %{p} @refl ] |
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102 | ] |
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103 | ] qed. |
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104 | |
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105 | lemma free_not_in_bounds : ∀x. if Zleb (pos one) x |
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106 | then Zltb x OZ |
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107 | else false = false. |
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108 | #x lapply (Zltb_to_Zleb_true x OZ) |
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109 | elim (Zltb_dec … x OZ) |
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110 | [ 1: #Hlt >Hlt #H lapply (H (refl ??)) elim x |
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111 | [ 2,3: #x' ] normalize in ⊢ (% → ?); |
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112 | [ 1: #Habsurd destruct (Habsurd) |
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113 | | 2,3: #_ @refl ] |
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114 | | 2: * #Hlt #Hle #_ >Hlt cases (Zleb (pos one) x) @refl ] |
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115 | qed. |
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116 | |
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117 | lemma free_not_valid : ∀x. if Zleb (pos one) x |
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118 | then Zleb x OZ |
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119 | else false = false. |
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120 | #x |
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121 | cut (Zle (pos one) x ∧ Zle x OZ → False) |
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122 | [ * #H1 #H2 lapply (transitive_Zle … H1 H2) #H @H ] #Hguard |
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123 | cut (Zleb (pos one) x = true ∧ Zleb x OZ = true → False) |
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124 | [ * #H1 #H2 @Hguard @conj /2 by Zleb_true_to_Zle/ ] |
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125 | cases (Zleb (pos one) x) cases (Zleb x OZ) |
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126 | /2 by refl/ #H @(False_ind … (H (conj … (refl ??) (refl ??)))) |
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127 | qed. |
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