[877] | 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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[905] | 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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[877] | 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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[892] | 472 | |
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| 473 | (* |
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[877] | 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)) ∧ |
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| 478 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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| 479 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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| 480 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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| 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)))) ∧ |
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| 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)) ∧ |
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| 486 | P (DA ? ACC_A) ∧ |
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| 487 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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| 488 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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| 489 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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| 490 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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| 491 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 492 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 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) ∧ |
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| 496 | P (RLC ? ACC_A) ∧ |
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| 497 | P (RR ? ACC_A) ∧ |
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| 498 | P (RRC ? ACC_A) ∧ |
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| 499 | P (SWAP ? ACC_A) ∧ |
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| 500 | P (RET ?) ∧ |
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| 501 | P (RETI ?) ∧ |
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| 502 | P (NOP ?) ∧ |
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| 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)) ∧ |
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| 514 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 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 ]]) |
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| 523 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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| 524 | (λd. P (MOVX ? d)) ∧ |
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| 525 | union_elim ? ? ( |
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| 526 | union_elim ? ? ( |
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| 527 | union_elim ? ? ( |
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| 528 | union_elim ? ? ( |
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| 529 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 530 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
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| 531 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
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| 532 | (product_elim ? ? [[dptr]] [[data16]])) |
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| 533 | (product_elim ? ? [[carry]] [[bit_addr]])) |
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| 534 | (product_elim ? ? [[bit_addr]] [[carry]]) |
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| 535 | (λd. P (MOV ? d)). |
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| 536 | % |
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| 537 | qed. |
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| 538 | |
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| 539 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
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| 540 | λP. (* |
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| 541 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
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| 542 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
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| 543 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
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| 544 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
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| 545 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
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| 546 | P (JMP INDIRECT_DPTR) ∧ |
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| 547 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
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| 548 | preinstruction_elim (λp. P (RealInstruction p)). *) |
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| 549 | % |
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| 550 | qed. |
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| 551 | |
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| 552 | |
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| 553 | axiom instruction_elim_complete: |
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| 554 | ∀P. instruction_elim P = true → ∀i. P i = true. |
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[892] | 555 | *) |
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[883] | 556 | (*definition eq_instruction ≝ |
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[877] | 557 | λi, j: instruction. |
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[883] | 558 | true.*) |
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| 559 | axiom eq_instruction: instruction → instruction → bool. |
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[885] | 560 | axiom eq_instruction_refl: ∀i. eq_instruction i i = true. |
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| 561 | |
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[877] | 562 | let rec vect_member |
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| 563 | (A: Type[0]) (n: nat) (eq: A → A → bool) |
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| 564 | (v: Vector A n) (a: A) on v: bool ≝ |
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| 565 | match v with |
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| 566 | [ VEmpty ⇒ false |
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| 567 | | VCons len hd tl ⇒ |
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| 568 | eq hd a ∨ (vect_member A ? eq tl a) |
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| 569 | ]. |
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[892] | 570 | |
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[877] | 571 | let rec list_addressing_mode_tags_elim_prop |
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| 572 | (n: nat) |
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| 573 | (l: Vector addressing_mode_tag (S n)) |
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[884] | 574 | on l: |
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| 575 | ∀P: l → Prop. |
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[877] | 576 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
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| 577 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
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| 578 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
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| 579 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
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| 580 | ∀x: l. P x ≝ |
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| 581 | match l return |
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| 582 | λy. |
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| 583 | match y with |
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| 584 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
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[884] | 585 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
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[877] | 586 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
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| 587 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
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| 588 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
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| 589 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
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| 590 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
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| 591 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
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| 592 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
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| 593 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
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| 594 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
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| 595 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
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| 596 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
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| 597 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
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| 598 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
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| 599 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
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| 600 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
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| 601 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
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| 602 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
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| 603 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
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| 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 |
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| 607 | [ VEmpty ⇒ λAbsurd. ⊥ |
---|
| 608 | | VCons len hd tl ⇒ λProof. ? |
---|
[884] | 609 | ] (refl ? (S n)). cases daemon. qed. (* |
---|
[877] | 610 | [ destruct(Absurd) |
---|
| 611 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
---|
| 612 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
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| 613 | # A15 # A16 # A17 # A18 # A19 # X |
---|
| 614 | cases X |
---|
[884] | 615 | # SUB cases daemon ] qed. |
---|
[877] | 616 | cases SUB |
---|
| 617 | [ # BYTE |
---|
| 618 | normalize |
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| 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. |
---|
[883] | 697 | *) |
---|
[877] | 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 | |
---|
[883] | 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). |
---|
[901] | 776 | |
---|
[883] | 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 | |
---|
[901] | 781 | (* |
---|
[883] | 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)))). |
---|
[894] | 789 | |
---|
[884] | 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. |
---|
[894] | 798 | *) |
---|
[884] | 799 | |
---|
[894] | 800 | |
---|
| 801 | |
---|
| 802 | |
---|
[893] | 803 | (* |
---|
[884] | 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. |
---|
[893] | 808 | *) |
---|
[884] | 809 | |
---|
[890] | 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 | |
---|
[901] | 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. |
---|
[890] | 829 | |
---|
[894] | 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 | |
---|
[901] | 836 | axiom fetch_assembly: |
---|
[892] | 837 | ∀pc,i,code_memory,assembled. |
---|
| 838 | assembled = assembly1 i → |
---|
[890] | 839 | let len ≝ length … assembled in |
---|
| 840 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
| 841 | let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in |
---|
| 842 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
| 843 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
| 844 | (eq_instruction instr i ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true. |
---|
[901] | 845 | (* #pc #i #code_memory #assembled cases i [8: *] |
---|
[890] | 846 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
[892] | 847 | [47,48,49: |
---|
| 848 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
---|
[890] | 849 | [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, |
---|
[892] | 850 | 59,60,63,64,65,66,67: #ARG]] |
---|
| 851 | [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, |
---|
| 852 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
---|
| 853 | [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, |
---|
| 854 | 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, |
---|
| 855 | 68,69,70,71: #ARG2]] |
---|
| 856 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
---|
| 857 | normalize in ⊢ (???% → ?) |
---|
[897] | 858 | [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
| 859 | normalize in ⊢ (???% → ?)] |
---|
[892] | 860 | #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2) |
---|
| 861 | try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4 |
---|
| 862 | change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??) |
---|
| 863 | whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?) |
---|
[897] | 864 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
---|
| 865 | [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, |
---|
| 866 | 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, |
---|
| 867 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
---|
[896] | 868 | whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
[901] | 869 | qed. *) |
---|
| 870 | |
---|
| 871 | let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝ |
---|
| 872 | match expected with |
---|
| 873 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
| 874 | | cons i tl ⇒ |
---|
| 875 | let fetched ≝ fetch code_memory pc in |
---|
| 876 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
| 877 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
| 878 | eq_instruction instr i = true ∧ fetch_many code_memory final_pc pc' tl]. |
---|
| 879 | |
---|
| 880 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
---|
| 881 | #A #a #b #EQ destruct // |
---|
[892] | 882 | qed. |
---|
[901] | 883 | |
---|
| 884 | lemma pair_destruct: ∀A,B,a1,a2,b1,b2. pair A B a1 a2 = 〈b1,b2〉 → a1=b1 ∧ a2=b2. |
---|
| 885 | #A #B #a1 #a2 #b1 #b2 #EQ destruct /2/ |
---|
| 886 | qed. |
---|
| 887 | |
---|
| 888 | axiom eq_bv_to_eq: ∀n.∀v1,v2: BitVector n. eq_bv … v1 v2 = true → v1=v2. |
---|
| 889 | |
---|
| 890 | lemma fetch_assembly_pseudo: |
---|
| 891 | ∀program,ppc,lookup_labels,lookup_datalabels. |
---|
| 892 | ∀pi,code_memory,len,assembled,instructions,pc. |
---|
[916] | 893 | let expansion ≝ jump_expansion ppc program in |
---|
[921] | 894 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi → |
---|
| 895 | Some … 〈len,assembled〉 = assembly_1_pseudoinstruction program ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi → |
---|
[901] | 896 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
| 897 | fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions. |
---|
| 898 | #program #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc |
---|
[921] | 899 | #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2 |
---|
| 900 | cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b |
---|
[901] | 901 | >EQ2a >EQ2b -EQ2a EQ2b; |
---|
| 902 | generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc |
---|
| 903 | generalize in match pc elim instructions |
---|
| 904 | [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl |
---|
| 905 | | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd |
---|
| 906 | generalize in match (fetch_assembly pc i code_memory … (refl …) H1) |
---|
| 907 | cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %) |
---|
| 908 | cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1 #K2 % // |
---|
| 909 | >(eq_bv_to_eq … K2) @IH @H2 ] |
---|
| 910 | qed. |
---|
| 911 | |
---|
[877] | 912 | (* This establishes the correspondence between pseudo program counters and |
---|
| 913 | program counters. It is at the heart of the proof. *) |
---|
| 914 | (*CSC: code taken from build_maps *) |
---|
| 915 | definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
---|
| 916 | λinstr_list. |
---|
| 917 | foldl ?? |
---|
| 918 | (λt. λi. |
---|
| 919 | match t with |
---|
| 920 | [ None ⇒ None ? |
---|
| 921 | | Some ppc_pc_map ⇒ |
---|
| 922 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
---|
| 923 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
---|
| 924 | let 〈label, i〉 ≝ i in |
---|
[921] | 925 | match construct_costs instr_list ppc program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with |
---|
[877] | 926 | [ None ⇒ None ? |
---|
| 927 | | Some pc_ignore ⇒ |
---|
| 928 | let 〈pc,ignore〉 ≝ pc_ignore in |
---|
| 929 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
---|
| 930 | ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list). |
---|
| 931 | |
---|
| 932 | definition tech_pc_sigma0: pseudo_assembly_program → option (nat × (BitVectorTrie Word 16)) ≝ |
---|
| 933 | λinstr_list. |
---|
| 934 | match sigma0 instr_list with |
---|
| 935 | [ None ⇒ None … |
---|
| 936 | | Some result ⇒ |
---|
| 937 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
| 938 | Some … pc_sigma_map ]. |
---|
| 939 | |
---|
| 940 | definition sigma_safe: pseudo_assembly_program → option (Word → Word) ≝ |
---|
| 941 | λinstr_list. |
---|
| 942 | match sigma0 instr_list with |
---|
| 943 | [ None ⇒ None ? |
---|
| 944 | | Some result ⇒ |
---|
| 945 | let 〈ppc,pc_sigma_map〉 ≝ result in |
---|
| 946 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
---|
| 947 | if gtb pc (2^16) then |
---|
| 948 | None ? |
---|
| 949 | else |
---|
| 950 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
---|
| 951 | |
---|
| 952 | axiom policy_ok: ∀p. sigma_safe p ≠ None …. |
---|
| 953 | |
---|
| 954 | definition sigma: pseudo_assembly_program → Word → Word ≝ |
---|
| 955 | λp. |
---|
| 956 | match sigma_safe p return λr:option (Word → Word). r ≠ None … → Word → Word with |
---|
| 957 | [ None ⇒ λabs. ⊥ |
---|
| 958 | | Some r ⇒ λ_.r] (policy_ok p). |
---|
| 959 | cases abs // |
---|
| 960 | qed. |
---|
| 961 | |
---|
| 962 | lemma length_append: |
---|
| 963 | ∀A.∀l1,l2:list A. |
---|
| 964 | |l1 @ l2| = |l1| + |l2|. |
---|
| 965 | #A #l1 elim l1 |
---|
| 966 | [ // |
---|
| 967 | | #hd #tl #IH #l2 normalize <IH //] |
---|
| 968 | qed. |
---|
| 969 | |
---|
| 970 | let rec does_not_occur (id:Identifier) (l:list labelled_instruction) on l: bool ≝ |
---|
| 971 | match l with |
---|
| 972 | [ nil ⇒ true |
---|
| 973 | | cons hd tl ⇒ notb (instruction_matches_identifier id hd) ∧ does_not_occur id tl]. |
---|
| 974 | |
---|
| 975 | lemma does_not_occur_None: |
---|
| 976 | ∀id,i,list_instr. |
---|
| 977 | does_not_occur id (list_instr@[〈None …,i〉]) = |
---|
| 978 | does_not_occur id list_instr. |
---|
| 979 | #id #i #list_instr elim list_instr |
---|
| 980 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH %] |
---|
| 981 | qed. |
---|
| 982 | |
---|
| 983 | let rec occurs_exactly_once (id:Identifier) (l:list labelled_instruction) on l : bool ≝ |
---|
| 984 | match l with |
---|
| 985 | [ nil ⇒ false |
---|
| 986 | | cons hd tl ⇒ |
---|
| 987 | if instruction_matches_identifier id hd then |
---|
| 988 | does_not_occur id tl |
---|
| 989 | else |
---|
| 990 | occurs_exactly_once id tl ]. |
---|
| 991 | |
---|
| 992 | lemma occurs_exactly_once_None: |
---|
| 993 | ∀id,i,list_instr. |
---|
| 994 | occurs_exactly_once id (list_instr@[〈None …,i〉]) = |
---|
| 995 | occurs_exactly_once id list_instr. |
---|
| 996 | #id #i #list_instr elim list_instr |
---|
| 997 | [ % | #hd #tl #IH whd in ⊢ (??%%) >IH >does_not_occur_None %] |
---|
| 998 | qed. |
---|
| 999 | |
---|
| 1000 | lemma index_of_internal_None: ∀i,id,instr_list,n. |
---|
| 1001 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
---|
| 1002 | index_of_internal ? (instruction_matches_identifier id) instr_list n = |
---|
| 1003 | index_of_internal ? (instruction_matches_identifier id) (instr_list@[〈None …,i〉]) n. |
---|
| 1004 | #i #id #instr_list elim instr_list |
---|
| 1005 | [ #n #abs whd in abs; cases abs |
---|
| 1006 | | #hd #tl #IH #n whd in ⊢ (% → ??%%); whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ?) |
---|
| 1007 | cases (instruction_matches_identifier id hd) whd in ⊢ (match % with [_ ⇒ ? | _ ⇒ ?] → ??%%) |
---|
| 1008 | [ #H % |
---|
| 1009 | | #H @IH whd in H; cases (occurs_exactly_once ??) in H ⊢ % |
---|
| 1010 | [ #_ % | #abs cases abs ]]] |
---|
| 1011 | qed. |
---|
| 1012 | |
---|
| 1013 | lemma address_of_word_labels_code_mem_None: ∀i,id,instr_list. |
---|
| 1014 | occurs_exactly_once id (instr_list@[〈None …,i〉]) → |
---|
| 1015 | address_of_word_labels_code_mem instr_list id = |
---|
| 1016 | address_of_word_labels_code_mem (instr_list@[〈None …,i〉]) id. |
---|
| 1017 | #i #id #instr_list #H whd in ⊢ (??%%) whd in ⊢ (??(??%?)(??%?)) |
---|
| 1018 | >(index_of_internal_None … H) % |
---|
| 1019 | qed. |
---|
| 1020 | |
---|
| 1021 | axiom tech_pc_sigma0_append: |
---|
[921] | 1022 | ∀preamble,instr_list,prefix,label,i,pc',code,ppc,pc,costs,costs'. |
---|
[877] | 1023 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
---|
[921] | 1024 | construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = Some … 〈pc',code〉 → |
---|
[877] | 1025 | tech_pc_sigma0 〈preamble,prefix@[〈label,i〉]〉 = Some … 〈pc',costs'〉. |
---|
| 1026 | |
---|
| 1027 | axiom tech_pc_sigma0_append_None: |
---|
[921] | 1028 | ∀preamble,instr_list,prefix,i,ppc,pc,costs. |
---|
[877] | 1029 | Some … 〈pc,costs〉 = tech_pc_sigma0 〈preamble,prefix〉 → |
---|
[921] | 1030 | construct_costs 〈preamble,instr_list〉 … ppc pc (λx.zero 16) (λx. zero 16) costs i = None … |
---|
[877] | 1031 | → False. |
---|
| 1032 | |
---|
[903] | 1033 | (* |
---|
[877] | 1034 | definition build_maps' ≝ |
---|
| 1035 | λpseudo_program. |
---|
| 1036 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
| 1037 | let result ≝ |
---|
| 1038 | foldl_strong |
---|
| 1039 | (option Identifier × pseudo_instruction) |
---|
| 1040 | (λpre. Σres:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 1041 | let pre' ≝ 〈preamble,pre〉 in |
---|
| 1042 | let 〈labels,pc_costs〉 ≝ res in |
---|
| 1043 | tech_pc_sigma0 pre' = Some … pc_costs ∧ |
---|
| 1044 | ∀id. occurs_exactly_once id pre → |
---|
| 1045 | lookup ?? id labels (zero …) = sigma pre' (address_of_word_labels_code_mem pre id)) |
---|
| 1046 | instr_list |
---|
| 1047 | (λprefix,i,tl,prf,t. |
---|
| 1048 | let 〈labels, pc_costs〉 ≝ t in |
---|
| 1049 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
| 1050 | let 〈label, i'〉 ≝ i in |
---|
| 1051 | let labels ≝ |
---|
| 1052 | match label with |
---|
| 1053 | [ None ⇒ labels |
---|
| 1054 | | Some label ⇒ |
---|
| 1055 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 1056 | insert ? ? label program_counter_bv labels |
---|
| 1057 | ] |
---|
| 1058 | in |
---|
| 1059 | match construct_costs 〈preamble,instr_list〉 program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
| 1060 | [ None ⇒ |
---|
| 1061 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
| 1062 | dummy |
---|
| 1063 | | Some construct ⇒ 〈labels, construct〉 |
---|
| 1064 | ] |
---|
| 1065 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 |
---|
| 1066 | in |
---|
| 1067 | let 〈labels, pc_costs〉 ≝ result in |
---|
| 1068 | let 〈pc, costs〉 ≝ pc_costs in |
---|
| 1069 | 〈labels, costs〉. |
---|
| 1070 | [3: whd % // #id normalize in ⊢ (% → ?) #abs @⊥ // |
---|
| 1071 | | whd cases construct in p3 #PC #CODE #JMEQ % |
---|
| 1072 | [ @(tech_pc_sigma0_append ??????????? (jmeq_to_eq ??? JMEQ)) | #id #Hid ] |
---|
| 1073 | | (* dummy case *) @⊥ |
---|
| 1074 | @(tech_pc_sigma0_append_None ?? prefix ???? (jmeq_to_eq ??? p3)) ] |
---|
| 1075 | [*: generalize in match (sig2 … t) whd in ⊢ (% → ?) |
---|
| 1076 | >p whd in ⊢ (% → ?) >p1 * #IH0 #IH1 >IH0 // ] |
---|
| 1077 | whd in ⊢ (??(????%?)?) -labels1; |
---|
| 1078 | cases label in Hid |
---|
| 1079 | [ #Hid whd in ⊢ (??(????%?)?) >IH1 -IH1 |
---|
| 1080 | [ >(address_of_word_labels_code_mem_None … Hid) |
---|
| 1081 | (* MANCA LEMMA: INDIRIZZO TROVATO NEL PROGRAMMA! *) |
---|
| 1082 | | whd in Hid >occurs_exactly_once_None in Hid // ] |
---|
| 1083 | | -label #label #Hid whd in ⊢ (??(????%?)?) |
---|
| 1084 | |
---|
| 1085 | ] |
---|
| 1086 | qed. |
---|
| 1087 | |
---|
| 1088 | lemma build_maps_ok: |
---|
| 1089 | ∀p:pseudo_assembly_program. |
---|
| 1090 | let 〈labels,costs〉 ≝ build_maps' p in |
---|
| 1091 | ∀pc. |
---|
| 1092 | (nat_of_bitvector … pc) < length … (\snd p) → |
---|
| 1093 | lookup ?? pc labels (zero …) = sigma p (\snd (fetch_pseudo_instruction (\snd p) pc)). |
---|
| 1094 | #p cases p #preamble #instr_list |
---|
| 1095 | elim instr_list |
---|
| 1096 | [ whd #pc #abs normalize in abs; cases (not_le_Sn_O ?) [#H cases (H abs) ] |
---|
| 1097 | | #hd #tl #IH |
---|
| 1098 | whd in ⊢ (match % with [ _ ⇒ ?]) |
---|
| 1099 | ] |
---|
| 1100 | qed. |
---|
| 1101 | *) |
---|
| 1102 | |
---|
[906] | 1103 | (* |
---|
[877] | 1104 | lemma rev_preserves_length: |
---|
| 1105 | ∀A.∀l. length … (rev A l) = length … l. |
---|
| 1106 | #A #l elim l |
---|
| 1107 | [ % |
---|
| 1108 | | #hd #tl #IH normalize >length_append normalize /2/ ] |
---|
| 1109 | qed. |
---|
| 1110 | |
---|
| 1111 | lemma rev_append: |
---|
| 1112 | ∀A.∀l1,l2. |
---|
| 1113 | rev A (l1@l2) = rev A l2 @ rev A l1. |
---|
| 1114 | #A #l1 elim l1 normalize // |
---|
| 1115 | qed. |
---|
| 1116 | |
---|
| 1117 | lemma rev_rev: ∀A.∀l. rev … (rev A l) = l. |
---|
| 1118 | #A #l elim l |
---|
| 1119 | [ // |
---|
| 1120 | | #hd #tl #IH normalize >rev_append normalize // ] |
---|
| 1121 | qed. |
---|
| 1122 | |
---|
| 1123 | lemma split_len_Sn: |
---|
| 1124 | ∀A:Type[0].∀l:list A.∀len. |
---|
| 1125 | length … l = S len → |
---|
| 1126 | Σl'.Σa. l = l'@[a] ∧ length … l' = len. |
---|
| 1127 | #A #l elim l |
---|
| 1128 | [ normalize #len #abs destruct |
---|
| 1129 | | #hd #tl #IH #len |
---|
| 1130 | generalize in match (rev_rev … tl) |
---|
| 1131 | cases (rev A tl) in ⊢ (??%? → ?) |
---|
| 1132 | [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ |
---|
| 1133 | | #a #l' #H <H normalize #EQ |
---|
| 1134 | %[@(hd::rev … l')] %[@a] % // |
---|
| 1135 | >length_append in EQ #EQ normalize in EQ; normalize; |
---|
| 1136 | generalize in match (injective_S … EQ) #EQ2 /2/ ]] |
---|
| 1137 | qed. |
---|
| 1138 | |
---|
| 1139 | lemma list_elim_rev: |
---|
| 1140 | ∀A:Type[0].∀P:list A → Type[0]. |
---|
| 1141 | P [ ] → (∀l,a. P l → P (l@[a])) → |
---|
| 1142 | ∀l. P l. |
---|
| 1143 | #A #P #H1 #H2 #l |
---|
| 1144 | generalize in match (refl … (length … l)) |
---|
| 1145 | generalize in ⊢ (???% → ?) #n generalize in match l |
---|
| 1146 | elim n |
---|
| 1147 | [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ] |
---|
| 1148 | | #m #IH #L #EQ |
---|
| 1149 | cases (split_len_Sn … EQ) #l' * #a * /3/ ] |
---|
| 1150 | qed. |
---|
| 1151 | |
---|
| 1152 | axiom is_prefix: ∀A:Type[0]. list A → list A → Prop. |
---|
| 1153 | axiom prefix_of_append: |
---|
| 1154 | ∀A:Type[0].∀l,l1,l2:list A. |
---|
| 1155 | is_prefix … l l1 → is_prefix … l (l1@l2). |
---|
| 1156 | axiom prefix_reflexive: ∀A,l. is_prefix A l l. |
---|
| 1157 | axiom nil_prefix: ∀A,l. is_prefix A [ ] l. |
---|
| 1158 | |
---|
| 1159 | record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }. |
---|
| 1160 | |
---|
| 1161 | definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝ |
---|
| 1162 | λA,P,H,x. match x with [ mk_Propify p ⇒ H p ]. |
---|
| 1163 | |
---|
| 1164 | definition app ≝ |
---|
| 1165 | λA:Type[0].λl1:Propify (list A).λl2:list A. |
---|
| 1166 | match l1 with |
---|
| 1167 | [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ]. |
---|
| 1168 | |
---|
| 1169 | lemma app_nil: ∀A,l1. app A l1 [ ] = l1. |
---|
| 1170 | #A * /3/ |
---|
| 1171 | qed. |
---|
| 1172 | |
---|
| 1173 | lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3). |
---|
| 1174 | #A * #l1 normalize // |
---|
| 1175 | qed. |
---|
| 1176 | |
---|
| 1177 | let rec foldli (A: Type[0]) (B: Propify (list A) → Type[0]) |
---|
| 1178 | (f: ∀prefix. B prefix → ∀x.B (app … prefix [x])) |
---|
| 1179 | (prefix: Propify (list A)) (b: B prefix) (l: list A) on l : |
---|
| 1180 | B (app … prefix l) ≝ |
---|
| 1181 | match l with |
---|
| 1182 | [ nil ⇒ ? (* b *) |
---|
| 1183 | | cons hd tl ⇒ ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
| 1184 | ]. |
---|
| 1185 | [ applyS b |
---|
| 1186 | | <(app_assoc ?? [hd]) @(foldli A B f (app … prefix [hd]) (f prefix b hd) tl) ] |
---|
| 1187 | qed. |
---|
| 1188 | |
---|
| 1189 | (* |
---|
| 1190 | let rec foldli (A: Type[0]) (B: list A → Type[0]) (f: ∀prefix. B prefix → ∀x. B (prefix@[x])) |
---|
| 1191 | (prefix: list A) (b: B prefix) (l: list A) on l : B (prefix@l) ≝ |
---|
| 1192 | match l with |
---|
| 1193 | [ nil ⇒ ? (* b *) |
---|
| 1194 | | cons hd tl ⇒ |
---|
| 1195 | ? (*foldli A B f (prefix@[hd]) (f prefix b hd) tl*) |
---|
| 1196 | ]. |
---|
| 1197 | [ applyS b |
---|
| 1198 | | applyS (foldli A B f (prefix@[hd]) (f prefix b hd) tl) ] |
---|
| 1199 | qed. |
---|
| 1200 | *) |
---|
| 1201 | |
---|
| 1202 | definition foldll: |
---|
| 1203 | ∀A:Type[0].∀B: Propify (list A) → Type[0]. |
---|
| 1204 | (∀prefix. B prefix → ∀x. B (app … prefix [x])) → |
---|
| 1205 | B (mk_Propify … []) → ∀l: list A. B (mk_Propify … l) |
---|
| 1206 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
| 1207 | |
---|
| 1208 | axiom is_pprefix: ∀A:Type[0]. Propify (list A) → list A → Prop. |
---|
| 1209 | axiom pprefix_of_append: |
---|
| 1210 | ∀A:Type[0].∀l,l1,l2. |
---|
| 1211 | is_pprefix A l l1 → is_pprefix A l (l1@l2). |
---|
| 1212 | axiom pprefix_reflexive: ∀A,l. is_pprefix A (mk_Propify … l) l. |
---|
| 1213 | axiom nil_pprefix: ∀A,l. is_pprefix A (mk_Propify … [ ]) l. |
---|
| 1214 | |
---|
| 1215 | |
---|
| 1216 | axiom foldll': |
---|
| 1217 | ∀A:Type[0].∀l: list A. |
---|
| 1218 | ∀B: ∀prefix:Propify (list A). is_pprefix ? prefix l → Type[0]. |
---|
| 1219 | (∀prefix,proof. B prefix proof → ∀x,proof'. B (app … prefix [x]) proof') → |
---|
| 1220 | B (mk_Propify … [ ]) (nil_pprefix …) → B (mk_Propify … l) (pprefix_reflexive … l). |
---|
| 1221 | #A #l #B |
---|
| 1222 | generalize in match (foldll A (λprefix. is_pprefix ? prefix l)) #HH |
---|
| 1223 | |
---|
| 1224 | |
---|
| 1225 | #H #acc |
---|
| 1226 | @foldll |
---|
| 1227 | [ |
---|
| 1228 | | |
---|
| 1229 | ] |
---|
| 1230 | |
---|
| 1231 | ≝ λA,B,f. foldli A B f (mk_Propify … [ ]). |
---|
| 1232 | |
---|
| 1233 | |
---|
| 1234 | (* |
---|
| 1235 | record subset (A:Type[0]) (P: A → Prop): Type[0] ≝ |
---|
| 1236 | { subset_wit:> A; |
---|
| 1237 | subset_proof: P subset_wit |
---|
| 1238 | }. |
---|
| 1239 | *) |
---|
| 1240 | |
---|
| 1241 | definition build_maps' ≝ |
---|
| 1242 | λpseudo_program. |
---|
| 1243 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
| 1244 | let result ≝ |
---|
| 1245 | foldll |
---|
| 1246 | (option Identifier × pseudo_instruction) |
---|
| 1247 | (λprefix. |
---|
| 1248 | Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 1249 | match prefix return λ_.Prop with [mk_Propify prefix ⇒ tech_pc_sigma0 〈preamble,prefix〉 ≠ None ?]) |
---|
| 1250 | (λprefix,t,i. |
---|
| 1251 | let 〈labels, pc_costs〉 ≝ t in |
---|
| 1252 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
| 1253 | let 〈label, i'〉 ≝ i in |
---|
| 1254 | let labels ≝ |
---|
| 1255 | match label with |
---|
| 1256 | [ None ⇒ labels |
---|
| 1257 | | Some label ⇒ |
---|
| 1258 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 1259 | insert ? ? label program_counter_bv labels |
---|
| 1260 | ] |
---|
| 1261 | in |
---|
| 1262 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
| 1263 | [ None ⇒ |
---|
| 1264 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
| 1265 | dummy |
---|
| 1266 | | Some construct ⇒ 〈labels, construct〉 |
---|
| 1267 | ] |
---|
| 1268 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 instr_list |
---|
| 1269 | in |
---|
| 1270 | let 〈labels, pc_costs〉 ≝ result in |
---|
| 1271 | let 〈pc, costs〉 ≝ pc_costs in |
---|
| 1272 | 〈labels, costs〉. |
---|
| 1273 | [ |
---|
| 1274 | | @⊥ |
---|
| 1275 | | normalize % // |
---|
| 1276 | ] |
---|
| 1277 | qed. |
---|
| 1278 | |
---|
| 1279 | definition build_maps' ≝ |
---|
| 1280 | λpseudo_program. |
---|
| 1281 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
---|
| 1282 | let result ≝ |
---|
| 1283 | foldl |
---|
| 1284 | (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 1285 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
| 1286 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t))) |
---|
| 1287 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 1288 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 1289 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 1290 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?) |
---|
| 1291 | (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
---|
| 1292 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
---|
| 1293 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)). |
---|
| 1294 | λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 1295 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 1296 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 1297 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? . |
---|
| 1298 | let 〈labels, pc_costs〉 ≝ t in |
---|
| 1299 | let 〈program_counter, costs〉 ≝ pc_costs in |
---|
| 1300 | let 〈label, i'〉 ≝ i in |
---|
| 1301 | let labels ≝ |
---|
| 1302 | match label with |
---|
| 1303 | [ None ⇒ labels |
---|
| 1304 | | Some label ⇒ |
---|
| 1305 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
---|
| 1306 | insert ? ? label program_counter_bv labels |
---|
| 1307 | ] |
---|
| 1308 | in |
---|
| 1309 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
---|
| 1310 | [ None ⇒ |
---|
| 1311 | let dummy ≝ 〈labels,pc_costs〉 in |
---|
| 1312 | dummy |
---|
| 1313 | | Some construct ⇒ 〈labels, construct〉 |
---|
| 1314 | ] |
---|
| 1315 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 ?(*instr_list*) |
---|
| 1316 | in |
---|
| 1317 | let 〈labels, pc_costs〉 ≝ result in |
---|
| 1318 | let 〈pc, costs〉 ≝ pc_costs in |
---|
| 1319 | 〈labels, costs〉. |
---|
| 1320 | [4: @(list_elim_rev ? |
---|
| 1321 | (λinstr_list. list ( |
---|
| 1322 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
---|
| 1323 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
---|
| 1324 | is_prefix ? instr_list_prefix' instr_list → |
---|
| 1325 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?))) |
---|
| 1326 | ?? instr_list) (* CSC: BAD ORDER FOR CODE EXTRACTION *) |
---|
| 1327 | [ @[ ] |
---|
| 1328 | | #l' #a #limage %2 |
---|
| 1329 | [ %[@a] #PREFIX #PREFIX_OK |
---|
| 1330 | | (* CSC: EVEN WORST CODE FOR EXTRACTION: WE SHOULD STRENGTHEN |
---|
| 1331 | THE INDUCTION HYPOTHESIS INSTEAD *) |
---|
| 1332 | elim limage |
---|
| 1333 | [ %1 |
---|
| 1334 | | #HD #TL #IH @(?::IH) cases HD #ELEM #K1 %[@ELEM] #K2 #K3 |
---|
| 1335 | @K1 @(prefix_of_append ???? K3) |
---|
| 1336 | ] |
---|
| 1337 | ] |
---|
| 1338 | |
---|
| 1339 | |
---|
| 1340 | |
---|
| 1341 | |
---|
| 1342 | cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt' |
---|
| 1343 | % [@ (LIST_PREFIX @ [i])] % |
---|
| 1344 | [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1 |
---|
| 1345 | | (* DOABLE IN PRINCIPLE *) |
---|
| 1346 | ] |
---|
| 1347 | | (* assert false case *) |
---|
| 1348 | |3: % [@ ([ ])] % [2: % | (* DOABLE *)] |
---|
| 1349 | | |
---|
[906] | 1350 | *) |
---|
[877] | 1351 | |
---|
[906] | 1352 | axiom assembly_ok: |
---|
[915] | 1353 | ∀program,assembled,costs,labels. |
---|
| 1354 | Some … 〈labels,costs〉 = build_maps program → |
---|
[906] | 1355 | Some … 〈assembled,costs〉 = assembly program → |
---|
| 1356 | let code_memory ≝ load_code_memory assembled in |
---|
[915] | 1357 | let preamble ≝ \fst program in |
---|
| 1358 | let datalabels ≝ construct_datalabels preamble in |
---|
| 1359 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
[906] | 1360 | let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in |
---|
| 1361 | ∀ppc,len,assembledi. |
---|
| 1362 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
[921] | 1363 | Some … 〈len,assembledi〉 = assembly_1_pseudoinstruction program ppc (sigma program ppc) lookup_labels lookup_datalabels pi → |
---|
[906] | 1364 | encoding_check code_memory (sigma program ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len)) assembledi ∧ |
---|
| 1365 | sigma program newppc = bitvector_of_nat … (nat_of_bitvector … (sigma program ppc) + len). |
---|
| 1366 | |
---|
[908] | 1367 | axiom bitvector_of_nat_nat_of_bitvector: |
---|
[915] | 1368 | ∀n,v. |
---|
| 1369 | bitvector_of_nat n (nat_of_bitvector n v) = v. |
---|
[908] | 1370 | |
---|
[924] | 1371 | axiom assembly_ok_to_expand_pseudo_instruction_ok: |
---|
| 1372 | ∀program,assembled,costs. |
---|
| 1373 | Some … 〈assembled,costs〉 = assembly program → |
---|
| 1374 | ∀ppc. |
---|
| 1375 | let code_memory ≝ load_code_memory assembled in |
---|
| 1376 | let preamble ≝ \fst program in |
---|
| 1377 | let data_labels ≝ construct_datalabels preamble in |
---|
| 1378 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
| 1379 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
| 1380 | let expansion ≝ jump_expansion ppc program in |
---|
| 1381 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
| 1382 | ∃instructions. |
---|
| 1383 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi. |
---|
| 1384 | |
---|
[908] | 1385 | lemma fetch_assembly_pseudo2: |
---|
[915] | 1386 | ∀program,assembled,costs,labels. |
---|
| 1387 | Some … 〈labels,costs〉 = build_maps program → |
---|
[908] | 1388 | Some … 〈assembled,costs〉 = assembly program → |
---|
[924] | 1389 | ∀ppc. |
---|
[915] | 1390 | let code_memory ≝ load_code_memory assembled in |
---|
| 1391 | let preamble ≝ \fst program in |
---|
| 1392 | let data_labels ≝ construct_datalabels preamble in |
---|
| 1393 | let lookup_labels ≝ λx. sigma program (address_of_word_labels_code_mem (\snd program) x) in |
---|
| 1394 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
[916] | 1395 | let expansion ≝ jump_expansion ppc program in |
---|
[908] | 1396 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
[924] | 1397 | ∃instructions. |
---|
| 1398 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program ppc) expansion pi ∧ |
---|
| 1399 | fetch_many code_memory (sigma program newppc) (sigma program ppc) instructions. |
---|
| 1400 | #program #assembled #costs #labels #BUILD_MAPS #ASSEMBLY #ppc |
---|
| 1401 | generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program assembled costs ASSEMBLY ppc) |
---|
[915] | 1402 | letin code_memory ≝ (load_code_memory assembled) |
---|
| 1403 | letin preamble ≝ (\fst program) |
---|
| 1404 | letin data_labels ≝ (construct_datalabels preamble) |
---|
| 1405 | letin lookup_labels ≝ (λx. sigma program (address_of_word_labels_code_mem (\snd program) x)) |
---|
| 1406 | letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?)) |
---|
[924] | 1407 | whd in ⊢ (% → %) |
---|
[908] | 1408 | generalize in match (assembly_ok … BUILD_MAPS ASSEMBLY ppc) |
---|
| 1409 | cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc |
---|
| 1410 | generalize in match (fetch_assembly_pseudo program ppc |
---|
[915] | 1411 | (λx. sigma program (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi |
---|
| 1412 | (load_code_memory assembled)); |
---|
[924] | 1413 | whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND |
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| 1414 | whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?); |
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[921] | 1415 | normalize nodelta in EXPAND; (* HERE *) |
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| 1416 | generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program ppc))) -H1; #H1 |
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| 1417 | >bitvector_of_nat_nat_of_bitvector in H1; #H1 |
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| 1418 | <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?) |
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[908] | 1419 | #H1 #H2 |
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| 1420 | cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2 |
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[921] | 1421 | generalize in match (H1 ?? (refl …) (refl …) ?) -H1; |
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[924] | 1422 | [ #K3 % [2: % [% | @K3]] | @K1 ] |
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[908] | 1423 | qed. |
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| 1424 | |
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[906] | 1425 | (* OLD? |
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[877] | 1426 | definition assembly_specification: |
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| 1427 | ∀assembly_program: pseudo_assembly_program. |
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| 1428 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
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| 1429 | λpseudo_assembly_program. |
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| 1430 | λcode_mem. |
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| 1431 | ∀pc: Word. |
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| 1432 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
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| 1433 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
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| 1434 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
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| 1435 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
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| 1436 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
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| 1437 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
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| 1438 | match pre_assembled with |
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| 1439 | [ None ⇒ True |
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| 1440 | | Some pc_code ⇒ |
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| 1441 | let 〈new_pc,code〉 ≝ pc_code in |
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| 1442 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
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| 1443 | |
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| 1444 | axiom assembly_meets_specification: |
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| 1445 | ∀pseudo_assembly_program. |
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| 1446 | match assembly pseudo_assembly_program with |
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| 1447 | [ None ⇒ True |
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| 1448 | | Some code_mem_cost ⇒ |
---|
| 1449 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
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| 1450 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
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| 1451 | ]. |
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| 1452 | (* |
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| 1453 | # PROGRAM |
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| 1454 | [ cases PROGRAM |
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| 1455 | # PREAMBLE |
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| 1456 | # INSTR_LIST |
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| 1457 | elim INSTR_LIST |
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| 1458 | [ whd |
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| 1459 | whd in ⊢ (∀_. %) |
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| 1460 | # PC |
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| 1461 | whd |
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| 1462 | | # INSTR |
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| 1463 | # INSTR_LIST_TL |
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| 1464 | # H |
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| 1465 | whd |
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| 1466 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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| 1467 | ] |
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| 1468 | | cases not_implemented |
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| 1469 | ] *) |
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[906] | 1470 | *) |
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[877] | 1471 | |
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| 1472 | definition status_of_pseudo_status: PseudoStatus → option Status ≝ |
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| 1473 | λps. |
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| 1474 | let pap ≝ code_memory … ps in |
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| 1475 | match assembly pap with |
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| 1476 | [ None ⇒ None … |
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| 1477 | | Some p ⇒ |
---|
| 1478 | let cm ≝ load_code_memory (\fst p) in |
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[909] | 1479 | let pc ≝ sigma pap (program_counter ? ps) in |
---|
[877] | 1480 | Some … |
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| 1481 | (mk_PreStatus (BitVectorTrie Byte 16) |
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| 1482 | cm |
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| 1483 | (low_internal_ram … ps) |
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| 1484 | (high_internal_ram … ps) |
---|
| 1485 | (external_ram … ps) |
---|
| 1486 | pc |
---|
| 1487 | (special_function_registers_8051 … ps) |
---|
| 1488 | (special_function_registers_8052 … ps) |
---|
| 1489 | (p1_latch … ps) |
---|
| 1490 | (p3_latch … ps) |
---|
| 1491 | (clock … ps)) ]. |
---|
| 1492 | |
---|
[909] | 1493 | (* |
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[877] | 1494 | definition write_at_stack_pointer': |
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| 1495 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
| 1496 | λM: Type[0]. |
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| 1497 | λs: PreStatus M. |
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| 1498 | λv: Byte. |
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| 1499 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
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| 1500 | let bit_zero ≝ get_index_v… nu O ? in |
---|
| 1501 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
| 1502 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
| 1503 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
| 1504 | if bit_zero then |
---|
| 1505 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
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| 1506 | v (low_internal_ram ? s) in |
---|
| 1507 | set_low_internal_ram ? s memory |
---|
| 1508 | else |
---|
| 1509 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
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| 1510 | v (high_internal_ram ? s) in |
---|
| 1511 | set_high_internal_ram ? s memory. |
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| 1512 | [ cases l0 % |
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| 1513 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
| 1514 | qed. |
---|
| 1515 | |
---|
| 1516 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
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| 1517 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
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| 1518 | ≝ |
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| 1519 | λticks_of. |
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| 1520 | λs. |
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| 1521 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
| 1522 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
| 1523 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
| 1524 | let s ≝ set_program_counter ? s pc in |
---|
| 1525 | match instr with |
---|
| 1526 | [ Instruction instr ⇒ |
---|
| 1527 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
| 1528 | | Comment cmt ⇒ s |
---|
| 1529 | | Cost cst ⇒ s |
---|
| 1530 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
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| 1531 | | Call call ⇒ |
---|
| 1532 | let a ≝ address_of_word_labels s call in |
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| 1533 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1534 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
| 1535 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
| 1536 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
| 1537 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1538 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
| 1539 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
| 1540 | set_program_counter ? s a |
---|
| 1541 | | Mov dptr ident ⇒ |
---|
| 1542 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
| 1543 | ]. |
---|
| 1544 | [ |
---|
| 1545 | |2,3,4: % |
---|
| 1546 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
| 1547 | | |
---|
| 1548 | | % |
---|
| 1549 | ] |
---|
| 1550 | cases not_implemented |
---|
| 1551 | qed. |
---|
[909] | 1552 | *) |
---|
[911] | 1553 | |
---|
| 1554 | axiom execute_1_pseudo_instruction_preserves_code_memory: |
---|
| 1555 | ∀ticks_of,ps. |
---|
| 1556 | code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps. |
---|
| 1557 | |
---|
[877] | 1558 | (* |
---|
| 1559 | lemma execute_code_memory_unchanged: |
---|
| 1560 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
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| 1561 | #ticks #ps whd in ⊢ (??? (??%)) |
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| 1562 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
| 1563 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
| 1564 | whd in ⊢ (??? (??%)) cases instr |
---|
| 1565 | [ #pre cases pre |
---|
| 1566 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1567 | cases (split ????) #z1 #z2 % |
---|
| 1568 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1569 | cases (split ????) #z1 #z2 % |
---|
| 1570 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1571 | cases (split ????) #z1 #z2 % |
---|
| 1572 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
| 1573 | [ #x1 whd in ⊢ (??? (??%)) |
---|
| 1574 | | *: cases not_implemented |
---|
| 1575 | ] |
---|
| 1576 | | #comment % |
---|
| 1577 | | #cost % |
---|
| 1578 | | #label % |
---|
| 1579 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
| 1580 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
| 1581 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
---|
| 1582 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
| 1583 | (* CSC: ??? *) |
---|
| 1584 | | #dptr #label (* CSC: ??? *) |
---|
| 1585 | ] |
---|
| 1586 | cases not_implemented |
---|
| 1587 | qed. |
---|
| 1588 | *) |
---|
| 1589 | |
---|
| 1590 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
| 1591 | ∀ps,ps': PseudoStatus. |
---|
| 1592 | code_memory … ps = code_memory … ps' → |
---|
| 1593 | match status_of_pseudo_status ps with |
---|
| 1594 | [ None ⇒ status_of_pseudo_status ps' = None … |
---|
| 1595 | | Some _ ⇒ ∃w. status_of_pseudo_status ps' = Some … w |
---|
| 1596 | ]. |
---|
[909] | 1597 | #ps #ps' #H whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
[877] | 1598 | generalize in match (refl … (assembly (code_memory … ps))) |
---|
| 1599 | cases (assembly ?) in ⊢ (???% → %) |
---|
| 1600 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
| 1601 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
[909] | 1602 | qed. |
---|
[877] | 1603 | |
---|
| 1604 | lemma main_thm: |
---|
| 1605 | ∀ticks_of. |
---|
[919] | 1606 | ∀ps,s,s''. |
---|
| 1607 | status_of_pseudo_status ps = Some … s → |
---|
| 1608 | status_of_pseudo_status (execute_1_pseudo_instruction ticks_of ps) = Some … s'' → |
---|
| 1609 | execute_1 s = s''. |
---|
[923] | 1610 | #ticks_of #ps #s #s'' |
---|
| 1611 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps)) |
---|
| 1612 | whd in ⊢ (? → ??%? → ??%? → ?) |
---|
[911] | 1613 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
[923] | 1614 | generalize in match (refl … (assembly (code_memory … ps))) |
---|
| 1615 | generalize in match (assembly (code_memory … ps)) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?) |
---|
| 1616 | cases (build_maps (code_memory … ps)) |
---|
| 1617 | [ cases (code_memory … ps) #preamble #instr_list whd in ⊢ (???% → ?) #EQ >EQ #H |
---|
| 1618 | #abs @⊥ normalize in abs; destruct (abs) ] |
---|
[925] | 1619 | * #labels #costs generalize in match (refl … (code_memory … ps)) |
---|
| 1620 | cases (code_memory … ps) in ⊢ (???% → %) #preamble #instr_list #EQ normalize nodelta; |
---|
[923] | 1621 | generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta; |
---|
| 1622 | [ #EQ >EQ #_ #abs @⊥ normalize in abs; destruct (abs) ] |
---|
| 1623 | * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta; |
---|
| 1624 | #H generalize in match (H ??? (refl …) (refl …)) -H; #H; |
---|
[919] | 1625 | #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1; |
---|
| 1626 | #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2; |
---|
| 1627 | change with |
---|
| 1628 | (execute_1 (set_program_counter ? (set_code_memory ?? ps (load_code_memory assembled)) ?) |
---|
| 1629 | = set_program_counter ? (set_code_memory ?? (execute_1_pseudo_instruction ticks_of ps) (load_code_memory assembled)) ?) |
---|
[923] | 1630 | whd in match (\snd 〈preamble,instr_list〉) in H; |
---|
| 1631 | whd in match (\fst 〈preamble,instr_list〉) in H; |
---|
| 1632 | whd in match (\snd 〈final_pc,assembled〉) in H; |
---|
[924] | 1633 | -s s'' labels costs final_ppc final_pc; |
---|
[925] | 1634 | letin ps' ≝ (execute_1_pseudo_instruction ticks_of ps) |
---|
| 1635 | (* NICE STATEMENT HERE *) |
---|
| 1636 | generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps' |
---|
| 1637 | whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) >EQ normalize nodelta; |
---|
| 1638 | cases (fetch_pseudo_instruction instr_list (program_counter … ps)) |
---|
| 1639 | #pi #newppc normalize nodelta; * #instructions * |
---|
| 1640 | cases pi normalize nodelta; |
---|
| 1641 | [ (* Instruction *) |
---|
| 1642 | | (* Comment *) #comment #H1 #H2 #EQ normalize in H1 generalize in match (option_destruct_Some ??? H1) #K1 |
---|
| 1643 | >K1 in H2 whd in ⊢ (% → ?) #H2 whd in ⊢ (??%?) |
---|
| 1644 | | (* Cost *) |
---|
| 1645 | | (* Jmp *) |
---|
| 1646 | | (* Call *) |
---|
| 1647 | | (* Mov *) |
---|
| 1648 | ] |
---|
| 1649 | qed. |
---|