[877] | 1 | include "ASM/Assembly.ma". |
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| 2 | include "ASM/Interpret.ma". |
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| 3 | |
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[905] | 4 | definition bit_elim_prop: ∀P: bool → Prop. Prop ≝ |
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| 5 | λP. |
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| 6 | P true ∧ P false. |
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| 7 | |
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| 8 | let rec bitvector_elim_prop_internal |
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| 9 | (n: nat) (P: BitVector n → Prop) (m: nat) on m: m ≤ n → BitVector (n - m) → Prop ≝ |
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| 10 | match m return λm. m ≤ n → BitVector (n - m) → Prop with |
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| 11 | [ O ⇒ λprf1. λprefix. P ? |
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| 12 | | S n' ⇒ λprf2. λprefix. bit_elim_prop (λbit. bitvector_elim_prop_internal n P n' ? ?) |
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| 13 | ]. |
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| 14 | [ applyS prefix |
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| 15 | | letin res ≝ (bit ::: prefix) |
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| 16 | < (minus_S_S ? ?) |
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| 17 | > (minus_Sn_m ? ?) |
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| 18 | [ @ res |
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| 19 | | @ prf2 |
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| 20 | ] |
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| 21 | | /2/ |
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| 22 | ]. |
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| 23 | qed. |
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| 24 | |
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| 25 | definition bitvector_elim_prop ≝ |
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| 26 | λn: nat. |
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| 27 | λP: BitVector n → Prop. |
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| 28 | bitvector_elim_prop_internal n P n ? ?. |
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| 29 | [ @ (le_n ?) |
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| 30 | | < (minus_n_n ?) |
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| 31 | @ [[ ]] |
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| 32 | ] |
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| 33 | qed. |
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| 34 | |
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| 35 | lemma bool_eq_internal_eq: |
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| 36 | ∀b, c. |
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| 37 | (λb. λc. (if b then c else (if c then false else true))) b c = true → b = c. |
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| 38 | #b #c |
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| 39 | cases b |
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| 40 | [ normalize // |
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| 41 | | normalize |
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| 42 | cases c |
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| 43 | [ normalize // |
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| 44 | | normalize // |
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| 45 | ] |
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| 46 | ] |
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| 47 | qed. |
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| 48 | |
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[877] | 49 | definition bit_elim: ∀P: bool → bool. bool ≝ |
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| 50 | λP. |
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| 51 | P true ∧ P false. |
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| 52 | |
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| 53 | let rec bitvector_elim_internal |
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| 54 | (n: nat) (P: BitVector n → bool) (m: nat) on m: m ≤ n → BitVector (n - m) → bool ≝ |
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| 55 | match m return λm. m ≤ n → BitVector (n - m) → bool with |
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| 56 | [ O ⇒ λprf1. λprefix. P ? |
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| 57 | | S n' ⇒ λprf2. λprefix. bit_elim (λbit. bitvector_elim_internal n P n' ? ?) |
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| 58 | ]. |
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| 59 | [ applyS prefix |
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| 60 | | letin res ≝ (bit ::: prefix) |
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| 61 | < (minus_S_S ? ?) |
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| 62 | > (minus_Sn_m ? ?) |
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| 63 | [ @ res |
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| 64 | | @ prf2 |
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| 65 | ] |
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| 66 | | /2/ |
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| 67 | ]. |
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| 68 | qed. |
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| 69 | |
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| 70 | definition bitvector_elim ≝ |
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| 71 | λn: nat. |
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| 72 | λP: BitVector n → bool. |
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| 73 | bitvector_elim_internal n P n ? ?. |
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| 74 | [ @ (le_n ?) |
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| 75 | | < (minus_n_n ?) |
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| 76 | @ [[ ]] |
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| 77 | ] |
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| 78 | qed. |
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| 79 | |
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| 80 | axiom vector_associative_append: |
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| 81 | ∀A: Type[0]. |
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| 82 | ∀n, m, o: nat. |
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| 83 | ∀v: Vector A n. |
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| 84 | ∀q: Vector A m. |
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| 85 | ∀r: Vector A o. |
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| 86 | ((v @@ q) @@ r) |
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| 87 | ≃ |
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| 88 | (v @@ (q @@ r)). |
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| 89 | |
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| 90 | lemma vector_cons_append: |
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| 91 | ∀A: Type[0]. |
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| 92 | ∀n: nat. |
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| 93 | ∀e: A. |
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| 94 | ∀v: Vector A n. |
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| 95 | e ::: v = [[ e ]] @@ v. |
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| 96 | # A # N # E # V |
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| 97 | elim V |
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| 98 | [ normalize % |
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| 99 | | # NN # AA # VV # IH |
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| 100 | normalize |
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| 101 | % |
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| 102 | ] |
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| 103 | qed. |
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| 104 | |
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| 105 | lemma super_rewrite2: |
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| 106 | ∀A:Type[0].∀n,m.∀v1: Vector A n.∀v2: Vector A m. |
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| 107 | ∀P: ∀m. Vector A m → Prop. |
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| 108 | n=m → v1 ≃ v2 → P n v1 → P m v2. |
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| 109 | #A #n #m #v1 #v2 #P #EQ <EQ in v2; #V #JMEQ >JMEQ // |
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| 110 | qed. |
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| 111 | |
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| 112 | lemma mem_middle_vector: |
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| 113 | ∀A: Type[0]. |
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| 114 | ∀m, o: nat. |
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| 115 | ∀eq: A → A → bool. |
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| 116 | ∀reflex: ∀a. eq a a = true. |
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| 117 | ∀p: Vector A m. |
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| 118 | ∀a: A. |
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| 119 | ∀r: Vector A o. |
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| 120 | mem A eq ? (p@@(a:::r)) a = true. |
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| 121 | # A # M # O # EQ # REFLEX # P # A |
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| 122 | elim P |
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| 123 | [ normalize |
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| 124 | > (REFLEX A) |
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| 125 | normalize |
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| 126 | # H |
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| 127 | % |
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| 128 | | # NN # AA # PP # IH |
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| 129 | normalize |
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| 130 | cases (EQ A AA) // |
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| 131 | @ IH |
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| 132 | ] |
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| 133 | qed. |
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| 134 | |
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| 135 | lemma mem_monotonic_wrt_append: |
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| 136 | ∀A: Type[0]. |
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| 137 | ∀m, o: nat. |
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| 138 | ∀eq: A → A → bool. |
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| 139 | ∀reflex: ∀a. eq a a = true. |
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| 140 | ∀p: Vector A m. |
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| 141 | ∀a: A. |
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| 142 | ∀r: Vector A o. |
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| 143 | mem A eq ? r a = true → mem A eq ? (p @@ r) a = true. |
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| 144 | # A # M # O # EQ # REFLEX # P # A |
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| 145 | elim P |
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| 146 | [ #R #H @H |
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| 147 | | #NN #AA # PP # IH #R #H |
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| 148 | normalize |
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| 149 | cases (EQ A AA) |
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| 150 | [ normalize % |
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| 151 | | @ IH @ H |
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| 152 | ] |
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| 153 | ] |
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| 154 | qed. |
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| 155 | |
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| 156 | lemma subvector_multiple_append: |
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| 157 | ∀A: Type[0]. |
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| 158 | ∀o, n: nat. |
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| 159 | ∀eq: A → A → bool. |
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| 160 | ∀refl: ∀a. eq a a = true. |
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| 161 | ∀h: Vector A o. |
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| 162 | ∀v: Vector A n. |
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| 163 | ∀m: nat. |
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| 164 | ∀q: Vector A m. |
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| 165 | bool_to_Prop (subvector_with A ? ? eq v (h @@ q @@ v)). |
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| 166 | # A # O # N # EQ # REFLEX # H # V |
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| 167 | elim V |
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| 168 | [ normalize |
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| 169 | # M # V % |
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| 170 | | # NN # AA # VV # IH # MM # QQ |
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| 171 | change with (bool_to_Prop (andb ??)) |
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| 172 | cut ((mem A EQ (O + (MM + S NN)) (H@@QQ@@AA:::VV) AA) = true) |
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| 173 | [ |
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| 174 | | # HH > HH |
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| 175 | > (vector_cons_append ? ? AA VV) |
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| 176 | change with (bool_to_Prop (subvector_with ??????)) |
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| 177 | @(super_rewrite2 A ((MM + 1)+ NN) (MM+S NN) ?? |
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| 178 | (λSS.λVS.bool_to_Prop (subvector_with ?? (O+SS) ?? (H@@VS))) |
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| 179 | ? |
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| 180 | (vector_associative_append A ? ? ? QQ [[AA]] VV)) |
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| 181 | [ >associative_plus // |
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| 182 | | @IH ] |
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| 183 | ] |
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| 184 | @(mem_monotonic_wrt_append) |
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| 185 | [ @ REFLEX |
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| 186 | | @(mem_monotonic_wrt_append) |
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| 187 | [ @ REFLEX |
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| 188 | | normalize |
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| 189 | > REFLEX |
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| 190 | normalize |
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| 191 | % |
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| 192 | ] |
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| 193 | ] |
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| 194 | qed. |
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| 195 | |
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| 196 | lemma vector_cons_empty: |
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| 197 | ∀A: Type[0]. |
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| 198 | ∀n: nat. |
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| 199 | ∀v: Vector A n. |
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| 200 | [[ ]] @@ v = v. |
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| 201 | # A # N # V |
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| 202 | elim V |
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| 203 | [ normalize % |
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| 204 | | # NN # HH # VV #H % |
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| 205 | ] |
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| 206 | qed. |
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| 207 | |
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| 208 | corollary subvector_hd_tl: |
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| 209 | ∀A: Type[0]. |
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| 210 | ∀o: nat. |
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| 211 | ∀eq: A → A → bool. |
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| 212 | ∀refl: ∀a. eq a a = true. |
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| 213 | ∀h: A. |
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| 214 | ∀v: Vector A o. |
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| 215 | bool_to_Prop (subvector_with A ? ? eq v (h ::: v)). |
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| 216 | # A # O # EQ # REFLEX # H # V |
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| 217 | > (vector_cons_append A ? H V) |
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| 218 | < (vector_cons_empty A ? ([[H]] @@ V)) |
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| 219 | @ (subvector_multiple_append A ? ? EQ REFLEX [[]] V ? [[ H ]]) |
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| 220 | qed. |
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| 221 | |
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| 222 | lemma eq_a_reflexive: |
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| 223 | ∀a. eq_a a a = true. |
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| 224 | # A |
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| 225 | cases A |
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| 226 | % |
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| 227 | qed. |
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| 228 | |
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| 229 | lemma is_in_monotonic_wrt_append: |
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| 230 | ∀m, n: nat. |
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| 231 | ∀p: Vector addressing_mode_tag m. |
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| 232 | ∀q: Vector addressing_mode_tag n. |
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| 233 | ∀to_search: addressing_mode. |
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| 234 | bool_to_Prop (is_in ? p to_search) → bool_to_Prop (is_in ? (q @@ p) to_search). |
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| 235 | # M # N # P # Q # TO_SEARCH |
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| 236 | # H |
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| 237 | elim Q |
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| 238 | [ normalize |
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| 239 | @ H |
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| 240 | | # NN # PP # QQ # IH |
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| 241 | normalize |
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| 242 | cases (is_a PP TO_SEARCH) |
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| 243 | [ normalize |
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| 244 | % |
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| 245 | | normalize |
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| 246 | normalize in IH |
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| 247 | @ IH |
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| 248 | ] |
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| 249 | ] |
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| 250 | qed. |
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| 251 | |
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| 252 | corollary is_in_hd_tl: |
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| 253 | ∀to_search: addressing_mode. |
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| 254 | ∀hd: addressing_mode_tag. |
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| 255 | ∀n: nat. |
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| 256 | ∀v: Vector addressing_mode_tag n. |
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| 257 | bool_to_Prop (is_in ? v to_search) → bool_to_Prop (is_in ? (hd:::v) to_search). |
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| 258 | # TO_SEARCH # HD # N # V |
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| 259 | elim V |
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| 260 | [ # H |
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| 261 | normalize in H; |
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| 262 | cases H |
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| 263 | | # NN # HHD # VV # IH # HH |
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| 264 | > vector_cons_append |
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| 265 | > (vector_cons_append ? ? HHD VV) |
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| 266 | @ (is_in_monotonic_wrt_append ? 1 ([[HHD]]@@VV) [[HD]] TO_SEARCH) |
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| 267 | @ HH |
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| 268 | ] |
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| 269 | qed. |
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| 270 | |
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| 271 | let rec list_addressing_mode_tags_elim |
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| 272 | (n: nat) (l: Vector addressing_mode_tag (S n)) on l: (l → bool) → bool ≝ |
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| 273 | 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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| 274 | (l → bool) → bool ] with |
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| 275 | [ VEmpty ⇒ true |
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| 276 | | VCons len hd tl ⇒ λP. |
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| 277 | let process_hd ≝ |
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| 278 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
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| 279 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
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| 280 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
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| 281 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
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| 282 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
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| 283 | | acc_a ⇒ λP.P ACC_A |
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| 284 | | acc_b ⇒ λP.P ACC_B |
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| 285 | | dptr ⇒ λP.P DPTR |
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| 286 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
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| 287 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
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| 288 | | acc_dptr ⇒ λP.P ACC_DPTR |
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| 289 | | acc_pc ⇒ λP.P ACC_PC |
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| 290 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
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| 291 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
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| 292 | | carry ⇒ λP.P CARRY |
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| 293 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
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| 294 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
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| 295 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
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| 296 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
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| 297 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
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| 298 | ] |
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| 299 | in |
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| 300 | andb (process_hd P) |
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| 301 | (match len return λx. x = len → bool with |
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| 302 | [ O ⇒ λprf. true |
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| 303 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
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| 304 | ]. |
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| 305 | try % |
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| 306 | [ 2: cases (sym_eq ??? prf); @tl |
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| 307 | | generalize in match H; generalize in match tl; cases prf; |
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| 308 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
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| 309 | #tl |
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| 310 | normalize in ⊢ (∀_: %. ?) |
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| 311 | # H |
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| 312 | whd |
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| 313 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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| 314 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
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| 315 | qed. |
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| 316 | |
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| 317 | definition product_elim ≝ |
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| 318 | λm, n: nat. |
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| 319 | λv: Vector addressing_mode_tag (S m). |
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| 320 | λq: Vector addressing_mode_tag (S n). |
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| 321 | λP: (v × q) → bool. |
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| 322 | list_addressing_mode_tags_elim ? v (λx. list_addressing_mode_tags_elim ? q (λy. P 〈x, y〉)). |
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| 323 | |
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| 324 | definition union_elim ≝ |
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| 325 | λA, B: Type[0]. |
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| 326 | λelimA: (A → bool) → bool. |
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| 327 | λelimB: (B → bool) → bool. |
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| 328 | λelimU: A ⊎ B → bool. |
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| 329 | elimA (λa. elimB (λb. elimU (inl ? ? a) ∧ elimU (inr ? ? b))). |
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[892] | 330 | |
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| 331 | (* |
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[877] | 332 | definition preinstruction_elim: ∀P: preinstruction [[ relative ]] → bool. bool ≝ |
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| 333 | λP. |
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| 334 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADD ? ACC_A addr)) ∧ |
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| 335 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (ADDC ? ACC_A addr)) ∧ |
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| 336 | list_addressing_mode_tags_elim ? [[ registr ; direct ; indirect ; data ]] (λaddr. P (SUBB ? ACC_A addr)) ∧ |
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| 337 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ; dptr ]] (λaddr. P (INC ? addr)) ∧ |
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| 338 | list_addressing_mode_tags_elim ? [[ acc_a ; registr ; direct ; indirect ]] (λaddr. P (DEC ? addr)) ∧ |
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| 339 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (MUL ? ACC_A addr)) ∧ |
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| 340 | list_addressing_mode_tags_elim ? [[acc_b]] (λaddr. P (DIV ? ACC_A addr)) ∧ |
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| 341 | list_addressing_mode_tags_elim ? [[ registr ; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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| 342 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CLR ? addr)) ∧ |
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| 343 | list_addressing_mode_tags_elim ? [[ acc_a ; carry ; bit_addr ]] (λaddr. P (CPL ? addr)) ∧ |
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| 344 | P (DA ? ACC_A) ∧ |
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| 345 | bitvector_elim 8 (λr. P (JC ? (RELATIVE r))) ∧ |
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| 346 | bitvector_elim 8 (λr. P (JNC ? (RELATIVE r))) ∧ |
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| 347 | bitvector_elim 8 (λr. P (JZ ? (RELATIVE r))) ∧ |
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| 348 | bitvector_elim 8 (λr. P (JNZ ? (RELATIVE r))) ∧ |
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| 349 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 350 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JNB ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 351 | bitvector_elim 8 (λr. (bitvector_elim 8 (λb: BitVector 8. P (JBC ? (BIT_ADDR b) (RELATIVE r))))) ∧ |
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| 352 | list_addressing_mode_tags_elim ? [[ registr; direct ]] (λaddr. bitvector_elim 8 (λr. P (DJNZ ? addr (RELATIVE r)))) ∧ |
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| 353 | P (RL ? ACC_A) ∧ |
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| 354 | P (RLC ? ACC_A) ∧ |
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| 355 | P (RR ? ACC_A) ∧ |
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| 356 | P (RRC ? ACC_A) ∧ |
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| 357 | P (SWAP ? ACC_A) ∧ |
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| 358 | P (RET ?) ∧ |
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| 359 | P (RETI ?) ∧ |
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| 360 | P (NOP ?) ∧ |
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| 361 | bit_elim (λb. P (XCHD ? ACC_A (INDIRECT b))) ∧ |
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| 362 | list_addressing_mode_tags_elim ? [[ carry; bit_addr ]] (λaddr. P (SETB ? addr)) ∧ |
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| 363 | bitvector_elim 8 (λaddr. P (PUSH ? (DIRECT addr))) ∧ |
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| 364 | bitvector_elim 8 (λaddr. P (POP ? (DIRECT addr))) ∧ |
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| 365 | union_elim ? ? (product_elim ? ? [[ acc_a ]] [[ direct; data ]]) |
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| 366 | (product_elim ? ? [[ registr; indirect ]] [[ data ]]) |
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| 367 | (λd. bitvector_elim 8 (λb. P (CJNE ? d (RELATIVE b)))) ∧ |
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| 368 | list_addressing_mode_tags_elim ? [[ registr; direct; indirect ]] (λaddr. P (XCH ? ACC_A addr)) ∧ |
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| 369 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ data ; registr ; direct ; indirect ]]) |
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| 370 | (product_elim ? ? [[direct]] [[ acc_a ; data ]]) |
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| 371 | (λd. P (XRL ? d)) ∧ |
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| 372 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 373 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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| 374 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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| 375 | (λd. P (ANL ? d)) ∧ |
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| 376 | union_elim ? ? (union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; data ; direct ; indirect ]]) |
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| 377 | (product_elim ? ? [[direct]] [[ acc_a ; data ]])) |
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| 378 | (product_elim ? ? [[carry]] [[ bit_addr ; n_bit_addr]]) |
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| 379 | (λd. P (ORL ? d)) ∧ |
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| 380 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ ext_indirect ; ext_indirect_dptr ]]) |
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| 381 | (product_elim ? ? [[ ext_indirect ; ext_indirect_dptr ]] [[acc_a]]) |
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| 382 | (λd. P (MOVX ? d)) ∧ |
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| 383 | union_elim ? ? ( |
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| 384 | union_elim ? ? ( |
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| 385 | union_elim ? ? ( |
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| 386 | union_elim ? ? ( |
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| 387 | union_elim ? ? (product_elim ? ? [[acc_a]] [[ registr ; direct ; indirect ; data ]]) |
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| 388 | (product_elim ? ? [[ registr ; indirect ]] [[ acc_a ; direct ; data ]])) |
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| 389 | (product_elim ? ? [[direct]] [[ acc_a ; registr ; direct ; indirect ; data ]])) |
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| 390 | (product_elim ? ? [[dptr]] [[data16]])) |
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| 391 | (product_elim ? ? [[carry]] [[bit_addr]])) |
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| 392 | (product_elim ? ? [[bit_addr]] [[carry]]) |
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| 393 | (λd. P (MOV ? d)). |
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| 394 | % |
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| 395 | qed. |
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| 396 | |
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| 397 | definition instruction_elim: ∀P: instruction → bool. bool ≝ |
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| 398 | λP. (* |
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| 399 | bitvector_elim 11 (λx. P (ACALL (ADDR11 x))) ∧ |
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| 400 | bitvector_elim 16 (λx. P (LCALL (ADDR16 x))) ∧ |
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| 401 | bitvector_elim 11 (λx. P (AJMP (ADDR11 x))) ∧ |
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| 402 | bitvector_elim 16 (λx. P (LJMP (ADDR16 x))) ∧ *) |
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| 403 | bitvector_elim 8 (λx. P (SJMP (RELATIVE x))). (* ∧ |
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| 404 | P (JMP INDIRECT_DPTR) ∧ |
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| 405 | list_addressing_mode_tags_elim ? [[ acc_dptr; acc_pc ]] (λa. P (MOVC ACC_A a)) ∧ |
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| 406 | preinstruction_elim (λp. P (RealInstruction p)). *) |
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| 407 | % |
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| 408 | qed. |
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| 409 | |
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| 410 | |
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| 411 | axiom instruction_elim_complete: |
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| 412 | ∀P. instruction_elim P = true → ∀i. P i = true. |
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[892] | 413 | *) |
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[883] | 414 | (*definition eq_instruction ≝ |
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[877] | 415 | λi, j: instruction. |
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[883] | 416 | true.*) |
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[885] | 417 | |
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[991] | 418 | definition eq_addressing_mode: addressing_mode → addressing_mode → bool ≝ |
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| 419 | λa, b: addressing_mode. |
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| 420 | match a with |
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| 421 | [ DIRECT d ⇒ |
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| 422 | match b with |
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| 423 | [ DIRECT e ⇒ eq_bv ? d e |
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| 424 | | _ ⇒ false |
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| 425 | ] |
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| 426 | | INDIRECT b' ⇒ |
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| 427 | match b with |
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| 428 | [ INDIRECT e ⇒ eq_b b' e |
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| 429 | | _ ⇒ false |
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| 430 | ] |
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| 431 | | EXT_INDIRECT b' ⇒ |
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| 432 | match b with |
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| 433 | [ EXT_INDIRECT e ⇒ eq_b b' e |
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| 434 | | _ ⇒ false |
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| 435 | ] |
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| 436 | | REGISTER bv ⇒ |
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| 437 | match b with |
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| 438 | [ REGISTER bv' ⇒ eq_bv ? bv bv' |
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| 439 | | _ ⇒ false |
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| 440 | ] |
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| 441 | | ACC_A ⇒ match b with [ ACC_A ⇒ true | _ ⇒ false ] |
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| 442 | | ACC_B ⇒ match b with [ ACC_B ⇒ true | _ ⇒ false ] |
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| 443 | | DPTR ⇒ match b with [ DPTR ⇒ true | _ ⇒ false ] |
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| 444 | | DATA b' ⇒ |
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| 445 | match b with |
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| 446 | [ DATA e ⇒ eq_bv ? b' e |
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| 447 | | _ ⇒ false |
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| 448 | ] |
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| 449 | | DATA16 w ⇒ |
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| 450 | match b with |
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| 451 | [ DATA16 e ⇒ eq_bv ? w e |
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| 452 | | _ ⇒ false |
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| 453 | ] |
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| 454 | | ACC_DPTR ⇒ match b with [ ACC_DPTR ⇒ true | _ ⇒ false ] |
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| 455 | | ACC_PC ⇒ match b with [ ACC_PC ⇒ true | _ ⇒ false ] |
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| 456 | | EXT_INDIRECT_DPTR ⇒ match b with [ EXT_INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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| 457 | | INDIRECT_DPTR ⇒ match b with [ INDIRECT_DPTR ⇒ true | _ ⇒ false ] |
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| 458 | | CARRY ⇒ match b with [ CARRY ⇒ true | _ ⇒ false ] |
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| 459 | | BIT_ADDR b' ⇒ |
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| 460 | match b with |
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| 461 | [ BIT_ADDR e ⇒ eq_bv ? b' e |
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| 462 | | _ ⇒ false |
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| 463 | ] |
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| 464 | | N_BIT_ADDR b' ⇒ |
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| 465 | match b with |
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| 466 | [ N_BIT_ADDR e ⇒ eq_bv ? b' e |
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| 467 | | _ ⇒ false |
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| 468 | ] |
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| 469 | | RELATIVE n ⇒ |
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| 470 | match b with |
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| 471 | [ RELATIVE e ⇒ eq_bv ? n e |
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| 472 | | _ ⇒ false |
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| 473 | ] |
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| 474 | | ADDR11 w ⇒ |
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| 475 | match b with |
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| 476 | [ ADDR11 e ⇒ eq_bv ? w e |
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| 477 | | _ ⇒ false |
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| 478 | ] |
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| 479 | | ADDR16 w ⇒ |
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| 480 | match b with |
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| 481 | [ ADDR16 e ⇒ eq_bv ? w e |
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| 482 | | _ ⇒ false |
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| 483 | ] |
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| 484 | ]. |
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| 485 | |
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| 486 | lemma eq_bv_refl: |
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| 487 | ∀n, b. |
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| 488 | eq_bv n b b = true. |
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| 489 | #n #b cases b // |
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| 490 | qed. |
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| 491 | |
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| 492 | lemma eq_b_refl: |
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| 493 | ∀b. |
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| 494 | eq_b b b = true. |
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| 495 | #b cases b // |
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| 496 | qed. |
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| 497 | |
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| 498 | lemma eq_addressing_mode_refl: |
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| 499 | ∀a. eq_addressing_mode a a = true. |
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| 500 | #a cases a try #arg1 try #arg2 try @eq_bv_refl try @eq_b_refl |
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| 501 | try normalize % |
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| 502 | qed. |
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| 503 | |
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| 504 | definition eq_sum: ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
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| 505 | λlt, rt, leq, req, left, right. |
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| 506 | match left with |
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| 507 | [ inl l ⇒ |
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| 508 | match right with |
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| 509 | [ inl l' ⇒ leq l l' |
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| 510 | | _ ⇒ false |
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| 511 | ] |
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| 512 | | inr r ⇒ |
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| 513 | match right with |
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| 514 | [ inr r' ⇒ req r r' |
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| 515 | | _ ⇒ false |
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| 516 | ] |
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| 517 | ]. |
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| 518 | |
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| 519 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
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| 520 | λlt, rt, leq, req, left, right. |
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| 521 | let 〈l, r〉 ≝ left in |
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| 522 | let 〈l', r'〉 ≝ right in |
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| 523 | leq l l' ∧ req r r'. |
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| 524 | |
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| 525 | definition eq_preinstruction: preinstruction [[relative]] → preinstruction [[relative]] → bool ≝ |
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| 526 | λi, j. |
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| 527 | match i with |
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| 528 | [ ADD arg1 arg2 ⇒ |
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| 529 | match j with |
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| 530 | [ ADD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 531 | | _ ⇒ false |
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| 532 | ] |
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| 533 | | ADDC arg1 arg2 ⇒ |
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| 534 | match j with |
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| 535 | [ ADDC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 536 | | _ ⇒ false |
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| 537 | ] |
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| 538 | | SUBB arg1 arg2 ⇒ |
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| 539 | match j with |
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| 540 | [ SUBB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 541 | | _ ⇒ false |
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| 542 | ] |
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| 543 | | INC arg ⇒ |
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| 544 | match j with |
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| 545 | [ INC arg' ⇒ eq_addressing_mode arg arg' |
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| 546 | | _ ⇒ false |
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| 547 | ] |
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| 548 | | DEC arg ⇒ |
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| 549 | match j with |
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| 550 | [ DEC arg' ⇒ eq_addressing_mode arg arg' |
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| 551 | | _ ⇒ false |
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| 552 | ] |
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| 553 | | MUL arg1 arg2 ⇒ |
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| 554 | match j with |
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| 555 | [ MUL arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 556 | | _ ⇒ false |
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| 557 | ] |
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| 558 | | DIV arg1 arg2 ⇒ |
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| 559 | match j with |
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| 560 | [ DIV arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 561 | | _ ⇒ false |
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| 562 | ] |
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| 563 | | DA arg ⇒ |
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| 564 | match j with |
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| 565 | [ DA arg' ⇒ eq_addressing_mode arg arg' |
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| 566 | | _ ⇒ false |
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| 567 | ] |
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| 568 | | JC arg ⇒ |
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| 569 | match j with |
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| 570 | [ JC arg' ⇒ eq_addressing_mode arg arg' |
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| 571 | | _ ⇒ false |
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| 572 | ] |
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| 573 | | JNC arg ⇒ |
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| 574 | match j with |
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| 575 | [ JNC arg' ⇒ eq_addressing_mode arg arg' |
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| 576 | | _ ⇒ false |
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| 577 | ] |
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| 578 | | JB arg1 arg2 ⇒ |
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| 579 | match j with |
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| 580 | [ JB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 581 | | _ ⇒ false |
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| 582 | ] |
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| 583 | | JNB arg1 arg2 ⇒ |
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| 584 | match j with |
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| 585 | [ JNB arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 586 | | _ ⇒ false |
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| 587 | ] |
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| 588 | | JBC arg1 arg2 ⇒ |
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| 589 | match j with |
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| 590 | [ JBC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 591 | | _ ⇒ false |
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| 592 | ] |
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| 593 | | JZ arg ⇒ |
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| 594 | match j with |
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| 595 | [ JZ arg' ⇒ eq_addressing_mode arg arg' |
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| 596 | | _ ⇒ false |
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| 597 | ] |
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| 598 | | JNZ arg ⇒ |
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| 599 | match j with |
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| 600 | [ JNZ arg' ⇒ eq_addressing_mode arg arg' |
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| 601 | | _ ⇒ false |
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| 602 | ] |
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| 603 | | CJNE arg1 arg2 ⇒ |
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| 604 | match j with |
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| 605 | [ CJNE arg1' arg2' ⇒ |
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| 606 | let prod_eq_left ≝ eq_prod [[acc_a]] [[direct; data]] eq_addressing_mode eq_addressing_mode in |
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| 607 | let prod_eq_right ≝ eq_prod [[registr; indirect]] [[data]] eq_addressing_mode eq_addressing_mode in |
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| 608 | let arg1_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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| 609 | arg1_eq arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 610 | | _ ⇒ false |
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| 611 | ] |
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| 612 | | DJNZ arg1 arg2 ⇒ |
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| 613 | match j with |
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| 614 | [ DJNZ arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 615 | | _ ⇒ false |
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| 616 | ] |
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| 617 | | CLR arg ⇒ |
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| 618 | match j with |
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| 619 | [ CLR arg' ⇒ eq_addressing_mode arg arg' |
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| 620 | | _ ⇒ false |
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| 621 | ] |
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| 622 | | CPL arg ⇒ |
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| 623 | match j with |
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| 624 | [ CPL arg' ⇒ eq_addressing_mode arg arg' |
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| 625 | | _ ⇒ false |
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| 626 | ] |
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| 627 | | RL arg ⇒ |
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| 628 | match j with |
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| 629 | [ RL arg' ⇒ eq_addressing_mode arg arg' |
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| 630 | | _ ⇒ false |
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| 631 | ] |
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| 632 | | RLC arg ⇒ |
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| 633 | match j with |
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| 634 | [ RLC arg' ⇒ eq_addressing_mode arg arg' |
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| 635 | | _ ⇒ false |
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| 636 | ] |
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| 637 | | RR arg ⇒ |
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| 638 | match j with |
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| 639 | [ RR arg' ⇒ eq_addressing_mode arg arg' |
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| 640 | | _ ⇒ false |
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| 641 | ] |
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| 642 | | RRC arg ⇒ |
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| 643 | match j with |
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| 644 | [ RRC arg' ⇒ eq_addressing_mode arg arg' |
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| 645 | | _ ⇒ false |
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| 646 | ] |
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| 647 | | SWAP arg ⇒ |
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| 648 | match j with |
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| 649 | [ SWAP arg' ⇒ eq_addressing_mode arg arg' |
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| 650 | | _ ⇒ false |
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| 651 | ] |
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| 652 | | SETB arg ⇒ |
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| 653 | match j with |
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| 654 | [ SETB arg' ⇒ eq_addressing_mode arg arg' |
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| 655 | | _ ⇒ false |
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| 656 | ] |
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| 657 | | PUSH arg ⇒ |
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| 658 | match j with |
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| 659 | [ PUSH arg' ⇒ eq_addressing_mode arg arg' |
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| 660 | | _ ⇒ false |
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| 661 | ] |
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| 662 | | POP arg ⇒ |
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| 663 | match j with |
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| 664 | [ POP arg' ⇒ eq_addressing_mode arg arg' |
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| 665 | | _ ⇒ false |
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| 666 | ] |
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| 667 | | XCH arg1 arg2 ⇒ |
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| 668 | match j with |
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| 669 | [ XCH arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 670 | | _ ⇒ false |
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| 671 | ] |
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| 672 | | XCHD arg1 arg2 ⇒ |
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| 673 | match j with |
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| 674 | [ XCHD arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
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| 675 | | _ ⇒ false |
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| 676 | ] |
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| 677 | | RET ⇒ match j with [ RET ⇒ true | _ ⇒ false ] |
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| 678 | | RETI ⇒ match j with [ RETI ⇒ true | _ ⇒ false ] |
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| 679 | | NOP ⇒ match j with [ NOP ⇒ true | _ ⇒ false ] |
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| 680 | | MOVX arg ⇒ |
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| 681 | match j with |
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| 682 | [ MOVX arg' ⇒ |
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| 683 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ext_indirect; ext_indirect_dptr]] eq_addressing_mode eq_addressing_mode in |
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| 684 | let prod_eq_right ≝ eq_prod [[ext_indirect; ext_indirect_dptr]] [[acc_a]] eq_addressing_mode eq_addressing_mode in |
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| 685 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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| 686 | sum_eq arg arg' |
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| 687 | | _ ⇒ false |
---|
| 688 | ] |
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| 689 | | XRL arg ⇒ |
---|
| 690 | match j with |
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| 691 | [ XRL arg' ⇒ |
---|
| 692 | let prod_eq_left ≝ eq_prod [[acc_a]] [[ data ; registr ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
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| 693 | let prod_eq_right ≝ eq_prod [[direct]] [[ acc_a ; data ]] eq_addressing_mode eq_addressing_mode in |
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| 694 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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| 695 | sum_eq arg arg' |
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| 696 | | _ ⇒ false |
---|
| 697 | ] |
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| 698 | | ORL arg ⇒ |
---|
| 699 | match j with |
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| 700 | [ ORL arg' ⇒ |
---|
| 701 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; data ; direct ; indirect ]] eq_addressing_mode eq_addressing_mode in |
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| 702 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
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| 703 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
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| 704 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
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| 705 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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| 706 | sum_eq arg arg' |
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| 707 | | _ ⇒ false |
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| 708 | ] |
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| 709 | | ANL arg ⇒ |
---|
| 710 | match j with |
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| 711 | [ ANL arg' ⇒ |
---|
| 712 | let prod_eq_left1 ≝ eq_prod [[acc_a]] [[ registr ; direct ; indirect ; data ]] eq_addressing_mode eq_addressing_mode in |
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| 713 | let prod_eq_left2 ≝ eq_prod [[direct]] [[ acc_a; data ]] eq_addressing_mode eq_addressing_mode in |
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| 714 | let prod_eq_left ≝ eq_sum ? ? prod_eq_left1 prod_eq_left2 in |
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| 715 | let prod_eq_right ≝ eq_prod [[carry]] [[ bit_addr ; n_bit_addr]] eq_addressing_mode eq_addressing_mode in |
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| 716 | let sum_eq ≝ eq_sum ? ? prod_eq_left prod_eq_right in |
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| 717 | sum_eq arg arg' |
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| 718 | | _ ⇒ false |
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| 719 | ] |
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| 720 | | MOV arg ⇒ |
---|
| 721 | match j with |
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| 722 | [ MOV arg' ⇒ |
---|
| 723 | let prod_eq_6 ≝ eq_prod [[acc_a]] [[registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
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| 724 | let prod_eq_5 ≝ eq_prod [[registr; indirect]] [[acc_a; direct; data]] eq_addressing_mode eq_addressing_mode in |
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| 725 | let prod_eq_4 ≝ eq_prod [[direct]] [[acc_a; registr; direct; indirect; data]] eq_addressing_mode eq_addressing_mode in |
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| 726 | let prod_eq_3 ≝ eq_prod [[dptr]] [[data16]] eq_addressing_mode eq_addressing_mode in |
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| 727 | let prod_eq_2 ≝ eq_prod [[carry]] [[bit_addr]] eq_addressing_mode eq_addressing_mode in |
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| 728 | let prod_eq_1 ≝ eq_prod [[bit_addr]] [[carry]] eq_addressing_mode eq_addressing_mode in |
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| 729 | let sum_eq_1 ≝ eq_sum ? ? prod_eq_6 prod_eq_5 in |
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| 730 | let sum_eq_2 ≝ eq_sum ? ? sum_eq_1 prod_eq_4 in |
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| 731 | let sum_eq_3 ≝ eq_sum ? ? sum_eq_2 prod_eq_3 in |
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| 732 | let sum_eq_4 ≝ eq_sum ? ? sum_eq_3 prod_eq_2 in |
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| 733 | let sum_eq_5 ≝ eq_sum ? ? sum_eq_4 prod_eq_1 in |
---|
| 734 | sum_eq_5 arg arg' |
---|
| 735 | | _ ⇒ false |
---|
| 736 | ] |
---|
| 737 | ]. |
---|
| 738 | |
---|
| 739 | lemma eq_sum_refl: |
---|
| 740 | ∀A, B: Type[0]. |
---|
| 741 | ∀leq, req. |
---|
| 742 | ∀s. |
---|
| 743 | ∀leq_refl: (∀t. leq t t = true). |
---|
| 744 | ∀req_refl: (∀u. req u u = true). |
---|
| 745 | eq_sum A B leq req s s = true. |
---|
| 746 | #A #B #leq #req #s #leq_refl #req_refl |
---|
| 747 | cases s whd in ⊢ (? → ??%?) // |
---|
| 748 | qed. |
---|
| 749 | |
---|
| 750 | lemma eq_prod_refl: |
---|
| 751 | ∀A, B: Type[0]. |
---|
| 752 | ∀leq, req. |
---|
| 753 | ∀s. |
---|
| 754 | ∀leq_refl: (∀t. leq t t = true). |
---|
| 755 | ∀req_refl: (∀u. req u u = true). |
---|
| 756 | eq_prod A B leq req s s = true. |
---|
| 757 | #A #B #leq #req #s #leq_refl #req_refl |
---|
| 758 | cases s whd in ⊢ (? → ? → ??%?) #l #r >leq_refl normalize @req_refl |
---|
| 759 | qed. |
---|
| 760 | |
---|
| 761 | lemma eq_preinstruction_refl: |
---|
| 762 | ∀i. |
---|
| 763 | eq_preinstruction i i = true. |
---|
| 764 | #i cases i try #arg1 try #arg2 |
---|
| 765 | try @eq_addressing_mode_refl |
---|
| 766 | [1,2,3,4,5,6,7,8,10,16,17,18,19,20: |
---|
| 767 | whd in ⊢ (??%?) |
---|
| 768 | try % |
---|
| 769 | >eq_addressing_mode_refl |
---|
| 770 | >eq_addressing_mode_refl % |
---|
| 771 | |13,15: |
---|
| 772 | whd in ⊢ (??%?) |
---|
| 773 | cases arg1 |
---|
| 774 | [*: |
---|
| 775 | #arg1_left normalize nodelta |
---|
| 776 | >eq_prod_refl [*: try % #argr @eq_addressing_mode_refl] |
---|
| 777 | ] |
---|
| 778 | |11,12: |
---|
| 779 | whd in ⊢ (??%?) |
---|
| 780 | cases arg1 |
---|
| 781 | [1: |
---|
| 782 | #arg1_left normalize nodelta |
---|
| 783 | >(eq_sum_refl …) |
---|
| 784 | [1: % | 2,3: #arg @eq_prod_refl ] |
---|
| 785 | @eq_addressing_mode_refl |
---|
| 786 | |2: |
---|
| 787 | #arg1_left normalize nodelta |
---|
| 788 | @eq_prod_refl [*: @eq_addressing_mode_refl ] |
---|
| 789 | |3: |
---|
| 790 | #arg1_left normalize nodelta |
---|
| 791 | >(eq_sum_refl …) [1: % |
---|
| 792 | |2,3: #arg @eq_prod_refl #arg @eq_addressing_mode_refl ] |
---|
| 793 | |4: |
---|
| 794 | #arg1_left normalize nodelta |
---|
| 795 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
| 796 | ] |
---|
| 797 | |14: |
---|
| 798 | whd in ⊢ (??%?) |
---|
| 799 | cases arg1 |
---|
| 800 | [ #arg1_left normalize nodelta |
---|
| 801 | @eq_sum_refl |
---|
| 802 | [1: #arg @eq_sum_refl |
---|
| 803 | [1: #arg @eq_sum_refl |
---|
| 804 | [1: #arg @eq_sum_refl |
---|
| 805 | [1: #arg @eq_prod_refl |
---|
| 806 | [*: @eq_addressing_mode_refl ] |
---|
| 807 | |2: #arg @eq_prod_refl |
---|
| 808 | [*: #arg @eq_addressing_mode_refl ] |
---|
| 809 | ] |
---|
| 810 | |2: #arg @eq_prod_refl |
---|
| 811 | [*: #arg @eq_addressing_mode_refl ] |
---|
| 812 | ] |
---|
| 813 | |2: #arg @eq_prod_refl |
---|
| 814 | [*: #arg @eq_addressing_mode_refl ] |
---|
| 815 | ] |
---|
| 816 | |2: #arg @eq_prod_refl |
---|
| 817 | [*: #arg @eq_addressing_mode_refl ] |
---|
| 818 | ] |
---|
| 819 | |2: #arg1_right normalize nodelta |
---|
| 820 | @eq_prod_refl [*: #arg @eq_addressing_mode_refl ] |
---|
| 821 | ] |
---|
| 822 | |*: |
---|
| 823 | whd in ⊢ (??%?) |
---|
| 824 | cases arg1 |
---|
| 825 | [*: #arg1 >eq_sum_refl |
---|
| 826 | [1,4: normalize @eq_addressing_mode_refl |
---|
| 827 | |2,3,5,6: #arg @eq_prod_refl |
---|
| 828 | [*: #arg @eq_addressing_mode_refl ] |
---|
| 829 | ] |
---|
| 830 | ] |
---|
| 831 | ] |
---|
| 832 | qed. |
---|
| 833 | |
---|
| 834 | definition eq_instruction: instruction → instruction → bool ≝ |
---|
| 835 | λi, j. |
---|
| 836 | match i with |
---|
| 837 | [ ACALL arg ⇒ |
---|
| 838 | match j with |
---|
| 839 | [ ACALL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 840 | | _ ⇒ false |
---|
| 841 | ] |
---|
| 842 | | LCALL arg ⇒ |
---|
| 843 | match j with |
---|
| 844 | [ LCALL arg' ⇒ eq_addressing_mode arg arg' |
---|
| 845 | | _ ⇒ false |
---|
| 846 | ] |
---|
| 847 | | AJMP arg ⇒ |
---|
| 848 | match j with |
---|
| 849 | [ AJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 850 | | _ ⇒ false |
---|
| 851 | ] |
---|
| 852 | | LJMP arg ⇒ |
---|
| 853 | match j with |
---|
| 854 | [ LJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 855 | | _ ⇒ false |
---|
| 856 | ] |
---|
| 857 | | SJMP arg ⇒ |
---|
| 858 | match j with |
---|
| 859 | [ SJMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 860 | | _ ⇒ false |
---|
| 861 | ] |
---|
| 862 | | JMP arg ⇒ |
---|
| 863 | match j with |
---|
| 864 | [ JMP arg' ⇒ eq_addressing_mode arg arg' |
---|
| 865 | | _ ⇒ false |
---|
| 866 | ] |
---|
| 867 | | MOVC arg1 arg2 ⇒ |
---|
| 868 | match j with |
---|
| 869 | [ MOVC arg1' arg2' ⇒ eq_addressing_mode arg1 arg1' ∧ eq_addressing_mode arg2 arg2' |
---|
| 870 | | _ ⇒ false |
---|
| 871 | ] |
---|
| 872 | | RealInstruction instr ⇒ |
---|
| 873 | match j with |
---|
| 874 | [ RealInstruction instr' ⇒ eq_preinstruction instr instr' |
---|
| 875 | | _ ⇒ false |
---|
| 876 | ] |
---|
| 877 | ]. |
---|
| 878 | |
---|
| 879 | lemma eq_instruction_refl: |
---|
| 880 | ∀i. eq_instruction i i = true. |
---|
| 881 | #i cases i |
---|
| 882 | [1,2,3,4,5,6: #arg1 @eq_addressing_mode_refl |
---|
| 883 | |7: #arg1 #arg2 |
---|
| 884 | whd in ⊢ (??%?) |
---|
| 885 | >eq_addressing_mode_refl |
---|
| 886 | >eq_addressing_mode_refl |
---|
| 887 | % |
---|
| 888 | |8: #arg @eq_preinstruction_refl |
---|
| 889 | ] |
---|
| 890 | qed. |
---|
| 891 | |
---|
[877] | 892 | let rec vect_member |
---|
| 893 | (A: Type[0]) (n: nat) (eq: A → A → bool) |
---|
| 894 | (v: Vector A n) (a: A) on v: bool ≝ |
---|
| 895 | match v with |
---|
| 896 | [ VEmpty ⇒ false |
---|
| 897 | | VCons len hd tl ⇒ |
---|
| 898 | eq hd a ∨ (vect_member A ? eq tl a) |
---|
| 899 | ]. |
---|
[892] | 900 | |
---|
[877] | 901 | let rec list_addressing_mode_tags_elim_prop |
---|
| 902 | (n: nat) |
---|
| 903 | (l: Vector addressing_mode_tag (S n)) |
---|
[884] | 904 | on l: |
---|
| 905 | ∀P: l → Prop. |
---|
[877] | 906 | ∀direct_a. ∀indirect_a. ∀ext_indirect_a. ∀register_a. ∀acc_a_a. |
---|
| 907 | ∀acc_b_a. ∀dptr_a. ∀data_a. ∀data16_a. ∀acc_dptr_a. ∀acc_pc_a. |
---|
| 908 | ∀ext_indirect_dptr_a. ∀indirect_dptr_a. ∀carry_a. ∀bit_addr_a. |
---|
| 909 | ∀n_bit_addr_a. ∀relative_a. ∀addr11_a. ∀addr16_a. |
---|
| 910 | ∀x: l. P x ≝ |
---|
| 911 | match l return |
---|
| 912 | λy. |
---|
| 913 | match y with |
---|
| 914 | [ O ⇒ λm: Vector addressing_mode_tag O. ∀prf: 0 = S n. True |
---|
[884] | 915 | | S y' ⇒ λl: Vector addressing_mode_tag (S y'). ∀prf: S y' = S n.∀P:l → Prop. |
---|
[877] | 916 | ∀direct_a: if vect_member … eq_a l direct then ∀x. P (DIRECT x) else True. |
---|
| 917 | ∀indirect_a: if vect_member … eq_a l indirect then ∀x. P (INDIRECT x) else True. |
---|
| 918 | ∀ext_indirect_a: if vect_member … eq_a l ext_indirect then ∀x. P (EXT_INDIRECT x) else True. |
---|
| 919 | ∀register_a: if vect_member … eq_a l registr then ∀x. P (REGISTER x) else True. |
---|
| 920 | ∀acc_a_a: if vect_member … eq_a l acc_a then P (ACC_A) else True. |
---|
| 921 | ∀acc_b_a: if vect_member … eq_a l acc_b then P (ACC_B) else True. |
---|
| 922 | ∀dptr_a: if vect_member … eq_a l dptr then P DPTR else True. |
---|
| 923 | ∀data_a: if vect_member … eq_a l data then ∀x. P (DATA x) else True. |
---|
| 924 | ∀data16_a: if vect_member … eq_a l data16 then ∀x. P (DATA16 x) else True. |
---|
| 925 | ∀acc_dptr_a: if vect_member … eq_a l acc_dptr then P ACC_DPTR else True. |
---|
| 926 | ∀acc_pc_a: if vect_member … eq_a l acc_pc then P ACC_PC else True. |
---|
| 927 | ∀ext_indirect_dptr_a: if vect_member … eq_a l ext_indirect_dptr then P EXT_INDIRECT_DPTR else True. |
---|
| 928 | ∀indirect_dptr_a: if vect_member … eq_a l indirect_dptr then P INDIRECT_DPTR else True. |
---|
| 929 | ∀carry_a: if vect_member … eq_a l carry then P CARRY else True. |
---|
| 930 | ∀bit_addr_a: if vect_member … eq_a l bit_addr then ∀x. P (BIT_ADDR x) else True. |
---|
| 931 | ∀n_bit_addr_a: if vect_member … eq_a l n_bit_addr then ∀x. P (N_BIT_ADDR x) else True. |
---|
| 932 | ∀relative_a: if vect_member … eq_a l relative then ∀x. P (RELATIVE x) else True. |
---|
| 933 | ∀addr11_a: if vect_member … eq_a l addr11 then ∀x. P (ADDR11 x) else True. |
---|
| 934 | ∀addr_16_a: if vect_member … eq_a l addr16 then ∀x. P (ADDR16 x) else True. |
---|
| 935 | ∀x:l. P x |
---|
| 936 | ] with |
---|
| 937 | [ VEmpty ⇒ λAbsurd. ⊥ |
---|
| 938 | | VCons len hd tl ⇒ λProof. ? |
---|
[884] | 939 | ] (refl ? (S n)). cases daemon. qed. (* |
---|
[877] | 940 | [ destruct(Absurd) |
---|
| 941 | | # A1 # A2 # A3 # A4 # A5 # A6 # A7 |
---|
| 942 | # A8 # A9 # A10 # A11 # A12 # A13 # A14 |
---|
| 943 | # A15 # A16 # A17 # A18 # A19 # X |
---|
| 944 | cases X |
---|
[884] | 945 | # SUB cases daemon ] qed. |
---|
[877] | 946 | cases SUB |
---|
| 947 | [ # BYTE |
---|
| 948 | normalize |
---|
| 949 | ]. |
---|
| 950 | |
---|
| 951 | |
---|
| 952 | (* let prepare_hd ≝ |
---|
| 953 | match hd with |
---|
| 954 | [ direct ⇒ λdirect_prf. ? |
---|
| 955 | | indirect ⇒ λindirect_prf. ? |
---|
| 956 | | ext_indirect ⇒ λext_indirect_prf. ? |
---|
| 957 | | registr ⇒ λregistr_prf. ? |
---|
| 958 | | acc_a ⇒ λacc_a_prf. ? |
---|
| 959 | | acc_b ⇒ λacc_b_prf. ? |
---|
| 960 | | dptr ⇒ λdptr_prf. ? |
---|
| 961 | | data ⇒ λdata_prf. ? |
---|
| 962 | | data16 ⇒ λdata16_prf. ? |
---|
| 963 | | acc_dptr ⇒ λacc_dptr_prf. ? |
---|
| 964 | | acc_pc ⇒ λacc_pc_prf. ? |
---|
| 965 | | ext_indirect_dptr ⇒ λext_indirect_prf. ? |
---|
| 966 | | indirect_dptr ⇒ λindirect_prf. ? |
---|
| 967 | | carry ⇒ λcarry_prf. ? |
---|
| 968 | | bit_addr ⇒ λbit_addr_prf. ? |
---|
| 969 | | n_bit_addr ⇒ λn_bit_addr_prf. ? |
---|
| 970 | | relative ⇒ λrelative_prf. ? |
---|
| 971 | | addr11 ⇒ λaddr11_prf. ? |
---|
| 972 | | addr16 ⇒ λaddr16_prf. ? |
---|
| 973 | ] |
---|
| 974 | in ? *) |
---|
| 975 | ]. |
---|
| 976 | [ 1: destruct(absd) |
---|
| 977 | | 2: # A1 # A2 # A3 # A4 # A5 # A6 |
---|
| 978 | # A7 # A8 # A9 # A10 # A11 # A12 |
---|
| 979 | # A13 # A14 # A15 # A16 # A17 # A18 |
---|
| 980 | # A19 * |
---|
| 981 | ]. |
---|
| 982 | |
---|
| 983 | |
---|
| 984 | match l return λx.match x with [O ⇒ λl: Vector … O. bool | S x' ⇒ λl: Vector addressing_mode_tag (S x'). |
---|
| 985 | (l → bool) → bool ] with |
---|
| 986 | [ VEmpty ⇒ true |
---|
| 987 | | VCons len hd tl ⇒ λP. |
---|
| 988 | let process_hd ≝ |
---|
| 989 | match hd return λhd. ∀P: hd:::tl → bool. bool with |
---|
| 990 | [ direct ⇒ λP.bitvector_elim 8 (λx. P (DIRECT x)) |
---|
| 991 | | indirect ⇒ λP.bit_elim (λx. P (INDIRECT x)) |
---|
| 992 | | ext_indirect ⇒ λP.bit_elim (λx. P (EXT_INDIRECT x)) |
---|
| 993 | | registr ⇒ λP.bitvector_elim 3 (λx. P (REGISTER x)) |
---|
| 994 | | acc_a ⇒ λP.P ACC_A |
---|
| 995 | | acc_b ⇒ λP.P ACC_B |
---|
| 996 | | dptr ⇒ λP.P DPTR |
---|
| 997 | | data ⇒ λP.bitvector_elim 8 (λx. P (DATA x)) |
---|
| 998 | | data16 ⇒ λP.bitvector_elim 16 (λx. P (DATA16 x)) |
---|
| 999 | | acc_dptr ⇒ λP.P ACC_DPTR |
---|
| 1000 | | acc_pc ⇒ λP.P ACC_PC |
---|
| 1001 | | ext_indirect_dptr ⇒ λP.P EXT_INDIRECT_DPTR |
---|
| 1002 | | indirect_dptr ⇒ λP.P INDIRECT_DPTR |
---|
| 1003 | | carry ⇒ λP.P CARRY |
---|
| 1004 | | bit_addr ⇒ λP.bitvector_elim 8 (λx. P (BIT_ADDR x)) |
---|
| 1005 | | n_bit_addr ⇒ λP.bitvector_elim 8 (λx. P (N_BIT_ADDR x)) |
---|
| 1006 | | relative ⇒ λP.bitvector_elim 8 (λx. P (RELATIVE x)) |
---|
| 1007 | | addr11 ⇒ λP.bitvector_elim 11 (λx. P (ADDR11 x)) |
---|
| 1008 | | addr16 ⇒ λP.bitvector_elim 16 (λx. P (ADDR16 x)) |
---|
| 1009 | ] |
---|
| 1010 | in |
---|
| 1011 | andb (process_hd P) |
---|
| 1012 | (match len return λx. x = len → bool with |
---|
| 1013 | [ O ⇒ λprf. true |
---|
| 1014 | | S y ⇒ λprf. list_addressing_mode_tags_elim y ? P ] (refl ? len)) |
---|
| 1015 | ]. |
---|
| 1016 | try % |
---|
| 1017 | [ 2: cases (sym_eq ??? prf); @tl |
---|
| 1018 | | generalize in match H; generalize in match tl; cases prf; |
---|
| 1019 | (* cases prf in tl H; : ??? WAS WORKING BEFORE *) |
---|
| 1020 | #tl |
---|
| 1021 | normalize in ⊢ (∀_: %. ?) |
---|
| 1022 | # H |
---|
| 1023 | whd |
---|
| 1024 | normalize in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
| 1025 | cases (is_a hd (subaddressing_modeel y tl H)) whd // ] |
---|
| 1026 | qed. |
---|
[883] | 1027 | *) |
---|
[877] | 1028 | |
---|
[883] | 1029 | definition load_code_memory_aux ≝ |
---|
| 1030 | fold_left_i_aux … ( |
---|
| 1031 | λi, mem, v. |
---|
| 1032 | insert … (bitvector_of_nat … i) v mem) (Stub Byte 16). |
---|
| 1033 | |
---|
| 1034 | axiom split_elim: |
---|
| 1035 | ∀A,l,m,v.∀P: (Vector A l) × (Vector A m) → Prop. |
---|
| 1036 | (∀vl,vm. v = vl@@vm → P 〈vl,vm〉) → P (split A l m v). |
---|
[901] | 1037 | |
---|
[985] | 1038 | example half_add_SO: |
---|
[883] | 1039 | ∀pc. |
---|
| 1040 | \snd (half_add 16 (bitvector_of_nat … pc) (bitvector_of_nat … 1)) = bitvector_of_nat … (S pc). |
---|
[985] | 1041 | cases daemon. |
---|
| 1042 | qed. |
---|
[883] | 1043 | |
---|
[901] | 1044 | (* |
---|
[883] | 1045 | axiom not_eqvb_S: |
---|
| 1046 | ∀pc. |
---|
| 1047 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S pc))). |
---|
| 1048 | |
---|
| 1049 | axiom not_eqvb_SS: |
---|
| 1050 | ∀pc. |
---|
| 1051 | (¬eq_bv 16 (bitvector_of_nat 16 pc) (bitvector_of_nat 16 (S (S pc)))). |
---|
[894] | 1052 | |
---|
[884] | 1053 | axiom bitvector_elim_complete: |
---|
| 1054 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv. |
---|
| 1055 | |
---|
| 1056 | lemma bitvector_elim_complete': |
---|
| 1057 | ∀n,P. bitvector_elim n P = true → ∀bv. P bv = true. |
---|
| 1058 | #n #P #H generalize in match (bitvector_elim_complete … H) #K #bv |
---|
| 1059 | generalize in match (K bv) normalize cases (P bv) normalize // #abs @⊥ // |
---|
| 1060 | qed. |
---|
[894] | 1061 | *) |
---|
[884] | 1062 | |
---|
[893] | 1063 | (* |
---|
[884] | 1064 | lemma andb_elim': |
---|
| 1065 | ∀b1,b2. (b1 = true) → (b2 = true) → (b1 ∧ b2) = true. |
---|
| 1066 | #b1 #b2 #H1 #H2 @andb_elim cases b1 in H1; normalize // |
---|
| 1067 | qed. |
---|
[893] | 1068 | *) |
---|
[884] | 1069 | |
---|
[890] | 1070 | let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
---|
| 1071 | (encoding: list Byte) on encoding: Prop ≝ |
---|
| 1072 | match encoding with |
---|
| 1073 | [ nil ⇒ final_pc = pc |
---|
| 1074 | | cons hd tl ⇒ |
---|
| 1075 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
---|
| 1076 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
---|
| 1077 | ]. |
---|
| 1078 | |
---|
[901] | 1079 | lemma encoding_check_append: ∀code_memory,final_pc,l1,pc,l2. |
---|
| 1080 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … final_pc) (l1@l2) → |
---|
| 1081 | let intermediate_pc ≝ pc + length … l1 in |
---|
| 1082 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … intermediate_pc) l1 ∧ |
---|
| 1083 | encoding_check code_memory (bitvector_of_nat … intermediate_pc) (bitvector_of_nat … final_pc) l2. |
---|
| 1084 | #code_memory #final_pc #l1 elim l1 |
---|
| 1085 | [ #pc #l2 whd in ⊢ (????% → ?) #H <plus_n_O whd whd in ⊢ (?%?) /2/ |
---|
| 1086 | | #hd #tl #IH #pc #l2 * #H1 #H2 >half_add_SO in H2; #H2 cases (IH … H2) <plus_n_Sm |
---|
| 1087 | #K1 #K2 % [2:@K2] whd % // >half_add_SO @K1 ] |
---|
| 1088 | qed. |
---|
[890] | 1089 | |
---|
[894] | 1090 | axiom bitvector_3_elim_prop: |
---|
| 1091 | ∀P: BitVector 3 → Prop. |
---|
| 1092 | P [[false;false;false]] → P [[false;false;true]] → P [[false;true;false]] → |
---|
| 1093 | P [[false;true;true]] → P [[true;false;false]] → P [[true;false;true]] → |
---|
| 1094 | P [[true;true;false]] → P [[true;true;true]] → ∀v. P v. |
---|
| 1095 | |
---|
[936] | 1096 | definition ticks_of_instruction ≝ |
---|
| 1097 | λi. |
---|
| 1098 | let trivial_code_memory ≝ assembly1 i in |
---|
| 1099 | let trivial_status ≝ load_code_memory trivial_code_memory in |
---|
| 1100 | \snd (fetch trivial_status (zero ?)). |
---|
| 1101 | |
---|
[901] | 1102 | axiom fetch_assembly: |
---|
[892] | 1103 | ∀pc,i,code_memory,assembled. |
---|
| 1104 | assembled = assembly1 i → |
---|
[890] | 1105 | let len ≝ length … assembled in |
---|
| 1106 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
| 1107 | let fetched ≝ fetch code_memory (bitvector_of_nat … pc) in |
---|
| 1108 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
| 1109 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
[936] | 1110 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' (bitvector_of_nat … (pc + len))) = true. |
---|
[901] | 1111 | (* #pc #i #code_memory #assembled cases i [8: *] |
---|
[890] | 1112 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
---|
[892] | 1113 | [47,48,49: |
---|
| 1114 | |*: #arg @(list_addressing_mode_tags_elim_prop … arg) whd try % -arg |
---|
[890] | 1115 | [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] | 1116 | 59,60,63,64,65,66,67: #ARG]] |
---|
| 1117 | [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, |
---|
| 1118 | 56,57,69,70,72,73,75: #arg2 @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2 |
---|
| 1119 | [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, |
---|
| 1120 | 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, |
---|
| 1121 | 68,69,70,71: #ARG2]] |
---|
| 1122 | [1,2,19,20: #arg3 @(list_addressing_mode_tags_elim_prop … arg3) whd try % -arg3 #ARG3] |
---|
| 1123 | normalize in ⊢ (???% → ?) |
---|
[897] | 1124 | [92,94,42,93,95: @split_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
---|
| 1125 | normalize in ⊢ (???% → ?)] |
---|
[892] | 1126 | #H >H * #H1 try (change in ⊢ (% → ?) with (? ∧ ?) * #H2) |
---|
| 1127 | try (change in ⊢ (% → ?) with (? ∧ ?) * #H3) whd in ⊢ (% → ?) #H4 |
---|
| 1128 | change in ⊢ (let fetched ≝ % in ?) with (fetch0 ??) |
---|
| 1129 | whd in ⊢ (let fetched ≝ ??% in ?) <H1 whd in ⊢ (let fetched ≝ % in ?) |
---|
[897] | 1130 | [17,18,19,20,21,22,23,24,25,26,31,34,35,36,37,38: <H3] |
---|
| 1131 | [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, |
---|
| 1132 | 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, |
---|
| 1133 | 69,70,73,74,78,80,81,84,85,95,98,101,102,103,104,105,106,107,108,109,110: <H2] |
---|
[896] | 1134 | whd >eq_instruction_refl >H4 @eq_bv_refl |
---|
[901] | 1135 | qed. *) |
---|
| 1136 | |
---|
| 1137 | let rec fetch_many code_memory final_pc pc expected on expected: Prop ≝ |
---|
| 1138 | match expected with |
---|
| 1139 | [ nil ⇒ eq_bv … pc final_pc = true |
---|
| 1140 | | cons i tl ⇒ |
---|
| 1141 | let fetched ≝ fetch code_memory pc in |
---|
| 1142 | let 〈instr_pc, ticks〉 ≝ fetched in |
---|
| 1143 | let 〈instr,pc'〉 ≝ instr_pc in |
---|
[936] | 1144 | eq_instruction instr i = true ∧ |
---|
| 1145 | ticks = (ticks_of_instruction i) ∧ |
---|
| 1146 | fetch_many code_memory final_pc pc' tl]. |
---|
[901] | 1147 | |
---|
| 1148 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
---|
| 1149 | #A #a #b #EQ destruct // |
---|
[892] | 1150 | qed. |
---|
[901] | 1151 | |
---|
[936] | 1152 | axiom eq_instruction_to_eq: ∀i1,i2. eq_instruction i1 i2 = true → i1 = i2. |
---|
| 1153 | |
---|
[901] | 1154 | lemma fetch_assembly_pseudo: |
---|
[987] | 1155 | ∀program.∀pol:policy program.∀ppc,lookup_labels,lookup_datalabels. |
---|
[901] | 1156 | ∀pi,code_memory,len,assembled,instructions,pc. |
---|
[987] | 1157 | let expansion ≝ pol ppc in |
---|
[921] | 1158 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (bitvector_of_nat ? pc) expansion pi → |
---|
[987] | 1159 | Some … 〈len,assembled〉 = assembly_1_pseudoinstruction program pol ppc (bitvector_of_nat ? pc) lookup_labels lookup_datalabels pi → |
---|
[901] | 1160 | encoding_check code_memory (bitvector_of_nat … pc) (bitvector_of_nat … (pc + len)) assembled → |
---|
| 1161 | fetch_many code_memory (bitvector_of_nat … (pc + len)) (bitvector_of_nat … pc) instructions. |
---|
[987] | 1162 | #program #pol #ppc #lookup_labels #lookup_datalabels #pi #code_memory #len #assembled #instructions #pc |
---|
[921] | 1163 | #EQ1 whd in ⊢ (???% → ?) <EQ1 whd in ⊢ (???% → ?) #EQ2 |
---|
| 1164 | cases (pair_destruct ?????? (option_destruct_Some … EQ2)) -EQ2; #EQ2a #EQ2b |
---|
[901] | 1165 | >EQ2a >EQ2b -EQ2a EQ2b; |
---|
| 1166 | generalize in match (pc + |flatten … (map … assembly1 instructions)|); #final_pc |
---|
| 1167 | generalize in match pc elim instructions |
---|
| 1168 | [ #pc whd in ⊢ (% → %) #H >H @eq_bv_refl |
---|
| 1169 | | #i #tl #IH #pc #H whd cases (encoding_check_append … H); -H; #H1 #H2 whd |
---|
| 1170 | generalize in match (fetch_assembly pc i code_memory … (refl …) H1) |
---|
| 1171 | cases (fetch code_memory (bitvector_of_nat … pc)) #newi_pc #ticks whd in ⊢ (% → %) |
---|
[936] | 1172 | cases newi_pc #newi #newpc whd in ⊢ (% → %) #K cases (conjunction_true … K) -K; #K1 |
---|
[985] | 1173 | cases (conjunction_true … K1) -K1; #K1 #K2 #K3 % try % |
---|
| 1174 | [ @K1 | @eqb_true_to_eq <(eq_instruction_to_eq … K1) @K2 | >(eq_bv_eq … K3) @IH @H2 ] |
---|
[901] | 1175 | qed. |
---|
| 1176 | |
---|
[906] | 1177 | (* |
---|
[877] | 1178 | lemma rev_preserves_length: |
---|
| 1179 | ∀A.∀l. length … (rev A l) = length … l. |
---|
| 1180 | #A #l elim l |
---|
| 1181 | [ % |
---|
| 1182 | | #hd #tl #IH normalize >length_append normalize /2/ ] |
---|
| 1183 | qed. |
---|
| 1184 | |
---|
| 1185 | lemma rev_append: |
---|
| 1186 | ∀A.∀l1,l2. |
---|
| 1187 | rev A (l1@l2) = rev A l2 @ rev A l1. |
---|
| 1188 | #A #l1 elim l1 normalize // |
---|
| 1189 | qed. |
---|
| 1190 | |
---|
| 1191 | lemma rev_rev: ∀A.∀l. rev … (rev A l) = l. |
---|
| 1192 | #A #l elim l |
---|
| 1193 | [ // |
---|
| 1194 | | #hd #tl #IH normalize >rev_append normalize // ] |
---|
| 1195 | qed. |
---|
| 1196 | |
---|
| 1197 | lemma split_len_Sn: |
---|
| 1198 | ∀A:Type[0].∀l:list A.∀len. |
---|
| 1199 | length … l = S len → |
---|
| 1200 | Σl'.Σa. l = l'@[a] ∧ length … l' = len. |
---|
| 1201 | #A #l elim l |
---|
| 1202 | [ normalize #len #abs destruct |
---|
| 1203 | | #hd #tl #IH #len |
---|
| 1204 | generalize in match (rev_rev … tl) |
---|
| 1205 | cases (rev A tl) in ⊢ (??%? → ?) |
---|
| 1206 | [ #H <H normalize #EQ % [@[ ]] % [@hd] normalize /2/ |
---|
| 1207 | | #a #l' #H <H normalize #EQ |
---|
| 1208 | %[@(hd::rev … l')] %[@a] % // |
---|
| 1209 | >length_append in EQ #EQ normalize in EQ; normalize; |
---|
| 1210 | generalize in match (injective_S … EQ) #EQ2 /2/ ]] |
---|
| 1211 | qed. |
---|
| 1212 | |
---|
| 1213 | lemma list_elim_rev: |
---|
| 1214 | ∀A:Type[0].∀P:list A → Type[0]. |
---|
| 1215 | P [ ] → (∀l,a. P l → P (l@[a])) → |
---|
| 1216 | ∀l. P l. |
---|
| 1217 | #A #P #H1 #H2 #l |
---|
| 1218 | generalize in match (refl … (length … l)) |
---|
| 1219 | generalize in ⊢ (???% → ?) #n generalize in match l |
---|
| 1220 | elim n |
---|
| 1221 | [ #L cases L [ // | #x #w #abs (normalize in abs) @⊥ // ] |
---|
| 1222 | | #m #IH #L #EQ |
---|
| 1223 | cases (split_len_Sn … EQ) #l' * #a * /3/ ] |
---|
| 1224 | qed. |
---|
| 1225 | |
---|
| 1226 | axiom is_prefix: ∀A:Type[0]. list A → list A → Prop. |
---|
| 1227 | axiom prefix_of_append: |
---|
| 1228 | ∀A:Type[0].∀l,l1,l2:list A. |
---|
| 1229 | is_prefix … l l1 → is_prefix … l (l1@l2). |
---|
| 1230 | axiom prefix_reflexive: ∀A,l. is_prefix A l l. |
---|
| 1231 | axiom nil_prefix: ∀A,l. is_prefix A [ ] l. |
---|
| 1232 | |
---|
| 1233 | record Propify (A:Type[0]) : Type[0] (*Prop*) ≝ { in_propify: A }. |
---|
| 1234 | |
---|
| 1235 | definition Propify_elim: ∀A. ∀P:Prop. (A → P) → (Propify A → P) ≝ |
---|
| 1236 | λA,P,H,x. match x with [ mk_Propify p ⇒ H p ]. |
---|
| 1237 | |
---|
| 1238 | definition app ≝ |
---|
| 1239 | λA:Type[0].λl1:Propify (list A).λl2:list A. |
---|
| 1240 | match l1 with |
---|
| 1241 | [ mk_Propify l1 ⇒ mk_Propify … (l1@l2) ]. |
---|
| 1242 | |
---|
| 1243 | lemma app_nil: ∀A,l1. app A l1 [ ] = l1. |
---|
| 1244 | #A * /3/ |
---|
| 1245 | qed. |
---|
| 1246 | |
---|
| 1247 | lemma app_assoc: ∀A,l1,l2,l3. app A (app A l1 l2) l3 = app A l1 (l2@l3). |
---|
| 1248 | #A * #l1 normalize // |
---|
| 1249 | qed. |
---|
| 1250 | *) |
---|
| 1251 | |
---|
[906] | 1252 | axiom assembly_ok: |
---|
[987] | 1253 | ∀program,pol,assembled. |
---|
| 1254 | let 〈labels,costs〉 ≝ build_maps program pol in |
---|
| 1255 | Some … 〈assembled,costs〉 = assembly program pol → |
---|
[906] | 1256 | let code_memory ≝ load_code_memory assembled in |
---|
[915] | 1257 | let preamble ≝ \fst program in |
---|
| 1258 | let datalabels ≝ construct_datalabels preamble in |
---|
[987] | 1259 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
[906] | 1260 | let lookup_datalabels ≝ λx. lookup ?? x datalabels (zero ?) in |
---|
| 1261 | ∀ppc,len,assembledi. |
---|
| 1262 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
[987] | 1263 | Some … 〈len,assembledi〉 = assembly_1_pseudoinstruction program pol ppc (sigma program pol ppc) lookup_labels lookup_datalabels pi → |
---|
| 1264 | encoding_check code_memory (sigma program pol ppc) (bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len)) assembledi ∧ |
---|
| 1265 | sigma program pol newppc = bitvector_of_nat … (nat_of_bitvector … (sigma program pol ppc) + len). |
---|
[906] | 1266 | |
---|
[908] | 1267 | axiom bitvector_of_nat_nat_of_bitvector: |
---|
[915] | 1268 | ∀n,v. |
---|
| 1269 | bitvector_of_nat n (nat_of_bitvector n v) = v. |
---|
[908] | 1270 | |
---|
[924] | 1271 | axiom assembly_ok_to_expand_pseudo_instruction_ok: |
---|
[987] | 1272 | ∀program,pol,assembled,costs. |
---|
| 1273 | Some … 〈assembled,costs〉 = assembly program pol → |
---|
[924] | 1274 | ∀ppc. |
---|
| 1275 | let code_memory ≝ load_code_memory assembled in |
---|
| 1276 | let preamble ≝ \fst program in |
---|
| 1277 | let data_labels ≝ construct_datalabels preamble in |
---|
[987] | 1278 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
[924] | 1279 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
[987] | 1280 | let expansion ≝ pol ppc in |
---|
[924] | 1281 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
| 1282 | ∃instructions. |
---|
[987] | 1283 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi. |
---|
[924] | 1284 | |
---|
[985] | 1285 | axiom pair_elim': |
---|
| 1286 | ∀A,B,C: Type[0]. |
---|
| 1287 | ∀T: A → B → C. |
---|
| 1288 | ∀p. |
---|
| 1289 | ∀P: A×B → C → Prop. |
---|
| 1290 | (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt)) → |
---|
| 1291 | P p (let 〈lft, rgt〉 ≝ p in T lft rgt). |
---|
| 1292 | |
---|
| 1293 | axiom pair_elim'': |
---|
| 1294 | ∀A,B,C,C': Type[0]. |
---|
| 1295 | ∀T: A → B → C. |
---|
| 1296 | ∀T': A → B → C'. |
---|
| 1297 | ∀p. |
---|
| 1298 | ∀P: A×B → C → C' → Prop. |
---|
| 1299 | (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt) (T' lft rgt)) → |
---|
| 1300 | P p (let 〈lft, rgt〉 ≝ p in T lft rgt) (let 〈lft, rgt〉 ≝ p in T' lft rgt). |
---|
| 1301 | |
---|
| 1302 | axiom split_elim': |
---|
| 1303 | ∀A: Type[0]. |
---|
| 1304 | ∀B: Type[1]. |
---|
| 1305 | ∀l, m, v. |
---|
| 1306 | ∀T: Vector A l → Vector A m → B. |
---|
| 1307 | ∀P: B → Prop. |
---|
| 1308 | (∀lft, rgt. v = lft @@ rgt → P (T lft rgt)) → |
---|
| 1309 | P (let 〈lft, rgt〉 ≝ split A l m v in T lft rgt). |
---|
| 1310 | |
---|
| 1311 | axiom split_elim'': |
---|
| 1312 | ∀A: Type[0]. |
---|
| 1313 | ∀B,B': Type[1]. |
---|
| 1314 | ∀l, m, v. |
---|
| 1315 | ∀T: Vector A l → Vector A m → B. |
---|
| 1316 | ∀T': Vector A l → Vector A m → B'. |
---|
| 1317 | ∀P: B → B' → Prop. |
---|
| 1318 | (∀lft, rgt. v = lft @@ rgt → P (T lft rgt) (T' lft rgt)) → |
---|
| 1319 | P (let 〈lft, rgt〉 ≝ split A l m v in T lft rgt) |
---|
| 1320 | (let 〈lft, rgt〉 ≝ split A l m v in T' lft rgt). |
---|
| 1321 | |
---|
[908] | 1322 | lemma fetch_assembly_pseudo2: |
---|
[987] | 1323 | ∀program,pol,assembled. |
---|
| 1324 | let 〈labels,costs〉 ≝ build_maps program pol in |
---|
| 1325 | Some … 〈assembled,costs〉 = assembly program pol → |
---|
[924] | 1326 | ∀ppc. |
---|
[915] | 1327 | let code_memory ≝ load_code_memory assembled in |
---|
| 1328 | let preamble ≝ \fst program in |
---|
| 1329 | let data_labels ≝ construct_datalabels preamble in |
---|
[987] | 1330 | let lookup_labels ≝ λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x) in |
---|
[915] | 1331 | let lookup_datalabels ≝ λx. lookup ? ? x data_labels (zero ?) in |
---|
[987] | 1332 | let expansion ≝ pol ppc in |
---|
[908] | 1333 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc in |
---|
[924] | 1334 | ∃instructions. |
---|
[987] | 1335 | Some ? instructions = expand_pseudo_instruction lookup_labels lookup_datalabels (sigma program pol ppc) expansion pi ∧ |
---|
| 1336 | fetch_many code_memory (sigma program pol newppc) (sigma program pol ppc) instructions. |
---|
| 1337 | #program #pol #assembled @pair_elim' #labels #costs #BUILD_MAPS #ASSEMBLY #ppc |
---|
| 1338 | generalize in match (assembly_ok_to_expand_pseudo_instruction_ok program pol assembled costs ASSEMBLY ppc) |
---|
[915] | 1339 | letin code_memory ≝ (load_code_memory assembled) |
---|
| 1340 | letin preamble ≝ (\fst program) |
---|
| 1341 | letin data_labels ≝ (construct_datalabels preamble) |
---|
[987] | 1342 | letin lookup_labels ≝ (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) |
---|
[915] | 1343 | letin lookup_datalabels ≝ (λx. lookup ? ? x data_labels (zero ?)) |
---|
[987] | 1344 | whd in ⊢ (% → %) generalize in match (assembly_ok program pol assembled) >BUILD_MAPS #XX generalize in match (XX ASSEMBLY ppc) -XX; |
---|
[908] | 1345 | cases (fetch_pseudo_instruction (\snd program) ppc) #pi #newppc |
---|
[987] | 1346 | generalize in match (fetch_assembly_pseudo program pol ppc |
---|
| 1347 | (λx. sigma program pol (address_of_word_labels_code_mem (\snd program) x)) (λx. lookup ?? x data_labels (zero ?)) pi |
---|
[915] | 1348 | (load_code_memory assembled)); |
---|
[924] | 1349 | whd in ⊢ ((∀_.∀_.∀_.∀_.%) → (∀_.∀_.%) → % → %) #H1 #H2 * #instructions #EXPAND |
---|
| 1350 | whd in H1:(∀_.∀_.∀_.∀_.? → ???% → ?) H2:(∀_.∀_.???% → ?); |
---|
[921] | 1351 | normalize nodelta in EXPAND; (* HERE *) |
---|
[987] | 1352 | generalize in match (λlen, assembled.H1 len assembled instructions (nat_of_bitvector … (sigma program pol ppc))) -H1; #H1 |
---|
[921] | 1353 | >bitvector_of_nat_nat_of_bitvector in H1; #H1 |
---|
| 1354 | <EXPAND in H1 H2; whd in ⊢ ((∀_.∀_.? → ???% → ?) → (∀_.∀_.???% → ?) → ?) |
---|
[908] | 1355 | #H1 #H2 |
---|
| 1356 | cases (H2 ?? (refl …)) -H2; #K1 #K2 >K2 |
---|
[921] | 1357 | generalize in match (H1 ?? (refl …) (refl …) ?) -H1; |
---|
[924] | 1358 | [ #K3 % [2: % [% | @K3]] | @K1 ] |
---|
[908] | 1359 | qed. |
---|
| 1360 | |
---|
[906] | 1361 | (* OLD? |
---|
[877] | 1362 | definition assembly_specification: |
---|
| 1363 | ∀assembly_program: pseudo_assembly_program. |
---|
| 1364 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
---|
| 1365 | λpseudo_assembly_program. |
---|
| 1366 | λcode_mem. |
---|
| 1367 | ∀pc: Word. |
---|
| 1368 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
---|
| 1369 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
---|
| 1370 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
---|
| 1371 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
---|
| 1372 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
---|
| 1373 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
---|
| 1374 | match pre_assembled with |
---|
| 1375 | [ None ⇒ True |
---|
| 1376 | | Some pc_code ⇒ |
---|
| 1377 | let 〈new_pc,code〉 ≝ pc_code in |
---|
| 1378 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
---|
| 1379 | |
---|
| 1380 | axiom assembly_meets_specification: |
---|
| 1381 | ∀pseudo_assembly_program. |
---|
| 1382 | match assembly pseudo_assembly_program with |
---|
| 1383 | [ None ⇒ True |
---|
| 1384 | | Some code_mem_cost ⇒ |
---|
| 1385 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
---|
| 1386 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
---|
| 1387 | ]. |
---|
| 1388 | (* |
---|
| 1389 | # PROGRAM |
---|
| 1390 | [ cases PROGRAM |
---|
| 1391 | # PREAMBLE |
---|
| 1392 | # INSTR_LIST |
---|
| 1393 | elim INSTR_LIST |
---|
| 1394 | [ whd |
---|
| 1395 | whd in ⊢ (∀_. %) |
---|
| 1396 | # PC |
---|
| 1397 | whd |
---|
| 1398 | | # INSTR |
---|
| 1399 | # INSTR_LIST_TL |
---|
| 1400 | # H |
---|
| 1401 | whd |
---|
| 1402 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
---|
| 1403 | ] |
---|
| 1404 | | cases not_implemented |
---|
| 1405 | ] *) |
---|
[906] | 1406 | *) |
---|
[877] | 1407 | |
---|
[942] | 1408 | definition internal_pseudo_address_map ≝ list (BitVector 8). |
---|
| 1409 | |
---|
| 1410 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
| 1411 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
| 1412 | |
---|
| 1413 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
| 1414 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
| 1415 | |
---|
| 1416 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
[987] | 1417 | ∀M:internal_pseudo_address_map.∀s:PseudoStatus.∀pol:policy (code_memory … s).∀addr:BitVector 7. |
---|
[942] | 1418 | member ? (eq_bv 8) (false:::addr) M = true → |
---|
| 1419 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
| 1420 | let pbl ≝ lookup ? 7 addr (low_internal_ram ? s) (zero 8) in |
---|
| 1421 | let pbu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) (low_internal_ram ? s) (zero 8) in |
---|
| 1422 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
| 1423 | let bu ≝ lookup ? 7 (\snd (half_add ? addr (bitvector_of_nat 7 1))) ram (zero 8) in |
---|
[987] | 1424 | bu@@bl = sigma (code_memory … s) pol (pbu@@pbl). |
---|
[942] | 1425 | |
---|
[949] | 1426 | (* changed from add to sub *) |
---|
[942] | 1427 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
[950] | 1428 | ∀T.∀M:internal_pseudo_address_map.∀s:PreStatus T.∀addr:BitVector 7. |
---|
[942] | 1429 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
[949] | 1430 | let 〈Saddr,flags〉 ≝ sub_7_with_carry addr (bitvector_of_nat 7 1) false in |
---|
| 1431 | let carr ≝ get_index_v ? ? flags 1 ? in |
---|
[942] | 1432 | carr = false → |
---|
[949] | 1433 | member ? (eq_bv 8) (false:::Saddr) M = false → |
---|
| 1434 | member ? (eq_bv 8) (false:::addr) M = false → |
---|
| 1435 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
| 1436 | // |
---|
| 1437 | qed. |
---|
[942] | 1438 | |
---|
| 1439 | definition addressing_mode_ok ≝ |
---|
[950] | 1440 | λT.λM:internal_pseudo_address_map.λs:PreStatus T. |
---|
[942] | 1441 | λaddr:addressing_mode. |
---|
| 1442 | match addr with |
---|
| 1443 | [ DIRECT d ⇒ |
---|
| 1444 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
| 1445 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
| 1446 | | INDIRECT i ⇒ |
---|
| 1447 | let d ≝ get_register ? s [[false;false;i]] in |
---|
| 1448 | ¬(member ? (eq_bv 8) d M) ∧ |
---|
| 1449 | ¬(member ? (eq_bv 8) (\fst (sub_8_with_carry d (bitvector_of_nat 8 1) false)) M) |
---|
| 1450 | | EXT_INDIRECT _ ⇒ true |
---|
| 1451 | | REGISTER _ ⇒ true |
---|
| 1452 | | ACC_A ⇒ true |
---|
| 1453 | | ACC_B ⇒ true |
---|
| 1454 | | DPTR ⇒ true |
---|
| 1455 | | DATA _ ⇒ true |
---|
| 1456 | | DATA16 _ ⇒ true |
---|
| 1457 | | ACC_DPTR ⇒ true |
---|
| 1458 | | ACC_PC ⇒ true |
---|
| 1459 | | EXT_INDIRECT_DPTR ⇒ true |
---|
| 1460 | | INDIRECT_DPTR ⇒ true |
---|
| 1461 | | CARRY ⇒ true |
---|
| 1462 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
| 1463 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
| 1464 | | RELATIVE _ ⇒ true |
---|
| 1465 | | ADDR11 _ ⇒ true |
---|
| 1466 | | ADDR16 _ ⇒ true ]. |
---|
[944] | 1467 | |
---|
| 1468 | definition next_internal_pseudo_address_map0 ≝ |
---|
[950] | 1469 | λT. |
---|
[944] | 1470 | λfetched. |
---|
| 1471 | λM: internal_pseudo_address_map. |
---|
[950] | 1472 | λs: PreStatus T. |
---|
[944] | 1473 | match fetched with |
---|
[942] | 1474 | [ Comment _ ⇒ Some ? M |
---|
| 1475 | | Cost _ ⇒ Some … M |
---|
| 1476 | | Jmp _ ⇒ Some … M |
---|
| 1477 | | Call _ ⇒ |
---|
| 1478 | Some … (\snd (half_add ? (get_8051_sfr … s SFR_SP) (bitvector_of_nat 8 1))::M) |
---|
| 1479 | | Mov _ _ ⇒ Some … M |
---|
| 1480 | | Instruction instr ⇒ |
---|
| 1481 | match instr with |
---|
| 1482 | [ ADD addr1 addr2 ⇒ |
---|
[950] | 1483 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
[942] | 1484 | Some ? M |
---|
| 1485 | else |
---|
| 1486 | None ? |
---|
| 1487 | | ADDC addr1 addr2 ⇒ |
---|
[950] | 1488 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
[942] | 1489 | Some ? M |
---|
| 1490 | else |
---|
| 1491 | None ? |
---|
| 1492 | | SUBB addr1 addr2 ⇒ |
---|
[950] | 1493 | if addressing_mode_ok T M s addr1 ∧ addressing_mode_ok T M s addr2 then |
---|
[942] | 1494 | Some ? M |
---|
| 1495 | else |
---|
| 1496 | None ? |
---|
| 1497 | | _ ⇒ (* TO BE COMPLETED *) Some ? M ]]. |
---|
[944] | 1498 | |
---|
| 1499 | |
---|
| 1500 | definition next_internal_pseudo_address_map ≝ |
---|
| 1501 | λM:internal_pseudo_address_map. |
---|
| 1502 | λs:PseudoStatus. |
---|
[950] | 1503 | next_internal_pseudo_address_map0 ? |
---|
| 1504 | (\fst (fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s))) M s. |
---|
[942] | 1505 | |
---|
| 1506 | definition status_of_pseudo_status: |
---|
[987] | 1507 | internal_pseudo_address_map → ∀ps:PseudoStatus. policy (code_memory … ps) → option Status ≝ |
---|
| 1508 | λM,ps,pol. |
---|
[877] | 1509 | let pap ≝ code_memory … ps in |
---|
[987] | 1510 | match assembly pap pol with |
---|
[877] | 1511 | [ None ⇒ None … |
---|
| 1512 | | Some p ⇒ |
---|
| 1513 | let cm ≝ load_code_memory (\fst p) in |
---|
[987] | 1514 | let pc ≝ sigma pap pol (program_counter ? ps) in |
---|
[942] | 1515 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
[951] | 1516 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
[877] | 1517 | Some … |
---|
| 1518 | (mk_PreStatus (BitVectorTrie Byte 16) |
---|
| 1519 | cm |
---|
[942] | 1520 | lir |
---|
| 1521 | hir |
---|
[877] | 1522 | (external_ram … ps) |
---|
| 1523 | pc |
---|
| 1524 | (special_function_registers_8051 … ps) |
---|
| 1525 | (special_function_registers_8052 … ps) |
---|
| 1526 | (p1_latch … ps) |
---|
| 1527 | (p3_latch … ps) |
---|
| 1528 | (clock … ps)) ]. |
---|
| 1529 | |
---|
[909] | 1530 | (* |
---|
[877] | 1531 | definition write_at_stack_pointer': |
---|
| 1532 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
| 1533 | λM: Type[0]. |
---|
| 1534 | λs: PreStatus M. |
---|
| 1535 | λv: Byte. |
---|
| 1536 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
| 1537 | let bit_zero ≝ get_index_v… nu O ? in |
---|
| 1538 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
| 1539 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
| 1540 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
| 1541 | if bit_zero then |
---|
| 1542 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1543 | v (low_internal_ram ? s) in |
---|
| 1544 | set_low_internal_ram ? s memory |
---|
| 1545 | else |
---|
| 1546 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
| 1547 | v (high_internal_ram ? s) in |
---|
| 1548 | set_high_internal_ram ? s memory. |
---|
| 1549 | [ cases l0 % |
---|
| 1550 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
| 1551 | qed. |
---|
| 1552 | |
---|
| 1553 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
| 1554 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
| 1555 | ≝ |
---|
| 1556 | λticks_of. |
---|
| 1557 | λs. |
---|
| 1558 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
| 1559 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
| 1560 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
| 1561 | let s ≝ set_program_counter ? s pc in |
---|
| 1562 | match instr with |
---|
| 1563 | [ Instruction instr ⇒ |
---|
| 1564 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
| 1565 | | Comment cmt ⇒ s |
---|
| 1566 | | Cost cst ⇒ s |
---|
| 1567 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
| 1568 | | Call call ⇒ |
---|
| 1569 | let a ≝ address_of_word_labels s call in |
---|
| 1570 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1571 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
| 1572 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
---|
| 1573 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
| 1574 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
| 1575 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
| 1576 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
| 1577 | set_program_counter ? s a |
---|
| 1578 | | Mov dptr ident ⇒ |
---|
| 1579 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
| 1580 | ]. |
---|
| 1581 | [ |
---|
| 1582 | |2,3,4: % |
---|
| 1583 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
| 1584 | | |
---|
| 1585 | | % |
---|
| 1586 | ] |
---|
| 1587 | cases not_implemented |
---|
| 1588 | qed. |
---|
[909] | 1589 | *) |
---|
[911] | 1590 | |
---|
| 1591 | axiom execute_1_pseudo_instruction_preserves_code_memory: |
---|
| 1592 | ∀ticks_of,ps. |
---|
| 1593 | code_memory … (execute_1_pseudo_instruction ticks_of ps) = code_memory … ps. |
---|
| 1594 | |
---|
[877] | 1595 | (* |
---|
| 1596 | lemma execute_code_memory_unchanged: |
---|
| 1597 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
| 1598 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
| 1599 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
| 1600 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
| 1601 | whd in ⊢ (??? (??%)) cases instr |
---|
| 1602 | [ #pre cases pre |
---|
| 1603 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1604 | cases (split ????) #z1 #z2 % |
---|
| 1605 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1606 | cases (split ????) #z1 #z2 % |
---|
| 1607 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
| 1608 | cases (split ????) #z1 #z2 % |
---|
| 1609 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
| 1610 | [ #x1 whd in ⊢ (??? (??%)) |
---|
| 1611 | | *: cases not_implemented |
---|
| 1612 | ] |
---|
| 1613 | | #comment % |
---|
| 1614 | | #cost % |
---|
| 1615 | | #label % |
---|
| 1616 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
| 1617 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
| 1618 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
---|
| 1619 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
| 1620 | (* CSC: ??? *) |
---|
| 1621 | | #dptr #label (* CSC: ??? *) |
---|
| 1622 | ] |
---|
| 1623 | cases not_implemented |
---|
| 1624 | qed. |
---|
| 1625 | *) |
---|
| 1626 | |
---|
| 1627 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
---|
[942] | 1628 | ∀M:internal_pseudo_address_map. |
---|
[877] | 1629 | ∀ps,ps': PseudoStatus. |
---|
[987] | 1630 | ∀pol. |
---|
| 1631 | ∀prf:code_memory … ps = code_memory … ps'. |
---|
| 1632 | let pol' ≝ ? in |
---|
| 1633 | match status_of_pseudo_status M ps pol with |
---|
| 1634 | [ None ⇒ status_of_pseudo_status M ps' pol' = None … |
---|
| 1635 | | Some _ ⇒ ∃w. status_of_pseudo_status M ps' pol' = Some … w |
---|
[877] | 1636 | ]. |
---|
[987] | 1637 | [2: <prf @pol] |
---|
| 1638 | #M #ps #ps' #pol #H normalize nodelta; whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
---|
| 1639 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
| 1640 | cases (assembly ??) in ⊢ (???% → %) |
---|
[877] | 1641 | [ #K whd whd in ⊢ (??%?) <H >K % |
---|
| 1642 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
---|
[909] | 1643 | qed. |
---|
[877] | 1644 | |
---|
[987] | 1645 | definition ticks_of0: ∀p:pseudo_assembly_program. policy p → Word → ? → nat × nat ≝ |
---|
| 1646 | λprogram: pseudo_assembly_program.λpol. |
---|
[929] | 1647 | λppc: Word. |
---|
[936] | 1648 | λfetched. |
---|
[929] | 1649 | match fetched with |
---|
| 1650 | [ Instruction instr ⇒ |
---|
| 1651 | match instr with |
---|
| 1652 | [ JC lbl ⇒ |
---|
[987] | 1653 | match pol ppc with |
---|
[929] | 1654 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1655 | | medium_jump ⇒ ? |
---|
| 1656 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1657 | ] |
---|
| 1658 | | JNC lbl ⇒ |
---|
[987] | 1659 | match pol ppc with |
---|
[929] | 1660 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1661 | | medium_jump ⇒ ? |
---|
| 1662 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1663 | ] |
---|
| 1664 | | JB bit lbl ⇒ |
---|
[987] | 1665 | match pol ppc with |
---|
[929] | 1666 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1667 | | medium_jump ⇒ ? |
---|
| 1668 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1669 | ] |
---|
| 1670 | | JNB bit lbl ⇒ |
---|
[987] | 1671 | match pol ppc with |
---|
[929] | 1672 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1673 | | medium_jump ⇒ ? |
---|
| 1674 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1675 | ] |
---|
| 1676 | | JBC bit lbl ⇒ |
---|
[987] | 1677 | match pol ppc with |
---|
[929] | 1678 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1679 | | medium_jump ⇒ ? |
---|
| 1680 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1681 | ] |
---|
| 1682 | | JZ lbl ⇒ |
---|
[987] | 1683 | match pol ppc with |
---|
[929] | 1684 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1685 | | medium_jump ⇒ ? |
---|
| 1686 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1687 | ] |
---|
| 1688 | | JNZ lbl ⇒ |
---|
[987] | 1689 | match pol ppc with |
---|
[929] | 1690 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1691 | | medium_jump ⇒ ? |
---|
| 1692 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1693 | ] |
---|
| 1694 | | CJNE arg lbl ⇒ |
---|
[987] | 1695 | match pol ppc with |
---|
[929] | 1696 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1697 | | medium_jump ⇒ ? |
---|
| 1698 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1699 | ] |
---|
| 1700 | | DJNZ arg lbl ⇒ |
---|
[987] | 1701 | match pol ppc with |
---|
[929] | 1702 | [ short_jump ⇒ 〈2, 2〉 |
---|
| 1703 | | medium_jump ⇒ ? |
---|
| 1704 | | long_jump ⇒ 〈4, 4〉 |
---|
| 1705 | ] |
---|
[936] | 1706 | | ADD arg1 arg2 ⇒ |
---|
| 1707 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
| 1708 | 〈ticks, ticks〉 |
---|
[929] | 1709 | | ADDC arg1 arg2 ⇒ |
---|
[936] | 1710 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
| 1711 | 〈ticks, ticks〉 |
---|
[929] | 1712 | | SUBB arg1 arg2 ⇒ |
---|
[936] | 1713 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
| 1714 | 〈ticks, ticks〉 |
---|
[929] | 1715 | | INC arg ⇒ |
---|
[936] | 1716 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
| 1717 | 〈ticks, ticks〉 |
---|
[929] | 1718 | | DEC arg ⇒ |
---|
[936] | 1719 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
| 1720 | 〈ticks, ticks〉 |
---|
[929] | 1721 | | MUL arg1 arg2 ⇒ |
---|
[936] | 1722 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
| 1723 | 〈ticks, ticks〉 |
---|
[929] | 1724 | | DIV arg1 arg2 ⇒ |
---|
[936] | 1725 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
| 1726 | 〈ticks, ticks〉 |
---|
[929] | 1727 | | DA arg ⇒ |
---|
[936] | 1728 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
| 1729 | 〈ticks, ticks〉 |
---|
[929] | 1730 | | ANL arg ⇒ |
---|
[936] | 1731 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
| 1732 | 〈ticks, ticks〉 |
---|
[929] | 1733 | | ORL arg ⇒ |
---|
[936] | 1734 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
| 1735 | 〈ticks, ticks〉 |
---|
[929] | 1736 | | XRL arg ⇒ |
---|
[936] | 1737 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
| 1738 | 〈ticks, ticks〉 |
---|
[929] | 1739 | | CLR arg ⇒ |
---|
[936] | 1740 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
| 1741 | 〈ticks, ticks〉 |
---|
[929] | 1742 | | CPL arg ⇒ |
---|
[936] | 1743 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
| 1744 | 〈ticks, ticks〉 |
---|
[929] | 1745 | | RL arg ⇒ |
---|
[936] | 1746 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
| 1747 | 〈ticks, ticks〉 |
---|
[929] | 1748 | | RLC arg ⇒ |
---|
[936] | 1749 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
| 1750 | 〈ticks, ticks〉 |
---|
[929] | 1751 | | RR arg ⇒ |
---|
[936] | 1752 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
| 1753 | 〈ticks, ticks〉 |
---|
[929] | 1754 | | RRC arg ⇒ |
---|
[936] | 1755 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
| 1756 | 〈ticks, ticks〉 |
---|
[929] | 1757 | | SWAP arg ⇒ |
---|
[936] | 1758 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
| 1759 | 〈ticks, ticks〉 |
---|
[929] | 1760 | | MOV arg ⇒ |
---|
[936] | 1761 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
| 1762 | 〈ticks, ticks〉 |
---|
[929] | 1763 | | MOVX arg ⇒ |
---|
[936] | 1764 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
| 1765 | 〈ticks, ticks〉 |
---|
[929] | 1766 | | SETB arg ⇒ |
---|
[936] | 1767 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
| 1768 | 〈ticks, ticks〉 |
---|
[929] | 1769 | | PUSH arg ⇒ |
---|
[936] | 1770 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
| 1771 | 〈ticks, ticks〉 |
---|
[929] | 1772 | | POP arg ⇒ |
---|
[936] | 1773 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
| 1774 | 〈ticks, ticks〉 |
---|
[929] | 1775 | | XCH arg1 arg2 ⇒ |
---|
[936] | 1776 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
| 1777 | 〈ticks, ticks〉 |
---|
[929] | 1778 | | XCHD arg1 arg2 ⇒ |
---|
[936] | 1779 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
| 1780 | 〈ticks, ticks〉 |
---|
[929] | 1781 | | RET ⇒ |
---|
[936] | 1782 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
| 1783 | 〈ticks, ticks〉 |
---|
[929] | 1784 | | RETI ⇒ |
---|
[936] | 1785 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
| 1786 | 〈ticks, ticks〉 |
---|
[929] | 1787 | | NOP ⇒ |
---|
[936] | 1788 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
| 1789 | 〈ticks, ticks〉 |
---|
[929] | 1790 | ] |
---|
| 1791 | | Comment comment ⇒ 〈0, 0〉 |
---|
| 1792 | | Cost cost ⇒ 〈0, 0〉 |
---|
[938] | 1793 | | Jmp jmp ⇒ 〈2, 2〉 |
---|
| 1794 | | Call call ⇒ 〈2, 2〉 |
---|
| 1795 | | Mov dptr tgt ⇒ 〈2, 2〉 |
---|
[929] | 1796 | ]. |
---|
| 1797 | cases not_implemented (* policy returned medium_jump for conditional jumping = impossible *) |
---|
| 1798 | qed. |
---|
| 1799 | |
---|
[987] | 1800 | definition ticks_of: ∀p:pseudo_assembly_program. policy p → Word → nat × nat ≝ |
---|
| 1801 | λprogram: pseudo_assembly_program.λpol. |
---|
[936] | 1802 | λppc: Word. |
---|
| 1803 | let 〈preamble, pseudo〉 ≝ program in |
---|
| 1804 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc in |
---|
[987] | 1805 | ticks_of0 program pol ppc fetched. |
---|
[930] | 1806 | |
---|
| 1807 | lemma get_register_set_program_counter: |
---|
| 1808 | ∀T,s,pc. get_register T (set_program_counter … s pc) = get_register … s. |
---|
| 1809 | #T #s #pc % |
---|
| 1810 | qed. |
---|
| 1811 | |
---|
| 1812 | lemma get_8051_sfr_set_program_counter: |
---|
| 1813 | ∀T,s,pc. get_8051_sfr T (set_program_counter … s pc) = get_8051_sfr … s. |
---|
| 1814 | #T #s #pc % |
---|
| 1815 | qed. |
---|
| 1816 | |
---|
| 1817 | lemma get_bit_addressable_sfr_set_program_counter: |
---|
| 1818 | ∀T,s,pc. get_bit_addressable_sfr T (set_program_counter … s pc) = get_bit_addressable_sfr … s. |
---|
| 1819 | #T #s #pc % |
---|
| 1820 | qed. |
---|
| 1821 | |
---|
| 1822 | lemma low_internal_ram_set_program_counter: |
---|
| 1823 | ∀T,s,pc. low_internal_ram T (set_program_counter … s pc) = low_internal_ram … s. |
---|
| 1824 | #T #s #pc % |
---|
| 1825 | qed. |
---|
| 1826 | |
---|
| 1827 | lemma get_arg_8_set_program_counter: |
---|
| 1828 | ∀n.∀l:Vector addressing_mode_tag (S n). ¬(is_in ? l ACC_PC) → |
---|
| 1829 | ∀T,s,pc,b.∀arg:l. |
---|
| 1830 | get_arg_8 T (set_program_counter … s pc) b arg = get_arg_8 … s b arg. |
---|
| 1831 | [2,3: cases arg; *; normalize //] |
---|
| 1832 | #n #l #H #T #s #pc #b * *; [11: #NH @⊥ //] #X try #Y % |
---|
| 1833 | qed. |
---|
| 1834 | |
---|
[940] | 1835 | lemma set_bit_addressable_sfr_set_code_memory: |
---|
| 1836 | ∀T, U: Type[0]. |
---|
| 1837 | ∀ps: PreStatus ?. |
---|
| 1838 | ∀code_mem. |
---|
| 1839 | ∀x. |
---|
| 1840 | ∀val. |
---|
| 1841 | set_bit_addressable_sfr ? (set_code_memory T U ps code_mem) x val = |
---|
| 1842 | set_code_memory T U (set_bit_addressable_sfr ? ps x val) code_mem. |
---|
| 1843 | #T #U #ps #code_mem #x #val |
---|
| 1844 | whd in ⊢ (??%?) |
---|
| 1845 | whd in ⊢ (???(???%?)) |
---|
| 1846 | cases (eqb ? 128) [ normalize nodelta cases not_implemented |
---|
| 1847 | | normalize nodelta |
---|
| 1848 | cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented |
---|
| 1849 | | normalize nodelta |
---|
| 1850 | cases (eqb (nat_of_bitvector 8 x) ?) [ normalize nodelta cases not_implemented |
---|
| 1851 | | normalize nodelta |
---|
| 1852 | cases (eqb ? 176) [ normalize nodelta % |
---|
| 1853 | | normalize nodelta |
---|
| 1854 | cases (eqb ? 153) [ normalize nodelta % |
---|
| 1855 | | normalize nodelta |
---|
| 1856 | cases (eqb ? 138) [ normalize nodelta % |
---|
| 1857 | | normalize nodelta |
---|
| 1858 | cases (eqb ? 139) [ normalize nodelta % |
---|
| 1859 | | normalize nodelta |
---|
| 1860 | cases (eqb ? 140) [ normalize nodelta % |
---|
| 1861 | | normalize nodelta |
---|
| 1862 | cases (eqb ? 141) [ normalize nodelta % |
---|
| 1863 | | normalize nodelta |
---|
| 1864 | cases (eqb ? 200) [ normalize nodelta % |
---|
| 1865 | | normalize nodelta |
---|
| 1866 | cases (eqb ? 202) [ normalize nodelta % |
---|
| 1867 | | normalize nodelta |
---|
| 1868 | cases (eqb ? 203) [ normalize nodelta % |
---|
| 1869 | | normalize nodelta |
---|
| 1870 | cases (eqb ? 204) [ normalize nodelta % |
---|
| 1871 | | normalize nodelta |
---|
| 1872 | cases (eqb ? 205) [ normalize nodelta % |
---|
| 1873 | | normalize nodelta |
---|
| 1874 | cases (eqb ? 135) [ normalize nodelta % |
---|
| 1875 | | normalize nodelta |
---|
| 1876 | cases (eqb ? 136) [ normalize nodelta % |
---|
| 1877 | | normalize nodelta |
---|
| 1878 | cases (eqb ? 137) [ normalize nodelta % |
---|
| 1879 | | normalize nodelta |
---|
| 1880 | cases (eqb ? 152) [ normalize nodelta % |
---|
| 1881 | | normalize nodelta |
---|
| 1882 | cases (eqb ? 168) [ normalize nodelta % |
---|
| 1883 | | normalize nodelta |
---|
| 1884 | cases (eqb ? 184) [ normalize nodelta % |
---|
| 1885 | | normalize nodelta |
---|
| 1886 | cases (eqb ? 129) [ normalize nodelta % |
---|
| 1887 | | normalize nodelta |
---|
| 1888 | cases (eqb ? 130) [ normalize nodelta % |
---|
| 1889 | | normalize nodelta |
---|
| 1890 | cases (eqb ? 131) [ normalize nodelta % |
---|
| 1891 | | normalize nodelta |
---|
| 1892 | cases (eqb ? 208) [ normalize nodelta % |
---|
| 1893 | | normalize nodelta |
---|
| 1894 | cases (eqb ? 224) [ normalize nodelta % |
---|
| 1895 | | normalize nodelta |
---|
| 1896 | cases (eqb ? 240) [ normalize nodelta % |
---|
| 1897 | | normalize nodelta |
---|
| 1898 | cases not_implemented |
---|
| 1899 | ]]]]]]]]]]]]]]]]]]]]]]]]]] |
---|
| 1900 | qed. |
---|
| 1901 | |
---|
| 1902 | lemma set_arg_8_set_code_memory: |
---|
| 1903 | ∀n:nat. |
---|
| 1904 | ∀l:Vector addressing_mode_tag (S n). |
---|
| 1905 | ¬(is_in ? l ACC_PC) → |
---|
| 1906 | ∀T: Type[0]. |
---|
| 1907 | ∀U: Type[0]. |
---|
| 1908 | ∀ps: PreStatus ?. |
---|
| 1909 | ∀code_mem. |
---|
| 1910 | ∀val. |
---|
| 1911 | ∀b: l. |
---|
| 1912 | set_arg_8 ? (set_code_memory T U ps code_mem) b val = |
---|
| 1913 | set_code_memory T U (set_arg_8 ? ps b val) code_mem. |
---|
| 1914 | [2,3: cases b; *; normalize //] |
---|
| 1915 | #n #l #prf #T #U #ps #code_mem #val * *; |
---|
| 1916 | [*: |
---|
| 1917 | [1,2,3,4,8,9,15,16,17,18,19: #x] |
---|
| 1918 | try #y whd in ⊢ (??(%)?) |
---|
| 1919 | whd in ⊢ (???(???(%)?)) |
---|
| 1920 | [1,2: |
---|
| 1921 | cases (split bool 4 4 ?) |
---|
| 1922 | #nu' #nl' |
---|
| 1923 | normalize nodelta |
---|
| 1924 | cases (split bool 1 3 nu') |
---|
| 1925 | #bit1' #ignore' |
---|
| 1926 | normalize nodelta |
---|
[949] | 1927 | cases (get_index_v bool 4 nu' 0 ?) |
---|
[940] | 1928 | [1,2,3,4: |
---|
| 1929 | normalize nodelta |
---|
| 1930 | try % |
---|
| 1931 | try (@(set_bit_addressable_sfr_set_code_memory T U ps code_mem x val)) |
---|
| 1932 | try (normalize in ⊢ (???(???%?))) |
---|
| 1933 | ] |
---|
| 1934 | |3,4: % |
---|
| 1935 | |*: |
---|
| 1936 | try normalize nodelta |
---|
| 1937 | normalize cases (not_implemented) |
---|
| 1938 | ] |
---|
| 1939 | ] |
---|
[949] | 1940 | |
---|
| 1941 | |
---|
[940] | 1942 | qed. |
---|
| 1943 | |
---|
| 1944 | axiom set_arg_8_set_program_counter: |
---|
| 1945 | ∀n:nat. |
---|
| 1946 | ∀l:Vector addressing_mode_tag (S n). |
---|
| 1947 | ¬(is_in ? l ACC_PC) → |
---|
| 1948 | ∀T: Type[0]. |
---|
| 1949 | ∀ps: PreStatus ?. |
---|
| 1950 | ∀pc. |
---|
| 1951 | ∀val. |
---|
| 1952 | ∀b: l. |
---|
| 1953 | set_arg_8 ? (set_program_counter T ps pc) b val = |
---|
| 1954 | set_program_counter T (set_arg_8 ? ps b val) pc. |
---|
| 1955 | [1,2: cases b; *; normalize //] |
---|
| 1956 | qed. |
---|
| 1957 | |
---|
| 1958 | |
---|
[930] | 1959 | lemma get_arg_8_set_code_memory: |
---|
| 1960 | ∀T1,T2,s,code_mem,b,arg. |
---|
| 1961 | get_arg_8 T1 (set_code_memory T2 T1 s code_mem) b arg = get_arg_8 … s b arg. |
---|
| 1962 | #T1 #T2 #s #code_mem #b #arg % |
---|
| 1963 | qed. |
---|
| 1964 | |
---|
| 1965 | lemma set_code_memory_set_flags: |
---|
| 1966 | ∀T1,T2,s,f1,f2,f3,code_mem. |
---|
| 1967 | set_code_memory T1 T2 (set_flags T1 s f1 f2 f3) code_mem = |
---|
| 1968 | set_flags … (set_code_memory … s code_mem) f1 f2 f3. |
---|
[935] | 1969 | #T1 #T2 #s #f1 #f2 #f3 #code_mem % |
---|
[930] | 1970 | qed. |
---|
| 1971 | |
---|
| 1972 | lemma set_program_counter_set_flags: |
---|
| 1973 | ∀T1,s,f1,f2,f3,pc. |
---|
| 1974 | set_program_counter T1 (set_flags T1 s f1 f2 f3) pc = |
---|
| 1975 | set_flags … (set_program_counter … s pc) f1 f2 f3. |
---|
[935] | 1976 | #T1 #s #f1 #f2 #f3 #pc % |
---|
[930] | 1977 | qed. |
---|
| 1978 | |
---|
| 1979 | lemma program_counter_set_flags: |
---|
| 1980 | ∀T1,s,f1,f2,f3. |
---|
| 1981 | program_counter T1 (set_flags T1 s f1 f2 f3) = program_counter … s. |
---|
[935] | 1982 | #T1 #s #f1 #f2 #f3 % |
---|
[930] | 1983 | qed. |
---|
| 1984 | |
---|
[948] | 1985 | lemma special_function_registers_8051_write_at_stack_pointer: |
---|
| 1986 | ∀T,s,x. |
---|
| 1987 | special_function_registers_8051 T (write_at_stack_pointer T s x) |
---|
| 1988 | = special_function_registers_8051 T s. |
---|
| 1989 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 1990 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 1991 | cases (get_index_v bool ????) % |
---|
| 1992 | qed. |
---|
| 1993 | |
---|
| 1994 | lemma get_8051_sfr_write_at_stack_pointer: |
---|
| 1995 | ∀T,s,x,y. get_8051_sfr T (write_at_stack_pointer T s x) y = get_8051_sfr T s y. |
---|
| 1996 | #T #s #x #y whd in ⊢ (??%%) // |
---|
| 1997 | qed. |
---|
| 1998 | |
---|
| 1999 | lemma code_memory_write_at_stack_pointer: |
---|
| 2000 | ∀T,s,x. code_memory T (write_at_stack_pointer T s x) = code_memory T s. |
---|
| 2001 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2002 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2003 | cases (get_index_v bool ????) % |
---|
| 2004 | qed. |
---|
| 2005 | |
---|
[950] | 2006 | axiom low_internal_ram_write_at_stack_pointer: |
---|
[988] | 2007 | ∀T1,T2,M,s1,s2,s3.∀pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
[950] | 2008 | get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP → |
---|
| 2009 | low_internal_ram ? s2 = low_internal_ram T2 s3 → |
---|
| 2010 | sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
| 2011 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
[988] | 2012 | bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) → |
---|
[950] | 2013 | low_internal_ram T1 |
---|
| 2014 | (write_at_stack_pointer ? |
---|
| 2015 | (set_8051_sfr ? |
---|
| 2016 | (write_at_stack_pointer ? |
---|
| 2017 | (set_8051_sfr ? |
---|
| 2018 | (set_low_internal_ram ? s1 |
---|
| 2019 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram ? s2))) |
---|
| 2020 | SFR_SP sp1) |
---|
| 2021 | bl) |
---|
| 2022 | SFR_SP sp2) |
---|
| 2023 | bu) |
---|
| 2024 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
---|
| 2025 | (low_internal_ram ? |
---|
| 2026 | (write_at_stack_pointer ? |
---|
| 2027 | (set_8051_sfr ? |
---|
| 2028 | (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl) |
---|
| 2029 | SFR_SP sp2) |
---|
| 2030 | pbu)). |
---|
| 2031 | |
---|
[951] | 2032 | axiom high_internal_ram_write_at_stack_pointer: |
---|
[988] | 2033 | ∀T1,T2,M,s1,s2,s3,pol.∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
---|
[951] | 2034 | get_8051_sfr ? s2 SFR_SP = get_8051_sfr ? s3 SFR_SP → |
---|
| 2035 | high_internal_ram ? s2 = high_internal_ram T2 s3 → |
---|
| 2036 | sp1 = \snd (half_add ? (get_8051_sfr ? s1 SFR_SP) (bitvector_of_nat 8 1)) → |
---|
| 2037 | sp2 = \snd (half_add ? sp1 (bitvector_of_nat 8 1)) → |
---|
[988] | 2038 | bu@@bl = sigma (code_memory … s2) pol (pbu@@pbl) → |
---|
[951] | 2039 | high_internal_ram T1 |
---|
| 2040 | (write_at_stack_pointer ? |
---|
| 2041 | (set_8051_sfr ? |
---|
| 2042 | (write_at_stack_pointer ? |
---|
| 2043 | (set_8051_sfr ? |
---|
| 2044 | (set_high_internal_ram ? s1 |
---|
| 2045 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram ? s2))) |
---|
| 2046 | SFR_SP sp1) |
---|
| 2047 | bl) |
---|
| 2048 | SFR_SP sp2) |
---|
| 2049 | bu) |
---|
| 2050 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
---|
| 2051 | (high_internal_ram ? |
---|
| 2052 | (write_at_stack_pointer ? |
---|
| 2053 | (set_8051_sfr ? |
---|
| 2054 | (write_at_stack_pointer ? (set_8051_sfr ? s3 SFR_SP sp1) pbl) |
---|
| 2055 | SFR_SP sp2) |
---|
| 2056 | pbu)). |
---|
[948] | 2057 | |
---|
[950] | 2058 | lemma set_program_counter_set_low_internal_ram: |
---|
| 2059 | ∀T,s,x,y. |
---|
| 2060 | set_program_counter T (set_low_internal_ram … s x) y = |
---|
| 2061 | set_low_internal_ram … (set_program_counter … s y) x. |
---|
| 2062 | // |
---|
| 2063 | qed. |
---|
| 2064 | |
---|
| 2065 | lemma set_clock_set_low_internal_ram: |
---|
| 2066 | ∀T,s,x,y. |
---|
| 2067 | set_clock T (set_low_internal_ram … s x) y = |
---|
| 2068 | set_low_internal_ram … (set_clock … s y) x. |
---|
| 2069 | // |
---|
| 2070 | qed. |
---|
| 2071 | |
---|
[951] | 2072 | lemma set_program_counter_set_high_internal_ram: |
---|
| 2073 | ∀T,s,x,y. |
---|
| 2074 | set_program_counter T (set_high_internal_ram … s x) y = |
---|
| 2075 | set_high_internal_ram … (set_program_counter … s y) x. |
---|
| 2076 | // |
---|
| 2077 | qed. |
---|
| 2078 | |
---|
| 2079 | lemma set_clock_set_high_internal_ram: |
---|
| 2080 | ∀T,s,x,y. |
---|
| 2081 | set_clock T (set_high_internal_ram … s x) y = |
---|
| 2082 | set_high_internal_ram … (set_clock … s y) x. |
---|
| 2083 | // |
---|
| 2084 | qed. |
---|
| 2085 | |
---|
| 2086 | lemma set_low_internal_ram_set_high_internal_ram: |
---|
| 2087 | ∀T,s,x,y. |
---|
| 2088 | set_low_internal_ram T (set_high_internal_ram … s x) y = |
---|
| 2089 | set_high_internal_ram … (set_low_internal_ram … s y) x. |
---|
| 2090 | // |
---|
| 2091 | qed. |
---|
| 2092 | |
---|
[948] | 2093 | lemma external_ram_write_at_stack_pointer: |
---|
| 2094 | ∀T,s,x. external_ram T (write_at_stack_pointer T s x) = external_ram T s. |
---|
| 2095 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2096 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2097 | cases (get_index_v bool ????) % |
---|
| 2098 | qed. |
---|
| 2099 | |
---|
| 2100 | lemma special_function_registers_8052_write_at_stack_pointer: |
---|
| 2101 | ∀T,s,x. |
---|
| 2102 | special_function_registers_8052 T (write_at_stack_pointer T s x) |
---|
| 2103 | = special_function_registers_8052 T s. |
---|
| 2104 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2105 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2106 | cases (get_index_v bool ????) % |
---|
| 2107 | qed. |
---|
| 2108 | |
---|
| 2109 | lemma p1_latch_write_at_stack_pointer: |
---|
| 2110 | ∀T,s,x. p1_latch T (write_at_stack_pointer T s x) = p1_latch T s. |
---|
| 2111 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2112 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2113 | cases (get_index_v bool ????) % |
---|
| 2114 | qed. |
---|
| 2115 | |
---|
| 2116 | lemma p3_latch_write_at_stack_pointer: |
---|
| 2117 | ∀T,s,x. p3_latch T (write_at_stack_pointer T s x) = p3_latch T s. |
---|
| 2118 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2119 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2120 | cases (get_index_v bool ????) % |
---|
| 2121 | qed. |
---|
| 2122 | |
---|
| 2123 | lemma clock_write_at_stack_pointer: |
---|
| 2124 | ∀T,s,x. clock T (write_at_stack_pointer T s x) = clock T s. |
---|
| 2125 | #T #s #x whd in ⊢ (??(??%)?) |
---|
| 2126 | cases (split ????) #nu #nl normalize nodelta; |
---|
| 2127 | cases (get_index_v bool ????) % |
---|
| 2128 | qed. |
---|
| 2129 | |
---|
[950] | 2130 | axiom get_index_v_set_index: |
---|
| 2131 | ∀T,n,x,m,p,y. get_index_v T n (set_index T n x m y p) m p = y. |
---|
| 2132 | |
---|
| 2133 | lemma get_8051_sfr_set_8051_sfr: |
---|
| 2134 | ∀T,s,x,y. get_8051_sfr T (set_8051_sfr ? s x y) x = y. |
---|
| 2135 | #T *; #A #B #C #D #E #F #G #H #I #J *; #y whd in ⊢ (??(??%?)?) |
---|
| 2136 | whd in match get_8051_sfr; normalize nodelta @get_index_v_set_index |
---|
| 2137 | qed. |
---|
| 2138 | |
---|
[959] | 2139 | lemma program_counter_set_8051_sfr: |
---|
| 2140 | ∀T,s,x,y. program_counter T (set_8051_sfr ? s x y) = program_counter ? s. |
---|
| 2141 | // |
---|
| 2142 | qed. |
---|
| 2143 | |
---|
[989] | 2144 | axiom get_arg_8_set_low_internal_ram: |
---|
| 2145 | ∀M,s,x,b,z. get_arg_8 M (set_low_internal_ram ? s x) b z = get_arg_8 ? s b z. |
---|
| 2146 | |
---|
[935] | 2147 | lemma eq_rect_Type1_r: |
---|
| 2148 | ∀A: Type[1]. |
---|
| 2149 | ∀a:A. |
---|
| 2150 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
| 2151 | #A #a #P #H #x #p |
---|
| 2152 | generalize in match H |
---|
| 2153 | generalize in match P |
---|
| 2154 | cases p |
---|
| 2155 | // |
---|
| 2156 | qed. |
---|
| 2157 | |
---|
[941] | 2158 | lemma split_eq_status: |
---|
| 2159 | ∀T. |
---|
| 2160 | ∀A1,A2,A3,A4,A5,A6,A7,A8,A9,A10. |
---|
| 2161 | ∀B1,B2,B3,B4,B5,B6,B7,B8,B9,B10. |
---|
| 2162 | A1=B1 → A2=B2 → A3=B3 → A4=B4 → A5=B5 → A6=B6 → A7=B7 → A8=B8 → A9=B9 → A10=B10 → |
---|
| 2163 | mk_PreStatus T A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 = |
---|
| 2164 | mk_PreStatus T B1 B2 B3 B4 B5 B6 B7 B8 B9 B10. |
---|
| 2165 | // |
---|
| 2166 | qed. |
---|
| 2167 | |
---|
[949] | 2168 | axiom split_append: |
---|
| 2169 | ∀A: Type[0]. |
---|
| 2170 | ∀m, n: nat. |
---|
| 2171 | ∀v, v': Vector A m. |
---|
| 2172 | ∀q, q': Vector A n. |
---|
| 2173 | let 〈v', q'〉 ≝ split A m n (v@@q) in |
---|
| 2174 | v = v' ∧ q = q'. |
---|
| 2175 | |
---|
| 2176 | axiom split_vector_singleton: |
---|
| 2177 | ∀A: Type[0]. |
---|
| 2178 | ∀n: nat. |
---|
| 2179 | ∀v: Vector A (S n). |
---|
| 2180 | ∀rest: Vector A n. |
---|
| 2181 | ∀s: Vector A 1. |
---|
| 2182 | ∀prf. |
---|
| 2183 | v = s @@ rest → |
---|
| 2184 | ((get_index_v A ? v 0 prf) ::: rest) = v. |
---|
| 2185 | |
---|
[985] | 2186 | example sub_minus_one_seven_eight: |
---|
[949] | 2187 | ∀v: BitVector 7. |
---|
| 2188 | false ::: (\fst (sub_7_with_carry v (bitvector_of_nat ? 1) false)) = |
---|
| 2189 | \fst (sub_8_with_carry (false ::: v) (bitvector_of_nat ? 1) false). |
---|
[985] | 2190 | cases daemon. |
---|
| 2191 | qed. |
---|
[949] | 2192 | |
---|
[950] | 2193 | lemma pair_destruct_1: |
---|
| 2194 | ∀A,B.∀a:A.∀b:B.∀c. 〈a,b〉 = c → a = \fst c. |
---|
| 2195 | #A #B #a #b *; /2/ |
---|
| 2196 | qed. |
---|
| 2197 | |
---|
| 2198 | lemma pair_destruct_2: |
---|
| 2199 | ∀A,B.∀a:A.∀b:B.∀c. 〈a,b〉 = c → b = \snd c. |
---|
| 2200 | #A #B #a #b *; /2/ |
---|
| 2201 | qed. |
---|
| 2202 | |
---|
| 2203 | (* |
---|
[945] | 2204 | lemma blah: |
---|
[949] | 2205 | ∀m: internal_pseudo_address_map. |
---|
[945] | 2206 | ∀s: PseudoStatus. |
---|
| 2207 | ∀arg: Byte. |
---|
[949] | 2208 | ∀b: bool. |
---|
| 2209 | addressing_mode_ok m s (DIRECT arg) = true → |
---|
| 2210 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
---|
| 2211 | get_arg_8 ? s b (DIRECT arg). |
---|
| 2212 | [2, 3: normalize % ] |
---|
| 2213 | #m #s #arg #b #hyp |
---|
| 2214 | whd in ⊢ (??%%) |
---|
| 2215 | @split_elim'' |
---|
| 2216 | #nu' #nl' #arg_nu_nl_eq |
---|
| 2217 | normalize nodelta |
---|
| 2218 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
---|
| 2219 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
---|
| 2220 | #get_index_v_eq |
---|
| 2221 | normalize nodelta |
---|
| 2222 | [2: |
---|
| 2223 | normalize nodelta |
---|
| 2224 | @split_elim'' |
---|
| 2225 | #bit_one' #three_bits' #bit_one_three_bit_eq |
---|
| 2226 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
---|
| 2227 | normalize nodelta |
---|
| 2228 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
---|
| 2229 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
---|
| 2230 | #Saddr #carr' #Saddr_carr_eq |
---|
| 2231 | normalize nodelta |
---|
| 2232 | #carr_hyp' |
---|
| 2233 | @carr_hyp' |
---|
| 2234 | [1: |
---|
| 2235 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
---|
| 2236 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
---|
| 2237 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
---|
| 2238 | #member_eq |
---|
| 2239 | normalize nodelta |
---|
| 2240 | [2: #destr destruct(destr) |
---|
| 2241 | |1: -carr_hyp'; |
---|
| 2242 | >arg_nu_nl_eq |
---|
| 2243 | <(split_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
---|
| 2244 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
---|
| 2245 | |2: @le_S @le_S @le_S @le_n |
---|
| 2246 | ] |
---|
| 2247 | cases (member (BitVector 8) ? (\fst ?) ?) |
---|
| 2248 | [1: #destr normalize in destr; destruct(destr) |
---|
| 2249 | |2: |
---|
| 2250 | ] |
---|
| 2251 | ] |
---|
| 2252 | |3: >get_index_v_eq in ⊢ (??%?) |
---|
| 2253 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
---|
| 2254 | >(split_vector_singleton … bit_one_three_bit_eq) |
---|
| 2255 | <arg_nu_nl_eq |
---|
| 2256 | whd in hyp:(??%?) |
---|
| 2257 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
---|
| 2258 | normalize nodelta [*: #ignore @sym_eq ] |
---|
| 2259 | ] |
---|
| 2260 | | |
---|
| 2261 | ]. |
---|
[950] | 2262 | *) |
---|
[949] | 2263 | (* |
---|
[945] | 2264 | map_address0 ... (DIRECT arg) = Some .. → |
---|
| 2265 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
---|
| 2266 | get_arg_8 (internal_ram ...) (DIRECT arg) |
---|
| 2267 | |
---|
| 2268 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
---|
| 2269 | then Some internal_pseudo_address_map M |
---|
| 2270 | else None internal_pseudo_address_map ) |
---|
| 2271 | =Some internal_pseudo_address_map M') |
---|
[946] | 2272 | *) |
---|
[945] | 2273 | |
---|
[877] | 2274 | lemma main_thm: |
---|
[988] | 2275 | ∀M,M',ps,s,s'',pol. |
---|
[942] | 2276 | next_internal_pseudo_address_map M ps = Some … M' → |
---|
[988] | 2277 | status_of_pseudo_status M ps pol = Some … s → |
---|
| 2278 | status_of_pseudo_status M' (execute_1_pseudo_instruction (ticks_of (code_memory … ps) pol) ps) ? = Some … s'' → |
---|
[926] | 2279 | ∃n. execute n s = s''. |
---|
[988] | 2280 | [2: >execute_1_pseudo_instruction_preserves_code_memory @pol] |
---|
| 2281 | #M #M' #ps #s #s'' #pol |
---|
| 2282 | generalize in match (fetch_assembly_pseudo2 (code_memory … ps) pol) |
---|
[942] | 2283 | whd in ⊢ (? → ? → ??%? → ??%? → ?) |
---|
[911] | 2284 | >execute_1_pseudo_instruction_preserves_code_memory |
---|
[988] | 2285 | generalize in match (refl … (assembly (code_memory … ps) pol)) |
---|
| 2286 | generalize in match (assembly (code_memory … ps) pol) in ⊢ (??%? → %) #ASS whd in ⊢ (???% → ?) |
---|
| 2287 | cases (build_maps (code_memory … ps) pol) |
---|
[985] | 2288 | #labels #costs change in ⊢ (? → ? → ??%? → ?) with (next_internal_pseudo_address_map0 ? ? ? ?) |
---|
[988] | 2289 | generalize in match pol; -pol; |
---|
| 2290 | @(match code_memory … ps return λx. ∀e:code_memory … ps = x.? with [pair preamble instr_list ⇒ ?]) [%] |
---|
| 2291 | #EQ0 #pol normalize nodelta; |
---|
[923] | 2292 | generalize in ⊢ (???(match % with [_ ⇒ ? | _ ⇒ ?]) → ?) *; normalize nodelta; |
---|
[942] | 2293 | [ #EQ >EQ #_ #_ #abs @⊥ normalize in abs; destruct (abs) ] |
---|
[923] | 2294 | * #final_ppc * #final_pc #assembled #EQ >EQ -EQ ASS; normalize nodelta; |
---|
[985] | 2295 | #H generalize in match (H ? (refl …)) -H; #H; |
---|
[942] | 2296 | #MAP |
---|
[919] | 2297 | #H1 generalize in match (option_destruct_Some ??? H1) -H1; #H1 <H1 -H1; |
---|
| 2298 | #H2 generalize in match (option_destruct_Some ??? H2) -H2; #H2 <H2 -H2; |
---|
| 2299 | change with |
---|
[926] | 2300 | (∃n. |
---|
[942] | 2301 | execute n |
---|
| 2302 | (set_low_internal_ram ? |
---|
| 2303 | (set_high_internal_ram ? |
---|
| 2304 | (set_program_counter ? |
---|
| 2305 | (set_code_memory ?? ps (load_code_memory assembled)) |
---|
[988] | 2306 | (sigma 〈preamble,instr_list〉 pol (program_counter ? ps))) |
---|
[942] | 2307 | (high_internal_ram_of_pseudo_high_internal_ram M ?)) |
---|
| 2308 | (low_internal_ram_of_pseudo_low_internal_ram M ?)) |
---|
| 2309 | = set_low_internal_ram ? |
---|
| 2310 | (set_high_internal_ram ? |
---|
| 2311 | (set_program_counter ? |
---|
| 2312 | (set_code_memory ?? (execute_1_pseudo_instruction ? ps) (load_code_memory assembled)) |
---|
[988] | 2313 | (sigma ???)) |
---|
[942] | 2314 | ?) |
---|
| 2315 | ?) |
---|
[923] | 2316 | whd in match (\snd 〈preamble,instr_list〉) in H; |
---|
| 2317 | whd in match (\fst 〈preamble,instr_list〉) in H; |
---|
| 2318 | whd in match (\snd 〈final_pc,assembled〉) in H; |
---|
[942] | 2319 | whd in match (\snd 〈preamble,instr_list〉) in MAP; |
---|
[924] | 2320 | -s s'' labels costs final_ppc final_pc; |
---|
[988] | 2321 | letin ps' ≝ (execute_1_pseudo_instruction (ticks_of 〈preamble,instr_list〉 pol) ps) |
---|
[925] | 2322 | (* NICE STATEMENT HERE *) |
---|
| 2323 | generalize in match (refl … ps') generalize in ⊢ (??%? → ?) normalize nodelta; -ps'; #ps' |
---|
[926] | 2324 | #K <K generalize in match K; -K; |
---|
| 2325 | (* STATEMENT WITH EQUALITY HERE *) |
---|
[936] | 2326 | whd in ⊢ (???(?%?) → ?) |
---|
| 2327 | whd in ⊢ (???% → ?) generalize in match (H (program_counter … ps)) -H; >EQ0 normalize nodelta; |
---|
[942] | 2328 | cases (fetch_pseudo_instruction instr_list (program_counter … ps)) in MAP ⊢ % |
---|
| 2329 | #pi #newppc normalize nodelta; #MAP * #instructions *; |
---|
| 2330 | cases pi in MAP; normalize nodelta; |
---|
| 2331 | [2,3: (* Comment, Cost *) #ARG #MAP #H1 #H2 #EQ %[1,3:@0] |
---|
| 2332 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP M'; |
---|
[930] | 2333 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
[985] | 2334 | #H2 >(eq_bv_eq … H2) >EQ % |
---|
[942] | 2335 | (* |6: (* Mov *) #arg1 #arg2 |
---|
| 2336 | #H1 #H2 #EQ %[@1] |
---|
| 2337 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
| 2338 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
| 2339 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
| 2340 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
| 2341 | >H2b >(eq_instruction_to_eq … H2a) |
---|
| 2342 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
---|
| 2343 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; whd in ⊢ (???% → ??%?) |
---|
| 2344 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
---|
| 2345 | normalize nodelta; |
---|
| 2346 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
---|
| 2347 | #result #flags |
---|
[985] | 2348 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % *) |
---|
[989] | 2349 | (* |5: (* Call *) #label #MAP |
---|
[948] | 2350 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP <MAP -MAP; |
---|
[988] | 2351 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
---|
[948] | 2352 | [ (* short *) #abs @⊥ destruct (abs) |
---|
| 2353 | |3: (* long *) #H1 #H2 #EQ %[@1] |
---|
[946] | 2354 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
| 2355 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
[988] | 2356 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
[946] | 2357 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
---|
| 2358 | >H2b >(eq_instruction_to_eq … H2a) |
---|
| 2359 | generalize in match EQ; -EQ; |
---|
[948] | 2360 | whd in ⊢ (???% → ??%?); |
---|
[950] | 2361 | generalize in match (refl … (half_add 8 (get_8051_sfr ? ps SFR_SP) (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry #new_sp #EQ1 normalize nodelta; |
---|
[985] | 2362 | >(eq_bv_eq … H2c) |
---|
[948] | 2363 | change with |
---|
| 2364 | ((?=let 〈ppc_bu,ppc_bl〉 ≝ split bool 8 8 newppc in ?) → |
---|
[988] | 2365 | (let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
---|
[948] | 2366 | generalize in match (refl … (split … 8 8 newppc)) cases (split bool 8 8 newppc) in ⊢ (??%? → %) #ppc_bu #ppc_bl #EQppc |
---|
[988] | 2367 | generalize in match (refl … (split … 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) cases (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc)) in ⊢ (??%? → %) #pc_bu #pc_bl #EQpc normalize nodelta; |
---|
[948] | 2368 | >get_8051_sfr_write_at_stack_pointer >get_8051_sfr_write_at_stack_pointer |
---|
[950] | 2369 | >get_8051_sfr_set_8051_sfr >get_8051_sfr_set_8051_sfr |
---|
| 2370 | generalize in match (refl … (half_add ? new_sp (bitvector_of_nat ? 1))) cases (half_add ???) in ⊢ (??%? → %) #carry' #new_sp' #EQ2 normalize nodelta; |
---|
[985] | 2371 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
[951] | 2372 | @split_eq_status; |
---|
[948] | 2373 | [ >code_memory_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2374 | >code_memory_write_at_stack_pointer % |
---|
[950] | 2375 | | >set_program_counter_set_low_internal_ram |
---|
| 2376 | >set_clock_set_low_internal_ram |
---|
| 2377 | @low_internal_ram_write_at_stack_pointer |
---|
[988] | 2378 | [ >EQ0 @pol | % | % |
---|
[950] | 2379 | | @(pair_destruct_2 … EQ1) |
---|
| 2380 | | @(pair_destruct_2 … EQ2) |
---|
| 2381 | | >(pair_destruct_1 ????? EQpc) |
---|
| 2382 | >(pair_destruct_2 ????? EQpc) |
---|
| 2383 | @split_elim #x #y #H <H -x y H; |
---|
| 2384 | >(pair_destruct_1 ????? EQppc) |
---|
| 2385 | >(pair_destruct_2 ????? EQppc) |
---|
| 2386 | @split_elim #x #y #H <H -x y H; |
---|
| 2387 | >EQ0 % ] |
---|
[951] | 2388 | | >set_low_internal_ram_set_high_internal_ram |
---|
| 2389 | >set_program_counter_set_high_internal_ram |
---|
| 2390 | >set_clock_set_high_internal_ram |
---|
| 2391 | @high_internal_ram_write_at_stack_pointer |
---|
[988] | 2392 | [ >EQ0 @pol | % | % |
---|
[951] | 2393 | | @(pair_destruct_2 … EQ1) |
---|
| 2394 | | @(pair_destruct_2 … EQ2) |
---|
| 2395 | | >(pair_destruct_1 ????? EQpc) |
---|
| 2396 | >(pair_destruct_2 ????? EQpc) |
---|
| 2397 | @split_elim #x #y #H <H -x y H; |
---|
| 2398 | >(pair_destruct_1 ????? EQppc) |
---|
| 2399 | >(pair_destruct_2 ????? EQppc) |
---|
| 2400 | @split_elim #x #y #H <H -x y H; |
---|
| 2401 | >EQ0 % ] |
---|
[948] | 2402 | | >external_ram_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2403 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2404 | >external_ram_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2405 | >external_ram_write_at_stack_pointer % |
---|
[988] | 2406 | | change with (? = sigma ?? (address_of_word_labels_code_mem (\snd (code_memory ? ps)) ?)) |
---|
[948] | 2407 | >EQ0 % |
---|
| 2408 | | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2409 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2410 | >special_function_registers_8051_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2411 | >special_function_registers_8051_write_at_stack_pointer % |
---|
| 2412 | | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2413 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2414 | >special_function_registers_8052_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2415 | >special_function_registers_8052_write_at_stack_pointer % |
---|
| 2416 | | >p1_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2417 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2418 | >p1_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2419 | >p1_latch_write_at_stack_pointer % |
---|
| 2420 | | >p3_latch_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2421 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2422 | >p3_latch_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2423 | >p3_latch_write_at_stack_pointer % |
---|
| 2424 | | >clock_write_at_stack_pointer whd in ⊢ (??%?) |
---|
| 2425 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2426 | >clock_write_at_stack_pointer whd in ⊢ (???%) |
---|
| 2427 | >clock_write_at_stack_pointer %] |
---|
[985] | 2428 | (*| (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
---|
[959] | 2429 | @pair_elim' #fst_5_addr #rest_addr #EQ1 |
---|
| 2430 | @pair_elim' #fst_5_pc #rest_pc #EQ2 |
---|
[946] | 2431 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
---|
| 2432 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
---|
| 2433 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
| 2434 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
---|
[959] | 2435 | @pair_elim' * #instr #newppc' #ticks #EQn |
---|
| 2436 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c >H2b >(eq_instruction_to_eq … H2a) whd in ⊢ (??%?) |
---|
[985] | 2437 | generalize in match EQ; -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
---|
[959] | 2438 | @pair_elim' #carry #new_sp change with (half_add ? (get_8051_sfr ? ps ?) ? = ? → ?) #EQ4 |
---|
| 2439 | @split_elim' #pc_bu #pc_bl >program_counter_set_8051_sfr XXX change with (newppc = ?) #EQ5 |
---|
| 2440 | @pair_elim' #carry' #new_sp' #EQ6 normalize nodelta; #EQx >EQx -EQx; |
---|
| 2441 | change in ⊢ (??(match ????% with [_ ⇒ ?])?) with (sigma … newppc) |
---|
| 2442 | @split_elim' #pc_bu' #pc_bl' #EQ7 change with (newppc' = ? → ?) |
---|
| 2443 | >get_8051_sfr_set_8051_sfr |
---|
| 2444 | |
---|
[985] | 2445 | whd in EQ:(???%) ⊢ ? >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
---|
[946] | 2446 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 newppc) in ?)=?) |
---|
| 2447 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 newppc))) |
---|
| 2448 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
---|
| 2449 | generalize in match (refl … (split bool 4 4 pc_bu)) |
---|
| 2450 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
---|
| 2451 | generalize in match (refl … (split bool 3 8 rest_addr)) |
---|
| 2452 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
---|
| 2453 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
---|
| 2454 | generalize in match |
---|
| 2455 | (refl … |
---|
| 2456 | (half_add 16 (sigma 〈preamble,instr_list〉 newppc) |
---|
| 2457 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
---|
| 2458 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
---|
| 2459 | @split_eq_status try % |
---|
| 2460 | [ change with (? = sigma ? (address_of_word_labels ps label)) |
---|
| 2461 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
---|
| 2462 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
---|
[985] | 2463 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] *)] |
---|
[948] | 2464 | |4: (* Jmp *) #label #MAP |
---|
| 2465 | generalize in match (option_destruct_Some ??? MAP) -MAP; #MAP >MAP -MAP; |
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[988] | 2466 | whd in ⊢ (???% → ?) cases (pol ?) normalize nodelta; |
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[948] | 2467 | [3: (* long *) #H1 #H2 #EQ %[@1] |
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[938] | 2468 | (* normalize in H1; !!!*) generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
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| 2469 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
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[988] | 2470 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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[938] | 2471 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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| 2472 | >H2b >(eq_instruction_to_eq … H2a) |
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| 2473 | generalize in match EQ; -EQ; |
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[985] | 2474 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
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[948] | 2475 | cases ps in EQ0 ⊢ %; #A1 #A2 #A3 #A4 #A5 #A6 #A7 #A8 #A9 #A10 #XXXX >XXXX % |
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| 2476 | |1: (* short *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
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[941] | 2477 | generalize in match |
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| 2478 | (refl ? |
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| 2479 | (sub_16_with_carry |
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[988] | 2480 | (sigma 〈preamble,instr_list〉 pol (program_counter … ps)) |
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| 2481 | (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)) |
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[941] | 2482 | false)) |
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| 2483 | cases (sub_16_with_carry ???) in ⊢ (??%? → %); #results #flags normalize nodelta; |
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| 2484 | generalize in match (refl … (split … 8 8 results)) cases (split ????) in ⊢ (??%? → %) #upper #lower normalize nodelta; |
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| 2485 | generalize in match (refl … (eq_bv … upper (zero 8))) cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; |
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| 2486 | #EQ1 #EQ2 #EQ3 #H1 [2: @⊥ destruct (H1)] |
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| 2487 | generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
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| 2488 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
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[988] | 2489 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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[941] | 2490 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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| 2491 | >H2b >(eq_instruction_to_eq … H2a) |
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| 2492 | generalize in match EQ; -EQ; |
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| 2493 | whd in ⊢ (???% → ?); |
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[985] | 2494 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) |
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[988] | 2495 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma ???) ? in ?) = ?) |
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| 2496 | generalize in match (refl … (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) (sign_extension lower))) |
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[941] | 2497 | cases (half_add ???) in ⊢ (??%? → %) #carry #newpc normalize nodelta #EQ4 |
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[988] | 2498 | @split_eq_status try % change with (newpc = sigma ?? (address_of_word_labels ps label)) |
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[941] | 2499 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
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| 2500 | | (* medium *) #H1 #H2 #EQ %[@1] generalize in match H1; -H1; |
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| 2501 | generalize in match |
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| 2502 | (refl … |
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[988] | 2503 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (address_of_word_labels_code_mem instr_list label)))) |
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[941] | 2504 | cases (split ????) in ⊢ (??%? → %) #fst_5_addr #rest_addr normalize nodelta; #EQ1 |
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| 2505 | generalize in match |
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| 2506 | (refl … |
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[988] | 2507 | (split … 5 11 (sigma 〈preamble,instr_list〉 pol (program_counter … ps)))) |
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[941] | 2508 | cases (split ????) in ⊢ (??%? → %) #fst_5_pc #rest_pc normalize nodelta; #EQ2 |
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| 2509 | generalize in match (refl … (eq_bv … fst_5_addr fst_5_pc)) |
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| 2510 | cases (eq_bv ???) in ⊢ (??%? → %) normalize nodelta; #EQ3 #TEQ [2: destruct (TEQ)] |
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| 2511 | generalize in match (option_destruct_Some ??? TEQ) -TEQ; #K1 >K1 in H2; whd in ⊢ (% → ?) |
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| 2512 | change in ⊢ (? →??%?) with (execute_1_0 ??) |
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[988] | 2513 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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[941] | 2514 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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| 2515 | >H2b >(eq_instruction_to_eq … H2a) |
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| 2516 | generalize in match EQ; -EQ; |
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| 2517 | whd in ⊢ (???% → ?); |
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[985] | 2518 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) whd in ⊢ (??%?) |
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[988] | 2519 | change with ((let 〈pc_bu,pc_bl〉 ≝ split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc) in ?)=?) |
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| 2520 | generalize in match (refl … (split bool 8 8 (sigma 〈preamble,instr_list〉 pol newppc))) |
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[941] | 2521 | cases (split ????) in ⊢ (??%? → %) #pc_bu #pc_bl normalize nodelta; #EQ4 |
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| 2522 | generalize in match (refl … (split bool 4 4 pc_bu)) |
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| 2523 | cases (split ????) in ⊢ (??%? → %) #nu #nl normalize nodelta; #EQ5 |
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| 2524 | generalize in match (refl … (split bool 3 8 rest_addr)) |
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| 2525 | cases (split ????) in ⊢ (??%? → %) #relevant1 #relevant2 normalize nodelta; #EQ6 |
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| 2526 | change with ((let 〈carry,new_pc〉 ≝ half_add ? (sigma … newppc) ? in ?) = ?) |
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| 2527 | generalize in match |
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| 2528 | (refl … |
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[988] | 2529 | (half_add 16 (sigma 〈preamble,instr_list〉 pol newppc) |
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[941] | 2530 | ((nu@@get_index' bool 0 3 nl:::relevant1)@@relevant2))) |
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[948] | 2531 | cases (half_add ???) in ⊢ (??%? → %) #carry #new_pc normalize nodelta; #EQ7 |
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[941] | 2532 | @split_eq_status try % |
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[988] | 2533 | [ change with (? = sigma ?? (address_of_word_labels ps label)) |
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[941] | 2534 | (* ARITHMETICS, BUT THE GOAL SEEMS FALSE *) |
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| 2535 | | whd in ⊢ (??%%) whd in ⊢ (??(?%?)?) whd in ⊢ (??(?(match ?(?%)? with [_ ⇒ ?])?)?) |
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[948] | 2536 | @(bitvector_3_elim_prop … (\fst (split bool 3 8 rest_addr))) %]] |
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[989] | 2537 | *) | (* Instruction *) -pi; whd in ⊢ (? → ??%? → ?) *; normalize nodelta; |
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[942] | 2538 | [1,2,3: (* ADD, ADDC, SUBB *) #arg1 #arg2 #MAP #H1 #H2 #EQ %[1,3,5:@1] |
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[930] | 2539 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
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| 2540 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
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[989] | 2541 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 pol (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
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[936] | 2542 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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| 2543 | >H2b >(eq_instruction_to_eq … H2a) |
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[942] | 2544 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); generalize in match MAP; -MAP; |
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[933] | 2545 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; |
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| 2546 | @(list_addressing_mode_tags_elim_prop … arg2) whd try % -arg2; #ARG2 |
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[989] | 2547 | normalize nodelta; #MAP; (* |
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[943] | 2548 | [1: change in ⊢ (? → %) with |
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| 2549 | ((let 〈result,flags〉 ≝ |
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| 2550 | add_8_with_carry |
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| 2551 | (get_arg_8 ? ps false ACC_A) |
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| 2552 | (get_arg_8 ? |
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| 2553 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
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| 2554 | false (DIRECT ARG2)) |
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[989] | 2555 | ? in ?) = ?) |
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| 2556 | [2,3: %] |
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| 2557 | change in ⊢ (???% → ?) with |
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| 2558 | (let 〈result,flags〉 ≝ add_8_with_carry ?(*(get_arg_8 ? ps false ACC_A)*) ?? in ?) |
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| 2559 | >get_arg_8_set_clock *) |
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| 2560 | [1,2: cases (addressing_mode_ok ???? ∧ addressing_mode_ok ????) in MAP ⊢ ? |
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| 2561 | [2,4: #abs @⊥ normalize in abs; destruct (abs) |
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| 2562 | |*:whd in ⊢ (??%? → ?) #H <(option_destruct_Some ??? H)] |
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| 2563 | [ change in ⊢ (? → %) with |
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| 2564 | ((let 〈result,flags〉 ≝ |
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| 2565 | add_8_with_carry |
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| 2566 | (get_arg_8 ? ps false ACC_A) |
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| 2567 | (get_arg_8 ? |
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| 2568 | (set_low_internal_ram ? ps (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps))) |
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| 2569 | false (DIRECT ARG2)) |
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| 2570 | ? in ?) = ?) |
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| 2571 | >get_arg_8_set_low_internal_ram |
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| 2572 | |
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| 2573 | cases (add_8_with_carry ???) |
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| 2574 | |
---|
| 2575 | [1,2,3,4,5,6,7,8: cases (add_8_with_carry ???) |*: cases (sub_8_with_carry ???)] |
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[931] | 2576 | #result #flags |
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[985] | 2577 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) % |
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| 2578 | (*| (* INC *) #arg1 #H1 #H2 #EQ %[@1] |
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[931] | 2579 | normalize in H1; generalize in match (option_destruct_Some ??? H1) #K1 >K1 in H2; whd in ⊢ (% → ?) |
---|
| 2580 | change in ⊢ (? → ??%?) with (execute_1_0 ??) |
---|
| 2581 | cases (fetch (load_code_memory assembled) (sigma 〈preamble,instr_list〉 (program_counter … ps))) * #instr #newppc' #ticks normalize nodelta; |
---|
[936] | 2582 | * * #H2a #H2b whd in ⊢ (% → ?) #H2c |
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| 2583 | >H2b >(eq_instruction_to_eq … H2a) |
---|
| 2584 | generalize in match EQ; -EQ; whd in ⊢ (???% → ??%?); |
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[931] | 2585 | @(list_addressing_mode_tags_elim_prop … arg1) whd try % -arg1; normalize nodelta; [1,2,3: #ARG] |
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[932] | 2586 | [1,2,3,4: cases (half_add ???) #carry #result |
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[934] | 2587 | | cases (half_add ???) #carry #bl normalize nodelta; |
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| 2588 | cases (full_add ????) #carry' #bu normalize nodelta ] |
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[985] | 2589 | #EQ >EQ -EQ; normalize nodelta; >(eq_bv_eq … H2c) -newppc'; |
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[934] | 2590 | [5: % |
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[940] | 2591 | |1: <(set_arg_8_set_code_memory 0 [[direct]] ? ? ? (set_clock pseudo_assembly_program |
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| 2592 | (set_program_counter pseudo_assembly_program ps newppc) |
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| 2593 | (\fst (ticks_of0 〈preamble,instr_list〉 |
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| 2594 | (program_counter pseudo_assembly_program ps) |
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| 2595 | (Instruction (INC Identifier (DIRECT ARG)))) |
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| 2596 | +clock pseudo_assembly_program |
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| 2597 | (set_program_counter pseudo_assembly_program ps newppc))) (load_code_memory assembled) result (DIRECT ARG)) |
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| 2598 | [2,3: // ] |
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| 2599 | <(set_arg_8_set_program_counter 0 [[direct]] ? ? ? ? ?) [2://] |
---|
| 2600 | whd in ⊢ (??%%) |
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| 2601 | cases (split bool 4 4 ARG) |
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| 2602 | #nu' #nl' |
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| 2603 | normalize nodelta |
---|
| 2604 | cases (split bool 1 3 nu') |
---|
| 2605 | #bit_1' #ignore' |
---|
| 2606 | normalize nodelta |
---|
| 2607 | cases (get_index_v bool 4 nu' ? ?) |
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| 2608 | [ normalize nodelta (* HERE *) whd in ⊢ (??%%) % |
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| 2609 | | |
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[985] | 2610 | ] *) |
---|