1 | include "ASM/Assembly.ma". |
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2 | include "ASM/Interpret.ma". |
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3 | |
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4 | (* RUSSEL **) |
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5 | |
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6 | include "basics/jmeq.ma". |
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7 | |
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8 | notation > "hvbox(a break ≃ b)" |
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9 | non associative with precedence 45 |
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10 | for @{ 'jmeq ? $a ? $b }. |
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11 | |
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12 | notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)" |
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13 | non associative with precedence 45 |
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14 | for @{ 'jmeq $t $a $u $b }. |
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15 | |
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16 | interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y). |
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17 | |
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18 | lemma eq_to_jmeq: |
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19 | ∀A: Type[0]. |
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20 | ∀x, y: A. |
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21 | x = y → x ≃ y. |
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22 | // |
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23 | qed. |
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24 | |
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25 | definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p. |
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26 | definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w]. |
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27 | |
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28 | coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?. |
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29 | coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?. |
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30 | |
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31 | axiom VOID: Type[0]. |
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32 | axiom assert_false: VOID. |
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33 | definition bigbang: ∀A:Type[0].False → VOID → A. |
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34 | #A #abs cases abs |
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35 | qed. |
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36 | |
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37 | coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?. |
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38 | |
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39 | lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p). |
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40 | #A #P #p cases p #w #q @q |
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41 | qed. |
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42 | |
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43 | lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y. |
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44 | #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) % |
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45 | qed. |
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46 | |
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47 | coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?. |
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48 | |
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49 | (* END RUSSELL **) |
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50 | |
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51 | |
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52 | definition bit_elim_prop: ∀P: bool → Prop. Prop ≝ |
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53 | λP. |
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54 | P true ∧ P false. |
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55 | |
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56 | let rec bitvector_elim_prop_internal |
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57 | (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝ |
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58 | match m return λm. m ≤ n → BitVector (n - m) → Prop with |
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59 | [ O ⇒ λprf1. λprefix. P ? |
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60 | | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?) |
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61 | ]. |
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62 | [ applyS prefix |
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63 | | letin res ≝ (bit ::: prefix) |
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64 | < (minus_S_S ? ?) |
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65 | > (minus_Sn_m ? ?) |
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66 | [ @ res |
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67 | | @ prf2 |
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68 | ] |
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69 | | /2/ |
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70 | ]. |
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71 | qed. |
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72 | |
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73 | definition bitvector_elim_prop ≝ |
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74 | λn: nat. |
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75 | λP: BitVector n → Prop. |
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76 | bitvector_elim_prop_internal n P n ? ?. |
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77 | [ @ (le_n ?) |
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78 | | < (minus_n_n ?) |
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79 | @ [[ ]] |
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80 | ] |
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81 | qed. |
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82 | |
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83 | lemma eq_b_eq: |
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84 | ∀b, c. |
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85 | eq_b b c = true → b = c. |
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86 | #b #c |
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87 | cases b |
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88 | cases c |
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89 | normalize // |
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90 | qed. |
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91 | |
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92 | lemma BitVector_O: ∀v:BitVector 0. v ≃ VEmpty bool. |
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93 | #v generalize in match (refl … 0) cases v in ⊢ (??%? → ?%%??) // |
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94 | #n #hd #tl #abs @⊥ // |
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95 | qed. |
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96 | |
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97 | lemma BitVector_Sn: ∀n.∀v:BitVector (S n). |
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98 | ∃hd.∃tl.v ≃ VCons bool n hd tl. |
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99 | #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → ??(λ_.??(λ_.?%%??))) |
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100 | [ #abs @⊥ // |
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101 | | #m #hd #tl #EQ <(injective_S … EQ) %[@hd] %[@tl] // ] |
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102 | qed. |
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103 | |
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104 | lemma eq_bv_eq: |
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105 | ∀n, v, q. |
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106 | eq_bv n v q = true → v = q. |
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107 | #n #v #q generalize in match v |
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108 | elim q |
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109 | [ #v #h @BitVector_O |
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110 | | #n #hd #tl #ih #v' #h |
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111 | cases (BitVector_Sn ? v') |
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112 | #hd' * #tl' #jmeq >jmeq in h; |
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113 | #new_h |
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114 | change in new_h with ((andb ? ?) = ?); |
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115 | cases(conjunction_true … new_h) |
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116 | #eq_heads #eq_tails |
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117 | whd in eq_heads:(??(??(%))?); |
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118 | cases(eq_b_eq … eq_heads) |
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119 | whd in eq_tails:(??(?????(%))?); |
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120 | change in eq_tails with (eq_bv ??? = ?); |
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121 | <(ih tl') // |
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122 | ] |
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123 | qed. |
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124 | |
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125 | lemma bool_eq_internal_eq: |
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126 | ∀b, c. |
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127 | (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c. |
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128 | #b #c |
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129 | cases b |
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130 | [ normalize // |
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131 | | normalize |
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132 | cases c |
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133 | [ normalize // |
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134 | | normalize // |
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135 | ] |
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136 | ] |
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137 | qed. |
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138 | |
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139 | lemma eq_bv_refl: |
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140 | ∀n,v. eq_bv n v v = true. |
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141 | #n #v |
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142 | elim v |
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143 | [ // |
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144 | | #n #hd #tl #ih |
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145 | normalize |
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146 | cases hd |
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147 | [ normalize |
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148 | @ ih |
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149 | | normalize |
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150 | @ ih |
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151 | ] |
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152 | ] |
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153 | qed. |
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154 | |
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155 | lemma eq_eq_bv: |
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156 | ∀n, v, q. |
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157 | v = q → eq_bv n v q = true. |
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158 | #n #v |
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159 | elim v |
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160 | [ #q #h <h normalize % |
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161 | | #n #hd #tl #ih #q #h >h // |
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162 | ] |
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163 | qed. |
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164 | |
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165 | let rec foldl_strong_internal |
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166 | (A: Type[0]) (P: list A → Type[0]) (l: list A) |
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167 | (H: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd])) |
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168 | (prefix: list A) (suffix: list A) (acc: P prefix) on suffix: |
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169 | l = prefix @ suffix → P(prefix @ suffix) ≝ |
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170 | match suffix return λl'. l = prefix @ l' → P (prefix @ l') with |
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171 | [ nil ⇒ λprf. ? |
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172 | | cons hd tl ⇒ λprf. ? |
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173 | ]. |
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174 | [ > (append_nil ?) |
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175 | @ acc |
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176 | | applyS (foldl_strong_internal A P l H (prefix @ [hd]) tl ? ?) |
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177 | [ @ (H prefix hd tl prf acc) |
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178 | | applyS prf |
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179 | ] |
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180 | ] |
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181 | qed. |
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182 | |
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183 | definition foldl_strong ≝ |
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184 | λA: Type[0]. |
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185 | λP: list A → Type[0]. |
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186 | λl: list A. |
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187 | λH: ∀prefix. ∀hd. ∀tl. l = prefix @ [hd] @ tl → P prefix → P (prefix @ [hd]). |
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188 | λacc: P [ ]. |
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189 | foldl_strong_internal A P l H [ ] l acc (refl …). |
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190 | |
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191 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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192 | λP. |
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193 | P true ∧ P false. |
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194 | |
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195 | let rec bitvector_elim_internal |
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196 | (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝ |
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197 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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198 | [ O ⇒ λprf1. λprefix. P ? |
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199 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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200 | ]. |
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201 | [ applyS prefix |
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202 | | letin res ≝ (bit ::: prefix) |
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203 | < (minus_S_S ? ?) |
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204 | > (minus_Sn_m ? ?) |
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205 | [ @ res |
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206 | | @ prf2 |
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207 | ] |
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208 | | /2/ |
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209 | ]. |
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210 | qed. |
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211 | |
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212 | definition bitvector_elim ≝ |
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213 | λn: nat. |
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214 | λP: BitVector n → bool. |
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215 | bitvector_elim_internal n P n ? ?. |
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216 | [ @ (le_n ?) |
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217 | | < (minus_n_n ?) |
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218 | @ [[ ]] |
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219 | ] |
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220 | qed. |
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221 | |
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222 | axiom vector_associative_append: |
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223 | ∀A: Type[0]. |
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224 | ∀n, m, o: nat. |
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225 | ∀v: Vector A n. |
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226 | ∀q: Vector A m. |
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227 | ∀r: Vector A o. |
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228 | ((v @@ q) @@ r) |
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229 | ≃ |
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230 | (v @@ (q @@ r)). |
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231 | |
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232 | lemma vector_cons_append: |
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233 | ∀A: Type[0]. |
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234 | ∀n: nat. |
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235 | ∀e: A. |
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236 | ∀v: Vector A n. |
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237 | e ::: v = [[ e ]] @@ v. |
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238 | # A # N # E # V |
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239 | elim V |
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240 | [ normalize % |
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241 | | # NN # AA # VV # IH |
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242 | normalize |
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243 | % |
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244 | ] |
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245 | qed. |
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246 | |
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247 | lemma super_rewrite2: |
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248 | ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m. |
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249 | ∀P: ∀m. Vector A m → Prop. |
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250 | n=m → v1 ≃ v2 → P n v1 → P m v2. |
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251 | #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ // |
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252 | qed. |
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253 | |
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254 | lemma mem_middle_vector: |
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255 | ∀A: Type[0]. |
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256 | ∀m, o: nat. |
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257 | ∀eq: A → A → bool. |
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258 | ∀reflex: ∀a. eq a a = true. |
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259 | ∀p: Vector A m. |
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260 | ∀a: A. |
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261 | ∀r: Vector A o. |
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262 | mem A eq ? (p@@(a:::r)) a = true. |
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263 | # A # M # O # EQ # REFLEX # P # A |
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264 | elim P |
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265 | [ normalize |
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266 | > (REFLEX A) |
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267 | normalize |
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268 | # H |
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269 | % |
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270 | | # NN # AA # PP # IH |
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271 | normalize |
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272 | cases (EQ A AA) // |
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273 | @ IH |
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274 | ] |
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275 | qed. |
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276 | |
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277 | lemma mem_monotonic_wrt_append: |
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278 | ∀A: Type[0]. |
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279 | ∀m, o: nat. |
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280 | ∀eq: A → A → bool. |
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281 | ∀reflex: ∀a. eq a a = true. |
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282 | ∀p: Vector A m. |
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283 | ∀a: A. |
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284 | ∀r: Vector A o. |
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285 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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286 | # A # M # O # EQ # REFLEX # P # A |
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287 | elim P |
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288 | [ #R #H @H |
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289 | | #NN #AA # PP # IH #R #H |
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290 | normalize |
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291 | cases (EQ A AA) |
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292 | [ normalize % |
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293 | | @ IH @ H |
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294 | ] |
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295 | ] |
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296 | qed. |
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297 | |
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298 | lemma subvector_multiple_append: |
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299 | ∀A: Type[0]. |
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300 | ∀o, n: nat. |
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301 | ∀eq: A → A → bool. |
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302 | ∀refl: ∀a. eq a a = true. |
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303 | ∀h: Vector A o. |
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304 | ∀v: Vector A n. |
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305 | ∀m: nat. |
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306 | ∀q: Vector A m. |
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307 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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308 | # A # O # N # EQ # REFLEX # H # V |
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309 | elim V |
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310 | [ normalize |
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311 | # M # V % |
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312 | | # NN # AA # VV # IH # MM # QQ |
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313 | change with (bool_to_Prop (andb ??)) |
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314 | cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true) |
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315 | [ |
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316 | | # HH > HH |
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317 | > (vector_cons_append ? ? AA VV) |
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318 | change with (bool_to_Prop (subvector_with ??????)) |
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319 | @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ?? |
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320 | (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS))) |
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321 | ? |
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322 | (vector_associative_append A ? ? ? QQ [[AA]] VV)) |
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323 | [ >associative_plus // |
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324 | | @IH ] |
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325 | ] |
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326 | @(mem_monotonic_wrt_append) |
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327 | [ @ REFLEX |
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328 | | @(mem_monotonic_wrt_append) |
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329 | [ @ REFLEX |
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330 | | normalize |
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331 | > REFLEX |
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332 | normalize |
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333 | % |
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334 | ] |
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335 | ] |
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336 | qed. |
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337 | |
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338 | lemma vector_cons_empty: |
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339 | ∀A: Type[0]. |
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340 | ∀n: nat. |
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341 | ∀v: Vector A n. |
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342 | [[ ]] @@ v = v. |
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343 | # A # N # V |
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344 | elim V |
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345 | [ normalize % |
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346 | | # NN # HH # VV #H % |
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347 | ] |
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348 | qed. |
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349 | |
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350 | corollary subvector_hd_tl: |
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351 | ∀A: Type[0]. |
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352 | ∀o: nat. |
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353 | ∀eq: A → A → bool. |
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354 | ∀refl: ∀a. eq a a = true. |
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355 | ∀h: A. |
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356 | ∀v: Vector A o. |
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357 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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358 | # A # O # EQ # REFLEX # H # V |
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359 | > (vector_cons_append A ? H V) |
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360 | < (vector_cons_empty A ? ([[H]] @@ V)) |
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361 | @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]]) |
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362 | qed. |
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363 | |
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364 | lemma eq_a_reflexive: |
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365 | ∀a. eq_a a a = true. |
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366 | # A |
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367 | cases A |
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368 | % |
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369 | qed. |
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370 | |
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371 | lemma is_in_monotonic_wrt_append: |
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372 | ∀m, n: nat. |
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373 | ∀p: Vector addressing_mode_tag m. |
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374 | ∀q: Vector addressing_mode_tag n. |
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375 | ∀to_search: addressing_mode. |
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376 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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377 | # M # N # P # Q # TO_SEARCH |
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378 | # H |
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379 | elim Q |
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380 | [ normalize |
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381 | @ H |
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382 | | # NN # PP # QQ # IH |
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383 | normalize |
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384 | cases (is_a PP TO_SEARCH) |
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385 | [ normalize |
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386 | % |
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387 | | normalize |
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388 | normalize in IH |
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389 | @ IH |
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390 | ] |
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391 | ] |
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392 | qed. |
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393 | |
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394 | corollary is_in_hd_tl: |
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395 | ∀to_search: addressing_mode. |
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396 | ∀hd: addressing_mode_tag. |
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397 | ∀n: nat. |
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398 | ∀v: Vector addressing_mode_tag n. |
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399 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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400 | # TO_SEARCH # HD # N # V |
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401 | elim V |
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402 | [ # H |
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403 | normalize in H; |
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404 | cases H |
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405 | | # NN # HHD # VV # IH # HH |
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406 | > vector_cons_append |
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407 | > (vector_cons_append ? ? HHD VV) |
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408 | @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH) |
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409 | @ HH |
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410 | ] |
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411 | qed. |
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412 | |
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413 | let rec list_addressing_mode_tags_elim |
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414 | (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝ |
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415 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
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416 | (l → bool) → bool ] with |
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417 | [ VEmpty ⇒ true |
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418 | | VCons len hd tl ⇒ λP. |
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419 | let process_hd ≝ |
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420 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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421 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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422 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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423 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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424 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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425 | | acc_a ⇒ λP.P ACC_A |
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426 | | acc_b ⇒ λP.P ACC_B |
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427 | | dptr ⇒ λP.P DPTR |
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428 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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429 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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430 | | acc_dptr ⇒ λP.P ACC_DPTR |
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431 | | acc_pc ⇒ λP.P ACC_PC |
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432 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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433 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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434 | | carry ⇒ λP.P CARRY |
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435 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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436 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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437 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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438 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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439 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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440 | ] |
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441 | in |
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442 | andb (process_hd P) |
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443 | (match len return λx. x = len → bool with |
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444 | [ O ⇒ λprf. true |
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445 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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446 | ]. |
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447 | try % |
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448 | [ 2: cases (sym_eq ??? prf); @tl |
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449 | | generalize in match H; generalize in match tl; cases prf; |
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450 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
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451 | #tl |
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452 | normalize in ⊢ (∀_: %. ?) |
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453 | # H |
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454 | whd |
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455 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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456 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
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457 | qed. |
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458 | |
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459 | definition product_elim ≝ |
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460 | λm, n: nat. |
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461 | λv: Vector addressing_mode_tag (S m). |
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462 | λq: Vector addressing_mode_tag (S n). |
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463 | λP: (v × q) → bool. |
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464 | list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)). |
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465 | |
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466 | definition union_elim ≝ |
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467 | λA, B: Type[0]. |
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468 | λelimA: (A → bool) → bool. |
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469 | λelimB: (B → bool) → bool. |
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470 | λelimU: A ⊎ B → bool. |
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471 | elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))). |
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472 | |
---|
473 | (* |
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474 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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475 | λP. |
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476 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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477 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
---|
478 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
---|
479 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
---|
480 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
---|
481 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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482 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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483 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
---|
484 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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485 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
---|
486 | P (DA ? ACC_A) ∧ |
---|
487 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
---|
488 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
---|
489 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
---|
490 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
---|
491 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
---|
492 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
---|
493 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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494 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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495 | P (RL ? ACC_A) ∧ |
---|
496 | P (RLC ? ACC_A) ∧ |
---|
497 | P (RR ? ACC_A) ∧ |
---|
498 | P (RRC ? ACC_A) ∧ |
---|
499 | P (SWAP ? ACC_A) ∧ |
---|
500 | P (RET ?) ∧ |
---|
501 | P (RETI ?) ∧ |
---|
502 | P (NOP ?) ∧ |
---|
503 | bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧ |
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504 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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505 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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506 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧ |
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507 | union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]]) |
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508 | (product_elim ? ? [[ registr; indirect ]] [[ data ]]) |
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509 | (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧ |
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510 | list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧ |
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511 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]]) |
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512 | (product_elim ? ? [[direct]] [[ acc_a ; data ]]) |
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513 | (λd. P (XRL ? d)) ∧ |
---|
514 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
---|
515 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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516 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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517 | (λd. P (ANL ? d)) ∧ |
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518 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]]) |
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519 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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520 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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521 | (λd. P (ORL ? d)) ∧ |
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522 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]]) |
---|
523 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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524 | (λd. P (MOVX ? d)) ∧ |
---|
525 | union_elim ? ? ( |
---|
526 | union_elim ? ? ( |
---|
527 | union_elim ? ? ( |
---|
528 | union_elim ? ? ( |
---|
529 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
---|
530 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
---|
531 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
---|
532 | (product_elim ? ? [[dptr]] [[data16]])) |
---|
533 | (product_elim ? ? [[carry]] [[bit_addr]])) |
---|
534 | (product_elim ? ? [[bit_addr]] [[carry]]) |
---|
535 | (λd. P (MOV ? d)). |
---|
536 | % |
---|
537 | qed. |
---|
538 | |
---|
539 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
---|
540 | λP. (* |
---|
541 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
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542 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
---|
543 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
---|
544 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
---|
545 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
---|
546 | P (JMP INDIRECT_DPTR) ∧ |
---|
547 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
---|
548 | preinstruction_elim (λp. P (RealInstruction p)). *) |
---|
549 | % |
---|
550 | qed. |
---|
551 | |
---|
552 | |
---|
553 | axiom instruction_elim_complete: |
---|
554 | ∀P. instruction_elim P = true → ∀i. P i = true. |
---|
555 | *) |
---|
556 | (*definition eq_instruction ≝ |
---|
557 | λi, j: instruction. |
---|
558 | true.*) |
---|
559 | axiom eq_instruction: instruction → instruction → bool. |
---|
560 | axiom eq_instruction_refl: ∀i. eq_instruction i i = true. |
---|
561 | |
---|
562 | let rec vect_member |
---|
563 | (A: Type[0]) (n: nat) (eq: A → A → bool) |
---|
564 | (v: Vector A n) (a: A) on v: bool ≝ |
---|
565 | match v with |
---|
566 | [ VEmpty ⇒ false |
---|
567 | | VCons len hd tl ⇒ |
---|
568 | eq hd a ∨ (vect_member A ? eq tl a) |
---|
569 | ]. |
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570 | |
---|
571 | let rec list_addressing_mode_tags_elim_prop |
---|
572 | (n: nat) |
---|
573 | (l: Vector addressing_mode_tag (S n)) |
---|
574 | on l: |
---|
575 | ∀P: l → Prop. |
---|
576 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
---|
577 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
---|
578 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
---|
579 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
---|
580 | ∀x: l. P x ≝ |
---|
581 | match l return |
---|
582 | λy. |
---|
583 | match y with |
---|
584 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
---|
585 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
---|
586 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
---|
587 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
---|
588 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
---|
589 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
---|
590 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
---|
591 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
---|
592 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
---|
593 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
---|
594 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
---|
595 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
---|
596 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
---|
597 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
---|
598 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
---|
599 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
---|
600 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
---|
601 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
---|
602 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
---|
603 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
---|
604 | ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True. |
---|
605 | ∀x:l. P x |
---|
606 | ] with |
---|
607 | [ VEmpty ⇒ λAbsurd. ⊥ |
---|
608 | | VCons len hd tl ⇒ λProof. ? |
---|
609 | ] (refl ? (S n)). cases daemon. qed. (* |
---|
610 | [ destruct(Absurd) |
---|
611 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
---|
612 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
---|
613 | # A15 # A16 # A17 # A18 # A19 # X |
---|
614 | cases X |
---|
615 | # SUB cases daemon ] qed. |
---|
616 | cases SUB |
---|
617 | [ # BYTE |
---|
618 | normalize |
---|
619 | ]. |
---|
620 | |
---|
621 | |
---|
622 | (* let prepare_hd ≝ |
---|
623 | match hd with |
---|
624 | [ direct ⇒ λdirect_prf. ? |
---|
625 | | indirect ⇒ λindirect_prf. ? |
---|
626 | | ext_indirect ⇒ λext_indirect_prf. ? |
---|
627 | | registr ⇒ λregistr_prf. ? |
---|
628 | | acc_a ⇒ λacc_a_prf. ? |
---|
629 | | acc_b ⇒ λacc_b_prf. ? |
---|
630 | | dptr ⇒ λdptr_prf. ? |
---|
631 | | data ⇒ λdata_prf. ? |
---|
632 | | data16 ⇒ λdata16_prf. ? |
---|
633 | | acc_dptr ⇒ λacc_dptr_prf. ? |
---|
634 | | acc_pc ⇒ λacc_pc_prf. ? |
---|
635 | | ext_indirect_dptr ⇒ λext_indirect_prf. ? |
---|
636 | | indirect_dptr ⇒ λindirect_prf. ? |
---|
637 | | carry ⇒ λcarry_prf. ? |
---|
638 | | bit_addr ⇒ λbit_addr_prf. ? |
---|
639 | | n_bit_addr ⇒ λn_bit_addr_prf. ? |
---|
640 | | relative ⇒ λrelative_prf. ? |
---|
641 | | addr11 ⇒ λaddr11_prf. ? |
---|
642 | | addr16 ⇒ λaddr16_prf. ? |
---|
643 | ] |
---|
644 | in ? *) |
---|
645 | ]. |
---|
646 | [ 1: destruct(absd) |
---|
647 | | 2: # A1 # A2 # A3 # A4 # A5 # A6 |
---|
648 | # A7 # A8 # A9 # A10 # A11 # A12 |
---|
649 | # A13 # A14 # A15 # A16 # A17 # A18 |
---|
650 | # A19 * |
---|
651 | ]. |
---|
652 | |
---|
653 | |
---|
654 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
---|
655 | (l → bool) → bool ] with |
---|
656 | [ VEmpty ⇒ true |
---|
657 | | VCons len hd tl ⇒ λP. |
---|
658 | let process_hd ≝ |
---|
659 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
---|
660 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
---|
661 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
---|
662 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
---|
663 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
---|
664 | | acc_a ⇒ λP.P ACC_A |
---|
665 | | acc_b ⇒ λP.P ACC_B |
---|
666 | | dptr ⇒ λP.P DPTR |
---|
667 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
---|
668 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
---|
669 | | acc_dptr ⇒ λP.P ACC_DPTR |
---|
670 | | acc_pc ⇒ λP.P ACC_PC |
---|
671 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
---|
672 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
---|
673 | | carry ⇒ λP.P CARRY |
---|
674 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
---|
675 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
---|
676 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
---|
677 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
---|
678 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
---|
679 | ] |
---|
680 | in |
---|
681 | andb (process_hd P) |
---|
682 | (match len return λx. x = len → bool with |
---|
683 | [ O ⇒ λprf. true |
---|
684 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
---|
685 | ]. |
---|
686 | try % |
---|
687 | [ 2: cases (sym_eq ??? prf); @tl |
---|
688 | | generalize in match H; generalize in match tl; cases prf; |
---|
689 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
---|
690 | #tl |
---|
691 | normalize in ⊢ (∀_: %. ?) |
---|
692 | # H |
---|
693 | whd |
---|
694 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
695 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
---|
696 | qed. |
---|
697 | *) |
---|
698 | (* |
---|
699 | lemma test: |
---|
700 | let i ≝ SJMP (RELATIVE (bitvector_of_nat 8 255)) in |
---|
701 | (let assembled ≝ assembly1 i in |
---|
702 | let code_memory ≝ load_code_memory assembled in |
---|
703 | let fetched ≝ fetch code_memory ? in |
---|
704 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
705 | eq_instruction (\fst instr_pc)) i = true. |
---|
706 | [2: @ zero |
---|
707 | | normalize |
---|
708 | ]*) |
---|
709 | |
---|
710 | lemma BitVectorTrie_O: |
---|
711 | ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0. |
---|
712 | #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??)) |
---|
713 | [ #w #_ %1 %[@w] % |
---|
714 | | #n #l #r #abs @⊥ // |
---|
715 | | #n #EQ %2 >EQ %] |
---|
716 | qed. |
---|
717 | |
---|
718 | lemma BitVectorTrie_Sn: |
---|
719 | ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n). |
---|
720 | #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%) |
---|
721 | [ #m #abs @⊥ // |
---|
722 | | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] // |
---|
723 | | #m #EQ %2 // ] |
---|
724 | qed. |
---|
725 | |
---|
726 | lemma lookup_prepare_trie_for_insertion_hit: |
---|
727 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n. |
---|
728 | lookup … b (prepare_trie_for_insertion … b v) a = v. |
---|
729 | #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize // |
---|
730 | qed. |
---|
731 | |
---|
732 | lemma lookup_insert_hit: |
---|
733 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n. |
---|
734 | lookup … b (insert … b v t) a = v. |
---|
735 | #A #a #v #n #b elim b -b -n // |
---|
736 | #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t) |
---|
737 | [ * #l * #r #JMEQ >JMEQ cases hd normalize // |
---|
738 | | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ] |
---|
739 | qed. |
---|
740 | |
---|
741 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
---|
742 | |
---|
743 | lemma lookup_prepare_trie_for_insertion_miss: |
---|
744 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n. |
---|
745 | (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a. |
---|
746 | #A #a #v #n #c elim c |
---|
747 | [ #b >(BitVector_O … b) normalize #abs @⊥ // |
---|
748 | | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
---|
749 | cases hd cases hd' normalize |
---|
750 | [2,3: #_ cases tl' // |
---|
751 | |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]] |
---|
752 | qed. |
---|
753 | |
---|
754 | lemma lookup_insert_miss: |
---|
755 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n. |
---|
756 | (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a. |
---|
757 | #A #a #v #n #c elim c -c -n |
---|
758 | [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF // |
---|
759 | | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
---|
760 | #t cases(BitVectorTrie_Sn … t) |
---|
761 | [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
---|
762 | [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH // |
---|
763 | | #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
---|
764 | [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize |
---|
765 | [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]] |
---|
766 | qed. |
---|
767 | |
---|
768 | definition load_code_memory_aux ≝ |
---|
769 | fold_left_i_aux … ( |
---|
770 | λi, mem, v. |
---|
771 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
---|
772 | |
---|
773 | axiom split_elim: |
---|
774 | ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop. |
---|
775 | (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v). |
---|
776 | |
---|
777 | axiom half_add_SO: |
---|
778 | ∀pc. |
---|
779 | \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc). |
---|
780 | |
---|
781 | (* |
---|
782 | axiom not_eqvb_S: |
---|
783 | ∀pc. |
---|
784 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))). |
---|
785 | |
---|
786 | axiom not_eqvb_SS: |
---|
787 | ∀pc. |
---|
788 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))). |
---|
789 | |
---|
790 | axiom bitvector_elim_complete: |
---|
791 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv. |
---|
792 | |
---|
793 | lemma bitvector_elim_complete': |
---|
794 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true. |
---|
795 | #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv |
---|
796 | generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ // |
---|
797 | qed. |
---|
798 | *) |
---|
799 | |
---|
800 | |
---|
801 | |
---|
802 | |
---|
803 | (* |
---|
804 | lemma andb_elim': |
---|
805 | ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true. |
---|
806 | #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize // |
---|
807 | qed. |
---|
808 | *) |
---|
809 | |
---|
810 | let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
811 | (encoding: list Byte) on encoding: Prop ≝ |
---|
812 | match encoding with |
---|
813 | [ nil ⇒ final_pc = pc |
---|
814 | | cons hd tl ⇒ |
---|
815 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
816 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
817 | ]. |
---|
818 | |
---|
819 | lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2. |
---|
820 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) → |
---|
821 | let intermediate_pc ≝ pc + length … l1 in |
---|
822 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧ |
---|
823 | encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2. |
---|
824 | #code_memory #final_pc #l1 elim l1 |
---|
825 | [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/ |
---|
826 | | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm |
---|
827 | #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ] |
---|
828 | qed. |
---|
829 | |
---|
830 | axiom bitvector_3_elim_prop: |
---|
831 | ∀P: BitVector 3 → Prop. |
---|
832 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
---|
833 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
---|
834 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
---|
835 | |
---|
836 | definition ticks_of_instruction ≝ |
---|
837 | λi. |
---|
838 | let trivial_code_memory ≝ assembly1 i in |
---|
839 | let trivial_status ≝ load_code_memory trivial_code_memory in |
---|
840 | \snd (fetch trivial_status (zero ?)). |
---|
841 | |
---|
842 | axiom fetch_assembly: |
---|
843 | ∀pc,i,code_memory,assembled. |
---|
844 | assembled = assembly1 i → |
---|
845 | let len ≝ length … assembled in |
---|
846 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
847 | let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in |
---|
848 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
849 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
850 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true. |
---|
851 | (* #pc #i #code_memory #assembled cases i [8: *] |
---|
852 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
853 | [47,48,49: |
---|
854 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
---|
855 | [2,3,5,7,10,12,16,17,18,21,25,26,27,30,31,32,37,38,39,40,41,42,43,44,45,48,51,58, |
---|
856 | 59,60,63,64,65,66,67: #ARG]] |
---|
857 | [4,5,6,7,8,9,10,11,12,13,22,23,24,27,28,39,40,41,42,43,44,45,46,47,48,49,50,51,52, |
---|
858 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
---|
859 | [1,2,4,7,9,10,12,13,15,16,17,18,20,22,23,24,25,26,27,28,29,30,31,32,33,36,37,38, |
---|
860 | 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65, |
---|
861 | 68,69,70,71: #ARG2]] |
---|
862 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
---|
863 | normalize in ⊢ (???% → ?) |
---|
864 | [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
865 | normalize in ⊢ (???% → ?)] |
---|
866 | #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2) |
---|
867 | try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4 |
---|
868 | change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??) |
---|
869 | whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?) |
---|
870 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
---|
871 | [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29, |
---|
872 | 30,31,32,33,34,35,36,37,38,39,40,43,45,48,49,52,53,54,55,56,57,60,61,62,65,66, |
---|
873 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
---|
874 | whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
875 | qed. *) |
---|
876 | |
---|
877 | let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝ |
---|
878 | match expected with |
---|
879 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
880 | | cons i tl ⇒ |
---|
881 | let fetched ≝ fetch code_memory pc in |
---|
882 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
883 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
884 | eq_instruction instr i = true ∧ |
---|
885 | ticks = (ticks_of_instruction i) ∧ |
---|
886 | fetch_many code_memory final_pc pc' tl]. |
---|
887 | |
---|
888 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
---|
889 | #A #a #b #EQ destruct // |
---|
890 | qed. |
---|
891 | |
---|
892 | lemma pair_destruct: ∀A,B,a1,a2,b1,b2. pair A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2. |
---|
893 | #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/ |
---|
894 | qed. |
---|
895 | |
---|
896 | axiom eq_bv_to_eq: ∀n.∀v1,v2: BitVector n. eq_bv … v1 v2 = true → v1=v2. |
---|
897 | |
---|
898 | axiom eq_instruction_to_eq: ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2. |
---|
899 | |
---|
900 | lemma fetch_assembly_pseudo: |
---|
901 | ∀program,ppc,lookup_labels,lookup_datalabels. |
---|
902 | ∀pi,code_memory,len,assembled,instructions,pc. |
---|
903 | let expansion ≝ jump_expansion ppc program in |
---|
904 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi → |
---|
905 | Some … 〈len,assembled〉 = assembly_1_pseudoinstruction program ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi → |
---|
906 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
907 | fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions. |
---|
908 | #program #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc |
---|
909 | #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2 |
---|
910 | cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b |
---|
911 | >EQ2a >EQ2b -EQ2a EQ2b; |
---|
912 | generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc |
---|
913 | generalize in match pc elim instructions |
---|
914 | [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl |
---|
915 | | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd |
---|
916 | generalize in match (fetch_assembly pc i code_memory … (refl …) H1) |
---|
917 | cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %) |
---|
918 | cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1 |
---|
919 | cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try % // |
---|
920 | [ @eqb_true_to_eq <(eq_instruction_to_eq … K1) // | >(eq_bv_to_eq … K3) @IH @H2 ] |
---|
921 | qed. |
---|
922 | |
---|
923 | (* This establishes the correspondence between pseudo program counters and |
---|
924 | program counters. It is at the heart of the proof. *) |
---|
925 | (*CSC: code taken from build_maps *) |
---|
926 | definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
927 | λinstr_list. |
---|
928 | foldl ?? |
---|
929 | (λt. λi. |
---|
930 | match t with |
---|
931 | [ None ⇒ None ? |
---|
932 | | Some ppc_pc_map ⇒ |
---|
933 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
934 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
935 | let 〈label, i〉 ≝ i in |
---|
936 | match construct_costs instr_list ppc program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with |
---|
937 | [ None ⇒ None ? |
---|
938 | | Some pc_ignore ⇒ |
---|
939 | let 〈pc,ignore〉 ≝ pc_ignore in |
---|
940 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
---|
941 | ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list). |
---|
942 | |
---|
943 | definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝ |
---|
944 | λinstr_list. |
---|
945 | match sigma0 instr_list with |
---|
946 | [ None ⇒ None … |
---|
947 | | Some result ⇒ |
---|
948 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
949 | Some … pc_sigma_map ]. |
---|
950 | |
---|
951 | definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝ |
---|
952 | λinstr_list. |
---|
953 | match sigma0 instr_list with |
---|
954 | [ None ⇒ None ? |
---|
955 | | Some result ⇒ |
---|
956 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
957 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
958 | if gtb pc (2^16) then |
---|
959 | None ? |
---|
960 | else |
---|
961 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
---|
962 | |
---|
963 | axiom policy_ok: ∀p. sigma_safe p ≠ None …. |
---|
964 | |
---|
965 | definition sigma: pseudo_assembly_program → Word → Word ≝ |
---|
966 | λp. |
---|
967 | match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
968 | [ None ⇒ λabs. ⊥ |
---|
969 | | Some r ⇒ λ_.r] (policy_ok p). |
---|
970 | cases abs // |
---|
971 | qed. |
---|
972 | |
---|
973 | lemma length_append: |
---|
974 | ∀A.∀l1,l2:list A. |
---|
975 | |l1 @ l2| = |l1| + |l2|. |
---|
976 | #A #l1 elim l1 |
---|
977 | [ // |
---|
978 | | #hd #tl #IH #l2 normalize <IH //] |
---|
979 | qed. |
---|
980 | |
---|
981 | let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝ |
---|
982 | match l with |
---|
983 | [ nil ⇒ true |
---|
984 | | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl]. |
---|
985 | |
---|
986 | lemma does_not_occur_None: |
---|
987 | ∀id,i,list_instr. |
---|
988 | does_not_occur id (list_instr@[〈None …,i〉]) = |
---|
989 | does_not_occur id list_instr. |
---|
990 | #id #i #list_instr elim list_instr |
---|
991 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %] |
---|
992 | qed. |
---|
993 | |
---|
994 | let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝ |
---|
995 | match l with |
---|
996 | [ nil ⇒ false |
---|
997 | | cons hd tl ⇒ |
---|
998 | if instruction_matches_identifier id hd then |
---|
999 | does_not_occur id tl |
---|
1000 | else |
---|
1001 | occurs_exactly_once id tl ]. |
---|
1002 | |
---|
1003 | lemma occurs_exactly_once_None: |
---|
1004 | ∀id,i,list_instr. |
---|
1005 | occurs_exactly_once id (list_instr@[〈None …,i〉]) = |
---|
1006 | occurs_exactly_once id list_instr. |
---|
1007 | #id #i #list_instr elim list_instr |
---|
1008 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %] |
---|
1009 | qed. |
---|
1010 | |
---|
1011 | lemma index_of_internal_None: ∀i,id,instr_list,n. |
---|
1012 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
---|
1013 | index_of_internal ? (instruction_matches_identifier id) instr_list n = |
---|
1014 | index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n. |
---|
1015 | #i #id #instr_list elim instr_list |
---|
1016 | [ #n #abs whd in abs; cases abs |
---|
1017 | | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?) |
---|
1018 | cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%) |
---|
1019 | [ #H % |
---|
1020 | | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ % |
---|
1021 | [ #_ % | #abs cases abs ]]] |
---|
1022 | qed. |
---|
1023 | |
---|
1024 | lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list. |
---|
1025 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
---|
1026 | address_of_word_labels_code_mem instr_list id = |
---|
1027 | address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id. |
---|
1028 | #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?)) |
---|
1029 | >(index_of_internal_None … H) % |
---|
1030 | qed. |
---|
1031 | |
---|
1032 | axiom tech_pc_sigma0_append: |
---|
1033 | ∀preamble,instr_list,prefix,label,i,pc',code,ppc,pc,costs,costs'. |
---|
1034 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
---|
1035 | construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 → |
---|
1036 | tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉. |
---|
1037 | |
---|
1038 | axiom tech_pc_sigma0_append_None: |
---|
1039 | ∀preamble,instr_list,prefix,i,ppc,pc,costs. |
---|
1040 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
---|
1041 | construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = None … |
---|
1042 | → False. |
---|
1043 | |
---|
1044 | (* |
---|
1045 | definition build_maps' ≝ |
---|
1046 | λpseudo_program. |
---|
1047 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
1048 | let result ≝ |
---|
1049 | foldl_strong |
---|
1050 | (option Identifier × pseudo_instruction) |
---|
1051 | (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
1052 | let pre' ≝ 〈preamble,pre〉 in |
---|
1053 | let 〈labels,pc_costs〉 ≝ res in |
---|
1054 | tech_pc_sigma0 pre' = Some … pc_costs ∧ |
---|
1055 | ∀id. occurs_exactly_once id pre → |
---|
1056 | lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id)) |
---|
1057 | instr_list |
---|
1058 | (λprefix,i,tl,prf,t. |
---|
1059 | let 〈labels, pc_costs〉 ≝ t in |
---|
1060 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
1061 | let 〈label, i'〉 ≝ i in |
---|
1062 | let labels ≝ |
---|
1063 | match label with |
---|
1064 | [ None ⇒ labels |
---|
1065 | | Some label ⇒ |
---|
1066 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
1067 | insert ? ? label program_counter_bv labels |
---|
1068 | ] |
---|
1069 | in |
---|
1070 | match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
1071 | [ None ⇒ |
---|
1072 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
1073 | dummy |
---|
1074 | | Some construct ⇒ 〈labels, construct〉 |
---|
1075 | ] |
---|
1076 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 |
---|
1077 | in |
---|
1078 | let 〈labels, pc_costs〉 ≝ result in |
---|
1079 | let 〈pc, costs〉 ≝ pc_costs in |
---|
1080 | 〈labels, costs〉. |
---|
1081 | [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ // |
---|
1082 | | whd cases construct in p3 #PC #CODE #JMEQ % |
---|
1083 | [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ] |
---|
1084 | | (* dummy case *) @⊥ |
---|
1085 | @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ] |
---|
1086 | [*: generalize in match (sig2 … t) whd in ⊢ (% → ?) |
---|
1087 | >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ] |
---|
1088 | whd in ⊢ (??(????%?)?) -labels1; |
---|
1089 | cases label in Hid |
---|
1090 | [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1 |
---|
1091 | [ >(address_of_word_labels_code_mem_None … Hid) |
---|
1092 | (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *) |
---|
1093 | | whd in Hid >occurs_exactly_once_None in Hid // ] |
---|
1094 | | -label #label #Hid whd in ⊢ (??(????%?)?) |
---|
1095 | |
---|
1096 | ] |
---|
1097 | qed. |
---|
1098 | |
---|
1099 | lemma build_maps_ok: |
---|
1100 | ∀p:pseudo_assembly_program. |
---|
1101 | let 〈labels,costs〉 ≝ build_maps' p in |
---|
1102 | ∀pc. |
---|
1103 | (nat_of_bitvector … pc) < length … (\snd p) → |
---|
1104 | lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)). |
---|
1105 | #p cases p #preamble #instr_list |
---|
1106 | elim instr_list |
---|
1107 | [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ] |
---|
1108 | | #hd #tl #IH |
---|
1109 | whd in ⊢ (match % with [ _ ⇒ ?]) |
---|
1110 | ] |
---|
1111 | qed. |
---|
1112 | *) |
---|
1113 | |
---|
1114 | (* |
---|
1115 | lemma rev_preserves_length: |
---|
1116 | ∀A.∀l. length … (rev A l) = length … l. |
---|
1117 | #A #l elim l |
---|
1118 | [ % |
---|
1119 | | #hd #tl #IH normalize >length_append normalize /2/ ] |
---|
1120 | qed. |
---|
1121 | |
---|
1122 | lemma rev_append: |
---|
1123 | ∀A.∀l1,l2. |
---|
1124 | rev A (l1@l2) = rev A l2 @ rev A l1. |
---|
1125 | #A #l1 elim l1 normalize // |
---|
1126 | qed. |
---|
1127 | |
---|
1128 | lemma rev_rev: ∀A.∀l. rev … (rev A l) = l. |
---|
1129 | #A #l elim l |
---|
1130 | [ // |
---|
1131 | | #hd #tl #IH normalize >rev_append normalize // ] |
---|
1132 | qed. |
---|
1133 | |
---|
1134 | lemma split_len_Sn: |
---|
1135 | ∀A:Type[0].∀l:list A.∀len. |
---|
1136 | length … l = S len → |
---|
1137 | Σl'.Σa. l = l'@[a] ∧ length … l' = len. |
---|
1138 | #A #l elim l |
---|
1139 | [ normalize #len #abs destruct |
---|
1140 | | #hd #tl #IH #len |
---|
1141 | generalize in match (rev_rev … tl) |
---|
1142 | cases (rev A tl) in ⊢ (??%? → ?) |
---|
1143 | [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ |
---|
1144 | | #a #l' #H <H normalize #EQ |
---|
1145 | %[@(hd::rev … l')] %[@a] % // |
---|
1146 | >length_append in EQ #EQ normalize in EQ; normalize; |
---|
1147 | generalize in match (injective_S … EQ) #EQ2 /2/ ]] |
---|
1148 | qed. |
---|
1149 | |
---|
1150 | lemma list_elim_rev: |
---|
1151 | ∀A:Type[0].∀P:list A → Type[0]. |
---|
1152 | P [ ] → (∀l,a. P l → P (l@[a])) → |
---|
1153 | ∀l. P l. |
---|
1154 | #A #P #H1 #H2 #l |
---|
1155 | generalize in match (refl … (length … l)) |
---|
1156 | generalize in ⊢ (???% → ?) #n generalize in match l |
---|
1157 | elim n |
---|
1158 | [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ] |
---|
1159 | | #m #IH #L #EQ |
---|
1160 | cases (split_len_Sn … EQ) #l' * #a * /3/ ] |
---|
1161 | qed. |
---|
1162 | |
---|
1163 | axiom is_prefix: ∀A:Type[0]. list A → list A → Prop. |
---|
1164 | axiom prefix_of_append: |
---|
1165 | ∀A:Type[0].∀l,l1,l2:list A. |
---|
1166 | is_prefix … l l1 → is_prefix … l (l1@l2). |
---|
1167 | axiom prefix_reflexive: ∀A,l. is_prefix A l l. |
---|
1168 | axiom nil_prefix: ∀A,l. is_prefix A [ ] l. |
---|
1169 | |
---|
1170 | record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }. |
---|
1171 | |
---|
1172 | definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝ |
---|
1173 | λA,P,H,x. match x with [ mk_Propify p ⇒ H p ]. |
---|
1174 | |
---|
1175 | definition app ≝ |
---|
1176 | λA:Type[0].λl1:Propify (list A).λl2:list A. |
---|
1177 | match l1 with |
---|
1178 | [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ]. |
---|
1179 | |
---|
1180 | lemma app_nil: ∀A,l1. app A l1 [ ] = l1. |
---|
1181 | #A * /3/ |
---|
1182 | qed. |
---|
1183 | |
---|
1184 | lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3). |
---|
1185 | #A * #l1 normalize // |
---|
1186 | qed. |
---|
1187 | |
---|
1188 | let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0]) |
---|
1189 | (f: ∀prefix. B prefix → ∀x.B (app … prefix [x])) |
---|
1190 | (prefix: Propify (list A)) (b: B prefix) (l: list A) on l : |
---|
1191 | B (app … prefix l) ≝ |
---|
1192 | match l with |
---|
1193 | [ nil ⇒ ? (* b *) |
---|
1194 | | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
1195 | ]. |
---|
1196 | [ applyS b |
---|
1197 | | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ] |
---|
1198 | qed. |
---|
1199 | |
---|
1200 | (* |
---|
1201 | let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x])) |
---|
1202 | (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝ |
---|
1203 | match l with |
---|
1204 | [ nil ⇒ ? (* b *) |
---|
1205 | | cons hd tl ⇒ |
---|
1206 | ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
1207 | ]. |
---|
1208 | [ applyS b |
---|
1209 | | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ] |
---|
1210 | qed. |
---|
1211 | *) |
---|
1212 | |
---|
1213 | definition foldll: |
---|
1214 | ∀A:Type[0].∀B: Propify (list A) → Type[0]. |
---|
1215 | (∀prefix. B prefix → ∀x. B (app … prefix [x])) → |
---|
1216 | B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l) |
---|
1217 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
1218 | |
---|
1219 | axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop. |
---|
1220 | axiom pprefix_of_append: |
---|
1221 | ∀A:Type[0].∀l,l1,l2. |
---|
1222 | is_pprefix A l l1 → is_pprefix A l (l1@l2). |
---|
1223 | axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l. |
---|
1224 | axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l. |
---|
1225 | |
---|
1226 | |
---|
1227 | axiom foldll': |
---|
1228 | ∀A:Type[0].∀l: list A. |
---|
1229 | ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0]. |
---|
1230 | (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') → |
---|
1231 | B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l). |
---|
1232 | #A #l #B |
---|
1233 | generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH |
---|
1234 | |
---|
1235 | |
---|
1236 | #H #acc |
---|
1237 | @foldll |
---|
1238 | [ |
---|
1239 | | |
---|
1240 | ] |
---|
1241 | |
---|
1242 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
1243 | |
---|
1244 | |
---|
1245 | (* |
---|
1246 | record subset (A:Type[0]) (P: A → Prop): Type[0] ≝ |
---|
1247 | { subset_wit:> A; |
---|
1248 | subset_proof: P subset_wit |
---|
1249 | }. |
---|
1250 | *) |
---|
1251 | |
---|
1252 | definition build_maps' ≝ |
---|
1253 | λpseudo_program. |
---|
1254 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
1255 | let result ≝ |
---|
1256 | foldll |
---|
1257 | (option Identifier × pseudo_instruction) |
---|
1258 | (λprefix. |
---|
1259 | Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
1260 | match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?]) |
---|
1261 | (λprefix,t,i. |
---|
1262 | let 〈labels, pc_costs〉 ≝ t in |
---|
1263 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
1264 | let 〈label, i'〉 ≝ i in |
---|
1265 | let labels ≝ |
---|
1266 | match label with |
---|
1267 | [ None ⇒ labels |
---|
1268 | | Some label ⇒ |
---|
1269 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
1270 | insert ? ? label program_counter_bv labels |
---|
1271 | ] |
---|
1272 | in |
---|
1273 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
1274 | [ None ⇒ |
---|
1275 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
1276 | dummy |
---|
1277 | | Some construct ⇒ 〈labels, construct〉 |
---|
1278 | ] |
---|
1279 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list |
---|
1280 | in |
---|
1281 | let 〈labels, pc_costs〉 ≝ result in |
---|
1282 | let 〈pc, costs〉 ≝ pc_costs in |
---|
1283 | 〈labels, costs〉. |
---|
1284 | [ |
---|
1285 | | @⊥ |
---|
1286 | | normalize % // |
---|
1287 | ] |
---|
1288 | qed. |
---|
1289 | |
---|
1290 | definition build_maps' ≝ |
---|
1291 | λpseudo_program. |
---|
1292 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
1293 | let result ≝ |
---|
1294 | foldl |
---|
1295 | (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
1296 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
1297 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t))) |
---|
1298 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
1299 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
1300 | is_prefix ? instr_list_prefix' instr_list → |
---|
1301 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?) |
---|
1302 | (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
1303 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
1304 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)). |
---|
1305 | λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
1306 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
1307 | is_prefix ? instr_list_prefix' instr_list → |
---|
1308 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? . |
---|
1309 | let 〈labels, pc_costs〉 ≝ t in |
---|
1310 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
1311 | let 〈label, i'〉 ≝ i in |
---|
1312 | let labels ≝ |
---|
1313 | match label with |
---|
1314 | [ None ⇒ labels |
---|
1315 | | Some label ⇒ |
---|
1316 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
1317 | insert ? ? label program_counter_bv labels |
---|
1318 | ] |
---|
1319 | in |
---|
1320 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
1321 | [ None ⇒ |
---|
1322 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
1323 | dummy |
---|
1324 | | Some construct ⇒ 〈labels, construct〉 |
---|
1325 | ] |
---|
1326 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*) |
---|
1327 | in |
---|
1328 | let 〈labels, pc_costs〉 ≝ result in |
---|
1329 | let 〈pc, costs〉 ≝ pc_costs in |
---|
1330 | 〈labels, costs〉. |
---|
1331 | [4: @(list_elim_rev ? |
---|
1332 | (λinstr_list. list ( |
---|
1333 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
1334 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
1335 | is_prefix ? instr_list_prefix' instr_list → |
---|
1336 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?))) |
---|
1337 | ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *) |
---|
1338 | [ @[ ] |
---|
1339 | | #l' #a #limage %2 |
---|
1340 | [ %[@a] #PREFIX #PREFIX_OK |
---|
1341 | | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN |
---|
1342 | THE INDUCTION HYPOTHESIS INSTEAD *) |
---|
1343 | elim limage |
---|
1344 | [ %1 |
---|
1345 | | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3 |
---|
1346 | @K1 @(prefix_of_append ???? K3) |
---|
1347 | ] |
---|
1348 | ] |
---|
1349 | |
---|
1350 | |
---|
1351 | |
---|
1352 | |
---|
1353 | cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt' |
---|
1354 | % [@ (LIST_PREFIX @ [i])] % |
---|
1355 | [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1 |
---|
1356 | | (* DOABLE IN PRINCIPLE *) |
---|
1357 | ] |
---|
1358 | | (* assert false case *) |
---|
1359 | |3: % [@ ([ ])] % [2: % | (* DOABLE *)] |
---|
1360 | | |
---|
1361 | *) |
---|
1362 | |
---|
1363 | axiom assembly_ok: |
---|
1364 | ∀program,assembled,costs,labels. |
---|
1365 | Some … 〈labels,costs〉 = build_maps program → |
---|
1366 | Some … 〈assembled,costs〉 = assembly program → |
---|
1367 | let code_memory ≝ load_code_memory assembled in |
---|
1368 | let preamble ≝ \fst program in |
---|
1369 | let datalabels ≝ construct_datalabels preamble in |
---|
1370 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
1371 | let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in |
---|
1372 | ∀ppc,len,assembledi. |
---|
1373 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1374 | Some … 〈len,assembledi〉 = assembly_1_pseudoinstruction program ppc (sigma program ppc) lookup_labels lookup_datalabels pi → |
---|
1375 | encoding_check code_memory (sigma program ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len)) assembledi ∧ |
---|
1376 | sigma program newppc = bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len). |
---|
1377 | |
---|
1378 | axiom bitvector_of_nat_nat_of_bitvector: |
---|
1379 | ∀n,v. |
---|
1380 | bitvector_of_nat n (nat_of_bitvector n v) = v. |
---|
1381 | |
---|
1382 | axiom assembly_ok_to_expand_pseudo_instruction_ok: |
---|
1383 | ∀program,assembled,costs. |
---|
1384 | Some … 〈assembled,costs〉 = assembly program → |
---|
1385 | ∀ppc. |
---|
1386 | let code_memory ≝ load_code_memory assembled in |
---|
1387 | let preamble ≝ \fst program in |
---|
1388 | let data_labels ≝ construct_datalabels preamble in |
---|
1389 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
1390 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
1391 | let expansion ≝ jump_expansion ppc program in |
---|
1392 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1393 | ∃instructions. |
---|
1394 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi. |
---|
1395 | |
---|
1396 | lemma fetch_assembly_pseudo2: |
---|
1397 | ∀program,assembled,costs,labels. |
---|
1398 | Some … 〈labels,costs〉 = build_maps program → |
---|
1399 | Some … 〈assembled,costs〉 = assembly program → |
---|
1400 | ∀ppc. |
---|
1401 | let code_memory ≝ load_code_memory assembled in |
---|
1402 | let preamble ≝ \fst program in |
---|
1403 | let data_labels ≝ construct_datalabels preamble in |
---|
1404 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
1405 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
1406 | let expansion ≝ jump_expansion ppc program in |
---|
1407 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
1408 | ∃instructions. |
---|
1409 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi ∧ |
---|
1410 | fetch_many code_memory (sigma program newppc) (sigma program ppc) instructions. |
---|
1411 | #program #assembled #costs #labels #BUILD_MAPS #ASSEMBLY #ppc |
---|
1412 | generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program assembled costs ASSEMBLY ppc) |
---|
1413 | letin code_memory ≝ (load_code_memory assembled) |
---|
1414 | letin preamble ≝ (\fst program) |
---|
1415 | letin data_labels ≝ (construct_datalabels preamble) |
---|
1416 | letin lookup_labels ≝ (λx. sigma program (address_of_word_labels_code_mem (\snd program) x)) |
---|
1417 | letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?)) |
---|
1418 | whd in ⊢ (% → %) |
---|
1419 | generalize in match (assembly_ok … BUILD_MAPS ASSEMBLY ppc) |
---|
1420 | cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc |
---|
1421 | generalize in match (fetch_assembly_pseudo program ppc |
---|
1422 | (λx. sigma program (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi |
---|
1423 | (load_code_memory assembled)); |
---|
1424 | whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND |
---|
1425 | whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?); |
---|
1426 | normalize nodelta in EXPAND; (* HERE *) |
---|
1427 | generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program ppc))) -H1; #H1 |
---|
1428 | >bitvector_of_nat_nat_of_bitvector in H1; #H1 |
---|
1429 | <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?) |
---|
1430 | #H1 #H2 |
---|
1431 | cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2 |
---|
1432 | generalize in match (H1 ?? (refl …) (refl …) ?) -H1; |
---|
1433 | [ #K3 % [2: % [% | @K3]] | @K1 ] |
---|
1434 | qed. |
---|
1435 | |
---|
1436 | (* OLD? |
---|
1437 | definition assembly_specification: |
---|
1438 | ∀assembly_program: pseudo_assembly_program. |
---|
1439 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
1440 | λpseudo_assembly_program. |
---|
1441 | λcode_mem. |
---|
1442 | ∀pc: Word. |
---|
1443 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
1444 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
1445 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
1446 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
1447 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
1448 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
1449 | match pre_assembled with |
---|
1450 | [ None ⇒ True |
---|
1451 | | Some pc_code ⇒ |
---|
1452 | let 〈new_pc,code〉 ≝ pc_code in |
---|
1453 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
1454 | |
---|
1455 | axiom assembly_meets_specification: |
---|
1456 | ∀pseudo_assembly_program. |
---|
1457 | match assembly pseudo_assembly_program with |
---|
1458 | [ None ⇒ True |
---|
1459 | | Some code_mem_cost ⇒ |
---|
1460 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
1461 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
1462 | ]. |
---|
1463 | (* |
---|
1464 | # PROGRAM |
---|
1465 | [ cases PROGRAM |
---|
1466 | # PREAMBLE |
---|
1467 | # INSTR_LIST |
---|
1468 | elim INSTR_LIST |
---|
1469 | [ whd |
---|
1470 | whd in ⊢ (∀_. %) |
---|
1471 | # PC |
---|
1472 | whd |
---|
1473 | | # INSTR |
---|
1474 | # INSTR_LIST_TL |
---|
1475 | # H |
---|
1476 | whd |
---|
1477 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
1478 | ] |
---|
1479 | | cases not_implemented |
---|
1480 | ] *) |
---|
1481 | *) |
---|
1482 | |
---|
1483 | definition internal_pseudo_address_map ≝ list (BitVector 8). |
---|
1484 | |
---|
1485 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
1486 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1487 | |
---|
1488 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
1489 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
1490 | |
---|
1491 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
1492 | ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀addr:BitVector 7. |
---|
1493 | member ? (eq_bv 8) (false:::addr) M = true → |
---|
1494 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1495 | let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in |
---|
1496 | let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in |
---|
1497 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
1498 | let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in |
---|
1499 | bu@@bl = sigma (code_memory … s) (pbu@@pbl). |
---|
1500 | |
---|
1501 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
1502 | ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀addr:BitVector 7. |
---|
1503 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
1504 | let 〈carr,Saddr〉 ≝ half_add ? addr (bitvector_of_nat 7 1) in |
---|
1505 | carr = false → |
---|
1506 | member ? (eq_bv 8) (false:::addr) M = false → |
---|
1507 | member ? (eq_bv 8) (false:::Saddr) M = false → |
---|
1508 | lookup ? 7 Saddr ram = lookup ? 7 Saddr (low_internal_ram … s). |
---|
1509 | |
---|
1510 | definition addressing_mode_ok ≝ |
---|
1511 | λM:internal_pseudo_address_map.λs:PseudoStatus. |
---|
1512 | λaddr:addressing_mode. |
---|
1513 | match addr with |
---|
1514 | [ DIRECT d ⇒ |
---|
1515 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1516 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1517 | | INDIRECT i ⇒ |
---|
1518 | let d ≝ get_register ? s [[false;false;i]] in |
---|
1519 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
1520 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
1521 | | EXT_INDIRECT _ ⇒ true |
---|
1522 | | REGISTER _ ⇒ true |
---|
1523 | | ACC_A ⇒ true |
---|
1524 | | ACC_B ⇒ true |
---|
1525 | | DPTR ⇒ true |
---|
1526 | | DATA _ ⇒ true |
---|
1527 | | DATA16 _ ⇒ true |
---|
1528 | | ACC_DPTR ⇒ true |
---|
1529 | | ACC_PC ⇒ true |
---|
1530 | | EXT_INDIRECT_DPTR ⇒ true |
---|
1531 | | INDIRECT_DPTR ⇒ true |
---|
1532 | | CARRY ⇒ true |
---|
1533 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1534 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
1535 | | RELATIVE _ ⇒ true |
---|
1536 | | ADDR11 _ ⇒ true |
---|
1537 | | ADDR16 _ ⇒ true ]. |
---|
1538 | |
---|
1539 | definition next_internal_pseudo_address_map ≝ |
---|
1540 | λM:internal_pseudo_address_map. |
---|
1541 | λs:PseudoStatus. |
---|
1542 | match \fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s)) with |
---|
1543 | [ Comment _ ⇒ Some ? M |
---|
1544 | | Cost _ ⇒ Some … M |
---|
1545 | | Jmp _ ⇒ Some … M |
---|
1546 | | Call _ ⇒ |
---|
1547 | Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M) |
---|
1548 | | Mov _ _ ⇒ Some … M |
---|
1549 | | Instruction instr ⇒ |
---|
1550 | match instr with |
---|
1551 | [ ADD addr1 addr2 ⇒ |
---|
1552 | if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then |
---|
1553 | Some ? M |
---|
1554 | else |
---|
1555 | None ? |
---|
1556 | | ADDC addr1 addr2 ⇒ |
---|
1557 | if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then |
---|
1558 | Some ? M |
---|
1559 | else |
---|
1560 | None ? |
---|
1561 | | SUBB addr1 addr2 ⇒ |
---|
1562 | if addressing_mode_ok M s addr1 ∧ addressing_mode_ok M s addr2 then |
---|
1563 | Some ? M |
---|
1564 | else |
---|
1565 | None ? |
---|
1566 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
1567 | |
---|
1568 | definition status_of_pseudo_status: |
---|
1569 | internal_pseudo_address_map → PseudoStatus → option Status ≝ |
---|
1570 | λM,ps. |
---|
1571 | let pap ≝ code_memory … ps in |
---|
1572 | match assembly pap with |
---|
1573 | [ None ⇒ None … |
---|
1574 | | Some p ⇒ |
---|
1575 | let cm ≝ load_code_memory (\fst p) in |
---|
1576 | let pc ≝ sigma pap (program_counter ? ps) in |
---|
1577 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
1578 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (low_internal_ram … ps) in |
---|
1579 | Some … |
---|
1580 | (mk_PreStatus (BitVectorTrie Byte 16) |
---|
1581 | cm |
---|
1582 | lir |
---|
1583 | hir |
---|
1584 | (external_ram … ps) |
---|
1585 | pc |
---|
1586 | (special_function_registers_8051 … ps) |
---|
1587 | (special_function_registers_8052 … ps) |
---|
1588 | (p1_latch … ps) |
---|
1589 | (p3_latch … ps) |
---|
1590 | (clock … ps)) ]. |
---|
1591 | |
---|
1592 | (* |
---|
1593 | definition write_at_stack_pointer': |
---|
1594 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
1595 | λM: Type[0]. |
---|
1596 | λs: PreStatus M. |
---|
1597 | λv: Byte. |
---|
1598 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
1599 | let bit_zero ≝ get_index_v… nu O ? in |
---|
1600 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
1601 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
1602 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
1603 | if bit_zero then |
---|
1604 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1605 | v (low_internal_ram ? s) in |
---|
1606 | set_low_internal_ram ? s memory |
---|
1607 | else |
---|
1608 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
1609 | v (high_internal_ram ? s) in |
---|
1610 | set_high_internal_ram ? s memory. |
---|
1611 | [ cases l0 % |
---|
1612 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
1613 | qed. |
---|
1614 | |
---|
1615 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
1616 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
1617 | ≝ |
---|
1618 | λticks_of. |
---|
1619 | λs. |
---|
1620 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
1621 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
1622 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
1623 | let s ≝ set_program_counter ? s pc in |
---|
1624 | match instr with |
---|
1625 | [ Instruction instr ⇒ |
---|
1626 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
1627 | | Comment cmt ⇒ s |
---|
1628 | | Cost cst ⇒ s |
---|
1629 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
1630 | | Call call ⇒ |
---|
1631 | let a ≝ address_of_word_labels s call in |
---|
1632 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1633 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1634 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
1635 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
1636 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
1637 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
1638 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
1639 | set_program_counter ? s a |
---|
1640 | | Mov dptr ident ⇒ |
---|
1641 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
1642 | ]. |
---|
1643 | [ |
---|
1644 | |2,3,4: % |
---|
1645 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
1646 | | |
---|
1647 | | % |
---|
1648 | ] |
---|
1649 | cases not_implemented |
---|
1650 | qed. |
---|
1651 | *) |
---|
1652 | |
---|
1653 | axiom execute_1_pseudo_instruction_preserves_code_memory: |
---|
1654 | ∀ticks_of,ps. |
---|
1655 | code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps. |
---|
1656 | |
---|
1657 | (* |
---|
1658 | lemma execute_code_memory_unchanged: |
---|
1659 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
1660 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
1661 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
1662 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
1663 | whd in ⊢ (??? (??%)) cases instr |
---|
1664 | [ #pre cases pre |
---|
1665 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1666 | cases (split ????) #z1 #z2 % |
---|
1667 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1668 | cases (split ????) #z1 #z2 % |
---|
1669 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
1670 | cases (split ????) #z1 #z2 % |
---|
1671 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
1672 | [ #x1 whd in ⊢ (??? (??%)) |
---|
1673 | | *: cases not_implemented |
---|
1674 | ] |
---|
1675 | | #comment % |
---|
1676 | | #cost % |
---|
1677 | | #label % |
---|
1678 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
1679 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
1680 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
---|
1681 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
1682 | (* CSC: ??? *) |
---|
1683 | | #dptr #label (* CSC: ??? *) |
---|
1684 | ] |
---|
1685 | cases not_implemented |
---|
1686 | qed. |
---|
1687 | *) |
---|
1688 | |
---|
1689 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
1690 | ∀M:internal_pseudo_address_map. |
---|
1691 | ∀ps,ps': PseudoStatus. |
---|
1692 | code_memory … ps = code_memory … ps' → |
---|
1693 | match status_of_pseudo_status M ps with |
---|
1694 | [ None ⇒ status_of_pseudo_status M ps' = None … |
---|
1695 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' = Some … w |
---|
1696 | ]. |
---|
1697 | #M #ps #ps' #H whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
1698 | generalize in match (refl … (assembly (code_memory … ps))) |
---|
1699 | cases (assembly ?) in ⊢ (???% → %) |
---|
1700 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
1701 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
1702 | qed. |
---|
1703 | |
---|
1704 | definition ticks_of0: pseudo_assembly_program → Word → ? → nat × nat ≝ |
---|
1705 | λprogram: pseudo_assembly_program. |
---|
1706 | λppc: Word. |
---|
1707 | λfetched. |
---|
1708 | match fetched with |
---|
1709 | [ Instruction instr ⇒ |
---|
1710 | match instr with |
---|
1711 | [ JC lbl ⇒ |
---|
1712 | match jump_expansion ppc program with |
---|
1713 | [ short_jump ⇒ 〈2, 2〉 |
---|
1714 | | medium_jump ⇒ ? |
---|
1715 | | long_jump ⇒ 〈4, 4〉 |
---|
1716 | ] |
---|
1717 | | JNC lbl ⇒ |
---|
1718 | match jump_expansion ppc program with |
---|
1719 | [ short_jump ⇒ 〈2, 2〉 |
---|
1720 | | medium_jump ⇒ ? |
---|
1721 | | long_jump ⇒ 〈4, 4〉 |
---|
1722 | ] |
---|
1723 | | JB bit lbl ⇒ |
---|
1724 | match jump_expansion ppc program with |
---|
1725 | [ short_jump ⇒ 〈2, 2〉 |
---|
1726 | | medium_jump ⇒ ? |
---|
1727 | | long_jump ⇒ 〈4, 4〉 |
---|
1728 | ] |
---|
1729 | | JNB bit lbl ⇒ |
---|
1730 | match jump_expansion ppc program with |
---|
1731 | [ short_jump ⇒ 〈2, 2〉 |
---|
1732 | | medium_jump ⇒ ? |
---|
1733 | | long_jump ⇒ 〈4, 4〉 |
---|
1734 | ] |
---|
1735 | | JBC bit lbl ⇒ |
---|
1736 | match jump_expansion ppc program with |
---|
1737 | [ short_jump ⇒ 〈2, 2〉 |
---|
1738 | | medium_jump ⇒ ? |
---|
1739 | | long_jump ⇒ 〈4, 4〉 |
---|
1740 | ] |
---|
1741 | | JZ lbl ⇒ |
---|
1742 | match jump_expansion ppc program with |
---|
1743 | [ short_jump ⇒ 〈2, 2〉 |
---|
1744 | | medium_jump ⇒ ? |
---|
1745 | | long_jump ⇒ 〈4, 4〉 |
---|
1746 | ] |
---|
1747 | | JNZ lbl ⇒ |
---|
1748 | match jump_expansion ppc program with |
---|
1749 | [ short_jump ⇒ 〈2, 2〉 |
---|
1750 | | medium_jump ⇒ ? |
---|
1751 | | long_jump ⇒ 〈4, 4〉 |
---|
1752 | ] |
---|
1753 | | CJNE arg lbl ⇒ |
---|
1754 | match jump_expansion ppc program with |
---|
1755 | [ short_jump ⇒ 〈2, 2〉 |
---|
1756 | | medium_jump ⇒ ? |
---|
1757 | | long_jump ⇒ 〈4, 4〉 |
---|
1758 | ] |
---|
1759 | | DJNZ arg lbl ⇒ |
---|
1760 | match jump_expansion ppc program with |
---|
1761 | [ short_jump ⇒ 〈2, 2〉 |
---|
1762 | | medium_jump ⇒ ? |
---|
1763 | | long_jump ⇒ 〈4, 4〉 |
---|
1764 | ] |
---|
1765 | | ADD arg1 arg2 ⇒ |
---|
1766 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
1767 | 〈ticks, ticks〉 |
---|
1768 | | ADDC arg1 arg2 ⇒ |
---|
1769 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
1770 | 〈ticks, ticks〉 |
---|
1771 | | SUBB arg1 arg2 ⇒ |
---|
1772 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
1773 | 〈ticks, ticks〉 |
---|
1774 | | INC arg ⇒ |
---|
1775 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
1776 | 〈ticks, ticks〉 |
---|
1777 | | DEC arg ⇒ |
---|
1778 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
1779 | 〈ticks, ticks〉 |
---|
1780 | | MUL arg1 arg2 ⇒ |
---|
1781 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
1782 | 〈ticks, ticks〉 |
---|
1783 | | DIV arg1 arg2 ⇒ |
---|
1784 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
1785 | 〈ticks, ticks〉 |
---|
1786 | | DA arg ⇒ |
---|
1787 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
1788 | 〈ticks, ticks〉 |
---|
1789 | | ANL arg ⇒ |
---|
1790 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
1791 | 〈ticks, ticks〉 |
---|
1792 | | ORL arg ⇒ |
---|
1793 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
1794 | 〈ticks, ticks〉 |
---|
1795 | | XRL arg ⇒ |
---|
1796 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
1797 | 〈ticks, ticks〉 |
---|
1798 | | CLR arg ⇒ |
---|
1799 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
1800 | 〈ticks, ticks〉 |
---|
1801 | | CPL arg ⇒ |
---|
1802 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
1803 | 〈ticks, ticks〉 |
---|
1804 | | RL arg ⇒ |
---|
1805 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
1806 | 〈ticks, ticks〉 |
---|
1807 | | RLC arg ⇒ |
---|
1808 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
1809 | 〈ticks, ticks〉 |
---|
1810 | | RR arg ⇒ |
---|
1811 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
1812 | 〈ticks, ticks〉 |
---|
1813 | | RRC arg ⇒ |
---|
1814 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
1815 | 〈ticks, ticks〉 |
---|
1816 | | SWAP arg ⇒ |
---|
1817 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
1818 | 〈ticks, ticks〉 |
---|
1819 | | MOV arg ⇒ |
---|
1820 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
1821 | 〈ticks, ticks〉 |
---|
1822 | | MOVX arg ⇒ |
---|
1823 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
1824 | 〈ticks, ticks〉 |
---|
1825 | | SETB arg ⇒ |
---|
1826 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
1827 | 〈ticks, ticks〉 |
---|
1828 | | PUSH arg ⇒ |
---|
1829 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
1830 | 〈ticks, ticks〉 |
---|
1831 | | POP arg ⇒ |
---|
1832 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
1833 | 〈ticks, ticks〉 |
---|
1834 | | XCH arg1 arg2 ⇒ |
---|
1835 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
1836 | 〈ticks, ticks〉 |
---|
1837 | | XCHD arg1 arg2 ⇒ |
---|
1838 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
1839 | 〈ticks, ticks〉 |
---|
1840 | | RET ⇒ |
---|
1841 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
1842 | 〈ticks, ticks〉 |
---|
1843 | | RETI ⇒ |
---|
1844 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
1845 | 〈ticks, ticks〉 |
---|
1846 | | NOP ⇒ |
---|
1847 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
1848 | 〈ticks, ticks〉 |
---|
1849 | ] |
---|
1850 | | Comment comment ⇒ 〈0, 0〉 |
---|
1851 | | Cost cost ⇒ 〈0, 0〉 |
---|
1852 | | Jmp jmp ⇒ 〈2, 2〉 |
---|
1853 | | Call call ⇒ 〈2, 2〉 |
---|
1854 | | Mov dptr tgt ⇒ 〈2, 2〉 |
---|
1855 | ]. |
---|
1856 | cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *) |
---|
1857 | qed. |
---|
1858 | |
---|
1859 | definition ticks_of: pseudo_assembly_program → Word → nat × nat ≝ |
---|
1860 | λprogram: pseudo_assembly_program. |
---|
1861 | λppc: Word. |
---|
1862 | let 〈preamble, pseudo〉 ≝ program in |
---|
1863 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in |
---|
1864 | ticks_of0 program ppc fetched. |
---|
1865 | |
---|
1866 | lemma get_register_set_program_counter: |
---|
1867 | ∀T,s,pc. get_register T (set_program_counter … s pc) = get_register … s. |
---|
1868 | #T #s #pc % |
---|
1869 | qed. |
---|
1870 | |
---|
1871 | lemma get_8051_sfr_set_program_counter: |
---|
1872 | ∀T,s,pc. get_8051_sfr T (set_program_counter … s pc) = get_8051_sfr … s. |
---|
1873 | #T #s #pc % |
---|
1874 | qed. |
---|
1875 | |
---|
1876 | lemma get_bit_addressable_sfr_set_program_counter: |
---|
1877 | ∀T,s,pc. get_bit_addressable_sfr T (set_program_counter … s pc) = get_bit_addressable_sfr … s. |
---|
1878 | #T #s #pc % |
---|
1879 | qed. |
---|
1880 | |
---|
1881 | lemma low_internal_ram_set_program_counter: |
---|
1882 | ∀T,s,pc. low_internal_ram T (set_program_counter … s pc) = low_internal_ram … s. |
---|
1883 | #T #s #pc % |
---|
1884 | qed. |
---|
1885 | |
---|
1886 | lemma get_arg_8_set_program_counter: |
---|
1887 | ∀n.∀l:Vector addressing_mode_tag (S n). ¬(is_in ? l ACC_PC) → |
---|
1888 | ∀T,s,pc,b.∀arg:l. |
---|
1889 | get_arg_8 T (set_program_counter … s pc) b arg = get_arg_8 … s b arg. |
---|
1890 | [2,3: cases arg; *; normalize //] |
---|
1891 | #n #l #H #T #s #pc #b * *; [11: #NH @⊥ //] #X try #Y % |
---|
1892 | qed. |
---|
1893 | |
---|
1894 | lemma set_bit_addressable_sfr_set_code_memory: |
---|
1895 | ∀T, U: Type[0]. |
---|
1896 | ∀ps: PreStatus ?. |
---|
1897 | ∀code_mem. |
---|
1898 | ∀x. |
---|
1899 | ∀val. |
---|
1900 | set_bit_addressable_sfr ? (set_code_memory T U ps code_mem) x val = |
---|
1901 | set_code_memory T U (set_bit_addressable_sfr ? ps x val) code_mem. |
---|
1902 | #T #U #ps #code_mem #x #val |
---|
1903 | whd in ⊢ (??%?) |
---|
1904 | whd in ⊢ (???(???%?)) |
---|
1905 | cases (eqb ? 128) [ normalize nodelta cases not_implemented |
---|
1906 | | normalize nodelta |
---|
1907 | cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented |
---|
1908 | | normalize nodelta |
---|
1909 | cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented |
---|
1910 | | normalize nodelta |
---|
1911 | cases (eqb ? 176) [ normalize nodelta % |
---|
1912 | | normalize nodelta |
---|
1913 | cases (eqb ? 153) [ normalize nodelta % |
---|
1914 | | normalize nodelta |
---|
1915 | cases (eqb ? 138) [ normalize nodelta % |
---|
1916 | | normalize nodelta |
---|
1917 | cases (eqb ? 139) [ normalize nodelta % |
---|
1918 | | normalize nodelta |
---|
1919 | cases (eqb ? 140) [ normalize nodelta % |
---|
1920 | | normalize nodelta |
---|
1921 | cases (eqb ? 141) [ normalize nodelta % |
---|
1922 | | normalize nodelta |
---|
1923 | cases (eqb ? 200) [ normalize nodelta % |
---|
1924 | | normalize nodelta |
---|
1925 | cases (eqb ? 202) [ normalize nodelta % |
---|
1926 | | normalize nodelta |
---|
1927 | cases (eqb ? 203) [ normalize nodelta % |
---|
1928 | | normalize nodelta |
---|
1929 | cases (eqb ? 204) [ normalize nodelta % |
---|
1930 | | normalize nodelta |
---|
1931 | cases (eqb ? 205) [ normalize nodelta % |
---|
1932 | | normalize nodelta |
---|
1933 | cases (eqb ? 135) [ normalize nodelta % |
---|
1934 | | normalize nodelta |
---|
1935 | cases (eqb ? 136) [ normalize nodelta % |
---|
1936 | | normalize nodelta |
---|
1937 | cases (eqb ? 137) [ normalize nodelta % |
---|
1938 | | normalize nodelta |
---|
1939 | cases (eqb ? 152) [ normalize nodelta % |
---|
1940 | | normalize nodelta |
---|
1941 | cases (eqb ? 168) [ normalize nodelta % |
---|
1942 | | normalize nodelta |
---|
1943 | cases (eqb ? 184) [ normalize nodelta % |
---|
1944 | | normalize nodelta |
---|
1945 | cases (eqb ? 129) [ normalize nodelta % |
---|
1946 | | normalize nodelta |
---|
1947 | cases (eqb ? 130) [ normalize nodelta % |
---|
1948 | | normalize nodelta |
---|
1949 | cases (eqb ? 131) [ normalize nodelta % |
---|
1950 | | normalize nodelta |
---|
1951 | cases (eqb ? 208) [ normalize nodelta % |
---|
1952 | | normalize nodelta |
---|
1953 | cases (eqb ? 224) [ normalize nodelta % |
---|
1954 | | normalize nodelta |
---|
1955 | cases (eqb ? 240) [ normalize nodelta % |
---|
1956 | | normalize nodelta |
---|
1957 | cases not_implemented |
---|
1958 | ]]]]]]]]]]]]]]]]]]]]]]]]]] |
---|
1959 | qed. |
---|
1960 | |
---|
1961 | lemma set_arg_8_set_code_memory: |
---|
1962 | ∀n:nat. |
---|
1963 | ∀l:Vector addressing_mode_tag (S n). |
---|
1964 | ¬(is_in ? l ACC_PC) → |
---|
1965 | ∀T: Type[0]. |
---|
1966 | ∀U: Type[0]. |
---|
1967 | ∀ps: PreStatus ?. |
---|
1968 | ∀code_mem. |
---|
1969 | ∀val. |
---|
1970 | ∀b: l. |
---|
1971 | set_arg_8 ? (set_code_memory T U ps code_mem) b val = |
---|
1972 | set_code_memory T U (set_arg_8 ? ps b val) code_mem. |
---|
1973 | [2,3: cases b; *; normalize //] |
---|
1974 | #n #l #prf #T #U #ps #code_mem #val * *; |
---|
1975 | [*: |
---|
1976 | [1,2,3,4,8,9,15,16,17,18,19: #x] |
---|
1977 | try #y whd in ⊢ (??(%)?) |
---|
1978 | whd in ⊢ (???(???(%)?)) |
---|
1979 | [1,2: |
---|
1980 | cases (split bool 4 4 ?) |
---|
1981 | #nu' #nl' |
---|
1982 | normalize nodelta |
---|
1983 | cases (split bool 1 3 nu') |
---|
1984 | #bit1' #ignore' |
---|
1985 | normalize nodelta |
---|
1986 | cases (get_index_v bool 4 nu' 1 ?) |
---|
1987 | [1,2,3,4: |
---|
1988 | normalize nodelta |
---|
1989 | try % |
---|
1990 | try (@(set_bit_addressable_sfr_set_code_memory T U ps code_mem x val)) |
---|
1991 | try (normalize in ⊢ (???(???%?))) |
---|
1992 | ] |
---|
1993 | |3,4: % |
---|
1994 | |*: |
---|
1995 | try normalize nodelta |
---|
1996 | normalize cases (not_implemented) |
---|
1997 | ] |
---|
1998 | ] |
---|
1999 | qed. |
---|
2000 | |
---|
2001 | axiom set_arg_8_set_program_counter: |
---|
2002 | ∀n:nat. |
---|
2003 | ∀l:Vector addressing_mode_tag (S n). |
---|
2004 | ¬(is_in ? l ACC_PC) → |
---|
2005 | ∀T: Type[0]. |
---|
2006 | ∀ps: PreStatus ?. |
---|
2007 | ∀pc. |
---|
2008 | ∀val. |
---|
2009 | ∀b: l. |
---|
2010 | set_arg_8 ? (set_program_counter T ps pc) b val = |
---|
2011 | set_program_counter T (set_arg_8 ? ps b val) pc. |
---|
2012 | [1,2: cases b; *; normalize //] |
---|
2013 | qed. |
---|
2014 | |
---|
2015 | |
---|
2016 | lemma get_arg_8_set_code_memory: |
---|
2017 | ∀T1,T2,s,code_mem,b,arg. |
---|
2018 | get_arg_8 T1 (set_code_memory T2 T1 s code_mem) b arg = get_arg_8 … s b arg. |
---|
2019 | #T1 #T2 #s #code_mem #b #arg % |
---|
2020 | qed. |
---|
2021 | |
---|
2022 | lemma set_code_memory_set_flags: |
---|
2023 | ∀T1,T2,s,f1,f2,f3,code_mem. |
---|
2024 | set_code_memory T1 T2 (set_flags T1 s f1 f2 f3) code_mem = |
---|
2025 | set_flags … (set_code_memory … s code_mem) f1 f2 f3. |
---|
2026 | #T1 #T2 #s #f1 #f2 #f3 #code_mem % |
---|
2027 | qed. |
---|
2028 | |
---|
2029 | lemma set_program_counter_set_flags: |
---|
2030 | ∀T1,s,f1,f2,f3,pc. |
---|
2031 | set_program_counter T1 (set_flags T1 s f1 f2 f3) pc = |
---|
2032 | set_flags … (set_program_counter … s pc) f1 f2 f3. |
---|
2033 | #T1 #s #f1 #f2 #f3 #pc % |
---|
2034 | qed. |
---|
2035 | |
---|
2036 | lemma program_counter_set_flags: |
---|
2037 | ∀T1,s,f1,f2,f3. |
---|
2038 | program_counter T1 (set_flags T1 s f1 f2 f3) = program_counter … s. |
---|
2039 | #T1 #s #f1 #f2 #f3 % |
---|
2040 | qed. |
---|
2041 | |
---|
2042 | lemma eq_rect_Type1_r: |
---|
2043 | ∀A: Type[1]. |
---|
2044 | ∀a:A. |
---|
2045 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
2046 | #A #a #P #H #x #p |
---|
2047 | generalize in match H |
---|
2048 | generalize in match P |
---|
2049 | cases p |
---|
2050 | // |
---|
2051 | qed. |
---|
2052 | |
---|
2053 | lemma split_eq_status: |
---|
2054 | ∀T. |
---|
2055 | ∀A1,A2,A3,A4,A5,A6,A7,A8,A9,A10. |
---|
2056 | ∀B1,B2,B3,B4,B5,B6,B7,B8,B9,B10. |
---|
2057 | A1=B1 → A2=B2 → A3=B3 → A4=B4 → A5=B5 → A6=B6 → A7=B7 → A8=B8 → A9=B9 → A10=B10 → |
---|
2058 | mk_PreStatus T A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 = |
---|
2059 | mk_PreStatus T B1 B2 B3 B4 B5 B6 B7 B8 B9 B10. |
---|
2060 | // |
---|
2061 | qed. |
---|
2062 | |
---|
2063 | lemma main_thm: |
---|
2064 | ∀M,M',ps,s,s''. |
---|
2065 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
2066 | status_of_pseudo_status M ps = Some … s → |
---|
2067 | status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps)) ps) = Some … s'' → |
---|
2068 | ∃n. execute n s = s''. |
---|
2069 | #M #M' #ps #s #s'' |
---|
2070 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps)) |
---|
2071 | whd in ⊢ (? → ? → ??%? → ??%? → ?) |
---|
2072 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
2073 | generalize in match (refl … (assembly (code_memory … ps))) |
---|
2074 | generalize in match (assembly (code_memory … ps)) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?) |
---|
2075 | cases (build_maps (code_memory … ps)) |
---|
2076 | [ cases (code_memory … ps) #preamble #instr_list whd in ⊢ (???% → ?) #EQ >EQ #H #MAP |
---|
2077 | #abs @⊥ normalize in abs; destruct (abs) ] |
---|
2078 | * #labels #costs whd in ⊢ (? → ? → ??%? → ?) generalize in match (refl … (code_memory … ps)) |
---|
2079 | cases (code_memory … ps) in ⊢ (???% → %) #preamble #instr_list #EQ0 normalize nodelta; |
---|
2080 | generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta; |
---|
2081 | [ #EQ >EQ #_ #_ #abs @⊥ normalize in abs; destruct (abs) ] |
---|
2082 | * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta; |
---|
2083 | #H generalize in match (H ??? (refl …) (refl …)) -H; #H; |
---|
2084 | #MAP |
---|
2085 | #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1; |
---|
2086 | #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2; |
---|
2087 | change with |
---|
2088 | (∃n. |
---|
2089 | execute n |
---|
2090 | (set_low_internal_ram ? |
---|
2091 | (set_high_internal_ram ? |
---|
2092 | (set_program_counter ? |
---|
2093 | (set_code_memory ?? ps (load_code_memory assembled)) |
---|
2094 | (sigma 〈preamble,instr_list〉 (program_counter ? ps))) |
---|
2095 | (high_internal_ram_of_pseudo_high_internal_ram M ?)) |
---|
2096 | (low_internal_ram_of_pseudo_low_internal_ram M ?)) |
---|
2097 | = set_low_internal_ram ? |
---|
2098 | (set_high_internal_ram ? |
---|
2099 | (set_program_counter ? |
---|
2100 | (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory assembled)) |
---|
2101 | (sigma ??)) |
---|
2102 | ?) |
---|
2103 | ?) |
---|
2104 | whd in match (\snd 〈preamble,instr_list〉) in H; |
---|
2105 | whd in match (\fst 〈preamble,instr_list〉) in H; |
---|
2106 | whd in match (\snd 〈final_pc,assembled〉) in H; |
---|
2107 | whd in match (\snd 〈preamble,instr_list〉) in MAP; |
---|
2108 | -s s'' labels costs final_ppc final_pc; |
---|
2109 | letin ps' ≝ (execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉) ps) |
---|
2110 | (* NICE STATEMENT HERE *) |
---|
2111 | generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps' |
---|
2112 | #K <K generalize in match K; -K; |
---|
2113 | (* STATEMENT WITH EQUALITY HERE *) |
---|
2114 | whd in ⊢ (???(?%?) → ?) |
---|
2115 | whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta; |
---|
2116 | cases (fetch_pseudo_instruction instr_list (program_counter … ps)) in MAP ⊢ % |
---|
2117 | #pi #newppc normalize nodelta; #MAP * #instructions *; |
---|
2118 | cases pi in MAP; normalize nodelta; |
---|
2119 | [2,3: (* Comment, Cost *) #ARG #MAP #H1 #H2 #EQ %[1,3:@0] |
---|
2120 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M'; |
---|
2121 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2122 | #H2 >(eq_bv_to_eq … H2) >EQ % |
---|
2123 | (* |6: (* Mov *) #arg1 #arg2 |
---|
2124 | #H1 #H2 #EQ %[@1] |
---|
2125 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2126 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2127 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2128 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2129 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2130 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
2131 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?) |
---|
2132 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
2133 | normalize nodelta; |
---|
2134 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
2135 | #result #flags |
---|
2136 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) % *) |
---|
2137 | (* |4,5: (* Jmp, Call *) #label |
---|
2138 | whd in ⊢ (???% → ?) cases (jump_expansion ??) normalize nodelta; |
---|
2139 | [3,6: (* long *) #H1 #H2 #EQ %[1,3:@1] |
---|
2140 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2141 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2142 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2143 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2144 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2145 | generalize in match EQ; -EQ; |
---|
2146 | [2: whd in ⊢ (???% → ??%?); |
---|
2147 | cases (half_add ???) #carry #new_sp normalize nodelta; |
---|
2148 | >(eq_bv_to_eq … H2c) |
---|
2149 | change with |
---|
2150 | ((?=let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 newppc in ?) → |
---|
2151 | (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?) |
---|
2152 | (* MISMATCH ON WHAT IS ON THE STACK!!!! *) |
---|
2153 | (* HOW TO PROVE THIS?? *) |
---|
2154 | ] |
---|
2155 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) |
---|
2156 | cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX |
---|
2157 | [2: % | ] |
---|
2158 | |4: (* short Call *) #abs @⊥ destruct (abs) |
---|
2159 | |1: (* short Jmp *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2160 | generalize in match |
---|
2161 | (refl ? |
---|
2162 | (sub_16_with_carry |
---|
2163 | (sigma 〈preamble,instr_list〉 (program_counter … ps)) |
---|
2164 | (sigma 〈preamble,instr_list〉 (address_of_word_labels_code_mem instr_list label)) |
---|
2165 | false)) |
---|
2166 | cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta; |
---|
2167 | generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta; |
---|
2168 | generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; |
---|
2169 | #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)] |
---|
2170 | generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2171 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2172 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2173 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2174 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2175 | generalize in match EQ; -EQ; |
---|
2176 | whd in ⊢ (???% → ?); |
---|
2177 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) |
---|
2178 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ??) ? in ?) = ?) |
---|
2179 | generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 newppc) (sign_extension lower))) |
---|
2180 | cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4 |
---|
2181 | @split_eq_status try % change with (newpc = sigma ? (address_of_word_labels ps label)) |
---|
2182 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2183 | | (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
2184 | generalize in match |
---|
2185 | (refl … |
---|
2186 | (split … 5 11 (sigma 〈preamble,instr_list〉 (address_of_word_labels_code_mem instr_list label)))) |
---|
2187 | cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1 |
---|
2188 | generalize in match |
---|
2189 | (refl … |
---|
2190 | (split … 5 11 (sigma 〈preamble,instr_list〉 (program_counter … ps)))) |
---|
2191 | cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2 |
---|
2192 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
2193 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
2194 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2195 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
2196 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
2197 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
2198 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2199 | generalize in match EQ; -EQ; |
---|
2200 | whd in ⊢ (???% → ?); |
---|
2201 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) whd in ⊢ (??%?) |
---|
2202 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?) |
---|
2203 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc))) |
---|
2204 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
2205 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
2206 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
2207 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
2208 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
2209 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
2210 | generalize in match |
---|
2211 | (refl … |
---|
2212 | (half_add 16 (sigma 〈preamble,instr_list〉 newppc) |
---|
2213 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
---|
2214 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
---|
2215 | @split_eq_status try % |
---|
2216 | [ change with (? = sigma ? (address_of_word_labels ps label)) |
---|
2217 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
2218 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
2219 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] |
---|
2220 | (*|5: (* Call *)*) *) |
---|
2221 | | (* Instruction *) -pi; *; normalize nodelta; |
---|
2222 | [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1] |
---|
2223 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
2224 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
2225 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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2226 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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2227 | >H2b >(eq_instruction_to_eq … H2a) |
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2228 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP; |
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2229 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; |
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2230 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
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2231 | normalize nodelta; #MAP;(* [1,2:whd in MAP:(??(match % with [ _ ⇒ ?])?)]*) |
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2232 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
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2233 | #result #flags |
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2234 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) % |
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2235 | | (* INC *) #arg1 #H1 #H2 #EQ %[@1] |
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2236 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
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2237 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
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2238 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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2239 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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2240 | >H2b >(eq_instruction_to_eq … H2a) |
---|
2241 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
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2242 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG] |
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2243 | [1,2,3,4: cases (half_add ???) #carry #result |
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2244 | | cases (half_add ???) #carry #bl normalize nodelta; |
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2245 | cases (full_add ????) #carry' #bu normalize nodelta ] |
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2246 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_to_eq … H2c) -newppc'; |
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2247 | [5: % |
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2248 | |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program |
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2249 | (set_program_counter pseudo_assembly_program ps newppc) |
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2250 | (\fst (ticks_of0 〈preamble,instr_list〉 |
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2251 | (program_counter pseudo_assembly_program ps) |
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2252 | (Instruction (INC Identifier (DIRECT ARG)))) |
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2253 | +clock pseudo_assembly_program |
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2254 | (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG)) |
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2255 | [2,3: // ] |
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2256 | <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://] |
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2257 | whd in ⊢ (??%%) |
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2258 | cases (split bool 4 4 ARG) |
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2259 | #nu' #nl' |
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2260 | normalize nodelta |
---|
2261 | cases (split bool 1 3 nu') |
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2262 | #bit_1' #ignore' |
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2263 | normalize nodelta |
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2264 | cases (get_index_v bool 4 nu' ? ?) |
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2265 | [ normalize nodelta (* HERE *) whd in ⊢ (??%%) % |
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2266 | | |
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2267 | ] |
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